0), no current is observed.\nMulti-walled CNTs, on the other hand, constitute a roll of multiple layers of graphene. Martel *et al.* report that no field effect was found, and that multi-walled CNTs are conductive. It is because the radius of a multi-walled CNTs is larger than those of single-walled, that the band-gap is very small.\nGraphene Nanoribbon\n-------------------\nA graphene nanoribbon (GNR) is a graphene sheet possessing a very small width *W**r*. A GNR can be obtained by cutting a CNT along its axis, with a scale ranging from 10-100nm. As with the CNT, the BCs of a GNR determine whether system behaves as a conductor or semiconductor. The zigzag BC corresponds to a conducting GNR, whereas the armchair BC can either yield a conducting or a semiconducting characteristics. Electrons can become trapped in a small confined region of the GNR with a certain BC, thus forming an energy band-gap. For the armchair structure, this band-gap should be (in theory) inversely proportional to the width *W**r* of the nanoribbon \n \n*E**g* ∝ 1/*W**r*. \nCurrent experiments, however, have shown that the band-gap observed does not rely upon the width and chiral index. It may be plausible that the lithography and etching techniques are inadequate to define the orientation precisely. Therefore, hybrid structures (combinations of armchair and zigzag GNRs) always co-exist within a sample. In any case, phonon scattering near the edge of the nanoribbon would also downgrade the mobility.\n[sensor] A schematic representation of how a graphene sensor may be used as a molecular detector. Graphene is very sensitive to the molecules attached to it, easily affecting its conductivity. In such a case, graphene can be used to detect small biological molecules (such as DNA), or even integrated into fire-alarms as a smoke detection system.\nApplications of Graphene Devices\n================================\nSome common applications of graphene to electronic devices are discussed below. These include graphene sensors, photonic and optical devices and capacitors. Details of graphene transistors will be discussed in section VI.\nGraphene Sensor\n---------------\nImpermeability is one of the most outstanding properties of graphene, since the carbon atoms are closely packed with one another. Furthermore, the conductivity of graphene is sensitive to impurities, and can change easily when other molecules or substrates become attached to it. The variation of such conductivity due to molecular attachment is the principal foundation for molecular sensors (cf. Figure [sensor]). Graphene, with its two-dimensional surface, provides a platform upon which tiny particles can attach. The deviation of electrical resistivity, caused by the attached molecules (impurities), can be measured by the Hall Effect. He *et al.* have successfully developed bio-sensors and gas sensors using such a method. In the near future, graphene is also likely to be a plausible candidate for magnetoresistance sensors.\nGraphene Photonic Devices\n-------------------------\nThere is no doubt that graphene film is an ideal material for photonic devices. Due to the fact that graphene’s atomic thickness has little absorption of light, this makes it almost transparent to all visible wavelengths (cf. section III.C). As such, there are a number of applications, including solar photonic devices, touch screens and graphene photonic transistors.\nOptical semiconducting devices are derived from the working principle of the photo-electric effect. Photons transfer their energy to electrons, which can only absorb photons of specific wavelength. Donors are generally doped so as to increase the Fermi level. The absorption range will therefore be reduced, thus enhancing the sensitivity to photon absorption or emission. Under the influence of an electric field, the emission of photons (due to recombination of the carriers) is referred to as electro-luminescence. Note that some recombination may also relate to the process of thermionic emission. Additionally, photo-conductivity is the process of photon absorption, whereby the conductivity is increased due to the excitation of additional carriers. This process leads to the creation of additional electrons and holes in the conduction and valence bands. As such, one could potentially apply this premise to that of light detectors. Indeed, Kim *et al.* further noticed that a graphene light detector can have a transit bandwidth reaching speeds of up to 1THz.\n[cap] Here, the graphene layers act as electrodes. This forms the basis of a capacitor which has a fast response time and large capacitance. Furthermore, the size of the capacitor can be minimised due to graphene’s atomic thickness. Whilst it is desirable to have a large capacitance *C*, the internal resistance *R* inside the capacitor is very small, due to the high conductivity of graphene itself. Thus we maintain a small time constant *τ* = *R**C* as low as 200 *μ*s. Furthermore, the cost of manufacturing the graphene capacitor is relatively inexpensive.\nBilayer Graphene Capacitor\n--------------------------\nThe working principle of the graphene bilayer capacitor is to store charge and bypass high-frequency signals. A simple graphene capacitor consists of two graphene layers, separated by a dielectric medium (cf. Figure [cap] for a schematic representation). Graphene is ideal for storing charge due to its high mobility. Furthermore, a single atomic layer is also capable of minimising the overall size of the capacitor. Interestingly, a graphene capacitor has ambipolar characteristics within the electrodes (the electrodes being either positive or negative), thus simplifying the manufacturing process. Miller *et al.* have also reported that the response time for a capacitor should be fast enough to react to high-frequency circuits. Therefore, the time constant (*τ* = *R**C*) is made as small as possible, where *R* is the internal resistance and *C* is capacitance. It is important to also note that *τ* is inversely proportional to cutoff frequency *f**c**u**t*. If one requires large capacitance *C* for charge storage, the internal resistance *R* inside the capacitor is required to be very small. Graphene, with its low resistance, is therefore an ideal material for the next generation of capacitors. Miller *et al.* further go on to show that graphene bilayer capacitors can perform extremely well at high frequencies. Thus, the aforementioned graphene devices may be very useful for ‘green’ energy storage.\n[gate] Schematic of a field-effect transistor (FET) is shown. An applied gate voltage *V**G* can control the accumulation of charge carriers inside the transistor channel. Hence, the flow of current from drain to source can be adjusted accordingly. The concept is analogous to that of a conventional tap - the flow of water representing current.\nDevelopments and Hurdles of Graphene Transistors\n================================================\nWe have already highlighted upon some of the major applications of graphene devices in previous section. However, they have also created a firm base for the next generation of the graphene electronics – graphene transistors. Transistors (which are made of semiconductors) are a key constituent of many electronic devices. These include inverting amplifiers, or logical controllers in integrated circuits. Generally speaking, a field-effect transistor (FET) consists of three terminals - gate, source and drain (cf. Figure [gate]). An electric current flows between the source and drain via a channel, and the current flow rate can be controlled via the gate voltage *V**G*. The gate oxide layer is a dielectric between the gate and channel. Impurities are also introduced via doping, to increase the concentration of charge-carriers on the channel. Two types of carriers: p-type (acceptor) and n-type (donor) are commonly used in industry. The materials constituting an FET (source-channel-drain) are typically of either n-p-n type or p-n-p type. The scale and speed of silicon-based transistors has undergone leaps in performance over recent decades. Nevertheless, the limit of transistor size has reached an optimal level, with new materials urgently being required to replace silicon. Graphene, a revolutionary new material, looks set to fill this void due to its outstanding electronic properties.\n[band] The energy bands for an insulator, semiconductor and conductor are all given. For the insulator, an energy band-gap *E**g* exists. The valence band is fully occupied by electrons, with no occupation of the conduction band. Here, electrons are incapable of moving freely between the bands - as per the definition of an insulator. In the case of a conductor, electrons can move freely within the conduction band (which is partially occupied), and hence current can flow. In particular, a small band-gap exists in the case of a semiconductor. Thus, only a small energy is required to excite an electron from the valance to conduction band. Semiconductors (such as silicon) are generally doped via donors or acceptors in order to make an electron/hole excite more easily across the gap. Using an applied electric field, one can also control the ON/OFF state of a semiconductor using this method.\nBand-gap of Graphene\n--------------------\n### Insulator, Semiconductor and Conductor\nThe electrical conductivity of a material can be split into three categories: insulators, semiconductors or conductors, according to its band structure (cf. Figure [band]). Electrons are not able to move freely within an insulator, as the valence band (fully occupied by electron states) and conduction band (unoccupied) are separated by an energy band-gap *E**g*. The semiconductor, however, is the basic building block for an ON/OFF switching device, as it possesses only a small energy band-gap. For example, silicon is a commonly used material within semiconductors, and is generally doped (via some donor for example) in order to make an electron excite more easily to the conduction band. In semiconductors, electrons can also be activated into the conduction band via the increase of temperature, or an applied electric field (up shift of Fermi level). Conductors normally relate to metals, as electrons are permitted to move freely in the conduction band. It is because metals have this partially filled conduction band, that electron excitation is gapless.\n### Methods of Band-gap Creation in Graphene\nFigure [semiband] shows that the electronic band structure of a semiconductor is parabolic, with a non-zero energy gap separating the conduction and valence bands. It is interesting to note that graphene has a linear cone band structure, and is gapless at the Dirac point, (cf. Figure [cone] and section III.D). Small thermal fluctuations, or the application of an electric field may excite electrons from the valence to conduction band within graphene. The Fermi energy will rise in the presence of an applied electric field (dependent upon the sign of the applied gate voltage), and thus exhibit a metallic characteristic. Additionally, there exists an electrical potential of the graphene sheet relative to some carrier reservoir – this is determined by the geometry of the device. The zero band-gap of graphene is clearly an obstacle with regards to its application in semiconducting devices, as no precise OFF state is provided. Accordingly, opening the band-gap is the most important task in making the graphene transistor become a practical reality.\nSeveral existing methods to create the gap include carbon nanotubes, graphene nanoribbons, deformed structures and bilayer graphene, to act as a transistor channel (cf. section IV). Graphene oxide is also capable of providing a small energy band-gap due to the change of electronic band structure. Another possibility is via deformation of the graphene layer, either by bending or physical strain, since theoretically, the band-gap properties depend upon the width and orientation of the graphene allotrope. A small band-gap in graphene can also be created via chemical doping, or by retaining some of the defects in the structure as mentioned by Coletti *et al.* and Terrones *et al.*. As previously discussed, doping in a silicon semiconductor is achieved by replacing some atoms of silicon with other elements. However, this is slightly different for the case of graphene. Due to the strong honeycomb structure, the dopants are generally placed upon the graphene surface, rather than by replacing a carbon atom itself. This leads to a structural defect, and thus changes the electronic band structure. This means that the band-gap can therefore be controlled. Some common dopants include gold, sulphur, boron and nitrogen. Boron nitride (BN) is one of the key compounds for the doping of graphene, and subsequent band-gap creation. Fan *et al.* have also discovered that BN, doped in a ‘random’ pattern upon the graphene surface, is capable of opening a band-gap. They report that the distribution of surface charges (electrons in *π* bond configuration) is altered by doping, hence forming a small band-gap. However, existing doping methods are sometimes not simple to control. However, Yan *et al.* have recently reported that doping via CVD methods can offer a more simple and stable means of creating this band-gap.\n[semiband] The band structure of a semiconductor is shown. Contrary to the linear energy-momentum relationship for graphene, this possesses parabolic shape. A small energy band-gap *E**g* exists, which is crucial for defining the ON/OFF logical state. The diagram also shows how an energy *h**ν* is required to excite an electron from the valence to conduction band, thus allowing a current to flow.\nAs discussed in section IV.A, the CNT is a traditional method of opening the band-gap, and is dependent upon the choice of boundary conditions (characterised by the chiral index). Bilayer graphene has a zero band-gap, although an applied electric field can be implemented to create such a gap (details of the band-gap in bilayer graphene can be referred to in ). Deformation of graphene structures has also been known to control the band-gap. Nevertheless, in all cases, developing a band-gap within graphene would normally reduce its mobility.\nSchwierz has stated that the mobility of GNRs only falls within the range of 200 − 1500 cm2V− 1s− 1. The decrease is due to dislocations present in the graphene crystalline structure, most likely induced during the manufacturing process. It is widely known that the speed of the carriers is proportional to their mobility - thus the tradeoff in creating a band-gap is to slow down the speed of the device. As mentioned by Schwierz, an ON/OFF switching state is not well-defined in current technologies, and much effort has been focussed upon this area. Alternatively, graphene may be suitable for high-frequency devices, which do not require a well-defined OFF state.\n### Band-gap on Substrate in Electric Field\nAs previously mentioned, the most general shape of suspended graphene consists of two sub-lattices (A and B), which can be displaced with respect to each other in a transverse direction (cf. Figure [Fig1ab]b). If graphene is coated on a substrate, this will give rise to some force (including the van der Waals) upon the interface. The sub-lattices should differ with up and down transverse displacements *u**z* along with the z-axis, assuming that an electric field *E**z* is applied. This means that symmetry between the up and down (out-of-plane) displacements is broken. For a single valley, we may estimate the value for the band-gap directly via the 2 × 2 Dirac Hamiltonian matrix:\n \n$$\\hat{H} = v\\_{F} {\\hat \\sigma} \\cdot {\\bf p} + \\sigma\\_z\\Delta(E\\_z),\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\sigma\\_z =\\begin{pmatrix}\n1 & & 0 \\\\\n0 & & -1\n \\end{pmatrix}$$ \nwhere Δ(*E**z*) describes a stress induced by the substrate, *E**z* is the electric field in z direction, and *σ**z* is one of the Pauli matrices. The eigenvalues of this matrix specify the Dirac spectrum $ \\epsilon(p)= \\pm \\sqrt{v\\_F^2 p^2 +\\Delta(E\\_z)^2}$, with a value for the band-gap equal to 2Δ(*E**z*).\nThe application of a transverse electrical field (of amplitude *E**z*) can also induce polarisation of sublattices (A and B). This is described by the same invariant term in the Hamiltonian as *σ**z*Δ(*E**z*). We have Δ(*E**z*) = *e**E**z**u**z*, and thus a band-gap equal to 2*e**u**z**E**z*. To have visible band-gap, the value of *E**z* is required to be large. The best way to realise this mechanism, is to use a ferroelectric material for the substrate. However, the deformation phenomena are of utmost importance here. Due to the transverse lattice displacements separating the sub-lattices (A and B), there arises a deformational field with a value of the order *D**δ**u**z*/*a*. Here, *D* is the deformation potential, ‘*a*’ is the inter-atomic spacing, and *δ**u**z* is the change of transverse displacement which arises due to the presence of the substrate. In the following estimation for the energy band-gap, we use the following parameters for graphene, $a=1.4\\AA$ and the deformation potential is equal to *D* = 6.8 eV. If we assume that *δ**u**z* is around 10% of the transverse displacement *u**z*, which is roughly equal to $0.4\\AA$ (the order of magnitude estimated for silicene ), this gives an energy band-gap *E**g* ≈ 190 meV. Thus, in the addition to deformation from the substrate, the application of *E**z* can contribute an additional force to the substrate, and may provide a further means of controling the band-gap. This value can increase or decrease linearly with the applied electric field. The phenomenon is general, and also arises in other two-dimensional crystals (cf. for example, silicene ).\nElectric Field Effect in Graphene\n---------------------------------\nGeim *et al.* have observed that electric field effects were found in two-dimensional graphene, and that such effects were stable at room temperature. Figure [bipolar] shows the schematic behavior of electrical conductivity in the presence of a gate voltage *V**G*. A minimum conductivity *σ*0 is also shown to exist at the Dirac point. Although there exists a zero charge density at this point, nevertheless, it exhibits metallic properties. A linear relationship between conductivity and gate voltage is found for an uncontaminated graphene film, with positive and negative slopes when *V**G* > 0 and *V**G* < 0 respectively. This indicates that conductivity is proportional to the DoS at the Fermi energy. An ambipolar characteristic within the electric field has also been discovered, and that either holes (*V**G* < 0) or electrons (*V**G* > 0) could both act as transport carriers.\nIn the early studies of Geim *et al.*, hole carriers were found to exist even at *V**G* = 0. Here, water vapor molecules attach to the surface of the film, thus changing the polarity. These water molecules can, however, be removed via the process of annealing. Recent reports also reveal that the mobility is stable when subject to a change of temperature. Only structural defects can affect the mobility of a sample - this would not reduce significantly, even in the presence of high electric fields. A monolayer graphene channel would also appear to remedy the short-channel effect present in silicon-based transistors. Such an improvement is one of the key foundations for a faster logical transistor.\nSchwierz also points out that a graphene transistor might have some special *I* − *V* characteristics, possessing two linear regions and one saturation region. When the drain-source voltage *V**D**S* is small, a linear relationship for the *I* − *V* curve is found. As *V**D**S* is increased further, a saturation of the *I* − *V* curve begins to emerge. If *V**D**S* exceeds a certain limit, a new linear region is yet again observed. Only a few studies with current saturation have been reported. A greater understanding of *I* − *V* characteristics would require addressing in the near future, as this saturation region would affect the performance of devices.\n[bipolar] The ambipolar characteristics of the electric field effect in graphene were first observed by Geim *et al.*. It is one of the key important properties that the conductivity *σ* increases with the absolute value of *V**G* (associated with the DoS). There is also a minimum quantum conductivity *σ*0, which exists in the absence of an applied gate voltage (i.e., *V**G* = 0).\nGraphene Oxide Transistor\n-------------------------\nA number of conventional graphene transistors, of various design, have been proposed in recent years. In principle, graphene can easily act as a transistor channel. Graphene oxide (GO) is a compound of both graphene and oxygen, a single layer of which is slightly thicker than a graphene monolayer. Jin *et al.* claim that the use of graphene oxide is one of the simplest ways to create an energy band-gap, thus turning graphene into a semiconductor. Graphene oxide is water soluble, and the fabrication process is not at all complicated. Recent studies have reported that graphene oxide can be tuned to either a semiconducting or insulating state, dependent upon the amount of oxygen, and bonding structure therein. A thin graphene oxide layer is generally coated on a silicon substrate, acting as a channel for the transistor. A schematic of a graphene oxide transistor is shown in Figure [tranoxide], and exhibits an ambipolar characteristic for a clean sample. It was also discovered that the drain-source current increases with the absolute value of gate voltage. The drain-source current was also found to be increasing with temperature. Despite the mobility of graphene oxide being much less than for graphene ( ∼ 100 cm2V− 1s− 1), the main advantage of this method is its simplicity, requiring no reduction process (cf. Jin *et al.* ).\n[tranoxide] Reduced graphene oxide (rGO) is a compound of both graphene and oxygen, possessing a very small band-gap. This acts as the channel inside the transistor. Despite the mobility of graphene oxide being less than that for pure graphene, it is one of the simplest ways to manufacture graphene transistors in industry.\nReduced graphene oxide (rGO) is obtained via a chemical process of removing oxygen from graphene oxide. However, reduction is not always perfect. Defects in the honeycomb structure are usually found in experiments, with residual oxygen compounds sometimes remaining on the boundary of samples. High temperatures can remove the oxide components, but are required to be in excess of 1000K. Like graphene oxide, reduced graphene oxide can act as the channel inside an FET. He *et al.* have applied a coating of reduced graphene oxide to polymer substrates (3-aminopropyltriethoxysilane PET substrate), discovering an ON/OFF switching ratio of approximately 3-4. This switching ratio is determined by how the current responds to the gate voltage - from the OFF to ON state, or vice versa. They further reveal that the conductivity of a transistor depends upon the thickness of the rGO electrodes - the mobility also degrading as a result of structural defects. However, even in the absence of a gate voltage, a non-zero drain-source current still exists, and thus current leakage may occur. He *et al.* also mention that a bending of reduced graphene oxide would widen the transition region of the ON/OFF state, thus reducing the switching ratio.\n[tranCNT] The CNT transistor is a well-developed area of research. A band-gap can be created by altering the boundary conditions. Normally, the performance of a CNT transistor will be characterised not only by the resistance *R*, capacitance *C*, and inductance *L* of the channel, but also the contact interfaces of the metal electrodes.\nCarbon Nanotube Transistor\n--------------------------\nA number of theories and experiments with CNT transistors have been undertaken for a few decades now, becoming a well-developed area of research. The CNT works as a transistor channel, capable of being tuned from conductor to semiconductor as mentioned by Tans *et al.*. Figure [tranCNT] shows a diagrammatic representation of such a CNT transistor. In theory, the CNT channel can perform long-range (ballistic) electrical transport. Avouris *et al.* have also proposed that the movement of electrons within the CNT structure is restricted to one dimension, thus limiting the amount of inelastic scattering. In brief, the performance of a CNT transistor can be characterised by its resistance *R*, capacitance *C* and inductance *L*. In the studies of Avouris *et al.*, they further mention that these physical quantities (*L*, *R* and *C*) simultaneously exhibit both classical and quantum behavior - these quantum effects are found to be more important for a very small transistor size. In fact, the quantum resistance *R**q* is of the scale *h*/4*e*2, whilst quantum capacitance *C**q* and inductance *L**q* in CNTs are associated with the DoS (cf. for details ). Capacitance and inductance, whether classical or quantum, depend upon the shape, size and boundary conditions of the CNT. Moreover, Schottky barriers are also found in CNT transistors, exhibiting a large resistance. These are due to the contact between the metal electrode and semiconducting CNT. Due to the differing work functions between the CNT and metal electrode, this leads to a consequent shift of the Fermi energy. Recently, it has been discovered that the shape of potential barrier, between the electrodes and CNT, is shallow and wide. They maintain quantum coherence of electron transport between the adjacent sections, for lengths even up to several micrometers.\nTheoretically, ballistic transport is found throughout the CNT transistor, with a large mean-free-path of the carriers. In the real case, like in metals and semiconductors, the electron transport is due to diffusion, thermal gradients and electric fields, together with the chiral index of CNT. The length of the CNT also determines the transport - the longer the nanotube, the more inelastic scattering that will occur. The drain-source current can be controlled via an electric field acting upon the CNT channel. Two types of CNT transistor (top gate and wrap-around gate) were developed recently. Avouris *et al.* have pointed out that the Schottky barriers in CNT-metal electrode contacts are very high when compared to those of silicon. Nonetheless, CNT transistors can work well upon inter-connection of intergated circuits, operating well even at room temperature. This is due to the small electron-phonon interaction inside the CNT. Overall, these transistors consume less power than their traditional silicon-based counterparts, thus proving to be an advantage for logic switches.\nFranklin *et al.* have simulated a short CNT transistor (with 9nm channel width), with reports of better performance and stable electrical properties in comparison to silicon-based devices possessing a 10nm channel width. The study claims that the short CNT channel provides a faster switching speed. However, in a silicon-based transistor, a shorter channel would deteriorate its quality. The overall stability of the drain-source current is compromised when the channel is of the scale of tens of nanometers (i.e., similar to that of the depletion layer). The electrical current through the transistor thus becomes unreliable. This is more commonly known as the short-channel effect. According to Franklin *et al.*, the short-channel effect can be minimised by a thin CNT. In addition, Jang *et al.* have also reported that CNTs provide a high switching ratio.\n[bilayer] The band structure of bilayer graphene has a parabolic shape, similar to that of a semiconductor. However, bilayer graphene is gapless, whereas the band-gap of a semiconductor is non-zero. In the presence of an electric field, a band-gap in graphene can be created and adjusted accordingly. More importantly, its bilayeral structure can reduce current leakage.\nBilayer Graphene Transistor\n---------------------------\nBilayer graphene consists of two individual graphene monolayers (cf. Figure [bilayer]). As expected, Sarma *et al.* have pointed out that the structure of bilayer graphene is similar to that of very thin graphite. Similar to monolayer graphene, the bilayer has a zero energy band-gap, but with a parabolic shape. A perpendicular electric field applied to the graphene bilayer, or impurities via doping can, however, create a band-gap. Furthermore, the magnitude of this electric field can tune the size of the gap. Ponomarenko *et al.* have also reported that the bilayer channel can be changed from conductor to insulator by tuning the applied electric field. However, one cannot control the band-gap easily via doping. The mobility in double layers is lower than that of a single layer, as stated by Wallace. Nonetheless, carriers in bilayer channels can still perform at high speed. Studies by Zhang *et al.* have discovered that the band-gap of bilayer graphene varies in the range of 0–298meV at room temperature. This narrow gap is suitable for tunable nanophotonic applications, allowing a certain range of the visible spectrum to pass through. Another advantage of bilayer graphene, is that the current leakage can be minimal. A more concise understanding as to precisely how the band-gap of bilayer or multilayer graphene is affected by an external electric field, will urgently be required in the near future.\n[trantunnel] The idea of the tunneling transistor is shown. Tunneling occurs during the transition of an electron from the conduction to valence band, as a gate voltage *V**G* is applied (the idea is similar to that of Landau-Zener tunneling). The gate voltage could alter the shape of the conduction band, and thus control the tunneling current between the two bands. A tunneling transistor generally consumes much less power than its traditional silicon-based counterpart.\nTunneling Graphene Transistor\n-----------------------------\nFigure [trantunnel] shows the ON/OFF states of a transistor, with and without an applied gate voltage *V**G*. As mentioned by Ionescu *et al.*, the application of a gate voltage could control tunneling between the two bands. A tunneling graphene transistor possesses a steep sub-threshold slope, hence providing a fast response time. The basic principle is that the applied gate voltage alters the shape (offset) of the conduction band, and hence the charge carriers from the conduction band could tunnel to the valence band. The choice of material for the channel inside is usually composed of either CNTs, GNRs or bilayer transistors. Ionescu *et al.* state that the current in tunneling transistors arises due to a leap of electrons between the conduction and valence bands, rather than via diffusive and thermal transport (the concept is similar to Landau-Zener tunneling ). In general, a tunneling transistor (via some quantum process) consumes much less power than a non-tunneling equivalent. Michetti *et al.* have also proposed that within the tunneling graphene transistor, the ON/OFF switching ratio could reach as high as 104, and that tunneling could occur even with a small applied gate voltage. Overall, the band-to-band tunneling in a graphene transistor is generally dependent upon the band-gap width, channel length, oxide thickness and gate voltage.\nZhao *et al.* have studied a nanoribbon tunneling transistor, claiming that an ambipolar behavior can be altered to an asymmetric type via doping - highly suited to logic switches. Their study also notes that a subthreshold slope was discovered of 14mV/dec (i.e., high switching ratio), thus consuming less electrical power. Another report by Zhang *et al.* has also revealed that the nanoribbon tunneling transistor could also operate at high speed. The tunneling probability is a function of the energy band-gap *P*(*E**g*), that is\n \n*P*(*E**g*) ∝ exp[ − *k**E**g*2*d*/(ℏ*v**F* *e**V**D**S*)], \nwhere *k* is a dimensionless constant, *d* refers to the channel width, *e* is an electric charge, while *V**D**S* is the drain-source voltage. The width of the ribbon would also affect tunneling probability. Zhang *et al.* further explain that a leakage current is present in the OFF state, most likely associated with some thermal emission. However, Yang *et al.*, in their study, claim that the nanoribbon tunneling transistor consumes energy which is 8-9 orders of magnitude lower than a silicon-based counterpart.\nVertical Tunneling Graphene Transistor\n--------------------------------------\nMalec *et al.* have investigated the vertical tunneling between graphene and adjacent metal layers. They found that the DoS in graphene would consequently affect the conductivity of the devices. Cobas *et al.* have also studied graphene as a dielectric sandwich, positioned between two metal plates. Quantum tunneling was found to act perpendicular between the two plates, and is dependent upon temperature. In 2012, Britnell *et al.* proposed a new type of transistor that stems from the idea of quantum tunneling between the insulating and graphene layers, thus effectively acting as a vertical transistor channel.\nThe multilayer tunneling transistor constitutes a few insulating layers of hexagonal boron nitride (hBN), positioned between the two graphene plane electrodes. Thus, the hBN is a kind of dielectric barrier which can prevent current leakage, thus consuming less energy.\n[tranvert] The multilayer graphene tunneling transistor, as proposed by Britnell *et al.*. Its innovative design leads to electrons tunnelling vertically, rather than horizontally across the channel. Both the gate and drain-source voltage are applied perpendicular to the layers, with charge carriers subsequently accumulating on graphene layers. Electrons can then tunnel between these layers.\nThe underlying mechanism is as follows (cf. Figure [tranvert] for a diagrammatic representation). The Fermi levels and potential barriers are controlled by an applied gate voltage *V**G*. An electron tunnels between the two graphene layers, only when an electric field (drain-source voltage) is applied, and is sufficiently high to overcome the barrier. When the gate voltage is applied between the silicon gate and the bottom graphene layer, charge carriers in the top and bottom graphene layers begin to accumulate. The Fermi level is raised, and the DoS increases at the same time. Together with the potential difference between the two graphene layers, electrons will be more likely to tunnel form one graphene layer to another. Not only does the applied gate voltage change the DoS, but also modifies the potential barriers between the graphene and insulating layers. Therefore, a tunneling current through hBN layers is induced, and can be tuned by the magnitude of *V**G*. Britnell *et al.* have also found that the transport carriers are holes. The report also reveals that a larger *V**G* and separation of the layers would also enhance the switching ratio. The tunneling nature would also allow the transistor to operate at high frequency, as well as with a high current-gain (cf. for details, Georgiou *et al.* ).\nRadio Frequency Transistor\n--------------------------\nOne can create a band-gap in graphene via any of the aforementioned methods. Even if the band-gap is open, generally, the switching ratio is very small. This is one of the major drawbacks of graphene logic transistors. However, this is no problem for an operational amplifier, or in high-frequency circuits. An amplifying transistor, in general, only requires an ON state. This is because an amplifier requires only a current/voltage input for its operation - thus no OFF state is required.. The high mobility of carriers in graphene renders a fast response to the input signal, which can be up to a few hundred GHz, (cf. for details ). More specifically, the cut-off frequency $f\\_{\\rm cut}$ and the maximum oscillation frequency $f\\_{\\rm osc}$ both determine the speed of a transistor. This maximum oscillation frequency refers to the unity power-gain. The cut-off frequency, however, is defined by a current-gain equal to one, and is proportional to the trans-conductance *g**r**f*.\n \n$$f\\_{\\rm cut} \\propto g\\_{\\rm rf}/L\\_{\\rm gate}.$$ \nwhere $L\\_{\\rm gate}$ is the transistor gate length. To reach a high value for *f**c**u**t*, one has to maximise $g\\_{\\rm rf}$ and minimise $L\\_{\\rm gate}$. Wu *et al.* have studied the shortest gate length down to 40nm with a cut-off frequency of 155 GHz. Zheng *et al.* have also found, via simulation, that the maximum cut-off frequency can reach up to 300 GHz. On the other hand, impurities and defects would not only downgrade the value of *f**c**u**t*, but the maximum oscillation frequency as well. Wu *et al.* also mention that the power-gain in the graphene transistor is particularly low because of the unstable saturation region, of which at present, not much is understood. They add that the maximum oscillation frequency $f\\_{\\rm osc}$ would generally be affected by the size and configuration of the device. Wu *et al.* have also reported that $f\\_{\\rm osc}$ can reach upwards of 20 GHz. It is important to note that the speed of the graphene transistor is influenced by the saturation of the drain current (cf. recent studies by ). As previously mentioned, the contact resistance between the metallic materials and graphene is hugely problematic, and even more serious than the undefined saturation drain current. Researchers are looking for further developments to minimise this resistance.\nFurther Developments and Conclusion\n===================================\nIndeed, a long road lies ahead to implement electronic graphene devices into industry. One of the main reasons is that we cannot maintain both the quantity and quality of graphene devices simultaneously throughout the manufacturing processes. For example, mechanical exfoliation can produce high quality graphene flakes, but only a small amount each time. Chemical vapor deposition and evaporation of silicon carbide are able to mass produce graphene substrates - however, these methods require very high temperatures which may sometimes burn out the graphene structure. It is of utmost importance to develop more precise fabrication techniques to reduce the impurities and defects of graphene products. Most current research is based in laboratories at huge financial expense. How to minimise the spending cost is the next issue to be overcome.\nIt is highly important to open up a band-gap for the graphene switching devices, of which existing methodologies include carbon nanotubes, graphene nanoribbons, deformed structures and bilayer graphene. However, this normally leads to a downgrading of the mobility. One of the major problems for the graphene transistor is current leakage, and the absence of a well-defined ON/OFF switching ratio. The tunneling transistor may be an innovative solution to this problem, in that it consumes less power and operates at a faster speed. Graphene can also be used in high-frequency amplifiers which do not require a well-defined switching ratio. A recent theory reveals that speeds could even reach upwards of 1THz for photonic devices, although one must be careful with regards to the huge contact resistance and unknown saturation regions of a graphene transistor. Some new concepts, such as the single electron transistor and quantum dot transistor, are of great interest, and are being applied to graphene in current research. Some research groups are now looking for more stable two-dimensional materials, such as graphyne and silicene, both of which also possess very good mobility. 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Our main interest is to understand the behavior of energy-critical configurations of the Landau-de Gennes *Q*-tensor model in the limit of vanishing correlation length. We demonstrate existence of configurations with a single Saturn-ring defect approaching the equator of the particle and no other line or point defects. We show this by analyzing asymptotics of energy minimizers under two symmetry constraints: rotational equivariance around the vertical axis and reflection across the horizontal plane. Energy blow-up at the ring defect is a significant obstacle to constructing well-behaved comparison maps needed to eliminate the possibility of point defects. The boundary estimates we develop to address this issue are new and should be applicable to a wider class of problems.\nIntroduction\n============\nThe study of defects in liquid crystals is well-motivated from physical considerations, and is also closely connected to many fundamental questions in analysis and geometry. The intimate connection between nematic liquid crystals and S2-valued harmonic maps is well-established through director-based models such as Oseen-Frank, and a comprehensive study of singularities in nematics will both exploit and expand the rich trove of analytical tools for studying geometrical variational problems. To better describe nematics in settings involving non-orientability, biaxiality, and the presence of line defects, physicists and mathematicians have turned to the tensorial Landau-de Gennes model, which is in some sense a relaxation of the non-convex constraints of director models. Indeed, much recent attention has concentrated on recovering the Oseen-Frank director and energy in the vanishing correlation length limit of Landau-de Gennes (see, e.g., .)\nIn this paper we revisit an important model problem, that of a spherical colloid particle immersed in a nematic which fills the exterior domain, approaching a constant uniaxial state at infinity. We work within the Landau-de Gennes framework, with homeotropic Dirichlet boundary conditions on the colloid surface. Physicists have long expected that there are two competing candidates for minimizers in this geometry: an orientable solution with dipolar symmetry, consisting of a single satellite point defect lying on the axis of symmetry, and a non-orientable director pattern having a circular “Saturn ring” singularity on the equatorial plane perpendicular to the symmetry axis. The latter configuration exhibits quadrupolar symmetry. In the first mathematical treatment of this problem, this expectation is confirmed in the case of very small colloids (for which the quadrupolar Saturn ring solution is minimizing,) or for very large colloids (in which, assuming axial symmetry, the dipolar satellite point defect prevails.) However, Saturn ring defects have been observed both experimentally and numerically in the physics literature (see, e.g., ) and appear to be energetically favorable in many settings, even for larger particles.\nMain results\n------------\nThe goal of this paper is to produce solutions of the spherical colloid problem which exhibit Saturn-ring defects in the limit of small correlation length. To do this, we minimize the Landau-de Gennes energy in a function space enforcing the expected (quadrupolar) symmetries of such a configuration. The symmetry hypothesis will ensure the existence of at least one ring defect on the horizontal plane; a much more difficult issue is to eliminate the possibility of other defects (rings or point defects). Additional ring defects can be excluded by carefully adapting lower bound techniques developed for the Ginzburg-Landau problem. Ruling out point defects, however, presents a new and significant analytical challenge. In general, determining the precise number of point defect in a three dimensional domain is a difficult task: compared to line defects, point defects carry a negligible amount of energy and are thus harder to detect using energy estimates. Moreover, unlike line defects, the number of point defects can not be deduced from topological considerations as even topologically trivial boundary conditions may give rise to an arbitrary number of point defects. Only very specific examples are known where the number of point defects can be determined (see e.g. ). In the present work this task is made even harder due to presence of a line defect approaching the boundary in that the boundary conditions are ``destroyed\" by energy blow-up at the ring defect. This considerably complicates the construction of well-behaved comparison maps. We overcome this obstacle by proving a very precise estimate in the blow-up region at the boundary. This estimate appears to be new, even within the context of some well-studied variational problems (such as Ginzburg-Landau with a weight).\nLet us now introduce the Landau-de Gennes functional, and the variational framework which we will use in our study. In nondimensional units the colloid particle is represented by the closed ball of radius one $B=\\lbrace {{\\vert \\cdot\\vert}}\\leq 1\\rbrace\\subset{{\\mathbb R}}^3$, so that the liquid crystal is contained in the domain $\\Omega={{\\mathbb R}}^3\\setminus B$. In these units the Landau-de Gennes energy depends on the nematic correlation length *ξ* > 0, and is given by\n \n$$\\begin{aligned}\n\\label{eq:energy}\nE\\_\\xi(Q)&=\\int\\_{\\Omega} \\left( {{\\vert \\nabla Q\\vert}}^2 +\\frac{1}{\\xi^2}f(Q)\\right) dx.\\end{aligned}$$ \nThe map *Q* takes values into the space $\\mathcal S\\_0$ of 3 × 3 symmetric matrices with zero trace and describes nematic alignment. The nematic potential is given by\n \n$$f(Q)=-\\frac 12{{\\vert Q\\vert}}^2-\\operatorname{tr}(Q^3)+\\frac 34 {{\\vert Q\\vert}}^4+C,$$ \nwhere the constant *C* is such that *f* satisfies\n \n$$\\label{eq:Ustar}\nf(Q)\\geq 0\\text{ with equality iff }Q\\in \\mathcal U\\_\\star:=\\left\\lbrace n\\otimes n -\\frac 13 I\\colon n\\in\\mathbb S^2\\right\\rbrace.$$ \nThe correlation length *ξ* is typically small and therefore we are going to be interested in the limit *ξ* → 0.\nAnchoring at the particle surface is assumed to be radial:\n \n$$\\label{eq:Qb}\nQ=Q\\_b:=e\\_r\\otimes e\\_r -\\frac 13 I\\qquad\\text{on }\\partial \\Omega,\\qquad e\\_r=\\frac{x}{{{\\vert x\\vert}}}.$$ \nAt infinity, the effect of the particle is not felt and the alignment is uniform, given by\n \n$$\\label{eq:Qinf}\nQ\\_\\infty:=e\\_3 \\otimes e\\_3 -\\frac 13 I.$$ \nMore precisely, this far field condition is enforced by considering configurations in the space $\\mathcal H$ given by\n \n$$\\label{eq:H}\n\\mathcal H :=Q\\_\\infty +\\dot {\\mathcal H},\\qquad\n\\dot {\\mathcal H} :=\\left\\lbrace Q\\in H^1\\_{loc}(\\Omega;\\mathcal S\\_0)\\colon \\int\\_\\Omega {{\\vert \\nabla Q\\vert}}^2 + \\int\\_\\Omega \\frac{{{\\vert Q\\vert}}^2}{{{\\vert x\\vert}}^2} <\\infty \\right\\rbrace.$$ \nWe denote by $\\mathcal H\\_b$ the space of such configurations that satisfy in addition the radial anchoring condition at the particle surface:\n \n$$\\label{eq:Hb}\n\\mathcal H\\_b = \n\\left\\lbrace Q\\in \\mathcal H\\colon Q\\text{ satisfies \\eqref{eq:Qb}}\n\\right\\rbrace.$$ \nWe seek to construct critical points of *E**ξ* with the quadrupolar symmetry of the Saturn ring configuration. This entails two symmetry constraints on the admissible *Q* ∈ H*b*:\n* rotation symmetry around the vertical axis,\n* and reflection symmetry across the equatorial plane.\nSee for a precise definition of each, in terms of group actions. We denote the space of maps in $\\mathcal H\\_b$ satisfying these two symmetry constraints by $\\mathcal H\\_{sym}$, i.e.\n \n$$\\begin{aligned}\n\\mathcal H\\_{sym} =\\lbrace Q\\in\\mathcal H\\_b\\text{ with quadrupolar symmetry}\\rbrace.\\end{aligned}$$ \nA more complete discussion of the space $\\mathcal H\\_{sym}$ will be given in Section [s:upper]. Minimizers of *E**ξ* in $\\mathcal H\\_{sym}$ do exist, because Hardy’s inequality ensures the coercivity of the energy. Moreover, they are critical points of *E**ξ* in the full space $\\mathcal H$: this is a consequence of the principle of symmetric criticality (see also ). Our main result shows that the energy of a symmetric minimizer concentrates, as *ξ* → 0, inside a Saturn ring shaped region around the particle. In the limit this region coincides with the equatorial circle\n \n$$\\mathcal C:=\\left\\lbrace (\\cos\\varphi,\\sin\\varphi,0)\\colon \\varphi\\in{{\\mathbb R}}\\right\\rbrace =\\partial B \\cap\\lbrace x\\_3=0\\rbrace.$$ \nSince the energy will blow up around $\\mathcal C$, the resulting limit configuration will not belong to $\\mathcal H\\_{sym}$. To define the limit space we cut out a small neighborhood of $\\mathcal C$, consider the exterior domain\n \n$$\\begin{aligned}\n\\Omega\\_\\delta^{ext} & = \\left\\lbrace x\\in \\Omega \\colon \\operatorname{dist}(x,\\mathcal C)>\\delta\\right\\rbrace,\\end{aligned}$$ \nand define the limit space $\\mathcal H^\\star\\_{sym}=\\cap\\_{\\delta >0}\\mathcal H^\\star\\_{sym}(\\Omega^{ext}\\_\\delta)$, where\n \n$$\\begin{aligned}\n\\mathcal H^\\star\\_{sym}(\\Omega^{ext}\\_\\delta)=\n\\Big\\lbrace Q & \\in H^1\\_{loc}(\\Omega^{ext}\\_\\delta;\\mathcal U\\_\\star) \\ \\text{with quadrupolar symmetry, s.t.} \\\\\n&\\int\\_{\\Omega^{ext}\\_\\delta}{{\\vert \\nabla Q\\vert}}^2 + \\int\\_{\\Omega^{ext}\\_\\delta}\\frac{{{\\vert Q-Q\\_\\infty\\vert}}^2}{{{\\vert x\\vert}}^2}<\\infty,\\quad\\text{and}\\quad\n Q=Q\\_b\\text{ for }{{\\vert x\\vert}}=1\\Big\\rbrace.\\end{aligned}$$ \nWe may now state our result asserting the existence and asymptotic behavior of solutions with quadrupolar symmetry:\n[thm:main] For any *ξ* ∈ (0, 1] let *Q**ξ* minimize *E**ξ* in $\\mathcal H\\_{sym}$. Then we have:\n* *upper bound:* there exists a universal constant *C* > 0 such that\n \n$$\\frac{1}{2\\pi}E\\_\\xi(Q\\_\\xi)\\leq \\pi\\ln\\frac 1\\xi + \\pi\\ln\\ln\\frac 1\\xi + C.$$\n* *lower bound:* there exists a universal constant *C* > 0 such that for any *δ* ∈ (0, 1),\n \n$$\\frac 1{2\\pi}E\\_\\xi(Q\\_\\xi;\\Omega\\_\\delta^{int})\\geq \\pi\\ln\\frac 1\\xi + \\pi\\ln\\ln\\frac 1\\xi - 2\\pi \\ln\\frac 1\\delta - C\\qquad\\text{as }\\xi\\to 0,$$ \nwhere $\\Omega\\_\\delta^{int}=\\Omega\\setminus\\Omega\\_\\delta^{ext}=\\Omega\\cap\\lbrace\\operatorname{dist}(\\cdot,\\mathcal C)\\leq\\delta\\rbrace$ is the *δ*-neighborhood of the ring defect $\\mathcal C$.\n* *limit configuration:* there is a subsequence *ξ* → 0 such that *Q**ξ* converges in $C^{1,\\alpha}\\_{loc}(\\overline\\Omega\\setminus\\mathcal C)$ to a map $Q\\_\\star\\in \\mathcal H^\\star\\_{sym}$ which is smooth in $\\overline\\Omega\\setminus\\mathcal C$, and uniaxial,\n \n$$Q\\_\\star(x)=n\\_\\star(x)\\otimes n\\_\\star(x)-\\frac 13 I,\\quad n(x)\\in\\mathbb S^2,$$ \nwhere *n*⋆ is a smooth S2-valued locally minimizing harmonic map in Ω. Furthermore, *n*⋆ satisfies the additional symmetry property that *n*⋆(*x*1, 0, *x*3) ⊥ *e*2.\n[r:limitsymmetry] Using cylindrical coordinates (*ρ*, *φ*, *z*) and corresponding orthonormal frame (*e**ρ*, *e**φ*, *e**z*), the additional symmetry statement in (*i**i**i*) amounts to *n*⋆(*ρ*, *φ*, *z*) ⊥ *e**φ*, i.e. *n*⋆(*ρ*, *φ*, *z*) lies in the azimuthal plane generated by *e**ρ* and *e**z*.\nIn proving Theorem [thm:main] we rely heavily on the symmetry constraint, which reduces the problem to two dimensions. Indeed, the relevant analogy is to a two-dimensional Ginzburg-Landau energy with a weight *w* = *ρ* arising from cylindrical symmetry. The asymptotic behavior of Ginzburg-Landau energies with weights have been studied in. The principal novelty of this work is that we must deal with *Q*-tensor-valued maps in an unbounded domain rather than complex-valued maps in a bounded domain. Points (*i*) and (*i**i*) of Theorem [thm:main] follow from careful adaptation of the techniques in, along with classical Ginzburg-Landau methods in, and more recent arguments for *Q*-tensor-valued maps in.\nThe most delicate part of Theorem [thm:main] is the statement (*i**i**i*) asserting that the limit is smooth everywhere away from the equatorial ring defect $\\mathcal C$. Proving this statement amounts to eliminating the possibility of point singularities appearing on the *z*-axis. As it was already pointed out, in general, energy minimization subject to topological constraints does not restrict the number of point defects. In the context of our problem, one negatively charged Saturn ring defect is sufficient to compensate the topological constraint imposed by the boundary conditions but so is, for example, a positively charged Saturn ring together with a pair of negatively charged point defects on the *z*-axis. In the process of ruling out existence of such pair of point defects, a crucial ingredient is to show that energy of the core of a ring defect would be the same up to terms of order *o*(1) in *ξ*, regardless of whether the ring is negatively or positively charged. In other words, the *O*(1) remainder in the energy asymptotics in Theorem [thm:main] does not depend on the sign of the ring defect’s charge.\nHere we encounter an additional difficulty not present in determining the *O*(1) core energy term for classical Ginzburg-Landau vortices. Indeed, in the authors crucially use the fact that the energy of Ginzburg-Landau vortices scales radially: at scale *r* ≫ *ξ*, the energy of a vortex goes as ln(*r*/*ξ*) = ln(1/*ξ*) + ln*r*, and the effect of phase winding is separated from the cost of core formation. Here, on the other hand, proximity to the boundary breaks this scaling invariance and influences the core shape, as seen by the presence of the lnln*ξ* term, and makes it much less clear that radial rescaling should reveal a universal *O*(1) core energy term.\nTo obtain the core energy estimate, we deform the minimizers in a very narrow wedge domain emanating tangentially from the limiting equatorial defect (see Lemma [l:Pxi]). This requires a sharp lower bound on the energy in a small disc tangential to the particle surface at its equator (see Lemma [l:finelowerboundhalfannulus]). The corresponding core energy estimate is new—to the best of our knowledge. Once the core energy is determined, the added energy cost of an anti-hedgehog pair may be computed thanks to ideas in (see Lemma [l:compareEtau]).\nBackground and relevant numerical results\n-----------------------------------------\nThe mathematical study of line defects in nematics was initiated in, in the singular limit as the correlation length *ξ* → 0, for domains and boundary values which induce defects along line segments. As mentioned earlier, global minimizers of the spherical colloid problem were first addressed mathematically in, in which the size of the colloid plays a determining role. As has long been known by physicists, equatorial ring defects can be observed even around large colloid particles, for example in the presence of external electric or magnetic fields or in confinement. The situation with a magnetic field was studied mathematically in, via a Landau-de Gennes energy modified to model interaction with a constant field. The main result of identifies the leading order term in an expansion of the energy, indicating the presence of an equatorial ring defect rather than a satellite point defect, provided the magnetic field is high enough *h* ≫ *ξ* |ln*ξ*|. In the complementary low magnetic field regime *h* ≪ *ξ* |ln*ξ*|, the lower bound established here in Theorem [thm:main] directly implies, in view of upper bounds established in, that minimizers cannot have quadrupolar symmetry, thus hinting at the presence of a satellite point defect. The asymptotics of that model are further and more precisely explored in.\nEven in the absence of external factors which appear to favor rings over satellite point defects, much physical evidence, both numerical and formal, suggests that there is a range of intermediate particle sizes for which configurations with Saturn ring defects may be stable and coexist with point defect configurations having lower energy.\nWe do not consider the important question of stability in this paper, but numerical simulations suggest that the solutions found here may be locally stable. To illustrate these observations, in Figs. [fig:configurationsa]-[fig:energy] we present the summary of simulations that reproduce and extend the results of. We numerically solved in COMSOL the equations for the gradient flow \n$$\\frac{\\partial Q}{\\partial t}=-\\frac{\\delta E\\_\\xi}{\\delta Q}$$ \n for the energy *E**ξ* defined in in the domain in the form of a large cylinder with a spherical void of radius 1 with the same center as that of the cylinder. The admissible *Q*-tensor fields have values in the set of symmetric traceless matrices and are rotationally equivariant with respect to *z*-axis that is also the axis of the cylinder. The *Q*-tensors are subject to the initial condition *Q*( ⋅ , 0) = *Q**i**n**i**t* and the boundary conditions and on the surfaces of the cylinder and the sphere, respectively.\nFollowing and assuming that (*ρ*, *φ*) are polar coordinates in a plane perpendicular to the axis of the cylinder, the simulations were run starting from two initial conditions\n \n$$\\begin{aligned}\nQ\\_{init}={e}\\_3\\otimes{e}\\_3-\\frac{1}{3}I\\qquad\\text{and}\\qquad\nQ\\_{init}={n}(\\psi)\\otimes{n}(\\psi)-\\frac{1}{3}I,\\end{aligned}$$ \nwhere ${\\bf n}(\\psi)=(\\cos{\\psi}\\cos{\\varphi},\\cos{\\psi}\\sin{\\varphi},\\sin{\\psi})$ and \n$$\\psi=2\\tan^{-1}\\left({\\frac{\\rho}{z}}\\right)-\\tan^{-1}\\left({\\frac{\\rho}{z+z\\_0}}\\right)-\\tan^{-1}\\left({\\frac{\\rho}{z+z\\_0^{-1}}}\\right).$$ \n Here the second choice of the initial condition represents an approximation of a nematic configuration with a hyperbolic point defect at distance *z*0 below the sphere’s center. Note that, for *ξ* < 0.005, the simulations leading to an equatorial Saturn ring were started from the critical point obtained for *ξ* = 0.005.\nFig. [fig:configurationsa]-[fig:configurationsb] shows the line fields of the nematic in (*r*, *z*)-coordinates when *ξ* = 1/70. For this choice of the correlation length, the critical point approached by the gradient flow simulation depends on the initial condition; the critical point in Fig. [fig:configurationsa] has dipolar symmetry, while the critical point Fig. [fig:configurationsb] is quadrupolar.\n.25 [fig:configurationsa]\n.26 [fig:configurationsb]\n.41 [fig:energy]\nFor larger values of *ξ*, once it exceeds a critical value *ξ**c* ≈ 0.017, the simulations converge to the equatorial Saturn ring configuration, regardless of the initial conditions. In fact, for the initial condition with a hyperbolic point defect, this defect expands first into a small ring below the south pole of the colloid. This ring then expands and travels up the surface of the colloid, eventually stopping at its equator.\nFor *ξ* < *ξ**c*, the dichotomy between the initial conditions persists for all values of *ξ* for which the simulations were run: the hyperbolic point defect expands into a small ring below the colloid and the constant initial condition leads to the equatorial Saturn ring. Even though the energy of the equatorial ring exceeds the energy of the small ring once *ξ* becomes sufficiently small, the equatorial ring remains stable. These observations are summarized in Fig. [fig:energy].\nAlthough we do not address the question of their minimality, in the present work we establish that Saturn ring-like critical points of the Landau-de Gennes energy indeed do exist for large particles. Here, rather than varying the radius of the particle, we use an equivalent description in which the size of the particle is fixed and the nematic correlation length is assumed to converge to zero.\nThis paper is organized as follows. In Sections [s:upper] and [s:lower], we establish the upper and lower bounds given by the statements (*i*) and (*i**i*) of Theorem [thm:main]. In Section [s:lim] we prove the statement (*i**i**i*) of Theorem [thm:main]; namely, a sequence of minimizers converges to a limiting map which is smooth away from the ring defect.\n### Acknowledgments\nS.A. and L.B. are supported by NSERC (Canada) Discovery Grants, D.G. was supported in part by NSF grant DMS-1729538, and X.L. by ANR project ANR-18-CE40-0023. We also thank the Erwin Schrödinger International Institute for Mathematics and Physics (Vienna) for their kind hospitality in December 2019, when this work was being completed.\nUpper bound\n===========\nTo obtain the upper bound, i.e. part (*i*) of Theorem [thm:main], we construct an admissible map $Q\\in\\mathcal H\\_{sym}$ whose energy has the expected behavior.\nLet us first describe more explicitly what quadrupolar symmetry means. Symmetries are formalized in terms of the equivariant action of the orthogonal group *O*(3) on maps $Q\\colon {{\\mathbb R}}^3\\to\\mathcal S\\_0$, given by\n \n(*R* ⋅ *Q*)(*x*) = *R**Q*(*R**t**x*)*R**t*, *R* ∈ *O*(3). \nThe energy *E**ξ* is invariant under this action, consistent with the physical requirement of frame invariance. Here we consider the subgroup\n \n*G**s**y**m* = {*R* ∈ *O*(3): *R**e*3 = ± *e*3}. \nThis is the largest subgroup of *O*(3) that maps the space $\\mathcal H$ into itself. Explicitly, *G**s**y**m* is generated by all rotations around the vertical axis *e*3 and by the reflection with respect to the horizontal plane ⟨*e*1, *e*2⟩, that is,\n \n$$\\begin{aligned}\n\\label{eq:groups}\nG\\_{sym}=\\left\\langle \\lbrace R\\_\\varphi\\rbrace\\_{\\varphi\\in{{\\mathbb R}}},S\\right\\rangle,\n\\quad\nR\\_\\varphi = \\left(\\begin{array}{ccc}\n\\cos\\varphi & -\\sin\\varphi & 0 \\\\\n\\sin\\varphi & \\cos\\varphi & 0 \\\\\n0&0&1\n\\end{array}\\right),\\quad\nS = \\left(\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0&0& -1\n\\end{array}\\right).\\end{aligned}$$ \nThe critical points we study in this work are minimizers of the energy among symmetric configurations belonging to the space\n \n$$\\label{eq:Hsym}\n\\mathcal H\\_{sym}=\\left\\lbrace Q\\in\\mathcal H\\_b\\colon R\\cdot Q=Q,\\;\\forall R\\in G\\_{sym}\\right\\rbrace,$$ \nIt is natural to use cylindrical coordinates to describe maps in the space $\\mathcal H\\_{sym}$, i.e. coordinates (*ρ*, *φ*, *z*) defined by\n \n$$x=R\\_\\varphi \\left(\\begin{array}{c} \\rho \\\\ 0 \\\\ z\\end{array}\\right) = \\rho R\\_\\varphi e\\_1 + z e\\_3.$$ \nIn these coordinates the domain Ω corresponds to {*ρ*2 + *z*2 > 1}, and maps $Q\\in\\mathcal H\\_{sym}$ can be written in the form\n \n$$\\label{eq:axialsym}\nQ(\\rho,\\varphi,z)=R\\_\\varphi \\widetilde Q(\\rho,z) R\\_{\\varphi}^t,$$ \nwhere in addition $\\widetilde Q(\\rho,z)=Q(\\rho,0,z)$ satisfies the mirror symmetry constraint\n \n$$\\label{eq:mirrorsym}\n\\widetilde Q(\\rho,-z)=S\\widetilde Q(\\rho,z)S^t.$$ \nWritten in cylindrical coordinates, the energy of a map $Q\\in \\mathcal H\\_{sym}$ takes the form\n \n$$\\begin{aligned}\n\\frac 1{2\\pi}E\\_\\xi(Q) & =\\widetilde E\\_\\xi(\\widetilde Q) := \\iint\\_{\\widetilde\\Omega}\\left({{\\vert \\nabla\\widetilde Q\\vert}}^2 + \\frac{1}{\\rho^2}\\Xi[\\widetilde Q] + \\frac{1}{\\xi^2}f(\\widetilde Q) \\right)\\rho\\, d\\rho dz,\\\\\n\\text{where }\\Xi[\\widetilde Q]&=\\vert \\partial\\_\\varphi[R\\_\\varphi\\widetilde Q R\\_\\varphi^t ]\\vert^2 =8 \\widetilde Q\\_{12}^2 + 2 \\widetilde Q\\_{13}^2+2\\widetilde Q\\_{23}^2 +2(\\widetilde Q\\_{11}-\\widetilde Q\\_{22})^2.\\end{aligned}$$ \nNote that $\\Xi[\\widetilde Q]\\leq 4\\, \\mathrm{dist}^2(\\widetilde Q,{{\\mathbb R}}Q\\_\\infty),$ where the distance is induced by the Frobenius norm. We will construct $\\widetilde Q$ in the region\n \n*D* : = {*ρ*2 + *z*2 > 1: *ρ*, *z* > 0}, \nwith boundary constraints \n$${\\left\\{\\begin{aligned}\n&\\widetilde Q(\\rho,z)=e\\_r\\otimes e\\_r -\\frac 13 I, & \\text{ for }&\\rho^2+z^2=1,\\\\\n&\\widetilde Q(\\rho,0)=S \\widetilde Q(\\rho,0) S^t, & \\text{ for }&\\rho>1,\n\\end{aligned}\\right.}$$ \n and then use the mirror symmetry to extend $\\tilde Q$ to the entire domain Ω. Note that $\\widetilde Q = S\\widetilde Q S^t$ implies that *e*3 is an eigenvector of $\\widetilde Q$ on the horizontal axis {*z* = 0} in order for the mirror symmetry not to create a jump of $\\tilde Q$.\nWe begin by observing that in all estimates in this and subsequent sections the letter *C* denotes a generic universal constant. To construct $\\widetilde Q$ we first divide *D* into 3 subdomains, $D=\\overline D\\_1 \\cup \\overline D\\_2 \\cup \\overline D\\_3,$ where \n$$D\\_1= D\\cap\\lbrace \\rho^2+z^2>2\\rbrace,\\quad\nD\\_2= (D \\setminus \\overline D\\_1) \\cap \\lbrace (\\rho-1)^2 + z^2 > 1/4\\rbrace,\\quad\nD\\_3= D\\setminus (\\overline D\\_1 \\cup \\overline D\\_2).$$ \n[scale=.7]\n(2,0) arc (0:90:2); (4,0) arc (0:90:4); (2,0) – (6,0); (0,2) – (0,6); (3,0) arc (0:105:1);\n(3.5,3.5) node *D*1; (1.5,2.5) node *D*2; (2.4,.4) node *D*3;\n(2,-.2) – (3,-.2); (2.5,-.2) node [below] 1/2; (-.2,2) – (-.2,4); (-.2,3) node [left] 1;\nLet\n \n$$\\widetilde Q \\equiv Q\\_\\infty = e\\_3\\otimes e\\_3-\\frac 13 I\\qquad\\text{ in }D\\_1,$$ \nso that\n \n$$\\label{eq:E\\_D1}\n\\widetilde E\\_\\xi(\\widetilde Q;D\\_1)=0.$$ \nThen we define a Lipschitz map $n\\colon \\partial D\\_2\\to\\mathbb S^2$ as follows. We first set\n \n$$\\begin{aligned}\nn&\\equiv e\\_3 \\quad\\text{ on }\\partial D\\_2 \\setminus \\left(\\partial D\\_3 \\cup \\lbrace \\rho^2+z^2=1\\rbrace\\right),\\\\\nn& = (\\rho,0,z) \\quad\\text{ on }\\partial D\\_2 \\cap \\lbrace \\rho^2+z^2=1\\rbrace. \\end{aligned}$$ \nTo define *n* on ∂*D*2 ∩ ∂*D*3 we use polar coordinates (*r*, *θ*) centered in (1, 0), that is, given by the relation *ρ* + *i**z* = 1 + *r**e**i**θ*, so that\n \n∂*D*2 ∩ ∂*D*3 = {*r* = 1/2, 0 < *θ* < *θ*0 : = *π*/2 + arcsin(1/4)}. \nWe set\n \n*n*(1/2, *θ*) = (sin*θ*, 0, cos*θ*) for 0 < *θ* < *π*/2, \nand in the remaining part of ∂*D*2 ∩ ∂*D*3 we interpolate linearly. More precisely, at the point (1/2, *θ*0), continuity of *n* imposes *n* = (cos*θ*1, 0, sin*θ*1) where *θ*1 = 2*θ*0 − *π* = 2arcsin(1/4), so we set\n \n*n*(1/2, *θ*) = (cos(2*θ* − *π*), 0, sin(2*θ* − *π*)) for *π*/2 < *θ* < *θ*0. \n[scale=1]\n(29:2) arc (29:90:2); (2,0) arc (0:29:2); (4,0) arc (0:90:4); (3,0) – (4,0); (2,0) – (3,0); (0,2) – (0,4); (3,0) arc (0:105:1);\n(0,2) node • – (0,2.5); (0,3) node • – (0,3.5); (0,4) node • – (0,4.5); (4,0) node • – ++(0,.5); (15:4) node • – ++(0,.5); (30:4) node • – ++(0,.5); (45:4) node • – ++(0,.5); (60:4) node • – ++(0,.5); (75:4) node • – ++(0,.5); (3,0) node • – ++(0,.5); (3.5,0) node • – ++(0,.5); (2,0)++(45:1) node • – ++(45:.5); (2,1) node • – ++(.5,0); (29:2) node • – ++(29:.5); (50:2) node • – ++(50:.5); (70:2) node • – ++(70:.5);\n(2.1,2.4) node *n*⌊∂*D*2;\nThe map $n\\colon \\partial D\\_2\\to\\mathbb S^2$ thus defined can be written in the form\n \n$$n=(\\cos\\varphi,0,\\sin\\varphi),\\qquad \\varphi\\in\\mathrm{Lip}(\\partial D\\_2,{{\\mathbb R}}).$$ \nThus, considering an *H*1 extension of *φ* to *D*2 we obtain $n\\colon D\\_2\\to\\mathbb S^2$ with ∫*D*2|∇*n*|2 ≤ *C* and moreover, thanks to Hardy’s inequality and since *n* = *e*3 on ∂*D*2 ∩ {*ρ* = 0}, we have also ∫*D*2*ρ*− 2|*n* − *e*3|2 ≤ *C*. Then we set\n \n$$\\widetilde Q=Q\\_n=n\\otimes n-\\frac 13 I\\qquad\\text{ in }D\\_2,$$ \nand, since *f*(*Q**n*) = 0 and Ξ[*Q**n*] ≤ *C*|*n* − *e*3|2, we have\n \n$$\\label{eq:E\\_D2}\n\\widetilde E\\_\\xi(\\widetilde Q;D\\_2)\\leq C.$$ \nNext we need to define $\\widetilde Q$ in *D*3. We introduce a parameter *σ* > 0 with *ξ* < *σ* < 1/8, and further divide *D*3 into 3 subdomains:\n \n$$\\begin{aligned}\nD\\_4&= D\\_3\\cap \\lbrace (\\rho - 1)^2+z^2 >(4\\sigma)^2\\rbrace, \\\\\nD\\_5&=(D\\_3\\setminus \\overline D\\_4)\\cap \\lbrace (\\rho - 1-2\\sigma)^2+z^2 >\\sigma^2\\rbrace,\\\\\n D\\_6 &=D\\_3\\setminus (\\overline D\\_4\\cup\\overline D\\_5).\\end{aligned}$$ \n[scale=.75]\n(0,0) arc (0:35:8); (4,0) arc (0:104.5:4); (0,0) – (4.4, 0);\n(2,0) arc (0:98:2);\n(1.5,0) arc (0:180:.5);\n(0,0) – ++ (106:4); (-1,2) node 1/2;\n(0,0) – ++(70:2); (.3,1.5) node 4*σ* ;\n(1,0) – ++ (70:.5); (1.25,.2) node *σ*;\n(1,0) node [below] *D*6;\n(2,2) node *D*4; (1,1) node *D*5;\nIn the polar coordinates (*r*, *θ*) centered at (1, 0), the domain *D*4 is given by\n \n*D*4 = {1 + *r**e**i**θ*: 4*σ* < *r* < 1/2, 0 < *θ* < *θ*0(*r*) : = *π*/2 + arcsin(*r*/2)}. \nIn *D*4 we define $\\widetilde Q$ as\n \n$$\\widetilde Q=Q\\_n = n\\otimes n -\\frac 13 I,$$ \nfor some map $n\\colon D\\_4\\to\\mathbb S^2$. The boundary conditions on {*ρ*2 + *z*2 = 1} impose\n \n*n*(*r*, *θ*0(*r*)) = (cos*θ*1(*r*), 0, sin*θ*1(*r*)), *θ*1(*r*) : = 2*θ*0(*r*) − *π*, \nand we define\n \n$$n(r,\\theta)=\\begin{cases}\n(\\sin\\theta,0,\\cos\\theta) &\\text{ for }0<\\theta<\\pi/2,\\\\\n(\\cos(2\\theta-\\pi),0,\\sin(2\\theta-\\pi))&\\text{ for }\\pi/2<\\theta<\\theta\\_0(r).\n\\end{cases}$$ \nThat way we have\n \n$$\\begin{aligned}\n\\int\\_{D\\_4}{{\\vert \\nabla \\widetilde Q\\vert}}^2 \\, \\rho \\, d\\rho\\, dz &\n=2 \\int\\_{D\\_4}{{\\vert \\nabla n\\vert}}^2 \\,\\rho \\, d\\rho\\, dz \\\\\n& = 2 \\int\\_{4\\sigma}^{1/2}\\int\\_0^{\\theta\\_0(r)}\\frac{1}{r^2}{{\\vert \\partial\\_\\theta n\\vert}}^2 (1+r\\cos\\theta)\\, r \\, d\\theta\\, dr \\\\\n&= 2 \\int\\_{4\\sigma}^{1/2}\\int\\_0^{\\pi/2}\\frac{1+r\\cos\\theta}{r} \\, d\\theta\\, dr\\\\\n&\\quad\n+ 8 \\int\\_{4\\sigma}^{1/2}\\int\\_{\\pi/2}^{\\theta\\_0(r)}\\frac{1+r\\cos\\theta}{r} \\, d\\theta\\, dr\\\\\n& \\leq \\pi\\ln\\frac 1\\sigma + C.\\end{aligned}$$ \nMoreover, in *D*4 we have $f(\\widetilde Q)=0$ and *ρ*− 2Ξ[*Q*] ≤ *C*, and therefore\n \n$$\\label{eq:E\\_D4}\n\\widetilde E\\_\\xi(\\widetilde Q;D\\_4)\\leq \\pi\\ln\\frac 1\\sigma +C.$$ \nIn the subdomain *D*5 we are again going to define $\\widetilde Q$ from a unit vector field $n\\colon D\\_5\\to\\mathbb S^2$. There we use polar coordinates (*s*, *ϕ*) centered at (1 + 2*σ*, 0), that is, given by *ρ* + *i**z* = 1 + 2*σ* + *s**e**i**ϕ*. In these coordinates the domain *D*5 is of the form\n \n$$D\\_5=\\left\\lbrace 1+2\\sigma + s e^{i\\phi}\\colon 0<\\phi<\\pi,\\, \\sigma<s<\\bar s(\\phi)\\right\\rbrace,$$ \nwhere $\\bar s(\\phi)$ is a Lipschitz function of *ϕ*, with $2\\sigma \\leq \\bar s \\leq 5\\sigma$. On the part of ∂*D*5 given by $\\lbrace s=\\bar s(\\phi)\\rbrace$, the values of *n* are given by the boundary conditions on {*ρ*2 + *z*2 = 1} and on ∂*D*4, and are of the form\n \n$$n(\\bar s(\\phi),\\phi)=(\\cos\\alpha(\\varphi), 0, \\sin\\alpha(\\varphi))\\qquad\\text{ for }\\phi\\in (0,\\pi),$$ \nfor Lipschitz function $\\alpha\\colon [0,\\pi]\\to{{\\mathbb R}}$, which satisfies *α*(0) = *π*/2 and *α*(*π*) = 0. On the part of ∂*D*5 given by {*s* = *σ*}, we set\n \n*n*(*σ*, *ϕ*) = (sin(*ϕ*/2), 0, cos(*ϕ*/2)) for *ϕ* ∈ (0, *π*). \nThen we define *n* in ∂*D*5 by interpolating in the *s* variable, i.e. we set\n \n$$\\begin{aligned}\nn(s,\\phi)&=(\\cos \\beta(s,\\phi),0,\\sin\\beta(s,\\phi)),\\\\\n\\beta(s,\\phi)&= \\frac1 2\\frac{\\bar s(\\phi)-s}{\\bar s(\\phi)-\\sigma}\\left(\\pi-\\phi\\right) + \\frac{s-\\sigma}{\\bar s(\\phi)-\\sigma}\\alpha(\\phi),\\qquad\\text{ for }\\phi \\in (0,\\pi).\\end{aligned}$$ \nNote that, by continuity, and since $n(\\sigma,0)=e\\_3 = n(\\bar s(0),0)$ and $n(\\sigma,\\pi)=e\\_1 = n(\\bar s(\\pi),\\pi)$, this forces the trace of *n* on ∂*D*5 ∩ {*z* = 0}, to be given by\n \n$$n(\\rho,0)=\\begin{cases}\ne\\_1 &\\text{ for }1<\\rho<1+\\sigma,\\\\\ne\\_3 &\\text{ for }1+3\\sigma <\\rho <1+4\\sigma.\n\\end{cases}$$ \nMoreover, since it is directly checked that |∂*s**n*| + |∂*ϕ**n*| ≤ *C*, we deduce\n \n$$\\int\\_{D\\_5}{{\\vert \\nabla n\\vert}}^2 \\leq C \\int\\_{\\sigma}^{5\\sigma} \\frac{ds}{s} \\leq C.$$ \nTherefore, setting $\\widetilde Q=Q\\_n$ in *D*5 we obtain\n \n$$\\label{eq:E\\_D5}\n\\widetilde E\\_\\xi(\\widetilde Q;D\\_5)\\leq C.$$ \nFinally, in *D*6 we define $\\widetilde Q$ in polar coordinates (*s*, *ϕ*) centered at (1 + 2*σ*, 0) as above, by\n \n$$\\begin{aligned}\n\\widetilde Q & = \\lambda(s)Q\\_n, \\qquad \nn =(\\sin(\\phi/2),0,\\cos(\\phi/2)),\\qquad\n\\lambda(s)=\\min(1,s/\\xi).\\end{aligned}$$ \nThen we have\n \n$$\\begin{aligned}\n\\int\\_{D\\_6}{{\\vert \\nabla\\widetilde Q\\vert}}^2 \\rho\\, d\\rho\\, dz &\n\\leq C +2 \\int\\_\\xi^\\sigma \\int\\_0^\\pi \\frac{{{\\vert \\partial\\_\\phi n\\vert}}^2}{s^2} (1+2\\sigma + s\\cos\\phi) s\\, d\\phi\\, ds \\\\\n& \\leq C + \\frac\\pi 2 (1+2\\sigma)\\ln\\frac\\sigma\\xi \\\\\n&\\leq \\frac \\pi 2 \\ln\\frac \\sigma\\xi + \\pi \\sigma\\ln\\frac 1\\xi + C,\\end{aligned}$$ \nand therefore, since $f(\\widetilde Q\\_n)=0$ for *s* > *ξ* and ≤ *C* for *s* < *ξ*, we deduce that\n \n$$\\label{eq:E\\_D6}\n\\widetilde E\\_\\xi(\\widetilde Q;D\\_6)\\leq \\frac \\pi 2 \\ln\\frac \\sigma\\xi + \\pi \\sigma\\ln\\frac 1\\xi +C.$$ \nGathering -, we obtain\n \n$$\\widetilde E\\_\\xi(\\widetilde Q;D)\\leq \\frac\\pi 2\\ln\\frac 1\\xi + \\frac{\\pi}{2}\\ln\\frac 1\\sigma + \\pi\\sigma\\ln\\frac 1\\xi + C.$$ \nOptimizing in *σ*, we are led to choosing\n \n$$\\sigma = \\frac{1}{2\\ln\\frac 1\\xi},$$ \nand conclude that\n \n$$\\widetilde E\\_\\xi(\\widetilde Q;D)\\leq \\frac\\pi 2\\ln\\frac 1\\xi + \\frac{\\pi}{2} \\ln\\ln \\frac 1\\xi + C,$$ \nwhich, upon applying the mirror symmetry, proves part (*i*) of Theorem [thm:main].\nLower bound\n===========\nIn this section we prove the lower bound. We use the notation $\\lesssim$ to denote inequality up to a universal multiplicative constant. We denote by $\\widetilde Q\\_\\xi$ the map defined by\n \n$$Q\\_\\xi(\\rho,\\varphi,z)=R\\_\\varphi \\widetilde Q\\_\\xi(\\rho,z) R\\_{\\varphi}^t,$$ \nwhich satisfies in addition the mirror symmetry. The map $\\widetilde Q\\_\\xi$ is thus defined in\n \n$$\\widetilde \\Omega = \\lbrace (\\rho,z)\\colon \\rho^2+z^2>1,\\, \\rho>0\\rbrace,$$ \nuniquely determined by its values in the region\n \n*D* = {*ρ*2 + *z*2 > 1: *ρ*, *z* > 0}, \nand minimizes\n \n$$\\widetilde E\\_\\xi(\\widetilde Q) = \\int\\_{D}\\left({{\\vert \\nabla\\widetilde Q\\vert}}^2 + \\frac{1}{\\rho^2}\\Xi[\\widetilde Q] + \\frac{1}{\\xi^2}f(\\widetilde Q) \\right)\\rho\\, d\\rho dz$$ \nunder the boundary constraints\n \n$$\\begin{aligned}\n\\widetilde Q(\\rho,z)&= e\\_r\\otimes e\\_r -\\frac 13 I\\qquad\\text{ for }\\rho^2+z^2=1,\\qquad\\text{where } e\\_r = (\\rho,0,z),\\\\\n\\text{ and }\\widetilde Q(\\rho,0)&=S \\widetilde Q(\\rho,0) S^t\\qquad\\text{ for }\\rho>1.\\end{aligned}$$ \nFor any $X\\in \\widetilde\\Omega$ we will denote by *B*(*X*, *r*) the disc of radius *r* > 0 centered at *X*.\nNote that our potential *f* satisfies\n \n$$\\label{eq:fdistU\\*}\nf(Q)\\lesssim \\operatorname{dist}^2(Q,\\mathcal U\\_\\star)\\lesssim f(Q)\\qquad\\text{for }{{\\vert Q\\vert}}\\lesssim 1,$$ \nhence we may fix *η* > 0 such that in the region $\\lbrace Q\\in\\mathcal S\\_0\\colon f(Q) < 2 \\eta\\rbrace$, the nearest neighbor projection *π* onto the smooth submanifold $\\mathcal U\\_\\star$ is well defined and smooth.\n[l:firstbounds] We have\n \n$${{\\Vert Q\\_\\xi\\Vert}}\\_{L^\\infty(\\Omega)}\\lesssim 1\\quad\\text{and}\\quad {{\\Vert \\nabla Q\\_\\xi\\Vert}}\\_{L^\\infty}\\lesssim \\frac 1\\xi.$$ \nThe *L*∞ bound is proved in, and the gradient bound follows from rescaled elliptic estimates.\nAway from the vertical axis of symmetry the energy is almost two-dimensional, and we exploit this observation to use *η*-compactness methods developed for the Ginzburg-Landau functional by Struwe.\n[l:boundpotentialsmallballs] For any *α* ∈ (0, 1] there exists *C**α* > 0 such that for any $X=(\\rho\\_0,z\\_0)\\in \\widetilde\\Omega\\cap\\lbrace \\rho\\geq \\frac 12\\rbrace$,\n \n$$\\int\\_{\\widetilde\\Omega\\cap B(X,\\xi^\\alpha)}\\left[ \\frac{1}{\\rho^2}\\Xi({\\widetilde Q\\_\\xi}) \n+\\frac{1}{\\xi^2} f(\\tilde Q) \\right]\\rho \\, d\\rho dz\\leq C\\_\\alpha.$$ \nFix any $X=(\\rho\\_0,z\\_0)\\in \\widetilde\\Omega\\cap\\lbrace \\rho\\geq \\frac 12\\rbrace$. Define \n$$F(r)=F(r; X):= r\\int\\_{\\partial B(X,r)\\cap \\widetilde{\\Omega}}\n \\left[ |\\nabla {\\widetilde Q\\_\\xi}|^2 + \\frac{1}{\\rho^2}\\Xi({\\widetilde Q\\_\\xi}) +\\frac{1}{\\xi^2} f({\\widetilde Q\\_\\xi})\\right] \\rho \\, ds\\_r,$$ \n where *d**s**r* denotes arclength measure on ∂*B*(*X*, *r*).\nFix any *β* ∈ (0, *α*). As in the proof of, by Fubini’s Theorem there exists *r**ξ* ∈ (*ξ**α*, *ξ**β*) for which\n \n$$\\begin{aligned}\n{{\\widetilde E}\\_\\xi}({\\widetilde Q\\_\\xi}) \\ge {{\\widetilde E}\\_\\xi}({\\widetilde Q\\_\\xi}; B(X,\\xi^\\beta) )\n \\ge \\int\\_{\\xi^\\alpha}^{\\xi^\\beta} F(r) \\frac{dr}{r} \n \\ge (\\beta-\\alpha) F(r\\_\\xi) \\ln \\frac 1\\xi.\\end{aligned}$$ \nIn particular,\n \n$$\\label{eq:Fbound}\nF(r\\_\\xi) \\le (\\beta-\\alpha) \\frac{ {{\\widetilde E}\\_\\xi}({\\widetilde Q\\_\\xi}; B(X,\\xi^\\beta)) }{ |\\ln\\xi|}\n \\le (\\beta-\\alpha) \\frac{ {{\\widetilde E}\\_\\xi}({\\widetilde Q\\_\\xi}) }{ |\\ln\\xi|}\\le C\\_1.$$ \nTo obtain the desired bound we require a version of the Pohozaev identity. We treat our problem as if it were two dimensional, with domain ${\\widetilde\\Omega}$ and a nonconstant weight function *w* = *ρ*. Consider a solution *u* of the equation \n$$-\\frac{1}{ w}\\nabla\\cdot w\\nabla u + \\partial\\_u g(x,u)=0 \\qquad x\\in {{\\mathbb R}}^2.$$ \n For a smooth vector field *Y*, we multiply by *Y* ⋅ ∇*u* and integrate by parts on $D\\subset{\\widetilde\\Omega}$, to obtain:\n \n$$\\begin{gathered}\n\\int\\_{D} \\left[ \\frac12 \\nabla\\cdot(w Y) |\\nabla u|^2 - w\\, (\\partial\\_j Y\\_k) \\partial\\_j u \\partial\\_k u \\right] dx \\\\\n\\qquad + \\int\\_{D} \\left[ g(x,u)\\nabla\\cdot (wY) \n+ \\partial\\_x g(x,u)\\cdot wY\\right] dx \\\\\n= \\int\\_{\\partial D} \\left[ g(x,y) Y\\cdot\\nu + \\frac12 Y\\cdot\\nu |\\nabla u|^2\n - (\\nabla u\\cdot \\nu)(Y\\cdot \\nabla u)\\right] w\\, ds.\\end{gathered}$$ \nIn our case, we choose the domain $D=D\\_r= B(X,r)\\cap\\widetilde{\\Omega}$, and the vector field *Y* = (*ρ* − *ρ*0, *z* − *z*0). The Euler-Lagrange equations have the form\n \n$$\\label{ELeq} -\\Delta {\\widetilde Q\\_\\xi}+ \\frac 12 \\partial\\_Q \\left(\\frac{1}{\\rho^2}\\Xi({\\widetilde Q\\_\\xi}) \n + \\frac{1}{ \\xi^{2}} f({\\widetilde Q\\_\\xi})\\right)= L({\\widetilde Q\\_\\xi}),$$ \nwhere $L=\\frac 16 |\\widetilde Q|^2 \\mathbf{I}$ is a Lagrange multiplier due to the vanishing trace condition. Since the test functions $Y\\cdot\\nabla {\\widetilde Q\\_\\xi}$ are trace-free, (as observed in,) $L({\\widetilde Q\\_\\xi})$ plays no role in the resulting identity, and so we may take $g(x,u)= g(\\rho, {\\widetilde Q\\_\\xi})= \\frac{1}{2\\rho^2}\\Xi({\\widetilde Q\\_\\xi}) \n + \\frac{1}{2\\xi^{2}} f({\\widetilde Q\\_\\xi})$. Substituting in the above identity, with $u=\\widetilde{Q}\\_{\\xi,ij}$ and summing over *i*, *j*, we obtain:\n \n$$\\begin{gathered}\n\\label{Id2}\nI\\_1:=\\int\\_{D\\_r} (3\\rho-\\rho\\_0) g(\\rho,{\\widetilde Q\\_\\xi})d\\rho\\, dz + \n \\int\\_{D\\_r} \\left[ \\frac12 |\\nabla{\\widetilde Q\\_\\xi}|^2 \n -\\frac{1}{\\rho^2} \\Xi({\\widetilde Q\\_\\xi}) \\right](\\rho-\\rho\\_0)\\, d\\rho\\, dz \\\\\n= \\int\\_{\\partial D\\_r} \\left[ g(\\rho,{\\widetilde Q\\_\\xi}) (Y\\cdot\\nu) + \\frac12 (Y\\cdot\\nu) |\\nabla {\\widetilde Q\\_\\xi}|^2\n - (\\nabla{\\widetilde Q\\_\\xi}\\cdot\\nu)(Y\\cdot\\nabla {\\widetilde Q\\_\\xi})\\right] \\rho\\, ds\\_r=: I\\_2.\\end{gathered}$$ \nRecalling *X* = (*ρ*0, *z*0) with $\\rho\\_0\\ge \\frac12$, we will apply the above identity (here and in the next lemma) for *r* ∈ (*ξ*, *ξ**β*), and so for *ξ* sufficiently small the domain *D**r* will be strongly starshaped, in the sense that $Y\\cdot \\nu\\ge \\frac{r}{4}$ on ∂*D**r*. Following, the right-hand side of may be estimated as:\n \n$$\\label{I2}\n I\\_2 \\le C\\, F(r) + C\\, r\\int\\_{\\partial{\\widetilde\\Omega}\\cap B(X,r)} |\\nabla Q\\_b|^2\\, ds\n \\le C\\, F(r) + O(r^2),$$ \nwith constant *C* independent of *X*, and recalling ${\\widetilde Q\\_\\xi}=Q\\_b$ on ∂*B*(0, 1). For the left-hand side, we note that in *D**r*, ∣*ρ* − *ρ*0∣ < *r* ≤ 3*ρ**ξ**β*, and hence \n$$I\\_1 \\ge 2\\int\\_{D\\_{r}} g(\\rho,{\\widetilde Q\\_\\xi}) \\, \\rho \\, d\\rho\\, dz - O(\\xi^\\beta |\\ln\\xi|).$$ \n Thus, we have for any *r* ∈ (*ξ*, *ξ**β*),\n \n$$\\label{nicebound}\n \\int\\_{D\\_{r}} g(\\rho,{\\widetilde Q\\_\\xi}) \\, \\rho \\, d\\rho\\, dz \\le C\\, F(r) + O(\\xi^\\beta |\\ln \\xi|).$$ \nChoosing *r* = *r**ξ* as in, and noting *B*(*X*, *ξ**α*) ⊂ *B*(*X*, *r**ξ*), the desired inequality is established.\nThe following is an adaptation of the *η*-compactness (*η*-ellipticity) condition to our setting.\n[l:pointwiseboundpotential] There exists *γ* > 0 such that, for any *α* ∈ (0, 1] there is *ξ*0(*α*) > 0 with the following property. If *ξ* ∈ (0, *ξ*0) and *r* ∈ [*ξ*, *ξ**α*] are such that\n \n$$F(r;X) \\leq \\gamma \\rho(X)\\qquad\\text{for some }X\\in \\widetilde\\Omega\\cap\\lbrace \\rho\\geq \\frac 12\\rbrace,$$ \nthen $f(\\widetilde Q)\\leq\\eta$ in $B\\_r(X)\\cap \\widetilde\\Omega$.\nThe proof is as in. Suppose the contrary: there exists *X*ʹ = (*ρ*ʹ, *z*ʹ) ∈ *B*(*X*, *r*) for which $f({\\widetilde Q\\_\\xi}(X'))>\\eta$. By Lemma [l:firstbounds], there exists *c* > 0 for which $f({\\widetilde Q\\_\\xi}(x))>\\eta/2$ for all *x* ∈ *B*(*X*ʹ, *c**ξ*). Thus, there is a constant *C*0 > 0, independent of *X*, *ξ*, for which\n \n$$\\label{contradict}\n\\int\\_{B(X',c\\xi)\\cap{\\widetilde\\Omega}} \\frac{1}{\\xi^2} f({\\widetilde Q\\_\\xi})\\, \\rho\\, d\\rho\\, dz \\ge C\\_0\\rho(X').$$ \nOn the other hand, by we then have \n$$C\\_0\\rho(X')\\le \\int\\_{B(X',c\\xi)\\cap{\\widetilde\\Omega}} \\frac{1}{\\xi^2} f({\\widetilde Q\\_\\xi})\\, \\rho\\, d\\rho\\, dz\n \\le \\int\\_{D\\_{r}} g(\\rho,{\\widetilde Q\\_\\xi}) \\, \\rho \\, d\\rho\\, dz \\le C\\, F(r) \\le\\gamma\\rho(X).$$ \n For any *γ* < *C*0 this is impossible, as ∣*ρ*(*X*ʹ) − *ρ*(*X*)∣ < *r* ≪ 1, and hence the conclusion must hold.\nAs in the Ginzburg-Landau case, we may now define the “bad balls” which contain the eventual defects:\n[l:boundedbadballs] There exist $M\\_0\\in\\mathbb N$ and *A*0 ≥ 1 such that\n \n$$\\lbrace f(\\widetilde Q\\_\\xi)>\\eta\\rbrace\\subset \\lbrace \\rho\\leq A\\_0\\rbrace,$$ \nand for any disjoint collection of balls $\\lbrace B(X\\_j,\\frac\\xi 5)\\rbrace\\_{j\\in J}$ with centers\n \n$$X\\_j\\in \\lbrace f(\\widetilde Q\\_\\xi)>\\eta\\rbrace \\cap \\lbrace \\rho\\geq \\frac 12\\rbrace,$$ \nthe cardinality of *J* must be ≤ *M*0.\nFor the first assertion, let $X'\\in S\\_\\xi:=\\{x\\in{\\widetilde\\Omega}: \\ f({\\widetilde Q\\_\\xi}(x))>\\eta)\\}$. As in the proof of Lemma [l:pointwiseboundpotential], there exists *c* > 0 for which $f({\\widetilde Q\\_\\xi}(x))>\\eta/2$ in *B*(*X*ʹ, *c**ξ*), with the same lower bound. Taking *r* = *r**ξ* as in, we obtain *C*0*ρ*(*X*ʹ) ≤ *C**F*(*r**ξ*) ≤ *C*ʹ, and hence *ρ*(*X*ʹ) is uniformly bounded in *ξ*.\nThe second assertion now follows as in the proof of. Indeed, since for all *X* ∈ *S**ξ*, $\\frac12\\le \\rho(X)\\le A\\_0$ by the first assertion, the value of *γ* in Lemma [l:pointwiseboundpotential] may be chosen independently of *X* ∈ *S**ξ* to give an upper bound which is uniform in *X*. Similarly, the lower bound on the potential term in balls *B*(*X*, *c**ξ*), may be chosen independently of *X* ∈ *S**ξ*. Hence, the identical arguments (based on Vitali covering of *S**ξ*) as in assure the bounded cardinality of the collection of “bad balls” $\\lbrace B(X\\_j,\\frac\\xi 5)\\rbrace\\_{j\\in J}$.\nRecall that given any Lipschitz simply connected bounded domain $R\\subset{{\\mathbb R}}^2$ and any continuous map $U\\colon \\partial R\\to \\mathcal U\\_\\star$ we can consider its homotopy class in $\\pi\\_1(\\mathcal U\\_\\star)\\approx\\mathbb Z/2\\mathbb Z$. A loop is trivial in $\\pi\\_1(\\mathcal U\\_\\star)$ if and only if it is orientable, i.e. it is of the form $\\gamma\\otimes \\gamma-\\frac 13 I$, for some continuous loop $\\gamma\\colon \\mathbb S^1\\to\\mathbb S^2$.\n[l:nontrivialrectangle] Consider for some *z*0 ∈ (0, 1/2] and *ρ*0 ≥ *A*0 the domain\n \n$$R=\\lbrace {{\\vert z\\vert}}<z\\_0,\\; \\rho< \\rho\\_0\\rbrace\\cap \\widetilde\\Omega.$$ \nand assume that $f(\\widetilde Q\\_\\xi)\\leq \\eta$ on ∂*R* and that $\\widetilde Q\\_\\xi$ restricted to ∂*R* is continuous. Then the projected map\n \n$$U\\_\\xi=\\pi(\\widetilde Q\\_\\xi)\\colon\\partial R\\to \\mathcal U\\_\\star,$$ \nhas non trivial homotopy class in $\\pi\\_1(\\mathcal U\\_\\star)$, that is, *U**ξ* is non-orientable.\n[Proof of Lemma [l:nontrivialrectangle]] Recall that $Q\\_\\xi\\in \\mathcal H\\_{sym}$, hence\n \n$$\\int\\_{\\widetilde\\Omega}\\left({{\\vert \\nabla\\widetilde Q\\_\\xi\\vert}}^2 + \\frac{{{\\vert \\widetilde Q\\_\\xi-Q\\_\\infty\\vert}}^2}{\\rho^2+z^2}\\right) \\rho d\\rho dz <\\infty.$$ \nIn particular, for any *ξ* ∈ (0, *ξ*0) and any *δ* > 0 we may choose *ρ*1 ≥ *ρ*0 such that\n \n$$\\int\\_{-1}^1 {{\\vert \\partial\\_z \\widetilde Q\\_\\xi(\\rho\\_1,z)\\vert}}^2 dz + \\int\\_{-1}^1{{\\vert \\widetilde Q\\_\\xi(\\rho\\_1,z)-Q\\_\\infty\\vert}}^2 dz <\\delta,$$ \nwhich implies\n \n$$\\label{eq:Qrho1}\n{{\\vert \\widetilde Q\\_\\xi(\\rho\\_1,z)-Q\\_\\infty\\vert}}^2\\lesssim\\delta\\quad\\forall z\\in [-1,1].$$ \nWe denote by *R*1 the domain\n \n$$R\\_1=\\widetilde\\Omega\\cap\\lbrace {{\\vert z\\vert}}<z\\_0,\\, \\rho <\\rho\\_1\\rbrace,$$ \nand define $V\\_\\xi=\\pi(\\widetilde Q\\_\\xi)\\colon\\partial R\\_1\\to\\mathcal U\\_\\star$. Since $\\pi(\\widetilde Q\\_\\xi)$ is well defined and has finite energy in *R*1 \\ *R*0, the maps *U**ξ* and *V**ξ* are homotopically equivalent. To prove Lemma [l:nontrivialrectangle] it thus suffices to prove that *V**ξ* is non orientable. Assume that *V**ξ* is orientable: there exists a continuous $n\\colon \\partial R\\_1\\to\\mathbb S^2$ such that\n \n$$V\\_\\xi = n\\otimes n -\\frac 13 I.$$ \nSince *n* is uniquely defined up to a sign and $V\\_\\xi=e\\_1\\otimes e\\_1-\\frac 13 I$ at (*ρ*, *z*) = (1, 0), we may assume that *n*(1, 0) = *e*1. The symmetry assumption implies that\n \n*n*(*ρ*, − *z*) = *τ**S**n*(*ρ*, *z*) for some *τ* ∈ { ± 1}. \nEvaluating this at (*ρ*, *z*) = (1, 0) gives *e*1 = *τ**S**e*1 = *τ**e*1, hence *τ* = 1. This implies that at (*ρ*, *z*) = (*ρ*1, 0) one must have *n*(*ρ*1, 0) = *S**n*(*ρ*1, 0), i.e. *n*(*ρ*1, 0) ⊥ *e*3, and therefore |*V**ξ*(*ρ*1, 0) − *Q*∞|2 = 2. For small enough *δ* this contradicts.\nThe next lemmas deal with universal lower bounds in annular regions.\n[l:lowerboundannulus] There exists *C* > 0 such that for any *ξ* > 0, for any annulus $\\omega\\subset{{\\mathbb R}}^2$ of the form\n \n$$\\omega = B(0,R)\\setminus B(0,r),\\qquad 0<r<R\\leq \\frac 12,$$ \nand any *H*1 map $Q\\colon\\overline \\omega\\to\\mathcal S\\_0$ satisfying *f*(*Q*) ≤ *η* in *ω* and such that $U=\\pi(Q)\\colon\\overline\\omega\\to\\mathcal U\\_\\star$ is continuous and nonorientable on ∂*B*(0, *R*), we have\n \n$$\\int\\_\\omega \\left({{\\vert \\nabla Q\\vert}}^2+\\frac{1}{\\xi^2}f(Q)\\right) \\geq \\pi \\ln \\frac{R}{r} - C \\xi\\left(\\frac 1 r-\\frac 1 R\\right).$$ \n[Proof of Lemma [l:lowerboundannulus]] This is proved in and, but for completeness we include here a proof, following the method in.\nLet $D={{\\vert Q-U\\vert}}=\\operatorname{dist}(Q,\\mathcal U\\_\\*)$, and denote by $P\\colon\\mathcal U\\_\\star\\to \\mathcal L(\\mathcal S\\_0)$ the smooth map given by $P(u)=P\\_{(T\\_u\\mathcal U\\_\\star)^\\perp}$ the orthogonal projection onto the normal space to $\\mathcal U\\_\\star$ at *u*. Note that by definition *Q* − *U* = *P*(*U*)(*Q* − *U*). Then for any direction *k* we compute\n \n$$\\begin{aligned}\n{{\\vert \\partial\\_k Q\\vert}}^2 &={{\\vert \\partial\\_k U\\vert}}^2 +{{\\vert \\partial\\_k (Q-U)\\vert}}^2 + 2\\partial\\_k U\\cdot \\partial\\_k (Q-U)\\\\\n& \\geq {{\\vert \\partial\\_k U\\vert}}^2 +{{\\vert \\partial\\_k D\\vert}}^2 + 2\\partial\\_k U\\cdot \\partial\\_k \\left[ P(U)(Q-U)\\right]\\\\\n&= {{\\vert \\partial\\_k U\\vert}}^2 +{{\\vert \\partial\\_k D\\vert}}^2 + 2\\partial\\_k U\\cdot \\partial\\_k \\left[ P(U) \\right] (Q-U) + 2\\partial\\_k U\\cdot P(U) \\partial\\_k(Q\\_U).\\end{aligned}$$ \nThe last term is zero because $\\partial\\_k U\\in T\\_U\\mathcal U\\_\\star$ and therefore *P*(*U*)∂*k**U* = 0. So we have\n \n$$\\begin{aligned}\n{{\\vert \\partial\\_k Q\\vert}}^2 \n& \\geq {{\\vert \\partial\\_k U\\vert}}^2 +{{\\vert \\partial\\_k D\\vert}}^2 - 2{{\\Vert DP(U)\\Vert}}D {{\\vert \\partial\\_k U\\vert}}^2 \\\\\n&\\geq (1-c D){{\\vert \\partial\\_k U\\vert}}^2 +{{\\vert \\partial\\_k D\\vert}}^2,\\end{aligned}$$ \nfor $c=2\\sup\\_{\\mathcal U\\_\\star}{{\\Vert DP(U)\\Vert}}$. This computation is very similar to and. Recalling now that *f*(*Q*) ≥ *α*2*D*2 for some *α* > 0 we find\n \n$$\\begin{aligned}\n{{\\vert \\partial\\_k Q\\vert}}^2+\\frac{1}{\\xi^2}f(Q) &\\geq (1-c D){{\\vert \\partial\\_k U\\vert}}^2 + {{\\vert \\partial\\_k D\\vert}}^2 +\\frac{\\alpha^2}{\\xi^2}D^2 \\\\\n& \\geq (1-cD){{\\vert \\partial\\_k U\\vert}}^2 + \\frac{\\alpha}{\\xi}{{\\vert \\partial\\_k [D^2]\\vert}}.\\end{aligned}$$ \nHence for any *s* ∈ (0, *R*) we have, letting *d*(*s*) = max∂*B*(0, *s*)*D*,\n \n$$\\begin{aligned}\n\\int\\_{\\partial B(0,s)}\\left({{\\vert \\nabla Q\\vert}}^2 +\\frac{1}{\\xi^2}f(Q) \\right) &\n\\geq (1-c d(s))\\int\\_{\\partial B(0,s)}{{\\vert \\partial\\_\\tau U\\vert}}^2\n + \\frac{\\alpha}{\\xi}d(s)^2\\end{aligned}$$ \nSince *U* is *H*1 in *ω* and non orientable on ∂*B*(0, *R*), it is continuous and nonorientable on ∂*B*(0, *s*) for a.e. *s* ∈ [*r*, *R*], and for such *s* we have (see e.g. )\n \n$$\\int\\_{\\partial B(0,s)}{{\\vert \\partial\\_\\tau U\\vert}}^2 \\geq \\frac{\\pi}{s},$$ \nand therefore\n \n$$\\begin{aligned}\n\\int\\_{\\partial B(0,s)}\\left({{\\vert \\nabla Q\\vert}}^2 +\\frac{1}{\\xi^2}f(Q) \\right) &\n\\geq \\frac{\\pi}{s}(1-c d(s)) + \\frac{\\alpha}{\\xi}d(s)^2 \\\\\n&\\geq \\inf\\_{d\\in [0,1]}\\left( \\frac{\\pi}{s}(1-c d) + \\frac{\\alpha}{\\xi}d^2\\right)\\\\\n&=\\frac\\pi s -\\frac{\\xi}{s^2}\\cdot \\begin{cases}\n\\frac{c^2\\pi^2}{4\\alpha} &\\quad\\text{if }\\frac{\\xi}{s}\\leq \\frac{c\\pi}{2\\alpha},\\\\\n\\frac{s}{\\xi}\\left(c\\pi -\\alpha\\frac{s}{\\xi}\\right) &\\quad\\text{if }\\frac\\xi s\\geq \\frac{c\\pi}{2\\alpha}\n\\end{cases}\\\\\n&\\geq \\frac \\pi s -\\frac{c^2\\pi^2}{4\\alpha}\\frac{\\xi}{s^2}.\\end{aligned}$$ \nIntegrating we deduce\n \n$$\\begin{aligned}\n\\int\\_\\omega \\left({{\\vert \\nabla Q\\vert}}^2 +\\frac{1}{\\xi^2}f(Q) \\right) &\n\\geq \\pi\\ln\\frac Rr -\\frac{c^2\\pi^2}{4\\alpha}\\xi\\left(\\frac 1r-\\frac 1R\\right).\\end{aligned}$$ \nUsing Lemma [l:lowerboundannulus] and the growing ball construction of Jerrard or Sandier (which is adapted to our setting in ), one obtains the following lower bound on perforated domains:\n[l:lowerboundperforated] There exists *C* > 0 such that for any *ξ* ∈ (0, 1], for any perforated domain $\\omega\\subset{{\\mathbb R}}^2$ of the form\n \n$$\\omega = B(0,R)\\setminus \\bigcup\\_{j=1}^N B(x\\_j,r),\\qquad B(x\\_j,r)\\subset B(0,\\frac R2)\\text{ disjoint}, \\; r\\geq \\xi,$$ \nand any *H*1 map $Q\\colon\\overline \\omega\\to\\mathcal S\\_0$ satisfying *f*(*Q*) ≤ *η* in *ω* and such that $U=\\pi(Q)\\colon\\overline\\omega\\to\\mathcal U\\_\\star$ is continuous and nonorientable on ∂*B*(0, *R*), we have\n \n$$\\int\\_\\omega \\left({{\\vert \\nabla Q\\vert}}^2+\\frac{1}{\\xi^2}f(Q)\\right) \\geq \\pi \\ln \\frac{R}{Nr} - C.$$ \nWe will also need a boundary version of Lemma [l:lowerboundannulus]:\n[l:lowerboundhalfannulus] There exists *C* > 0 such that for any *ξ* > 0, for any annulus $\\omega\\subset{{\\mathbb R}}^2$ of the form\n \n$$\\omega = B(0,R)\\setminus B(0,r),\\qquad 0<r<R\\leq \\frac 12,$$ \nand any *H*1 map $Q\\colon\\overline \\omega\\to\\mathcal S\\_0$ satisfying *f*(*Q*) ≤ *η* in *ω* and such that $U=\\pi(Q)\\colon\\overline\\omega\\to\\mathcal U\\_\\star$ is continuous and nonorientable on ∂*B*(0, *R*), if in addition *Q* = *U*0 on the left half *ω*− = *ω* ∩ {(*x*1, *x*2): *x*1 < 0} of the annulus, for some Lipschitz $U\\_0\\colon {{\\mathbb R}}^2\\to \\mathcal U\\_\\star$ with |∇*U*0| ≤ 1, then for any function *w* such that *w*(*x*) ≥ 1 − |*x*|2,\n \n$$\\int\\_{\\omega\\_+} \\left({{\\vert \\nabla Q\\vert}}^2+\\frac{1}{\\xi^2}f(Q)\\right)w(x)\\, dx \\geq 2\\pi \\ln \\frac{R}{r} - C \\left(1+ \\frac{\\xi}{r}\\right),$$ \nwhere *ω*+ = *ω* ∩ {(*x*1, *x*2): *x*1 > 0} is the right half of the annulus.\n[Proof of Lemma [l:lowerboundhalfannulus]] We claim that\n \n$$\\begin{aligned}\n\\label{eq:halfgeod}\n\\int\\_{\\partial B(0,s)\\cap\\omega\\_+}{{\\vert \\partial\\_\\tau U\\vert}}^2\\geq \\frac{2\\pi}{s}-16\\qquad\\text{for a.e. }s\\in [r,R].\\end{aligned}$$ \nThen, going through the computations in the proof of Lemma [l:lowerboundannulus] gives\n \n$$\\begin{aligned}\n\\int\\_{\\omega\\_+}\\left({{\\vert \\nabla Q\\vert}}^2 +\\frac{1}{\\xi^2}f(Q) \\right)w(x)\\,dx &\n\\geq \\int\\_{r}^{R}\\left(\\frac{2\\pi}{s} -\\frac{c^2\\pi^2}{\\alpha}\\frac{\\xi}{s^2}-16\\right)(1-s^2)\\, s\\, ds\n\\\\\n&\\geq 2\\pi\\ln\\frac Rr -C\\left(\\frac{c^2\\pi^2}{\\alpha}\\frac\\xi r +1\\right),\\end{aligned}$$ \nwhich proves Lemma [l:lowerboundhalfannulus], provided we show. We fix *s* ∈ [*r*, *R*] such that *U*⌊∂*B*(0, *s*) is in *H*1 and non-orientable. Since ${{\\mathbb R}}^2$ and $\\partial B(0,s)\\cap\\overline\\omega\\_+$ are simply connected we may orient *U*0 and *U* on these domains, i.e. write\n \n$$\\begin{aligned}\nU\\_0&=n\\_0\\otimes n\\_0-\\frac 13 I, & n&\\colon{{\\mathbb R}}^2\\to\\mathbb S^2,\\\\\nU(se^{i\\theta})&=n\\_s(\\theta)\\otimes n\\_s(\\theta)-\\frac 13 I & n\\_s &\\colon [-\\frac\\pi 2,\\frac\\pi 2]\\to\\mathbb S^2.\\end{aligned}$$ \nSince $U(se^{-i\\frac\\pi 2})=U\\_0(se^{-i\\frac\\pi 2})$ we may, up to switching the orientation of *n*0, assume that\n \n$$n\\_s(-\\frac\\pi 2)=n\\_0(se^{-i\\frac\\pi 2}).$$ \nSince $U(se^{i\\frac\\pi 2})=U\\_0(se^{i\\frac\\pi 2})$ we have $n\\_s(\\frac\\pi 2)=\\pm n\\_0(se^{\\frac i\\pi 2})$, and because *U* is non-orientable it must be\n \n$$n\\_s(\\frac\\pi 2)=-n\\_0(se^{i\\frac\\pi 2}).$$ \nNote also that, as ${{\\vert \\nabla n\\_0\\vert}}=\\frac 1{\\sqrt 2}{{\\vert \\nabla U\\_0\\vert}}\\leq 1$, we have\n \n$${{\\vert n\\_0(se^{-i\\frac\\pi 2})-n\\_0(e^{i\\frac\\pi 2})\\vert}}\\leq 2s\\leq 1,$$ \nwhich implies that\n \n$$\\operatorname{dist}\\_{\\mathbb S^2}(n\\_0(se^{-i\\frac\\pi 2}),-n\\_0(se^{i\\frac\\pi 2}))\\geq \\pi- 4 s$$ \nHence we find\n \n$$\\begin{aligned}\n\\int\\_{\\partial B(0,s)\\cap\\omega\\_+}{{\\vert \\partial\\_\\tau U\\vert}}^2 &=\\frac 1s \\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}2{{\\vert n\\_s'(\\theta)\\vert}}^2\\,d\\theta \\\\\n&\\geq \\frac 1s \\frac 2{\\pi} \\left(\\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}{{\\vert n\\_s'(\\theta)\\vert}}d\\theta\\right)^2\\\\\n&\\geq \\frac 1s \\frac{2}{\\pi}\\operatorname{dist}\\_{\\mathbb S^2}(n\\_s(-\\frac\\pi 2),n\\_s(\\frac\\pi 2))^2 \\\\\n&\\geq\\frac 1s \\frac{2\\pi^2 -16\\pi s + 32 s^2}{\\pi}\\\\\n&\\geq \\frac{2\\pi}{s}-16,\\end{aligned}$$ \nthus proving.\nIn Section [s:lim] we will also need the following refinement of Lemma [l:lowerboundhalfannulus]. In particular, we show that (to *O*(1)) we obtain the same lower bound in a smaller domain which pulls away from the boundary of the half-disk along a curve *x*1 = *λ**x*2*β* with *λ* > 0 and *β* > 1.\n[l:finelowerboundhalfannulus] There exists *C* > 0 such that the following holds. Let *ξ* > 0, $\\omega\\subset{{\\mathbb R}}^2$ of the form\n \n$$\\omega = B(0,R)\\setminus B(0,r),\\qquad 0<r<R\\leq \\frac 12,$$ \n$Q\\colon\\overline \\omega\\to\\mathcal S\\_0$ an *H*1 map satisfying *f*(*Q*) ≤ *η* in *ω*, such that $U=\\pi(Q)\\colon\\overline\\omega\\to\\mathcal U\\_\\star$ is continuous and nonorientable on ∂*B*(0, *R*), and *Q* = *U*0 on the left half *ω*− = *ω* ∩ {(*x*1, *x*2): *x*1 < 0} of the annulus, for some Lipschitz $U\\_0\\colon {{\\mathbb R}}^2\\to \\mathcal U\\_\\star$ with |∇*U*0| ≤ 1. Let also *w* be a function such that *w*(*x*) ≥ 1 − |*x*|2. Then, in the right half annulus *ω*+ = *ω* ∩ {(*x*1, *x*2): *x*1 > 0}, we have the lower bound provided by Lemma [l:lowerboundhalfannulus]. If *Q* also satisfies a matching upper bound, in the sense that\n \n$$\\begin{aligned}\n\\int\\_{\\omega\\_+} \\left({{\\vert \\nabla Q\\vert}}^2+\\frac{1}{\\xi^2}f(Q)\\right)w(x)\\, dx \\leq 2\\pi \\ln \\frac{R}{r} +K,\\end{aligned}$$ \nfor some *K* > 0, then the lower bound is also valid in the slightly smaller domain\n \n$$\\tilde\\omega\\_\\beta =\\omega\\cap \\lbrace (x\\_1,x\\_2)\\colon x\\_1>\\lambda x\\_2^\\beta\\rbrace,\\qquad\\lambda>0,\\beta>1,$$ \nin the sense that\n \n$$\\int\\_{\\tilde\\omega\\_\\beta} \\left({{\\vert \\nabla Q\\vert}}^2+\\frac{1}{\\xi^2}f(Q)\\right)w(x)\\, dx \\geq 2\\pi \\ln \\frac{R}{r} - C \\left(1+ \\frac{\\xi}{r}\\right) -CK-\\frac{C\\lambda}{\\beta-1}.$$ \n[Proof of Lemma [l:finelowerboundhalfannulus]] We use the same notations as in the proof of Lemma [l:lowerboundhalfannulus], and refine the lower bound obtained on each slice ∂*B*(0, *s*) ∩ *ω*+. The crucial computation is\n \n$$\\begin{aligned}\n\\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}\\left\\vert{{\\vert n'\\_s(\\theta)\\vert}}-1\\right\\vert^2\\, d\\theta &= \\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}{{\\vert n'\\_s(\\theta)\\vert}}^2\\, d\\theta -2\\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}{{\\vert n'\\_s(\\theta)\\vert}}\\, d\\theta +\\pi \\\\\n&\\leq \\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}{{\\vert n'\\_s(\\theta)\\vert}}^2\\, d\\theta -2\\operatorname{dist}\\_{\\mathbb S^2}(n\\_s(-\\frac\\pi 2),n\\_s(\\frac\\pi 2)) +\\pi \\\\\n&\\leq \\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}{{\\vert n'\\_s(\\theta)\\vert}}^2\\, d\\theta -2(\\pi-4s) +\\pi\\\\\n&\\leq \\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}{{\\vert n'\\_s(\\theta)\\vert}}^2\\, d\\theta -\\pi +8s.\\end{aligned}$$ \nThis can be interpreted as a stability estimate for geodesics in $\\mathbb S^2$: if the lower bound for the energy of a curve (minimized by constant speed geodesics) is almost saturated, then this curve’s speed must be close to constant. Plugging this back into the estimates performed in Lemma [l:lowerboundannulus], we deduce\n \n$$\\begin{aligned}\n&\\int\\_{\\partial B(0,s)\\cap\\omega\\_+}\\left({{\\vert \\nabla Q\\vert}}^2 +\\frac{1}{\\xi^2}f(Q) \\right)\\\\\n&\\geq (1-cd(s))\\left(\\frac{2\\pi}{s}-16 + \\frac{2}{s} \\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}\\left\\vert{{\\vert n'\\_s(\\theta)\\vert}}-1\\right\\vert^2\\, d\\theta\\right)+\\frac{\\alpha}{\\xi}d(s)^2,\\end{aligned}$$ \nwhere recall that *c* > 0 depends only on $\\mathcal U\\_\\star$ and $0\\leq d(s)\\lesssim \\eta$. Hence, possibly lowering *η* we can assume $cd(s)\\leq\\frac 12$, and arguing as in Lemma [l:lowerboundannulus] we obtain\n \n$$\\begin{aligned}\n&\\int\\_{\\partial B(0,s)\\cap\\omega\\_+}\\left({{\\vert \\nabla Q\\vert}}^2 +\\frac{1}{\\xi^2}f(Q) \\right)\\\\\n&\\geq (1-cd(s))\\frac{2\\pi}{s} +\\frac{\\alpha}{\\xi}d(s)^2 -16 + \\frac{1}{s} \\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}\\left\\vert{{\\vert n'\\_s(\\theta)\\vert}}-1\\right\\vert^2\\, d\\theta \\\\\n&\\geq \\frac{2\\pi}{s}-\\frac{c^2\\pi^2}{\\alpha}\\frac{\\xi}{s^2} -16 + \\frac{1}{s} \\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}\\left\\vert{{\\vert n'\\_s(\\theta)\\vert}}-1\\right\\vert^2\\, d\\theta.\\end{aligned}$$ \nIntegrating we deduce\n \n$$\\begin{aligned}\n\\int\\_{\\omega\\_+}\\left({{\\vert \\nabla Q\\vert}}^2 +\\frac{1}{\\xi^2}f(Q) \\right)w(x)\\,dx &\n\\geq 2\\pi\\ln\\frac Rr -C\\left(\\frac\\xi r +1\\right) \\\\\n&\\quad + \\int\\_{r}^{R} \\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}\\left\\vert{{\\vert n'\\_s(\\theta)\\vert}}-1\\right\\vert^2 d\\theta \\frac{ds}{s}.\\end{aligned}$$ \nCombining this with the assumption that we have a matching upper bound gives\n \n$$\\begin{aligned}\n\\label{eq:dnclosetoconstant}\n \\int\\_{r}^{R} \\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}\\left\\vert{{\\vert n'\\_s(\\theta)\\vert}}-1\\right\\vert^2 d\\theta \\frac{ds}{s} &\\leq K + C\\left(\\frac\\xi r +1\\right).\\end{aligned}$$ \nWe will use this to show that the part that we “forget” by integrating over $\\tilde\\omega\\_\\beta$ instead of *ω*+ is bounded. Indeed we have\n \n$$\\begin{aligned}\n\\int\\_{\\partial B(0,s)\\cap\\tilde\\omega\\_\\beta}{{\\vert \\partial\\_\\tau U\\vert}}^2 \n& = \\int\\_{\\partial B(0,s)\\cap\\omega\\_+}{{\\vert \\partial\\_\\tau U\\vert}}^2 \n-\\int\\_{\\partial B(0,s)\\cap\\omega\\_+\\setminus \\tilde\\omega\\_\\beta}{{\\vert \\partial\\_\\tau U\\vert}}^2 \\\\\n& \\geq \n\\frac{2\\pi}{s}-16 \\\\\n&\\quad\n-\\frac{2}{s}\\int\\_{-\\frac\\pi 2}^{-\\frac\\pi 2 + 2\\lambda s^{\\beta-1}}{{\\vert n\\_s'(\\theta)\\vert}}^2\\,d\\theta -\\frac{2}{s}\\int\\_{\\frac\\pi 2-2\\lambda s^{\\beta-1}}^{\\frac\\pi 2}{{\\vert n\\_s'(\\theta)\\vert}}^2\\,d\\theta \\\\\n&\\geq \\frac{2\\pi}{s}-16 -\\frac{2}{s}\\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}\\left\\vert{{\\vert n'\\_s(\\theta)\\vert}}-1\\right\\vert^2 d\\theta - 4\\lambda s^{\\beta-2}.\\end{aligned}$$ \nNotice that since *β* > 1 the last term is summable with respect to *s* small. Hence upon arguing as above we find\n \n$$\\begin{aligned}\n\\int\\_{\\tilde\\omega\\_\\beta}\\left({{\\vert \\nabla Q\\vert}}^2 +\\frac{1}{\\xi^2}f(Q) \\right)w(x)\\,dx &\\geq 2\\pi\\ln\\frac Rr -C\\left(\\frac\\xi r +1\\right) -\\frac{4\\lambda}{\\beta-1}\\\\\n&\\quad -2\\int\\_{r}^{R} \\int\\_{-\\frac\\pi 2}^{\\frac\\pi 2}\\left\\vert{{\\vert n'\\_s(\\theta)\\vert}}-1\\right\\vert^2 d\\theta \\frac{ds}{s},\\end{aligned}$$ \nand combining this with enables us to conclude.\nWe are finally ready to prove the lower bound on the energy stated in Theorem [thm:main].\n[Proof of Part (ii) of Theorem [thm:main]] We start by fixing a Lipschitz $\\mathcal U\\_\\star$-valued extension *U*0 of $\\widetilde Q\\_\\xi$ outside of $\\widetilde\\Omega$. In particular, for any circle ∂*B*(*a*, *r*) such that $f(\\widetilde Q\\_\\xi)<\\eta$ on $\\partial B(a,r)\\cap \\widetilde \\Omega$, it makes sense to consider $U\\_\\xi=\\pi(\\widetilde Q\\_\\xi)$ on ∂*B*(*a*, *r*) (even if part of this circle lies outside of $\\widetilde \\Omega$). Moreover *U**ξ* is orientable on ∂*B*(*a*, *r*) if and only if it is orientable on $\\partial (B(a,r)\\cap\\widetilde\\Omega)$.\nThanks to Lemma [l:boundedbadballs] and arguing as in, there exist *λ* > 0 and a family of disjoint balls *B*(*x**j**ξ*, *λ**ξ*) such that\n \n$$\\begin{aligned}\n&\\lbrace f(\\widetilde Q\\_\\xi)>\\eta\\rbrace\\cap\\lbrace\\rho\\geq \\frac 12\\rbrace\\subset\\bigcup\\_{j=1}^{J\\_\\xi} B(x\\_j^\\xi,\\lambda\\xi),\\qquad J\\_\\xi\\leq M\\_0,\\\\\n&{{\\vert x\\_i^\\xi-x\\_j^\\xi\\vert}}>8\\lambda\\xi \\qquad\\forall i\\neq j,\\\\\n&x\\_i^\\xi\\in\\lbrace z=0\\rbrace \\quad\\text{or}\\quad \\operatorname{dist}(x\\_i^\\xi,\\lbrace z=0\\rbrace)>8\\lambda\\xi\\qquad\\forall i,\\\\\n& x\\_i^\\xi\\in\\partial\\widetilde\\Omega \\quad\\text{or}\\quad \\operatorname{dist}(x\\_i^\\xi,\\partial\\widetilde\\Omega)>8\\lambda\\xi\\qquad\\forall i\\end{aligned}$$ \nLet us now consider a sequence *ξ* = *ξ**n* → 0. To keep notation simple we will not write explicitly the dependence on *n*, and will not relabel subsequences. We may extract converging subsequences of the *x**j**ξ* that lie inside $\\lbrace {{\\vert z\\vert}}\\leq \\frac 12,\\rho\\leq A\\_0\\rbrace$, and we denote by *a*1, …, *a**K* those of the limit points that lie in the axis {*z* = 0}, and $z\\_1=\\min(\\frac 12,\\frac d2)$, where *d* is the minimal distance of any other limit point to the axis {*z* = 0} or of any of the *a**j*’s to the other *a**j*’s. Then for any *δ* ∈ (0, *z*1) the balls *B*(*a**j*, *δ*) are disjoint and we have\n \n$$\\lbrace f(\\widetilde Q\\_\\xi)>\\eta\\rbrace\\cap\\lbrace{{\\vert z\\vert}}\\leq z\\_1\\rbrace\\subset\\bigcup\\_{j=1}^K B(a\\_j,\\delta)\\qquad\\text{for small enough }\\xi.$$ \nBy Fubini’s theorem we may find *z*0 ∈ [*z*1/2, *z*1] and *ρ*0 ≥ *A*0 such that $\\widetilde Q\\_\\xi$ restricted to ∂*R* is continuous, where $R=\\lbrace {{\\vert z\\vert}}<z\\_0,\\rho<\\rho\\_0\\rbrace\\cap\\widetilde \\Omega$. By the above $f(\\widetilde Q\\_\\xi)\\leq\\eta$ on ∂*R* for small enough *ξ*, so we may apply Lemma [l:nontrivialrectangle] to deduce that the $U\\_\\xi=\\pi(\\widetilde Q\\_\\xi)$ is non-orientable on ∂*R*. As a consequence, *U**ξ* must be non-orientable on ∂*B*(*a**j*, *z*1/2) for at least one index *j*.\nRelabeling we assume *j* = 1. We claim that *a*1 must be the leftmost possible point *a* = (1, 0). Assume indeed that *ρ*(*a*1) > 1 and fix *δ* ∈ (0, *z*1) such that *ρ*(*a*1) − *δ* > 1. Then applying Lemma [l:lowerboundperforated] on *B*(*a*1, *δ*) \\ ⋃*j**B*(*x**j**ξ*, *λ**ξ*) for small enough *ξ* we deduce that\n \n$$\\widetilde E\\_\\xi(\\widetilde Q\\_\\xi)\\geq (\\rho(a\\_1)-\\delta)\\pi\\ln \\frac \\delta\\xi -C,$$ \nbut since *ρ*(*a*1) − *δ* > 1 this implies that $\\widetilde E\\_\\xi(\\widetilde Q\\_\\xi)-\\pi\\ln\\frac 1\\xi\\to\\infty$ as *ξ* → 0, thus contradicting the upper bound obtained in Section [s:upper].\nGathering the above, we have that *a* = (1, 0) is a limit of some *x**j**ξ* and for any *δ* > 0, if *ξ* is small enough then $U\\_\\xi=\\pi(\\widetilde Q\\_\\xi)$ is well defined and non-orientable on ∂*B*(*a*, *δ*). Of the above defined *x**j**ξ*, we now consider only those which converge to *a*.\nSince there is a bounded number of *x**j**ξ*, arguing as in one may find a bounded *μ* > 1 and radii *σ**i**ξ* such that\n \n$$\\begin{aligned}\n& \\xi = \\sigma\\_0^\\xi \\ll \\cdots \\ll \\sigma\\_L^\\xi = \\delta,\\qquad L\\leq M\\_0,\\end{aligned}$$ \nand each ball *B*(*σ**i**ξ*, *λ**ξ*) is contained either in *B*(*a*, *μ**σ*0) or for some ℓ ≥ 1 in the annulus *B*(*a*, *μ**σ*ℓ) \\ *B*(*a*, *σ*ℓ) for ℓ ≥ 1. Inside each annulus *A*ℓ = *B*(*a*, *σ*ℓ) \\ *B*(*a*, *μ**σ*ℓ − 1) the map *U**ξ* is well defined and its homotopy class on *C**s* = ∂*B*(*a*, *s*) is constant for *s* ∈ [*μ**σ*ℓ − 1, *σ*ℓ]. We will refer to this constant homototpy class as the homotopy class of *U**ξ* on the annulus *A*ℓ. For ℓ = *L* we know that *U**ξ* is non-orientable on *A**L*.\nWe claim that there exists ℓ0 ∈ {1, …, *L*} such that *U**ξ* is orientable on *A*ℓ. Otherwise, *U**ξ* is non-orientable on all annuli *A*ℓ which lie outside *B*(*a*, *μ**ξ*). Thus, flattening the boundary $\\partial\\widetilde\\Omega$ we may for small enough *δ* apply Lemma [l:lowerboundhalfannulus] on each annulus. We deduce the lower bound\n \n$$\\begin{aligned}\n\\widetilde E(\\widetilde Q\\_\\xi;\\widetilde\\Omega\\cap B(a,\\delta)\\setminus B(a,\\mu\\xi))&\\geq \\sum\\_{\\ell=1}^L 2\\pi\\ln\\frac{\\sigma\\_\\ell}{\\mu\\sigma\\_{\\ell-1}} -C \\\\\n& \\geq 2\\pi\\ln\\frac{\\delta}{\\xi} -C,\\end{aligned}$$ \nand this contradicts the upper bound obtained in Section [s:upper].\nLet us then fix the largest ℓ0 ∈ {1, …, *L* − 1} such that *U**ξ* is orientable on *A*ℓ, and set $\\hat\\sigma\\_\\xi = \\sigma\\_{\\ell\\_0}^\\xi$. In particular *U**ξ* is non-orientable on *A*ℓ for all ℓ ≥ ℓ0 + 1, hence arguing as above we find the lower bound\n \n$$\\begin{aligned}\n\\widetilde E(\\widetilde Q\\_\\xi;\\widetilde\\Omega\\cap B(a,\\delta)\\setminus B(a,2\\mu\\hat \\sigma\\_\\xi))&\\geq \\sum\\_{\\ell=\\ell\\_0+1}^L 2\\pi\\ln\\frac{\\sigma\\_\\ell}{\\mu\\sigma\\_{\\ell-1}} -C \\nonumber\\\\\n& \\geq 2\\pi\\ln\\frac{\\delta}{\\hat\\sigma\\_\\xi} -C.\\label{eq:lowerboundlargeannulus}\\end{aligned}$$ \nMoreover, since *U**ξ* is orientable on $\\partial B(a,\\hat\\sigma\\_\\xi)$ and non-orientable on $\\partial B(a,\\mu\\hat\\sigma\\_\\xi)$, there must be at least one *x**j**ξ* such that $B(x\\_j^\\xi,\\lambda\\xi)\\subset B(a,\\mu\\hat\\sigma\\_\\xi)\\setminus B(a,\\hat\\sigma\\_\\xi)$ and *U**ξ* is non-orientable on ∂*B*(*x**j**ξ*, *λ**ξ*).\nNow we consider only those *x**j**ξ* which are in $B(a,\\mu\\hat\\sigma\\_\\xi)\\setminus B(a,\\hat\\sigma\\_\\xi)$. Considering the balls $B(x\\_j^\\xi,c\\_0\\hat\\sigma\\_\\xi)$ for some small enough *c*0 > 0 (depending only on the number of *x**j**ξ*’s) and applying the procedure in, we obtain a bounded number $c\\_0\\leq\\gamma\\leq \\frac{1}{16}$ and a collection of balls $B(y\\_j^\\xi,\\gamma\\hat\\sigma\\_\\xi)$ satisfying the following:\n \n$$\\begin{aligned}\n&\\lbrace f(\\widetilde Q\\_\\xi)>\\eta\\rbrace \\cap B(a,\\mu\\hat\\sigma\\_\\xi)\\setminus B(a,\\hat\\sigma\\_\\xi) \\subset\\bigcup\\_{j=1}^{\\hat J} B(y\\_j^\\xi,\\gamma\\hat\\sigma\\_\\xi),\\qquad\\hat J\\leq M\\_0,\\\\\n&{{\\vert y\\_i^\\xi-y\\_j^\\xi\\vert}}>8\\gamma\\hat\\sigma\\_\\xi\\qquad\\forall i\\neq j,\\\\\n& y\\_j^\\xi\\in\\lbrace z=0\\rbrace\\quad\\text{or}\\quad\\operatorname{dist}(y\\_j^\\xi,\\lbrace z=0\\rbrace)>8\\gamma\\hat\\sigma\\_\\xi.\\end{aligned}$$ \nSince all balls $B(y\\_j^\\xi,2\\gamma\\hat\\sigma\\_\\xi)$ are contained in the annulus $B(a,2\\mu\\hat\\sigma\\_\\xi)\\setminus B(a,\\hat\\sigma\\_\\xi/2)$ and for small enough *ξ* by the above *U**ξ* is non-orientable on $\\partial B(a,2\\mu\\hat\\sigma\\_\\xi)$ and orientable on $\\partial B(a,\\hat\\sigma\\_\\xi/2)$, we deduce that *U**ξ* must be non-orientable on $\\partial B(y\\_j^\\xi,2\\gamma\\hat\\sigma\\_\\xi)$ for at least one *y**j**ξ*. Relabeling we assume this is *y*1*ξ*.\nWe may apply Lemma [l:lowerboundperforated] to obtain a lower bound on the energy in $B(y\\_1^\\xi,2\\gamma\\hat\\sigma\\_\\xi)$. This ball may happen to intersect $\\partial\\widetilde\\Omega$, but considering the fixed Lipschitz extension *U*0 os $\\widetilde Q\\_\\xi$ outside $\\widetilde\\Omega$ we still have the same lower bound.\nLet us first assume that this *y*1*ξ* does not lie on the axis {*z* = 0}. Then, taking into accound that $\\rho\\geq 1-\\hat\\sigma\\_\\xi^2$ we obtain\n \n$$\\begin{aligned}\n\\widetilde E\\_\\xi(\\widetilde Q\\_\\xi;B(y\\_1^\\xi,2\\gamma\\hat\\sigma\\_\\xi)&\\geq (1-\\hat\\sigma\\_\\xi^2)\\pi\\ln\\frac{\\hat\\sigma\\_\\xi}{\\xi} -C.\\end{aligned}$$ \nBy symmetry, if *y*1*ξ* lies above the axis {*z* = 0} then the same lower bound will be obtained below the axis, so we deduce\n \n$$\\begin{aligned}\n\\widetilde E\\_\\xi(\\widetilde Q\\_\\xi;\\widetilde\\Omega\\cap B(a,2\\mu\\hat\\sigma\\_\\xi)\\setminus B(a,\\hat\\sigma\\_\\xi/2))&\n\\geq 2(1-\\hat\\sigma\\_\\xi^2)\\pi\\ln\\frac{\\hat\\sigma\\_\\xi}{\\xi}-O(1).\\end{aligned}$$ \nAdding this to the lower bound on $B(a,\\delta)\\setminus B(a,2\\mu\\hat\\sigma\\_\\xi)$, we obtain\n \n$$\\begin{aligned}\n\\widetilde E\\_\\xi(\\widetilde Q\\_\\xi)&\\geq 2\\pi \\ln\\frac\\delta{\\hat\\sigma\\_\\xi} + 2\\pi (1-\\hat\\sigma\\_\\xi^2)\\ln\\frac{\\hat\\sigma\\_\\xi}{\\xi} - O(1) \\\\\n&\\geq 2\\pi (1-\\hat\\sigma\\_\\xi^2)\\ln\\frac 1\\xi - O(1).\\end{aligned}$$ \nSince $\\hat\\sigma\\_\\xi\\to 0$ this contradicts the upper bound from Section [s:upper].\nThus the point *y*1*ξ* must lie on the axis {*z* = 0}, hence $\\rho\\geq 1+\\hat\\sigma\\_\\xi/2$ on $B(y\\_1^\\xi,2\\gamma\\hat\\sigma\\_\\xi)$, and applying Lemma [l:lowerboundperforated] we have\n \n$$\\begin{aligned}\n\\widetilde E\\_\\xi(\\widetilde Q\\_\\xi;B(y\\_1^\\xi,2\\gamma\\hat\\sigma\\_\\xi))&\\geq (1+\\frac 12\\hat\\sigma\\_\\xi)\\pi\\ln\\frac{\\hat\\sigma\\_\\xi}{\\xi} -C.\\end{aligned}$$ \nTogether with the lower bound on $B(a,\\delta)\\setminus B(a,2\\mu\\hat\\sigma\\_\\xi)$, this implies\n \n$$\\begin{aligned}\n\\widetilde E\\_\\xi(\\widetilde Q\\_\\xi;\\widetilde\\Omega\\cap B(a,\\delta))&\\geq 2\\pi\\ln\\frac{\\delta}{\\hat\\sigma\\_\\xi} +\\pi(1+\\frac 12\\hat\\sigma\\_\\xi)\\ln\\frac{\\hat\\sigma\\_\\xi}{\\xi} -C \\\\\n&\\geq \\pi\\ln\\frac 1\\xi +\\pi\\ln\\frac{1}{\\hat\\sigma\\_\\xi} + \\frac\\pi 2 \\hat\\sigma\\_\\xi\\ln\\frac 1\\xi -2\\pi\\ln\\frac 1\\delta -C.\\end{aligned}$$ \nNow notice that\n \n$$\\begin{aligned}\n\\inf\\_{0<\\sigma<1}\\left\\lbrace \\pi\\ln\\frac 1\\sigma + \\frac\\pi 2\\sigma\\ln\\frac 1\\xi\\right\\rbrace\\end{aligned}$$ \nis attained at $\\sigma=2/\\ln(\\frac 1\\xi)$, so the above lower bound implies\n \n$$\\widetilde E\\_\\xi(\\widetilde Q\\_\\xi;\\widetilde\\Omega\\cap B(a,\\delta))\n\\geq \\pi\\ln\\frac 1\\xi +\\pi\\ln\\ln\\frac 1\\xi -2\\pi\\ln\\frac 1\\delta -C.$$ \nThis is the lower bound we have been after. Now recall we have been arguing on an arbitrary sequence *ξ* = *ξ**n* → 0 and taking subsequences, so what this proves is that for any *δ* ∈ (0, *δ*0),\n \n$$\\liminf\\_{{\\varepsilon}\\to 0}\\left(\\widetilde E(\\widetilde Q\\_\\xi;\\widetilde\\Omega\\cap B(a,\\delta))-\\pi\\ln\\frac 1\\xi -\\pi\\ln\\ln\\frac 1\\xi\\right) \\geq -2\\pi\\ln\\frac 1\\delta - C,$$ \nand this is part (*i**i*) of Theorem [thm:main].\n[r:lowerbound] In the above proof, we obtain lower bounds on $B(a,\\delta)\\setminus B(a,2\\mu\\hat\\sigma\\_\\xi)$ and on $B(y\\_1^\\xi,2\\gamma\\hat\\sigma\\_\\xi)$ and realize that their sum corresponds to the upper bound obtained in Section [s:upper]. Therefore in those sets the lower bounds must be sharp, in the sense that matching upper bounds are valid. In particular we have\n \n$$\\widetilde E\\_\\xi(\\widetilde Q\\_\\xi;\\widetilde\\Omega\\cap B(a,\\delta)\\setminus B(a,2\\mu\\hat\\sigma\\_\\xi))\\leq 2\\pi\\ln\\frac{\\delta}{\\hat\\sigma\\_\\xi}+C,$$ \nand find ourselves in the situation of Lemma [l:finelowerboundhalfannulus]. As a consequence, a lower bound similar to the one in part (*i**i*) of Theorem [thm:main] is valid on the slightly smaller set\n \n*D**δ**i**n**t* ∩ {*ρ* ≥ 1 + *λ**z**β*}, \nfor any *λ* > 0 and *β* > 1. Combining this with the upper bound obtained in Section [s:upper] then yields\n \n$$\\widetilde E\\_\\xi(\\widetilde Q\\_\\xi;D^{ext}\\_\\delta\\cup\\lbrace \\rho\\leq 1+\\lambda z^\\beta\\rbrace)\\leq \n2\\pi\\ln\\frac 1\\delta + C(\\beta,\\lambda)\\qquad\\forall \\delta\\in (0,1),\\,\\lambda>0,\\,\\beta>1.$$ \nLimit configuration\n===================\nIn this section we prove part (*i**i**i*) of Theorem [thm:main]. Although in the previous sections we have already established tight upper and lower bounds on the energy away from the equator defect, extracting precise information about the structure of the minimizers and the absence of point singularities will be a multi-step process:\n* In § [ss:limconv] we use standard arguments to establish strong *H*1 convergence away from the ring defect.\n* In § [ss:limsym] we then establish the additional symmetry property that the director points inside the azimuthal plane (in the sense of Remark [r:limitsymmetry]). The argument of implies that as long as this symmetry is satisfied at the boundary, it should also be satisfied inside the domain. Applying this result in our context is however not straightforward: the limiting map is only minimizing away from the ring defect and it is necessary to cut out a small disc around the defect. The boundary conditions are not fixed there and they need to be carefully estimated using the rigidity imposed by the energy asymptotics (in the spirit of Lemma [l:finelowerboundhalfannulus]).\n* Finally, in § [ss:limsmooth], we use an argument of to rule out point defects which are not topologically necessary. That argument requires even more precise estimates on the boundary conditions. We can only rule out point defects provided that the ring defect is negatively charged—otherwise a positively charged ring defect would have to be compensated by a pair of point defects. The possibility of a positively charged ring defect is eventually eliminated by establishing that in a small region around the ring, it could not—compared to a negatively charged ring—improve the energy by more than *o*(1) as *ξ* → 0. Effectively, this amounts to showing that the core energy of a positively or negatively charged ring defect are the same.\nStrong convergence\n------------------\nWe start by establishing strong *H*1 convergence along a subsequence.\n[l:H1conv] There is a subsequence *ξ* → 0 and *Q*⋆ such that for all *δ* > 0, the sequence *Q**ξ* converges strongly in *H**l**o**c*1(Ω*δ**e**x**t*) to $Q\\_\\star\\in\\mathcal H^\\star\\_{sym}$ and\n \n$$\\begin{aligned}\nE(Q\\_\\xi;\\Omega^{ext}\\_\\delta)\\longrightarrow E(Q\\_\\star;\\Omega^{ext}\\_\\delta)\\quad\\text{and}\\quad\n\\frac{1}{\\xi^2}\\int\\_{\\Omega^{ext}\\_\\delta}f(Q\\_\\xi)\\longrightarrow 0,\\end{aligned}$$ \nas *ξ* → 0. Moreover, in any relatively open subset $U\\subset\\overline\\Omega$ where *Q*⋆ is smooth, we have *Q**ξ* → *Q*⋆ in *C**l**o**c*1, *α*(*U*) for all *α* ∈ (0, 1).\n[Proof of Lemma [l:H1conv]] For any *δ* > 0, the upper and lower bound imply\n \n$$\\frac 1{2\\pi} E(Q\\_\\xi;\\Omega\\_\\delta^{ext})\\leq 2\\pi\\ln\\frac 1\\delta +C.$$ \nBy a diagonal argument, this bound implies the existence a limit map $Q\\_\\star\\in \\mathcal H^\\star\\_{sym}$ such that, up to a subsequence, *Q**ξ* converges weakly to *Q*⋆ in *H**l**o**c*1(Ω*δ**e**x**t*) for every *δ* > 0. We denote by *T**δ* the part of ∂Ω*δ**e**x**t* that lies inside Ω, namely in cylindrical coordinates\n \n*T**δ* = Ω ∩ {(*ρ* − 1)2 + *z*2 = *δ*2}. \nBy Fubini’s theorem we may (upon extracting a further subsequence) fix *δ* arbitrarily small such that\n \n$$\\label{eq:QTdelta}\nE\\_\\xi(Q\\_\\xi;T\\_\\delta)+E\\_\\star(Q\\_\\star;T\\_\\delta)\\lesssim \\frac 1\\delta \\left(E\\_\\xi(Q\\_\\xi;\\Omega^{ext}\\_{\\delta/2}) + E\\_\\star(Q\\_\\star;\\Omega^{ext}\\_{\\delta/2})\\right)\\lesssim C(\\delta).$$ \nIn particular the trace of *Q**ξ* on *T**δ* is bounded in *H*1(*T**δ*), and converges therefore also weakly in *H*1(*T**δ*). The map *Q**ξ* is of the form\n \n$$Q\\_\\xi(\\rho,\\varphi,z)=R\\_\\varphi\\widetilde Q\\_\\xi(\\rho,z) R\\_\\varphi^t,$$ \nand $\\widetilde Q\\_\\xi$ converges weakly in $H^1\\_{loc}(\\widetilde \\Omega^{ext}\\_\\delta)$ and in $H^1(\\widetilde T\\_\\delta)$ to $\\widetilde Q\\_\\star$ such that\n \n$$Q\\_\\star(\\rho,\\varphi,z)=R\\_\\varphi\\widetilde Q\\_\\star(\\rho,z) R\\_\\varphi^t.$$ \n(Here $\\widetilde \\Omega^{ext}\\_\\delta$ and $\\widetilde T\\_\\delta$ denote the two-dimensional sections of Ω*δ**e**x**t* and *T**δ* corresponding to the (*ρ*, *z*) coordinates.) Consider a map $Q\\_0\\in\\mathcal H\\_{sym}^\\star(\\Omega^{ext}\\_\\delta)$ satisfying\n \n$$E\\_\\star(Q\\_0;\\Omega^{ext}\\_\\delta)=\\min\\left\\lbrace E\\_\\star(Q;\\Omega^{ext}\\_\\delta)\\colon Q\\in\\mathcal H\\_{sym}^\\star(\\Omega^{ext}\\_\\delta),\\,Q=Q\\_\\star\\text{ on }T\\_\\delta\\right\\rbrace.$$ \nSince $\\widetilde Q\\_\\xi$ converges weakly in $H^1(\\widetilde T\\_\\delta)$ to $ (\\widetilde Q\\_\\star)\\_{\\lfloor \\widetilde T\\_\\delta}=(\\widetilde Q\\_0)\\_{\\lfloor \\widetilde T\\_\\delta}$ and $\\widetilde T\\_\\delta$ is a one-dimensional curve, on this curve $\\widetilde Q\\_\\xi$ converges to $\\widetilde Q\\_0$ in *C*0, *α*(*T**δ*) for any *α* ∈ (0, 1/2). We claim that this allows to construct a map $\\overline Q\\_\\xi\\in \\mathcal H\\_{sym}$ such that\n \n$$\\label{eq:overlineQ}\n\\overline Q\\_\\xi=Q\\_\\xi\\quad\\text{in }\\Omega^{int}\\_\\delta=\\Omega\\setminus\\Omega\\_\\delta^{ext}\\qquad\\text{ and } E\\_\\xi(\\overline Q\\_\\xi;\\Omega^{ext}\\_\\delta)-E\\_\\star(Q\\_0;\\Omega^{ext}\\_\\delta)\\to 0\\quad\\text{as }\\xi\\to 0.$$ \nIn order to define $\\overline Q\\_\\xi$ in Ω*δ**e**x**t* we introduce the cross-sectional domains\n \n*D* = {(*ρ*, *z*): *ρ*, *z* > 0, *ρ*2 + *z*2 > 1}, \nand\n \n$$\\label{eq:Dext}\n\\begin{aligned} D^{ext}\\_\\delta&=\\left\\lbrace (\\rho,z) \\colon \\ (\\rho-1)^2+z^2>\\delta^2, \\ z>0 \\right\\rbrace, \\\\\nD^{int}\\_\\delta \n &= \\left\\{(\\rho,z) \\colon \\ (\\rho-1)^2+z^2\\le \\delta^2, \\ z>0 \\right\\rbrace\n = D\\setminus D^{ext}\\_\\delta.\n\\end{aligned}$$ \nWe use polar coordinates (*r*, *θ*) centered at (*ρ*, *z*) = (1, 0). In these coordinates the domains *D* and *D**δ**e**x**t* are given by\n \n$$\\begin{gathered}\n\\nonumber\nD=\\{ (r,\\theta): \\ 0<\\theta<\\theta\\_0(r), \\ r>0\\}, \\quad\nD^{ext}\\_\\delta = \\{ (r,\\theta): \\ 0<\\theta<\\theta\\_0(r), \\ r>\\delta\\}\\\\\n \\label{eq:Ddelta}\n \\text{where} \\ \\theta\\_0(r)=\\begin{cases}\n \\frac\\pi 2 +\\arcsin\\frac r2, &\\text{if $r\\le\\sqrt{2}$,}\\\\\n \\frac\\pi 2 +\\arcsin\\frac{1}{r}, &\\text{if $r>\\sqrt{2}$.} \\end{cases}\\end{gathered}$$ \nThen, the desired map $\\overline Q\\_\\xi$ will have the form\n \n$$\\begin{aligned}\n\\overline Q\\_\\xi(\\rho,\\varphi,z)&=R\\_\\varphi \\widehat Q\\_\\xi(\\rho,z)R\\_\\varphi^t,\\end{aligned}$$ \nand it suffices to define $\\widehat Q\\_\\xi$ in the domain *D**δ**e**x**t*.\nThe standard idea is to introduce a thin slice {*δ* < *r* < (1 + *λ*)*δ*} where we interpolate from $\\widehat Q\\_\\xi =\\widetilde Q\\_\\xi$ at *r* = *δ* to $\\widehat Q\\_\\xi=\\widetilde Q\\_0$ at *r* = (1 + *λ*)*δ*, and choose *λ* in order that the energy of $\\widehat Q\\_\\xi$ in that thin slice be negligible. Then we extend by $\\widetilde Q\\_0$ (slightly rescaled) in *D*(1 + *λ*)*δ*. Here, however, we need to be more careful because we have to preserve the boundary condition at *ρ*2 + *z*2 = 1, i.e., in polar coordinates at $\\theta=\\theta\\_0(r)=\\frac\\pi 2+\\arcsin\\frac r2$. Hence we interpolate instead in a deformed slice *S**λ* of the form\n \n$$S\\_\\lambda = \\lbrace \\delta < r < (1+\\tilde\\lambda(\\theta))\\delta,\\, 0<\\theta<\\theta\\_0(\\delta)\\rbrace,\\qquad\n\\tilde\\lambda(\\theta)=(\\theta\\_0(\\delta)-\\theta)\\lambda,$$ \nfor some *λ* > 0 to be chosen later. Next we define $\\widehat Q\\_\\xi$ in the deformed slice *S**λ*. As noted in, with a simple linear interpolation we might fail at controlling the potential part of the energy $\\int f(\\widehat Q\\_\\xi)$, and we need to proceed in two steps as in, namely first interpolate linearly between $\\widetilde Q\\_\\xi$ and its projection $\\pi(\\widetilde Q\\_\\xi)$ onto $\\mathcal U\\_\\star$, and then interpolate geodesically between $\\pi(\\widetilde Q\\_\\xi)$ and $\\widetilde Q\\_0$ inside $\\mathcal U\\_\\star$. To this end we denote by\n \n$$\\gamma\\colon [0,1]\\times V\\to\\mathcal U\\_\\star,\\qquad V\\text{ is a neighborhood of }\\lbrace (U,U)\\colon U\\in\\mathcal U\\_\\star\\rbrace\\text{ in }\\mathcal U\\_\\star\\times\\mathcal U\\_\\star,$$ \nthe smooth map such that *t* ↦ *γ*(*t*; *U*1, *U*2) is the constant speed geodesic from *U*1 to *U*2 in $\\mathcal U\\_\\star$, which is indeed unique, well-defined and depends smoothly on *U*1, *U*2 provided *U*1, *U*2 are close enough to each other. This map satisfies the bounds\n \n$${{\\vert \\partial\\_t\\gamma(t;U\\_1,U\\_2)\\vert}}\\lesssim{{\\vert U\\_1-U\\_2\\vert}},\\qquad {{\\vert \\partial\\_U\\gamma\\vert}}\\lesssim 1.$$ \nIn *S**λ* we define\n \n$$\\begin{aligned}\n&\\widehat Q(r,\\theta)=\\widetilde Q\\_\\xi(\\delta,\\theta)+\\mu\\_1(r,\\theta)\\left(\\pi(\\widetilde Q\\_\\xi(\\delta,\\theta))-\\widetilde Q\\_\\xi(\\delta,\\theta)\\right) \\quad\\text{for }\\delta <r <\\delta + \\frac{\\tilde\\lambda(\\theta)}{2}\\delta,\\\\\n&\\text{where }\\mu\\_1(r,\\theta) =\\frac{2}{\\tilde\\lambda(\\theta)\\delta}(r-\\delta),\\\\\n&\\widehat Q(r,\\theta)=\n\\gamma\\left(\\mu\\_2(r,\\theta);\\pi(\\widetilde Q\\_\\xi(\\delta,\\theta),\\widetilde Q\\_0(\\delta,\\theta)\\right)\\quad\\text{for }\\delta + \\frac{\\tilde\\lambda(\\theta)}{2}\\delta<r <\\delta +\\tilde\\lambda(\\theta)\\delta,\\\\\n&\\text{where }\\mu\\_2(r,\\theta)= \\frac{2}{\\tilde\\lambda(\\theta)\\delta}(r-\\delta-\\frac{\\tilde\\lambda(\\theta)}{2}\\delta).\\end{aligned}$$ \nNote that $\\pi(\\widetilde Q\\_\\xi(\\delta,\\theta))$ is well-defined for small *ξ* because $\\widetilde Q\\_\\xi$ converges uniformly to $\\widetilde Q\\_0$ on $\\widetilde T\\_\\delta$. Direct computations then show that\n \n$$\\begin{aligned}\n{{\\vert \\partial\\_r \\widehat Q\\vert}}&\\lesssim \\frac{1}{\\delta\\lambda}\\frac{1}{\\theta\\_0(\\delta)-\\theta}\\left({{\\vert \\pi(\\widetilde Q\\_\\xi(\\delta,\\theta))-\\widetilde Q\\_\\xi(\\delta,\\theta)\\vert}}\n+ {{\\vert \\pi(\\widetilde Q\\_\\xi(\\delta,\\theta))-\\widetilde Q\\_0(\\delta,\\theta)\\vert}}\\right)\n\\\\\n&\\lesssim \\frac{1}{\\delta\\lambda}\\frac{1}{\\theta\\_0(\\delta)-\\theta}{{\\vert \\widetilde Q\\_\\xi(\\delta,\\theta)-\\widetilde Q\\_0(\\delta,\\theta)\\vert}}\\\\\n{{\\vert \\partial\\_\\theta\\widehat Q\\vert}}&\\lesssim{{\\vert \\partial\\_\\theta\\widetilde Q(\\delta,\\theta)\\vert}}+{{\\vert \\partial\\_\\theta\\widetilde Q\\_0(\\delta,\\theta)\\vert}}\\\\\n&\\quad +\\frac{1}{\\theta\\_0(\\delta)-\\theta}\\left({{\\vert \\pi(\\widetilde Q\\_\\xi(\\delta,\\theta))-\\widetilde Q\\_\\xi(\\delta,\\theta)\\vert}}\n+ {{\\vert \\pi(\\widetilde Q\\_\\xi(\\delta,\\theta))-\\widetilde Q\\_0(\\delta,\\theta)\\vert}}\\right),\\\\\n&\\lesssim \n{{\\vert \\partial\\_\\theta\\widetilde Q(\\delta,\\theta)\\vert}}+{{\\vert \\partial\\_\\theta\\widetilde Q\\_0(\\delta,\\theta)\\vert}}\n+\\frac{1}{\\theta\\_0(\\delta)-\\theta}{{\\vert \\widetilde Q\\_\\xi(\\delta,\\theta)-\\widetilde Q\\_0(\\delta,\\theta)\\vert}}\\end{aligned}$$ \nNext, we denote by *σ* = *σ*(*ξ*) the quantity\n \n$$\\sigma={{\\Vert \\widetilde Q\\_\\xi-\\widetilde Q\\_0\\Vert}}\\_{C^{1/4}(\\widetilde T\\_\\delta)}\\longrightarrow 0\\qquad\\text{as }\\xi\\to 0,$$ \nand notice that since the boundary conditions on *ρ*2 + *z*2 = 1 ensure that $\\widetilde Q\\_\\xi(\\delta,\\theta\\_0(r))=\\widetilde Q\\_0(\\delta,\\theta\\_0(r))$, we have\n \n$${{\\vert \\widetilde Q\\_\\xi(\\delta,\\theta)-\\widetilde Q\\_0(\\delta,\\theta)\\vert}}\\leq \\sigma \\cdot(\\theta\\_0(\\delta)-\\theta)^{\\frac 14}.$$ \nTherefore, the above estimates on the derivatives of $\\widehat Q$ imply\n \n$$\\begin{aligned}\n{{\\vert \\nabla \\widehat Q\\vert}}^2&={{\\vert \\partial\\_r\\widehat Q\\vert}}^2+\\frac{1}{r^2}{{\\vert \\partial\\_\\theta \\widehat Q\\vert}}^2\\\\\n&\\lesssim \\frac{1}{\\delta^2\\lambda^2}\\frac{1}{(\\theta\\_0(\\delta)-\\theta)^{\\frac 32}}\\sigma^2 +\\frac{1}{\\delta^2}\\left({{\\vert \\partial\\_\\theta\\widetilde Q(\\delta,\\theta)\\vert}}^2+{{\\vert \\partial\\_\\theta\\widetilde Q\\_0(\\delta,\\theta)\\vert}}^2 \\right),\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n\\int\\_{S\\_\\lambda} {{\\vert \\nabla \\widehat Q\\vert}}^2 &= \\int\\_0^{\\theta\\_0(\\delta)}\\int\\_{\\delta}^{\\delta+\\delta\\lambda(\\theta\\_0(\\delta)-\\theta)}rdr {{\\vert \\nabla \\widehat Q\\vert}}^2 d\\theta \\\\\n&\\lesssim \\delta^2 \\lambda \\int\\_0^{\\theta\\_0(\\delta)}(\\theta\\_0(\\delta)-\\theta)\\\\\n&\\hspace{3em}\\cdot\\left[\n\\frac{1}{\\delta^2\\lambda^2}\\frac{1}{(\\theta\\_0(\\delta)-\\theta)^{\\frac 32}}\\sigma^2 +\\frac{1}{\\delta^2}\\left({{\\vert \\partial\\_\\theta\\widetilde Q(\\delta,\\theta)\\vert}}^2+{{\\vert \\partial\\_\\theta\\widetilde Q\\_0(\\delta,\\theta)\\vert}}^2 \\right)\n\\right]d\\theta\\\\\n&\\lesssim \\frac{\\sigma^2}{\\lambda} +\\lambda C(\\delta).\\end{aligned}$$ \nFor the last inequality we used the fact that, thanks to, the *L*2 norms of $\\partial\\_\\theta \\widetilde Q\\_\\xi(\\delta,\\theta)$ and $\\partial\\_\\theta \\widetilde Q\\_0(\\delta,\\theta)$ are bounded for *δ* fixed. Note moreover that the definition of $\\widehat Q$ ensures that\n \n$$f(\\widehat Q)\\lesssim \\operatorname{dist}^2(\\widehat Q,\\mathcal U\\_\\star)\\leq\\operatorname{dist}^2(\\widetilde Q\\_\\xi(\\delta,\\theta),\\mathcal U\\_\\star)\\lesssim f(\\widetilde Q\\_\\xi(\\delta,\\theta)),$$ \nthanks to the nonegeneracy property of the potential *f*. Using this pointwise inequality and we infer that\n \n$$\\frac{1}{\\xi^2}\\int\\_{S\\_\\lambda}f(\\widehat Q\\_\\xi)\\lesssim \\lambda C(\\delta).$$ \nFinally, since *ρ* ≈ 1 and $\\Xi[\\widehat Q]\\lesssim 1$ in *S**λ* we deduce from the above that\n \n$$\\widetilde E\\_\\xi(\\widehat Q\\_\\xi;S\\_\\lambda)\\lesssim \\frac{\\sigma^2}{\\lambda} +\\lambda C(\\delta).$$ \nChoosing *λ* = *σ* then implies that\n \n$$\\widetilde E\\_\\xi(\\widehat Q\\_\\xi;S\\_\\lambda)\\lesssim C(\\delta)\\sigma\\longrightarrow 0\\quad\\text{as }\\xi\\to 0.$$ \nNext we define $\\widehat Q\\_\\xi$ in *D**δ**e**x**t* \\ *S**λ* by setting\n \n$$\\begin{aligned}\n\\widehat Q\\_\\xi(r,\\theta)&=\\widetilde Q\\_0\\left(\\delta+\\frac{1}{1-\\tilde\\lambda(\\theta)}(r-\\delta-\\tilde\\lambda(\\theta)\\delta),\\theta\\right)\\qquad \\text{for }\\delta+\\tilde\\lambda(\\theta)\\delta < r < 2\\delta,\\\\\n\\widehat Q\\_\\xi &=\\widetilde Q\\_0\\qquad\\text{in }D\\_{2\\delta}.\\end{aligned}$$ \nThen we have\n \n$$\\begin{aligned}\n{{\\vert \\widetilde E\\_\\xi(\\widehat Q\\_\\xi;D^{ext}\\_\\delta\\setminus S\\_\\lambda)-\\widetilde E\\_\\star(\\widetilde Q\\_0;D^{ext}\\_\\delta)\\vert}}&={{\\vert \\widetilde E\\_\\star(\\widehat Q\\_\\xi;D^{ext}\\_\\delta\\setminus S\\_\\lambda)-\\widetilde E\\_\\star(\\widetilde Q\\_0;D^{ext}\\_\\delta)\\vert}}\\\\\n&\\lesssim \\lambda \\widetilde E\\_\\star(\\widetilde Q\\_0;D^{ext}\\_\\delta)\\longrightarrow 0,\\end{aligned}$$ \nas *ξ* → 0, since *λ* = *σ* → 0. Combining this and the fact that $\\widetilde E\\_\\xi(\\widehat Q\\_\\xi;S\\_\\lambda)\\to 0$, we deduce that\n \n$$\\widetilde E\\_\\xi(\\widehat Q\\_\\xi;D^{ext}\\_\\delta)-\\widetilde E\\_\\star(\\widetilde Q\\_0;D^{ext}\\_\\delta)\\longrightarrow 0,$$ \nwhich implies the desired estimate. By minimality of *Q**ξ* we then have\n \n$$E\\_\\xi(Q\\_\\xi;\\Omega^{ext}\\_\\delta)\\leq E\\_\\xi(\\overline Q\\_\\xi;\\Omega^{ext}\\_\\delta)\\leq E\\_\\star(Q\\_0;\\Omega^{ext}\\_\\delta)+o(1),$$ \nand by lower semicontinuity we deduce\n \n$$\\begin{aligned}\nE\\_\\star(Q\\_\\star;\\Omega^{ext}\\_\\delta)&\\leq\\liminf E\\_\\xi(Q\\_\\xi;\\Omega^{ext}\\_\\delta) \\leq \\limsup E\\_\\xi(\\overline Q\\_\\xi;\\Omega^{ext}\\_\\delta)\\leq E\\_\\star( Q\\_0;\\Omega^{ext}\\_\\delta).\\end{aligned}$$ \nIt follows that *Q*⋆ minimizes *E*⋆( ⋅ ; Ω*δ**e**x**t*) among all maps $Q\\in\\mathcal H^\\star\\_{sym}(\\Omega^{ext}\\_\\delta)$ such that *Q* = *Q*⋆ on *T**δ*, and that all above inequalities are in fact equalities. Thus we have\n \n$$\\int\\_{\\Omega^{ext}\\_\\delta}{{\\vert \\nabla Q\\_\\xi\\vert}}^2\\longrightarrow \\int\\_{\\Omega^{ext}\\_\\delta}{{\\vert \\nabla Q\\_\\star\\vert}}^2\\quad\\text{ and }\\quad \\frac{1}{\\xi^2}\\int\\_{\\Omega^{ext}\\_\\delta}f(Q\\_\\xi)\\longrightarrow 0.$$ \nIn particular, together with the weak convergence, this implies that ∇*Q**ξ* converges strongly in *L*2(Ω*δ**e**x**t*) towards ∇*Q*⋆, that is, the convergence is in fact strong.\nThe local *C*1, *α* convergence away from singularities follows from the analysis in. There the authors consider minimizers which are not subject to the constraint of axial symmetry, but they only use the minimizing property to obtain the strong *H*1 convergence, which we have just obtained, so their results apply directly.\nReduction to a director in a cross-section\n------------------------------------------\nThe limiting *Q*-tensor *Q*\\* inherits the symmetries of the space H*s**y**m*, but it also exhibits further symmetry by virtue of energy minimization. Here we show that the map *Q*⋆ may be represented by a uniaxial tensor with a unit director field *n* = (*n*1(*ρ*, *z*), 0, *n*3(*ρ*, *z*)), expressed in cylindrical coordinates in a cross-section of Ω.\nIn the sequel, whenever we refer to *ξ* → 0, we will always mean convergence along the subsequence obtained in Lemma [l:H1conv].\nSince the limit *Q*⋆ is symmetric, it is characterized by a map defined in the two-dimensional domain *D*. To describe *Q*⋆ further, it will also be convenient to introduce the following notations"},"data_id":{"kind":"string","value":"arxiv_0002826"}}},{"rowIdx":692,"cells":{"content":{"kind":"string","value":" On the impact of galaxy bias uncertainties on primordial non-Gaussianity constraints\n=====================================================================================\nIntroduction\n============\nOne of the main goals of modern cosmology is to determine the statistical properties of the density fluctuations of the primordial universe and gain insights into the physics of the mechanisms that generated them during the epoch of inflation. Current observational constraints are compatible with the simplest models of inflation that involve a single scalar field slowly rolling down its potential. A key prediction of these models is that the resulting fluctuations should be Gaussian distributed, and consequently, studies of primordial non-Gaussianity (PNG) have become a major focus in theoretical and observational cosmology given its power to discriminate between single-field models and more elaborate models involving multiple fields (see Ref. for an overview). A popular characterization of PNG is that of the so-called *local* type, in which the primordial gravitational (Bardeen) potential $\\phi(\\vx)$ is expanded as [eq:fnl] () = G() + fnl, where $\\phi\\_{\\rm G}$ is a Gaussian distributed random field, $\\left<\\cdots\\right>$ denotes ensemble average and the parameter ${f\\_\\textnormal{\\textsc{nl}}}$ quantifies the departures from non-Gaussianity. The current tightest bounds on ${f\\_\\textnormal{\\textsc{nl}}}$ come from the analysis of three-point statistics of the cosmic microwave background (CMB) by the Planck satellite, which constrain ${f\\_\\textnormal{\\textsc{nl}}}= -0.9 \\pm 5.1\\ (1\\sigma)$.\nThe next major improvements on the precision of ${f\\_\\textnormal{\\textsc{nl}}}$, $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$, are expected to come from analyses of the statistics of the galaxy distribution. For Gaussian distributed primordial fluctuations, on sufficiently large scales, the galaxy number density contrast *δ**g* can be written as $\\delta\\_g(\\vx) = b\\_1 \\delta\\_m(\\vx) + \\eps(\\vx)$, where *δ**m* denotes matter density contrast fluctuations, $\\eps$ is a noise field and *b*1 is the linear local-in-matter-density galaxy bias parameter that describes the response of galaxy number counts to long-wavelength total matter fluctuations (see Ref. for a review on galaxy bias). In the presence of local PNG, the galaxy distribution gets another contribution $\\delta\\_g(\\vx) \\supset b\\_\\phi {f\\_\\textnormal{\\textsc{nl}}}\\phi(\\vq)$, where *b**ϕ* is the galaxy bias parameter that describes the response of galaxy number counts to long-wavelength primordial gravitational potential perturbations with local PNG ($\\vq$ is the Lagrangian coordinate associated with the Eulerian coordinate $\\vx$). This ${f\\_\\textnormal{\\textsc{nl}}}$ contribution can therefore be used to place bounds on local PNG using galaxy observations. For example, Ref. showed that the galaxy power spectrum *P**g**g*(*k*) (the Fourier transform of the two-point correlation function; *k* denotes wavenumber) acquires a specific scale-dependent signature $\\propto b\\_1b\\_\\phi {f\\_\\textnormal{\\textsc{nl}}}/ k^2$ that becomes important on the largest observable scales. The galaxy bispectrum *B**g**g**g*(*k*1, *k*2, *k*3) (the Fourier transform of the three-point correlation function) is also a notoriously good probe of ${f\\_\\textnormal{\\textsc{nl}}}$, and it is sensitive to it also via the non-zero primordial matter bispectrum that it induces (we discuss the ${f\\_\\textnormal{\\textsc{nl}}}$ contributions to the power spectrum and bispectrum in more detail in Sec. [sec:model]). Current constraints on local PNG from galaxy surveys are of order $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}} \\sim 50$, but a number of forecast studies have been suggesting that $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}} \\sim \\mathcal{O}(1)$ or below could be achieved with upcoming surveys.\nA main theoretical uncertainty in searches for local PNG using the galaxy distribution concerns the galaxy bias parameters. They are functions of redshift and of galaxy properties such as their total mass, stellar mass, luminosity, etc. They formally describe the response of galaxy formation to the long-wavelength environment, and as a result, they are extremely challenging to predict given the many astrophysical processes involved in galaxy formation and evolution. In observational searches for local PNG, one should therefore fit for the bias parameters simultaneously with ${f\\_\\textnormal{\\textsc{nl}}}$, but naturally, the wider the priors adopted for the bias parameters, the weaker the constraining power on ${f\\_\\textnormal{\\textsc{nl}}}$. This motivates work on theoretical predictions of galaxy bias in order to determine the range of values a given observed galaxy sample is expected to take and/or determine relations between the various bias parameters to reduce the dimensionality of the parameter spaces explored in constraint analyses. A widely popular example of such relations is the so-called universality relation between *b**ϕ* and *b*1. Concretely, considering gravity-only dynamics in the formation of dark matter haloes and further assuming that their mass function is universal, it can be shown that *b**ϕ* = 2*δ**c*(*b*1 − 1), where *δ**c* = 1.686 is the threshold overdensity for spherical collapse. The adoption of such a relation is crucial to constrain ${f\\_\\textnormal{\\textsc{nl}}}$. For example, in the contribution to the galaxy power spectrum $\\propto b\\_1b\\_\\phi {f\\_\\textnormal{\\textsc{nl}}}/ k^2$, the value of *b*1 can be constrained by the smaller-scale part of the power spectrum, but then, if *b**ϕ* is allowed complete freedom, it becomes impossible to constrain ${f\\_\\textnormal{\\textsc{nl}}}$ (cf. upper left panel of Fig. [fig:contributions] below).\nEffectively all recent constraint and forecast studies on ${f\\_\\textnormal{\\textsc{nl}}}$ using the galaxy distribution adopt some form for the relation *b**ϕ*(*b*1). The most popular case is the use of the universality relation mentioned above, but this relation has been shown to represent only an approximation of the bias values measured for haloes in *N*-body simulations. More specifically, the simulation results are better described by a variant of the relation *b**ϕ* = *q*2*δ**c*(*b*1 − 1), where *q* ∈ [0.5, 0.9] (the exact value depends on redshift, halo mass and halo finding criterion). Another variant of the universality relation, *b**ϕ* = 2*δ**c*(*b*1 − *p*) with *p* = 1.6 was put forward by Ref. , who argued that it was a more appropriate description for recently formed haloes, which could host quasars (see also Ref. ). More recently, Ref. found using hydrodynamical simulations of galaxy formation with the IllustrisTNG model that *b**ϕ* = 2*δ**c*(*b*1 − *p*) with *p* ≈ 0.55 describes well the bias relation for galaxies selected by their stellar mass. Naturally, uncertainties on galaxy bias in general, and on the *b**ϕ*(*b*1) relation in particular, will propagate to the resulting inferred values of ${f\\_\\textnormal{\\textsc{nl}}}$ and $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$. This is a fact that has been acknowledged is past literature, but that has not been the subject of detailed and dedicated work, despite its utmost importance for PNG constraints. Our main goal in this paper is to take a few steps forward in understanding how such uncertainties can impact the inferred ${f\\_\\textnormal{\\textsc{nl}}}$ and how to mitigate them.\nSpecifically in this paper, we work with an idealized forecast setup to illustrate how galaxy bias uncertainties affect the constraints on ${f\\_\\textnormal{\\textsc{nl}}}$ obtained with a combination of multitracer galaxy power spectrum and bispectrum data. We explore two treatments of the galaxy bias *b**ϕ*: (i) one called parametrization 1, in which we assume *b**ϕ* = 2*δ**c*(*b*1 − *p*) and treat *p* as a free parameter, and (ii) another called parametrization 2, in which instead of constraining ${f\\_\\textnormal{\\textsc{nl}}}$, we constrain directly the product of $b\\_\\phi{f\\_\\textnormal{\\textsc{nl}}}$. In the latter approach, one is less interested in the actual value of ${f\\_\\textnormal{\\textsc{nl}}}$, but more on distinguishing it from zero as $b\\_\\phi{f\\_\\textnormal{\\textsc{nl}}}\\neq 0$ implies ${f\\_\\textnormal{\\textsc{nl}}}\\neq 0$, which is sufficient to distinguish single-field from multifield inflation. With parametrization 1, we will see that the shape of the likelihood becomes appreciably bimodal if *p* is allowed complete freedom. This complicates the interpretation of marginalized constraints on ${f\\_\\textnormal{\\textsc{nl}}}$, but we will see that priors $\\Delta p \\lesssim 0.5$ for fiducial values of *p* = 0.55 are sufficient to return unbiased constraints. Regarding parametrization 2, which is completely independent of any assumptions on the PNG bias parameters, we will find that the significance of the detection of local PNG (i.e. ${f\\_\\textnormal{\\textsc{nl}}}\\neq 0$) is similar to that from parametrization 1, when using the galaxy power spectrum. However, with parametrization 2, the addition of the galaxy bispectrum contributes much less significantly to improving the constraints, compared to the case with parametrization 1.\nOur goal in the remainder of this paper is not to draw precise quantitative estimates of $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$, but rather to outline and discuss some strategies to deal with galaxy bias uncertainties. These can and should be straightforwardly implemented in more complete forecast pipelines to derive more robust and survey-specific bounds on ${f\\_\\textnormal{\\textsc{nl}}}$. The outline of this paper is as follows. In Sec. [sec:model], we display the expressions of the multitracer galaxy power spectrum, galaxy bispectrum and corresponding covariance matrix that we use in our analysis. In Sec. [sec:results], we describe our forecast methodology, and present and discuss our numerical constraint results. Finally, we summarize and conclude in Sec. [sec:summary]. In App. [app:bgggderivation], we comment on a few aspects of the derivation of the galaxy bispectrum, and in App. [app:cov], we analyse the impact that varying levels of completeness of the calculation of the covariance matrix have on ${f\\_\\textnormal{\\textsc{nl}}}$ constraints.\nThe galaxy power spectrum and bispectrum with local PNG\n=======================================================\nIn this section we display the expressions of the galaxy power spectrum, galaxy bispectrum and corresponding covariances that we consider in our analysis.\nWe work with the following expression for the rest-frame galaxy number density constrast $\\delta\\_g(\\vx, z)$, [eq:biasexp] g(, z) &=& b1(z)m(, z) + b2(z) [m(, z)]2 + bK2(z)[Kij(, z)]2 + () + ()m(, z) \n&+& fnl, where *δ**m* is a long-wavelength total matter density fluctuation, *K**i**j* = [∂*i*∂*j*/∇2 − *δ**i**j*/3]*δ**m* is a long-wavelength tidal field and *ϕ* is the long-wavelength primordial gravitational potential (these terms and their products should be understood as *renormalized operators* in this galaxy bias expansion ). The density and tidal fields are evaluated at the evolved Eulerian position $\\vx$, whereas the primordial potential *ϕ* is evaluated at the initial Lagrangian position $\\vq$. The parameters *b*1, *b*2, *b**K*2, *b**ϕ*, *b**ϕ**δ* are the *galaxy bias parameters* that describe the response of the galaxy number density to the presence of the corresponding long-wavelength perturbations that each multiplies (see Ref. for a review); the bias parameters are a function of redshift, as well as of galaxy properties like stellar mass or luminosity. The stochastic terms $\\eps, \\eps\\_\\delta, \\eps\\_\\phi$ encapsulate the dependence of galaxy number counts on the smaller-wavelength properties of the environment where the galaxies form.\nThe terms in Eq. ([eq:biasexp]) are all that are needed to self-consistently derive the leading-order galaxy power spectrum and bispectrum (see also Sec. 7 of Ref. for a review of the contribution of PNG to the galaxy bias expansion). For simplicity, we skip modelling redshift space distortions (RSD) and always work in real space. Further, we skip considering so-called relativistic effects that can affect galaxy statistics with the same scale dependence as ${f\\_\\textnormal{\\textsc{nl}}}$. The form of their contribution is however known and it depends on the magnification and time-evolution bias parameters, which can both be estimated from the data. We note that these and other simplifications (such as ignoring observational systematics) can have an impact on the resulting ${f\\_\\textnormal{\\textsc{nl}}}$ and $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$ values, but here we are more interested in the relative impact of galaxy bias uncertainties on the constraints, which depends less sensitively on the level of completion of the rest of the analysis. We retain also only terms that are linear in ${f\\_\\textnormal{\\textsc{nl}}}$, which are the most relevant given current observational bounds, $|{f\\_\\textnormal{\\textsc{nl}}}| \\lesssim 5$.\nMultitracer galaxy power spectrum\n---------------------------------\nThe galaxy power spectrum *P**g**g*(*k*) is defined as $(2\\pi)^3P\\_{gg}(k)\\delta\\_D(\\vk + \\vk') = \\big<\\delta\\_g({\\vk})\\delta\\_g({\\vk'})\\big>$, where $\\delta\\_g(\\vk)$ is the Fourier transform of $\\delta\\_g(\\vx)$ and *δ**D* is the Dirac delta function (from hereon we skip writing explicitly the redshift dependence in the arguments to lighten the expressions). We make use of the multitracer framework for the galaxy power spectrum, in which a given galaxy sample in a given redshift bin is split into two subsamples with different bias parameters. The multitracer technique helps to reduce sample variance errors on the largest observed scales as both galaxy samples share the same large-scale modes; this allows to efficiently constrain ${f\\_\\textnormal{\\textsc{nl}}}$ via the relative amplitude of the clustering of the two samples. We will consider the case of two subsamples for simplicity, sample A and sample B, and consider the following estimators for their auto- and cross-power spectra [eq:pggestimators] ggAA(k) &=& k d3’ gA(’)gA(-’), \nggAB(k) &=& k d3’ gA(’)gB(-’), \nggBB(k) &=& k d3’ gB(’)gB(-’), where *V**s* is the survey volume and $\\int\\_k {\\rm d}^3\\vk'$ represents integrating over a spherical shell in Fourier space with radius *k*, width Δ*k* and volume *V**k* = 4*π**k*2Δ*k*; the superscripts $^{\\rm A,B}$ indicate the galaxy subsample, i.e., $\\hat{P}\\_{gg}^{\\rm AB}$ represents the estimator of the cross-power spectrum. Plugging the Fourier transform of Eq. ([eq:biasexp]) in the expectation value of the above equations, and retaining only terms to leading order in perturbation theory and ${f\\_\\textnormal{\\textsc{nl}}}$ yields [eq:pggmodelAA]<ggAA(k)> PAAgg(k) &=& 2 Pmm(k) + 2b1AbA fnlPm(k) + PAA, \n[eq:pggmodelAB]<ggAB(k)> PABgg(k) &=& b1Ab1B Pmm(k) + fnlPm(k), \n[eq:pggmodelBB]<ggBB(k)> PBBgg(k) &=& 2 Pmm(k) + 2b1BbB fnlPm(k) + PBB, where *P**m**m*(*k*) denotes the linear matter power spectrum and *P**m**ϕ* the cross matter-potential power spectrum. The linear matter density perturbations are related to the primordial potential as $\\delta\\_m^{(1)}(\\vk) = \\mathcal{M}(k) \\phi(\\vk)$, where the superscript (1) indicates the linear perturbation theory contribution to the total density field *δ**m* = *δ**m*(1) + *δ**m*(2) + ⋯, and M(*k*) = (2/3)*k*2*T**m*(*k*)/(Ω*m*0*H*02), with *T**m* being the matter transfer function, Ω*m*0 the present-day fractional cosmic mean matter density and *H*0 the present-day Hubble expansion rate. Hence, *P**m**ϕ* = *P**m**m*(*k*)/M(*k*). Finally, $P\\_{\\eps\\eps}$ is the power spectrum of the noise, which we assume to be Poissonian $P\\_{\\eps\\eps} = 1/\\bar{n}\\_g$, where *n̄**g* is the mean observed galaxy number density; note that we ignore the cross-power spectrum of the noise fields $P^{\\rm AB}\\_{\\eps\\eps}$.\nThe galaxy auto power spectrum is shown in the upper left panel of Fig. [fig:contributions]. On scales $k \\lesssim 0.01\\ h/{\\rm Mpc}$, *T**m* tends to a constant and the ${f\\_\\textnormal{\\textsc{nl}}}$ contribution becomes ∝ 1/*k*2, as shown by the brown line. The ratio of the total galaxy power spectrum (black line) and the contributions assuming Gaussian primordial fluctuations (grey) is therefore scale-dependent on large scales, which is why this signature from ${f\\_\\textnormal{\\textsc{nl}}}$ is popularly referred to as the *scale-dependent bias* effect.\nThe upper left panel of Fig. [fig:contributions] illustrates also neatly the critical importance of galaxy bias uncertainties. The local PNG signature is proportional to $b\\_1b\\_\\phi{f\\_\\textnormal{\\textsc{nl}}}$, and hence, in order to constrain ${f\\_\\textnormal{\\textsc{nl}}}$, one needs either additional data and/or prior information on the bias parameters. If prior information on the amplitude of *P**m**m* is available, then the value of *b*1 can be constrained from the smaller-scale part of the power spectrum where the ${f\\_\\textnormal{\\textsc{nl}}}$ contribution is negligible. Nonetheless, even if *b*1 is perfectly determined, one is still left with a perfect degeneracy between *b**ϕ* and ${f\\_\\textnormal{\\textsc{nl}}}$. The way of breaking this degeneracy that is most popular in the literature[1](#fn1) consists in relating *b**ϕ* and *b*1 via the universality relation mentioned in the previous section, *b**ϕ* = 2*δ**c*(*b*1 − 1). There is however no guarantee that observed galaxies obey this relation. Indeed, as mentioned already in Sec. [sec:introduction], studies have shown that the universality relation is not exactly satisfied even for haloes in gravity-only simulations. Further, *b**ϕ* = 2*δ**c*(*b*1 − 1.6) has been argued to be a better description of haloes that have recently undergone a major merger, and *b**ϕ* = 2*δ**c*(*b*1 − 0.55) has been put forward as a more adequate description of simulated stellar-mass selected galaxies. These results suggest that the exact *b**ϕ*(*b*1) relation is very likely tracer-dependent (i.e., different for different galaxy types) and currently fairly uncertain. This makes it a pressing matter to study and develop ways to incorporate and mitigate uncertainties around the *b**ϕ*(*b*1) relation in ${f\\_\\textnormal{\\textsc{nl}}}$ constraints; that is the subject of Sec. [sec:results] below.\n![The different panels show, as labeled, the galaxy auto-power spectrum and the galaxy bispectrum in a squeezed configuration (k_1=k_2 > k_3, k_3 = 4k_{\\rm F} \\approx 0.0027h/{\\rm Mpc}; k_{\\rm F} is the fundamental mode of a survey with 100 {\\rm Gpc^3}/h^3), equilateral configurations (k_1=k_2=k_3) and folded configurations (k_1=2k_2=2k_3). The result shown is the leading-order one in perturbation theory and {f_\\textnormal{\\textsc{nl}}} at z=1, and for the parameters of sample A in Tab. [table:samples]. In each panel, the black line shows the total contribution and the grey line shows the result expected for primordial Gaussian fluctuations ({f_\\textnormal{\\textsc{nl}}}= 0). The remaining curves show the various contributions \\propto {f_\\textnormal{\\textsc{nl}}}, as labeled (the legend in the lower right panel applies to all bispectrum panels).](fig_paper_contributions_PggBggg.png \"fig:\") [fig:contributions]\nGalaxy bispectrum\n-----------------\nThe galaxy bispectrum is defined as $(2\\pi)^3 B\\_{ggg}(k\\_1, k\\_2, k\\_3)\\delta\\_D({\\vk\\_{123}}) = \\big<\\delta\\_g({\\vk\\_1})\\delta\\_g({\\vk\\_2})\\delta\\_g({\\vk\\_3})\\big>$ (in our notation, $\\vk\\_{123} = \\vk\\_1 + \\vk\\_2 + \\vk\\_3$), and we consider the following estimator, [eq:bgggestimator] ggg(k1,k2,k3) = k1 d3a k2 d3b k3 d3c g(a)g(b)g(c)D(abc), where *V*123 = 8*π*2*k*1*k*2*k*3Δ*k*1Δ*k*2Δ*k*3. In our analysis we work with the bispectrum of only one of the samples used in the multitracer power spectrum part for simplicity. We use the bispectrum of one of the samples rather than the two samples combined to reduce the number of free bias parameters. In App. [app:bgggderivation] we comment on a few aspects of the derivation of the galaxy bispectrum (see also Refs. for a number of past works on the galaxy bispectrum in local PNG cosmologies). The final result can be written as [eq:bgggmodel] <ggg(k1,k2,k3)> Bggg(k1,k2,k3) = BgggG(k1,k2,k3) + BgggNG(k1,k2,k3), where [eq:bgggmodelG] &&BgggG(k1,k2,k3) = b13Bmmm(k1, k2, k3) + + B \n&+&+ \nis the contribution due to nonlinear gravitational clustering that is present even if ${f\\_\\textnormal{\\textsc{nl}}}= 0$, and [eq:bgggmodelNG] &&BgggNG(k1,k2,k3) = [ 2b13fnl(k3) + 2bfnlP \n&+& b12bfnlPmm(k1)Pmm(k2) (12( + ) + 2F2(k1, k2, 12)( + )) \n&+& (b12b + b1b2b + 2b1bK2b (122 - 13))fnlPmm(k1)Pmm(k2) ( + ) + (2 perm.) ] \nis the contribution due to local PNG that is $\\propto {f\\_\\textnormal{\\textsc{nl}}}$.[2](#fn2) In the equations above, *μ**a**b* is the cosine of the angle between the *k**a* and *k**b* legs of the triangle, and likewise to the case of the power spectrum, we work to leading order in the bispectrum (i.e., second order in perturbation theory) and in ${f\\_\\textnormal{\\textsc{nl}}}$. Further, $B\\_{\\eps\\eps\\eps}$ and $P\\_{\\eps\\eps\\_\\delta}$ denote, respectively, the bispectrum of the noise field $\\eps$ and the cross-power spectrum of the fields $\\eps$ and $\\eps\\_\\delta$. Assuming Poissonian noise these are given by $B\\_{\\eps\\eps\\eps} = 1/\\bar{n}\\_g^2$ and $P\\_{\\eps\\eps\\_\\delta} = b\\_1/(2\\bar{n}\\_g)$. To derive the second term on the right-hand side of Eq. ([eq:bgggmodelNG]), we have also assumed Poissonian statistics to relate $P\\_{\\eps\\eps\\_\\phi} = (b\\_\\phi/b\\_1)P\\_{\\eps\\eps\\_\\delta}$. Further, *B**m**m**m* denotes the tree-level matter bispectrum given by [eq:bmmm] Bmmm(k1, k2, k3) = 2F2(1, 2) Pmm(k1)Pmm(k2) + (2 perm.), where [eq:f2] F2(1, 2) F2(k1, k2, 12) = 57 + + 27 122 is the second-order perturbation theory mode-coupling kernel.\nIn Fig. [fig:contributions], the upper right, lower left and lower right panels show, respectively, the galaxy bispectrum for squeezed ($k\\_1=k\\_2 > k\\_3\\approx0.0027h/{\\rm Mpc}$), equilateral (*k*1 = *k*2 = *k*3) and folded (*k*1 = 2*k*2 = 2*k*3) configurations. A first noteworthy and well-known aspect is that the local PNG contributions are most prominent in the squeezed-limit. This is shown by the larger difference between the black and grey lines in the upper right panel. Note that the local PNG contribution is sizeable in the squeezed bispectrum for all modes *k*1 = *k*2 probed, which is why increasing the maximum wavenumber analysed $k\\_{\\rm max}$ can help improve significantly ${f\\_\\textnormal{\\textsc{nl}}}$ constraints with the bispectrum, but less so with the power spectrum. Further, contrary to the power spectrum case, in the galaxy bispectrum the ${f\\_\\textnormal{\\textsc{nl}}}$ contribution is not perfectly degenerate with *b**ϕ*. This degeneracy is broken by the terms $\\propto b\\_1^3 {f\\_\\textnormal{\\textsc{nl}}}$ (blue lines in Fig. [fig:contributions]) and $\\propto b\\_1^2b\\_{\\phi\\delta}{f\\_\\textnormal{\\textsc{nl}}}$ (red lines in Fig. [fig:contributions]). In the latter case, *b**ϕ**δ* can also be calculated assuming the universality of the halo mass function: [eq:univbphidelta] b = b- b1 + 1 + c[b2 - (8/21)(b1-1)]. The same caveats about the validity of the universality relation for *b**ϕ* apply likewise to *b**ϕ**δ*. To the best of our knowledge, however, this *b**ϕ**δ* expression has never been tested neither for halos in gravity-only simulations nor simulated galaxies in hydrodynamical simulations. In our forecasts below, when we study deviations from the universality relation for *b**ϕ*, we shall still assume the universality relation for *b**ϕ**δ*. The latter should be eventually tested with *N*-body simulations, but we note that this term contributes a smaller amount compared to terms ∝ *b**ϕ* to the constraining power on ${f\\_\\textnormal{\\textsc{nl}}}$, and hence, uncertainties on *b**ϕ**δ* are not as critical.\nCovariance of the multitracer galaxy power spectrum and bispectrum\n------------------------------------------------------------------\nIn our forecast analysis we work with the following Gaussian likelihood function [eq:likelihood] (), where $\\vD$ is the data vector, ${\\operatorname{\\rm \\bf Cov}}$ its covariance matrix and $\\vM$ is a theoretical prediction for the data vector that depends on parameters $\\vtheta$. The data vector consists of hypothetical measurements of the real-space multitracer power spectrum and bispectrum [eq:datavector] = { ggAA, ggAB, ggBB, gggAAA}; we will work with the bispectrum of subsample A, which is the higher number density one (cf. Sec. [sec:methodology]). Our theoretical predictions $\\vM$ are evaluated using Eqs. ([eq:pggmodelAA]), ([eq:pggmodelAB]), ([eq:pggmodelBB]), ([eq:bgggmodel]), ([eq:bgggmodelG]) and ([eq:bgggmodelNG]); we construct the data vector using the same equations evaluated at a fiducial cosmology. The covariance matrix of the data vector contains three main blocks:\n \n$$\\label{eq:cov\\_def}\n{\\operatorname{\\rm \\bf Cov}} = \n\\begin{pmatrix}\n\\operatorname{\\rm \\bf Cov}^{PP} & \\operatorname{\\rm \\bf Cov}^{BP} \\\\[1.5ex]\n\\cdots & \\operatorname{\\rm \\bf Cov}^{BB} \\\\[1.5ex]\n\\end{pmatrix}\n\\,\\,,$$ \nwhere $\\operatorname{\\rm \\bf Cov}^{PP}$, $\\operatorname{\\rm \\bf Cov}^{BP}$ and $\\operatorname{\\rm \\bf Cov}^{BB}$ denote, respectively, the covariance of the power spectrum part of the data vector, the cross-covariance of the bispectrum and power spectrum, and the covariance of the bispectrum part; the dots indicate the block is equal to the corresponding symmetric block. In App. [app:cov], we outline the derivation of the covariance matrix, and comment on the relative impact of the various contributions to it (see also Refs. for works on bispectrum covariances). In this section, we limit ourselves to displaying only the final expressions. For $\\operatorname{\\rm \\bf Cov}^{PP}$, we consider only the contribution from the disconnected part of the four-point function (the so-called Gaussian term); for the case of the multitracer power spectrum, the result is a 3 × 3 block diagonal matrix:\n \n$$\\label{eq:covPP}\n{\\operatorname{\\rm \\bf Cov}}^{PP}(k\\_1, k\\_2) = 2 \\frac{(2\\pi)^3\\delta\\_{k\\_1k\\_2}}{V\\_sV\\_{k\\_1}}\n\\begin{pmatrix}\n[P\\_{gg}^{AA}(k\\_1)]^2 & P\\_{gg}^{AB}(k\\_1)P\\_{gg}^{AA}(k\\_1) & [P\\_{gg}^{AB}(k\\_1)]^2 \\\\[1.5ex]\n\\cdots & \\frac12\\left[P\\_{gg}^{AA}(k\\_1)P\\_{gg}^{BB}(k\\_1) + [P\\_{gg}^{AB}(k\\_1)]^2\\right] & P\\_{gg}^{BB}(k\\_1)P\\_{gg}^{AB}(k\\_1) \\\\[1.5ex]\n\\cdots & \\cdots & [P\\_{gg}^{BB}(k\\_1)]^2\n\\end{pmatrix}\n\\,\\,.$$ \nWe evaluate the bispectrum covariance block as $\\operatorname{\\rm \\bf Cov}^{BB} = \\operatorname{\\rm \\bf Cov}^{BB}\\_{PPP} + \\operatorname{\\rm \\bf Cov}^{BB}\\_{BB}$, with (in our notation, the superscripts in $\\operatorname{\\rm \\bf Cov}$ indicate which estimators we are taking the covariance of, whereas the subscripts label the various contributions[3](#fn3)) [eq:covBBPPP] BBPPP(k1, k2, k3,k1’, k2’, k3’) = TT’ Pgg(k1)Pgg(k2)Pgg(k3) and [eq:covBBBB] BBBB(k1, k2, k3,k1’, k2’, k3’) &=& Bggg(k1’, k2, k3)Bggg(k1, k2’, k3’) k1k1’ + (8 perm.), \nwhere *U*(*k*1, *k*1ʹ) = 16*π*3*k*2*k*3*k*2ʹ*k*3ʹΔ*k*2Δ*k*3Δ*k*2ʹΔ*k*3ʹΔ*k*1, *δ**T**T*ʹ is a Kronecker delta function that is non-zero only if both triangles *T* = {*k*1, *k*2, *k*3} and *T*ʹ = {*k*1ʹ, *k*2ʹ, *k*3ʹ} are the same, and $S\\_{\\rm shape} = 6,2,1$ for equilateral, isosceles and scalene triangles, respectively; in Eq. ([eq:covBBBB]), the permutations are all 9 that link each side of one triangle to each side of the other. Finally, in our main results we will neglect the cross-covariance term, $\\operatorname{\\rm \\bf Cov}^{BP} = 0$; its contribution is analysed and discussed in App. [app:cov].\nResults\n=======\nIn this section we present our main results on the impact of local PNG galaxy bias uncertainties on ${f\\_\\textnormal{\\textsc{nl}}}$ constraints. We begin in Sec. [sec:methodology] by describing our idealized forecast setup and the two galaxy bias parametrizations that we explore in this paper. The numerical forecast results are then shown and discussed in Secs. [sec:numerics1] and [sec:numerics2].\nMethodology and forecast setup\n------------------------------\nlcccccccccccccc & Sample & *n̄**g* $\\left[{h^3/{\\rm Mpc^3}}\\right]$ & *b*1 & *b*2 & *b**K*2 & *b**ϕ* & *b**ϕ**δ* & \n& A & 1.74 × 10− 3 & 1.58 & − 0.62 & − 0.17 & 3.49 & 1.48 & \n& B & 1.07 × 10− 4 & 2.37 & 0.66 & − 0.39 & 6.15 & 5.01 & \n[table:samples]\nIn our forecast study we consider a galaxy sample at redshift *z* = 1. For the multitracer power spectrum part, we split this galaxy sample into two subsamples A and B as follows. We assume the galaxy sample follows a stellar mass function described by a Schechter function [eq:smf] (M\\*) = \\* ()- exp, with $\\phi\\_\\* = 0.0074\\ h^3/{\\rm Mpc^3}/{\\rm dex}$, *M*\\*ʹ = 1011 *M*⊙/*h* and *α* = 0.38; we have verified this is sufficiently realistic in that it matches well the stellar mass function of the IllustrisTNG simulations at *z* = 1. We then define the two subsamples via a cut in stellar mass *M*\\*: subsample A with *M*\\* ∈ [5 × 1010; 2 × 1011]*M*⊙/*h* and subsample B with *M*\\* > 2 × 1011*M*⊙/*h*. The mean number density *n̄**g* and linear bias parameter *b*1 of each sample are obtained by integrating, respectively, [eq:ng] |ng = M\\* , b1 = M\\* b1T(M\\*) over the corresponding mass ranges. In the equation for *b*1, $b\\_1^{\\rm T}(M\\_\\*)$ denotes the linear halo bias fitting formula of Ref. evaluated in terms of stellar mass according to the prescription described in Ref. . We evaluate the higher-order bias terms *b*2, *b**K*2 using, respectively, the polynomial fit of Ref. *b*2 = 0.412 − 2.143*b*1 + 0.929*b*12 + 0.008*b*13, and the Lagrangian linear-in-matter-density prediction *b**K*2 = − (2/7)(*b*1 − 1). The fiducial values of *b**ϕ* are obtained with the variant of the universality relation *b**ϕ* = 2*δ**c*(*b*1 − *p*) with *p* = 0.55, which is roughly the value preferred by the stellar-mass selected objects in the IllustrisTNG model. We use Eq. ([eq:univbphidelta]) to evaluate *b**ϕ**δ*. These specifications of the two subsamples are summarized in Tab. [table:samples]. For the bispectrum part of the analysis, we consider the bispectrum of subsample A, which has higher number density.[4](#fn4)\nFor the fiducial cosmology we assume a spatially flat $\\Lambda{\\rm CDM}$ model with present-day baryon density Ω*b*0 = 0.0486, cold dark matter density Ω*c*0 = 0.2603, dimensionless Hubble parameter *h* = 0.6774, scalar spectral index *n**s* = 0.967 and primordial scalar amplitude parameter A*s* = 2.068 × 10− 9 at a pivot scale $k\\_{\\rm pivot} = 0.05\\ /{\\rm Mpc}$. We neglect the impact of neutrino masses, Ω*m*0 = Ω*b*0 + Ω*c*0. We further consider a fiducial value of ${f\\_\\textnormal{\\textsc{nl}}}=3$, which is within the range currently allowed by CMB observations ${f\\_\\textnormal{\\textsc{nl}}}= -0.9 \\pm 5.1\\ (1\\sigma)$ ; we choose a non-zero value for ${f\\_\\textnormal{\\textsc{nl}}}$ to entertain the possibility of non-zero detections in our analysis, which is the most interesting scenario. For simplicity and to remain focused on the impact of galaxy bias uncertainties, we will vary only the cosmological parameters ${f\\_\\textnormal{\\textsc{nl}}}$ and A*s*. In real data analysis, one would always adopt priors on the remaining cosmological parameters from other data sets anyway. We evaluate all the linear power spectra and transfer functions with the CAMB code.\nOur assumed survey volume is $V\\_s = 100{\\rm Gpc^3}/h^3$, which is roughly the comoving volume of a full-sky survey spanning *z* = 1 ± 0.5. We consider 30 *k* bins between $k\\_{\\rm min} = k\\_{\\rm F} = \\pi/V\\_s^{1/3}$ and $k\\_{\\rm max} = 0.2\\ h/{\\rm Mpc}$ uniformly distributed in log-space. This value of $k\\_{\\rm max}$ is approximately that above which our leading-order treatment of the galaxy power spectrum and bispectrum becomes inadequate. In power spectrum only analysis (denoted by *P**g**g*-only from hereon), the value of $k\\_{\\rm max}$ is not as important as in combined analysis with the bispectrum (denoted by *P**g**g* + *B**g**g**g* from hereon); this is because the latter can still be appreciably sensitive to ${f\\_\\textnormal{\\textsc{nl}}}$ on small scales for squeezed configurations with large *k*1, *k*2 and small *k*3 (cf. upper right panel of Fig. [fig:contributions]). We have checked that with the more conservative option $k\\_{\\rm max} = 0.1\\ h/{\\rm Mpc}$, the *P**g**g* + *B**g**g**g* analysis had effectively the same constraining power on ${f\\_\\textnormal{\\textsc{nl}}}$ as the *P**g**g*-only analysis. We therefore opted for a higher $k\\_{\\rm max}$ value to study the consequences of galaxy bias uncertainties when the constraining power of the data is stronger. With this *k*-binning, we evaluate the bispectrum at all possible triangles with $k\\_{\\rm min} \\leq k\\_1, k\\_2, k\\_3 \\leq k\\_{\\rm max}$, which yields a total of 873 triangles.\nIn our numerical results, we discuss the impact of galaxy bias uncertainties on local PNG constraints under the following two characterizations of the parameter space:\n* In this case, *b**ϕ* is related to *b*1 via the variant of the universality relation *b**ϕ* = 2*δ**c*(*b*1 − *p*), with *p* being treated as a free parameter. We will be specifically interested on the impact of different assumed priors on *p*. The parameter space that we wish to constrain is 11 dimensional and given by [eq:params1] = {fnl, b1A, b1B, PA, PB, p, s, b2A, bK2A, PA, BA }, where *δ*A*s* is defined as $\\mathcal{A}\\_s = \\mathcal{A}\\_{s, \\rm fidu}\\left[1 + \\delta\\_{\\mathcal{A}\\_s}\\right]$, with $\\mathcal{A}\\_{s, \\rm fidu}$ being the fiducial value. For *P**g**g*-only analyses we vary just the first six of these parameters; we keep *δ*A*s* fixed, which would otherwise be perfectly degenerate with $b\\_1^{\\rm A}, b\\_1^{\\rm B}$. Note that we assume the value of *p* to be the same for both subsamples in the multitracer analysis. Important to recall is also the fact that we still assume the universality prediction of Eq. ([eq:univbphidelta]) to derive *b**ϕ**δ*, although this term contributes subdominantly compared to others.\n* In this case, instead of attempting to make assumptions on *b**ϕ* to constrain ${f\\_\\textnormal{\\textsc{nl}}}$, one fits instead directly for the products of ${f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi$ and ${f\\_\\textnormal{\\textsc{nl}}}b\\_{\\phi\\delta}$ that contribute to the galaxy power spectrum and bispectrum. The parameter space then becomes 13 dimensional and given by [eq:params2] = {[fnlbA], [fnlbB], b1A, b1B, PA, PB, fnl, s, [fnlbA], b2A, bK2A, PA, BA }. For *P**g**g*-only analyses we vary just the first six of these parameters; this parametrization is completely independent of any assumptions on *b**ϕ*, but one loses the ability to constraint the absolute value of ${f\\_\\textnormal{\\textsc{nl}}}$. The value of this approach lies instead on the possibility to *model-independently* detect non-zero $[{f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi^{\\rm A}]$ or $[{f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi^{\\rm B}]$, which would imply ${f\\_\\textnormal{\\textsc{nl}}}\\neq 0$, and consequently, rule out single-field models of inflation. For *P**g**g* + *B**g**g**g* analysis, the absolute value of ${f\\_\\textnormal{\\textsc{nl}}}$ can still be formally constrained because of the term $\\propto b\\_1^3{f\\_\\textnormal{\\textsc{nl}}}$ in Eq. ([eq:bgggmodelNG]) (although with much reduced constraining power, as we will see in Sec. [sec:numerics2]).[5](#fn5)\nFinally, we sample the parameter space with the EMCEE Python implementation of the affine-invariant Markov Chain Monte Carlo (MCMC) sampler proposed in Ref. . We assume wide uninformative linear priors on all parameters and use 32 *walkers* with a nominal chain convergence criterion that the chain size must be 100 times the autocorrelation time and the latter having varied less than 1% the last time it was calculated (every few thousand samples, in our case). We have also visually inspected the contour plots and found them to be stable enough for our conclusions well before nominal convergence was achieved. We have further rerun the constraints with the walkers initialized at different starting points, which returned the same constraints. As another cross-check, we have further compared the outcome of EMCEE with that of a two-dimensional grid search for the parameters ${f\\_\\textnormal{\\textsc{nl}}}$ and *p* in parametrization 1 (which as we will see next displays a non-trivial bimodal shape of the likelihood), which we found to be effectively indistinguishable.\nResults from parametrization 1: *b**ϕ* = 2*δ**c*(*b*1 − *p*)\n------------------------------------------------------------\nTo build intuition for the shape of the constraints on ${f\\_\\textnormal{\\textsc{nl}}}$ with the *b**ϕ* = 2*δ**c*(*b*1 − *p*) parametrization, we study first the results from constraints in which only ${f\\_\\textnormal{\\textsc{nl}}}$ and *p* are varied and the remaining parameters are kept fixed at their fiducial values. In Fig. [fig:intuition0Pgg], this is shown for the *P**g**g*-only constraints. The left panel shows that the two-dimensional distribution is appreciably bimodal with one peak at ${f\\_\\textnormal{\\textsc{nl}}}> 0$ and *p* < *b*1*A*, *b*2*A*, and another at ${f\\_\\textnormal{\\textsc{nl}}}< 0$ and *p* > *b*1*A*, *b*2*A*. This is as expected since the multitracer galaxy power spectrum is weakly sensitive to a simultaneous change of sign of ${f\\_\\textnormal{\\textsc{nl}}}$ and *b*1 − *p*; for a single galaxy sample, the power spectrum would be completely unchanged under this change of sign. This shape of the distribution can be understood by inspecting the degeneracy directions from the three spectra in our multitracer power spectrum analysis. Concretely, Eqs. ([eq:pggmodelAA]), ([eq:pggmodelAB]) and ([eq:pggmodelBB]) remain unchanged if ${f\\_\\textnormal{\\textsc{nl}}}$ and *p* vary, respectively, according to the relations [eq:degdirs] Degeneracy from PggAA :: p &=& b1, fiduA -, \nDegeneracy from PggAB :: p &= & -, \nDegeneracy from PggBB :: p &=& b1, fiduB -, where the subscript $\\_{\\rm fidu}$ indicates the fiducial value. These relations are shown by the dashed lines in the left panel of Fig. [fig:intuition0Pgg]. All three degeneracy directions asymptote to *p* → ± ∞ as ${f\\_\\textnormal{\\textsc{nl}}}\\to 0$. Hence, given that the likelihood is not negligibly small at ${f\\_\\textnormal{\\textsc{nl}}}= 0$, this implies that *p* cannot be constrained in our *P**g**g* analysis.[6](#fn6) On the other hand, an increase in ${f\\_\\textnormal{\\textsc{nl}}}$ can be compensated by *p* values that approach $b\\_1^{\\rm A}$ and $b\\_1^{\\rm B}$. Crucially, however, due to the different values of *b*1 of the two subsamples, this compensation is not perfect and ${f\\_\\textnormal{\\textsc{nl}}}$ can be constrained: in our concrete idealized setup, the two-dimensional 1*σ* contours span $-2 \\lesssim {f\\_\\textnormal{\\textsc{nl}}}\\lesssim 7$.\nIt follows also from the above discussion that single-tracer power spectrum analyses cannot constrain ${f\\_\\textnormal{\\textsc{nl}}}$ if *p* is varied freely. Similarly, multitracer analyses with the subsamples allowed to take on different values of *p* cannot also constrain ${f\\_\\textnormal{\\textsc{nl}}}$, unless prior information is added on the *p* parameters. This has important ramifications to galaxy selection strategies for multitracer analyses. Here, we have assumed *p* to be constant for samples selected by stellar-mass, which is in accordance with the results obtained in Ref. using simulations with the IllustrisTNG model. However, the combination of samples with varying selection criteria is also a perfectly viable option (say, a stellar-mass selected sample A and active galactic nuclei as sample B), but which forcibly requires assuming different *p* values for the various samples. We do not investigate this scenario explicitly in this paper (which is nonetheless qualitatively similar to cases discussed next with priors on *p*), although we stress that uncertainties around the *b**ϕ*(*b*1) relation should be a factor to take into consideration in the design of multitracer analyses.\n![Constraints on {f_\\textnormal{\\textsc{nl}}} and p from P_{gg}-only analysis and keeping all the remaining parameters fixed at the fiducial values. The left panel shows the shape of the likelihood function (shaded purple) and the 1\\sigma and 2\\sigma confidence contour levels (black lines) for the case without assumed priors on p. The dashed lines mark the {f_\\textnormal{\\textsc{nl}}}-p degeneracy directions of the spectra in the multitracer analysis, as labeled (cf. Eqs. ([eq:degdirs])). The right panel shows the marginalized constraints on {f_\\textnormal{\\textsc{nl}}} for different assumed priors on p, as labeled. The dotted lines indicate the fiducial parameter values. ](fig_paper_intuition_fNL_p_Pgg.png \"fig:\") [fig:intuition0Pgg]\n![Same as Fig. [fig:intuition0Pgg], but for combined P_{gg} + B_{ggg} analyses.](fig_paper_intuition_fNL_p_PggBggg.png \"fig:\") [fig:intuition0PggBggg]\nThe bimodal shape of the two-dimensional constraints on ${f\\_\\textnormal{\\textsc{nl}}}$ and *p* demands special care in the interpretation of the corresponding one-dimensional marginalized constraints. The latter are shown for ${f\\_\\textnormal{\\textsc{nl}}}$ by the black solid line in the right panel of Fig. [fig:intuition0Pgg]. The curve is strongly peaked around ${f\\_\\textnormal{\\textsc{nl}}}= 0$ since the degeneracy directions that leave *p* unconstrained assign most of the volume of the distribution there (the volume is not simply infinite because our wide prior *p* ∈ [ − 100, + 100] makes it finite). The 1*σ* range around the maximum of this distribution excludes even the fiducial value ${f\\_\\textnormal{\\textsc{nl}}}= 3$. The green and magenta lines in the right panel of Fig. [fig:intuition0Pgg] show the result from constraints assuming stronger priors on *p*: $p \\in p\\_{\\rm fidu} \\pm 2$ and $p \\in p\\_{\\rm fidu} \\pm 0.5$, respectively. As expected, the stronger priors reduce the volume of the distribution around ${f\\_\\textnormal{\\textsc{nl}}}= 0$, which progressively centers the marginalized constraints around the fiducial ${f\\_\\textnormal{\\textsc{nl}}}$ value. We have also explicitly checked (not shown) that a prior $p \\in p\\_{\\rm fidu} \\pm 0.15$ yields effectively the same constraints as those from keeping *p* fixed at the fiducial value (orange line in the right panel of Fig. [fig:intuition0Pgg]).\n![Triangle constraints plot from combined P_{gg}+B_{ggg} analyses with the parametrization 1, b_\\phi = 2\\delta_c(b_1-p). The panels in the diagonal show one-dimensional marginalized constraints and those in the off-diagonal show the 1\\sigma and 2\\sigma confidence levels of two-dimensional marginalized constraints. The different colors indicate different assumed priors on p, as labeled (only distinguishable along the {f_\\textnormal{\\textsc{nl}}} direction; the different priors on p also naturally have an impact along the p direction, but which are only hardly visible on the scale of the plot). The values of P_\\eps^{\\rm A}, P_\\eps^{\\rm B}, P_{\\eps\\eps_\\delta}^{\\rm A} have dimensions of power spectra and B_{\\eps\\eps\\eps}^{\\rm A} of bispectra.](fig_paper_triangle_constraints_parametrization_0_PggBggg.png \"fig:\") [fig:triangle0PggBggg]\nFigure [fig:intuition0PggBggg] shows the same as Fig. [fig:intuition0Pgg], but for the combined *P**g**g* + *B**g**g**g* analysis. The same qualitative features are present and the only differences are the quantitatively tighter constraints provided by the bispectrum information. In particular, both the 1*σ* and 2*σ* contours are disjoint in the *P**g**g* + *B**g**g**g* case, but in the *P**g**g*-only case, the 1*σ* contours are the only disjoint ones.\nIt is also interesting to discuss the result depicted by the blue line in the right panels of Figs. [fig:intuition0Pgg] and [fig:intuition0PggBggg], which shows the constraints from keeping *p* fixed but equal to the wrong value *p* = 1 (that of the universality relation). The two main consequences of choosing *p* = 1 when the fiducial is *p* = 0.55 are (i) a broadening of the distribution because the values of *b**ϕ* become smaller and this reduces the constraining power; and (ii) a shift in the position of the maximum of the distribution towards higher ${f\\_\\textnormal{\\textsc{nl}}}$. In our idealized forecast setup, the corresponding 1*σ* bounds still encompass our fiducial value of ${f\\_\\textnormal{\\textsc{nl}}}= 3$, but the result illustrates nonetheless the potential dangers of keeping *p* fixed to a wrong value (see e.g. Refs. for examples of constraint studies where two different values of *p* are used, *p* = 1 and *p* = 1.6).\nThe discussion above for the case in which only ${f\\_\\textnormal{\\textsc{nl}}}$ and *p* are varied in the constraints facilitates the interpretation of the results when all parameters in Eq. ([eq:params1]) are free. These are shown by the triangle constraints plot of Fig. [fig:triangle0PggBggg]. A first point to note is that the constraints on all parameters except ${f\\_\\textnormal{\\textsc{nl}}}$ and *p* are insensitive to the different priors on *p*. Further, the figure displays also the well known degeneracy between A*s* and the galaxy bias parameter *b*1, which is perfect in power spectrum analysis, but it is broken by the bispectrum information. The constraints on the ${f\\_\\textnormal{\\textsc{nl}}}-p$ plane are similar to those discussed already in Figs. [fig:intuition0Pgg] and [fig:intuition0PggBggg], although just slightly looser because of small degeneracies that arise with the remaining parameters.\nThe left panel of Fig. [fig:sigmas] summarizes the marginalized 1*σ* bounds on ${f\\_\\textnormal{\\textsc{nl}}}$ for *P**g**g*-only and combined *P**g**g* + *B**g**g**g* analyses, and for different priors on *p* with the parametrization 1, *b**ϕ* = 2*δ**c*(*b*1 − *p*). As discussed above, when *p* is allowed to vary freely, both *P**g**g*-only and *P**g**g* + *B**g**g**g* cases yield biased 1*σ* constraints that do not encompass the fiducial value (the 2*σ* intervals do encompass ${f\\_\\textnormal{\\textsc{nl}}}= 3$; not shown). The prior $p \\in p\\_{\\rm fidu} \\pm 0.5$ is however sufficient to return 1*σ* constraints that are nearly indistinguishable from those obtained assuming perfect knowledge of the *p* parameter of the galaxies. Also as discussed above, the adoption of the universality relation value *p* = 1 shifts the distributions slightly upwards and degrades slightly the constraining power of the data: the net effect are constraints on ${f\\_\\textnormal{\\textsc{nl}}}$ shifted from the fiducial value by ≈ 0.3*σ* and ≈ 0.7*σ* for the *P**g**g*-only and combined *P**g**g* + *B**g**g**g* analyses, respectively.\nResults from parametrization 2: ${f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi$\n----------------------------------------------------------------------\n[fig:sigmas]\n![Same as Fig. [fig:triangle0PggBggg], but for the P_{gg}-only and combined P_{gg}+B_{ggg} analysis with parametrization 2, {f_\\textnormal{\\textsc{nl}}}b_\\phi. The width of the constraints on b_1^{\\rm A} and b_1^{\\rm B} in the P_{gg} case is barely noticeable in the scale of the plot. Recall, in the P_{gg}-only case, we do not treat \\mathcal{A}_s as a free parameter.](fig_paper_triangle_constraints_parametrization_1_PggBggg.png \"fig:\") [fig:triangle1PggBggg]\nWe turn our attention now to the constraints obtained under parametrization 2 (cf. Eq. ([eq:params2])). These are shown in the triangle constraints plot of Fig. [fig:triangle1PggBggg]; the right panel of Fig. [fig:sigmas] summarizes the one-dimensional marginalized 1*σ* bounds on the four parameters relevant to the detection of local PNG.\nIn the *P**g**g*-only case, our idealized setup would be able to distinguish ${f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi^{\\rm A}$ and ${f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi^{\\rm B}$ from zero with approximately 1.25*σ* and 1.14*σ* significance, respectively. This is in line with the significance of approximately 1.30*σ* with which the *P**g**g*-only analysis would be able to distinguish non-zero ${f\\_\\textnormal{\\textsc{nl}}}$ in parametrization 1 with perfect knowledge of the *b**ϕ*(*b*1) relation (cf. orange triangle symbol in the left panel of Fig. [fig:sigmas]). This is as expected and indicates that, in multitracer power spectrum analyses, detections of non-zero ${f\\_\\textnormal{\\textsc{nl}}}$ that completely circumvent assumptions on the *b**ϕ*(*b*1) relation are possible and retain the same significance as when assuming perfect knowledge on the galaxy bias parameters; the only price to pay is that the exact value of ${f\\_\\textnormal{\\textsc{nl}}}$ is left undetermined.\nInterestingly, even though unfortunately, the addition of the bispectrum information does not tighten the constraints on ${f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi^{\\rm A}$ and ${f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi^{\\rm B}$ (cf. comparable red and black symbols in the right panel of Fig. [fig:sigmas]). This is in contrast with the tightening of the constraints on ${f\\_\\textnormal{\\textsc{nl}}}$ that the addition of the bispectrum attains with parametrization 1 (cf. circle vs. triangle symbols in the left panel of Fig. [fig:sigmas]). The reason behind the lack of constraining power of the bispectrum on local PNG with parametrization 2 can be traced back to a strong degeneracy that arises between ${f\\_\\textnormal{\\textsc{nl}}}$ and ${f\\_\\textnormal{\\textsc{nl}}}b\\_{\\phi\\delta}$, as shown in the corresponding panel of Fig. [fig:triangle1PggBggg]. Concretely, in parametrization 1, the constraining power of the bispectrum on local PNG comes from the six terms $\\propto {f\\_\\textnormal{\\textsc{nl}}}$ shown in Fig. [fig:contributions]. On the other hand, in the case of parametrization 2, the same number of terms exists, but the terms $\\propto b\\_1^3{f\\_\\textnormal{\\textsc{nl}}}$ and $\\propto b\\_1^2 [{f\\_\\textnormal{\\textsc{nl}}}b\\_{\\phi\\delta}]$ are not constrained by the power spectrum part of the data vector and are therefore free to compensate each others’ effects on the bispectrum. This degeneracy is in fact quite strong, given the similar scale dependence that these terms display on small-scales in squeezed configurations (cf. blue and red curves in the upper right panel of Fig. [fig:contributions]), which is where most of the constraining power lies. This works to effectively reduce the contribution from local PNG to the galaxy bispectrum and make it less constraining.[7](#fn7)\nNote however that the addition of bispectrum information is always useful in general to break the degeneracy between the *b*1 parameters and A*s*; the latter is assumed fixed at the fiducial value in our *P**g**g*-only analysis. This does not have however a particularly strong impact on the local PNG constraints, which are only very weakly correlated with *b*1 (cf. the similar constraints on ${f\\_\\textnormal{\\textsc{nl}}}$ from *P**g**g*-only and *P**g**g* + *B**g**g**g* analyses, despite markedly different constraints on *b*1 in Fig. [fig:triangle1PggBggg]).\nSummary & Conclusions\n=====================\nOne of the main open questions in cosmology concerns the degree of non-Gaussianity of the distribution of the energy fluctuations in the primordial Universe. The simplest single-field models of inflation predict Gaussian distributed fluctuations, and hence, any detection of primordial non-Gaussianity (PNG) would rule out these simpler models and open the door to more elaborate multifield constructions. The next major breakthroughs on observational searches for PNG are expected to come from analyses of the statistics of the large-scale galaxy distribution and are focused on so-called local-type PNG, parametrized by the amplitude of ${f\\_\\textnormal{\\textsc{nl}}}$ in Eq. ([eq:fnl]). The tightest current bounds from the CMB set ${f\\_\\textnormal{\\textsc{nl}}}= -0.9 \\pm 5.1\\ (1\\sigma)$, but upcoming large-scale structure surveys are expected to be able to probe $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}} \\sim \\mathcal{O}(1)$.\nA major challenge in using galaxies to constrain cosmology concerns the uncertainties associated with the bias parameters of the observed galaxy samples. In the case of local PNG, these parameters give rise to strong degeneracies with ${f\\_\\textnormal{\\textsc{nl}}}$ that drastically reduce the constraining power of the data. The most popular way of breaking these degeneracies involves establishing relations between the most relevant galaxy bias parameters, like the relation between the galaxy bias parameters *b*1 and *b**ϕ* (cf. Eq. ([eq:biasexp])). In the context of gravity-only dynamics and assuming universality of the halo mass function, these two bias parameters are related by *b**ϕ* = 2*δ**c*(*b*1 − *p*) with *p* = 1. There is, however, no reason to expect this to be a good description for actually observed galaxies. In fact Refs. showed that the *b**ϕ*(*b*1) relation is sensitive to the formation time of the host haloes (with *p* = 1.6 being a more adequate description of recent mergers), and Ref. showed that *p* = 0.55 provides a better description of stellar mass selected galaxies simulated with the IllustrisTNG model. This uncertainty on the *b**ϕ*(*b*1) relation will invariably impact the bounds on ${f\\_\\textnormal{\\textsc{nl}}}$. Our goal in this paper was to take a few steps in the direction of determining which strategies are available to make ${f\\_\\textnormal{\\textsc{nl}}}$ constraints more robust to such galaxy bias uncertainties.\nIn this paper, we worked with an idealized forecast setup with galaxies at *z* = 1 in a volume $V\\_s = 100{\\rm Gpc}^3/h^3$ (cf. Sec. [sec:methodology]) to analyse the impact that different assumptions on *b**ϕ* and its relation to *b*1 can have on local PNG constraints. We have considered both multitracer power spectrum data (called *P**g**g*-only), as well as its combination with the galaxy bispectrum (called *P**g**g* + *B**g**g**g*). We have focused on two parametrizations: (i) parametrization 1, in which *b**ϕ* = 2*δ**c*(*b*1 − *p*) and *p* is treated as a free parameter, and (ii) parametrization 2, in which one fits directly for products of ${f\\_\\textnormal{\\textsc{nl}}}$ and the local PNG bias parameters. The latter parametrization bypasses the need for any assumptions on the *b**ϕ*(*b*1) relation, but makes it harder to pin down the exact value of ${f\\_\\textnormal{\\textsc{nl}}}$[8](#fn8). Nonetheless, it can still be useful to distinguish ${f\\_\\textnormal{\\textsc{nl}}}$ from zero, and therefore, discriminate between single-field and multifield inflation. Our main findings can be summarized as follows:\n* The constraints on the ${f\\_\\textnormal{\\textsc{nl}}}-p$ plane are bimodal and the marginalized bounds on ${f\\_\\textnormal{\\textsc{nl}}}$ depend sensitively on the assumed priors on *p* (cf. Figs. [fig:intuition0Pgg] and [fig:intuition0PggBggg]). In our setup, priors $p \\in p\\_{\\rm fidu} \\pm 0.5$ can yield constraints similar to those with *p* fixed to the fiducial value $p\\_{\\rm fidu} = 0.55$ (cf. Fig. [fig:sigmas]).\n* Fixing *p* to the wrong value affects both ${f\\_\\textnormal{\\textsc{nl}}}$ and $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$. In our setup for a fiducial value *p* = 0.55, the adoption of the universality relation *p* = 1 in the combined *P**g**g* + *B**g**g**g* analysis shifts the marginalized constraints on ${f\\_\\textnormal{\\textsc{nl}}}$ upwards by ≈ 0.7*σ* (cf. Fig.[fig:sigmas]).\n* In *P**g**g*-only analysis, the significance of the detection of ${f\\_\\textnormal{\\textsc{nl}}}\\neq 0$ with parametrization 1 and ${f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi \\neq 0$ with parametrization 2 is the same (cf. Fig. [fig:sigmas]). This shows that the detection of local PNG using galaxies could be made completely independently of the *b**ϕ*(*b*1) relation.\n* The addition of the galaxy bispectrum improves constraints with parametrization 1, but not with parametrization 2 because the sensitivity of the bispectrum to local PNG is reduced by a strong degeneracy that arises between ${f\\_\\textnormal{\\textsc{nl}}}$ and $[{f\\_\\textnormal{\\textsc{nl}}}b\\_{\\phi\\delta}]$ (cf. discussion in Sec. [sec:numerics2]).\nOur work shows that constraints on ${f\\_\\textnormal{\\textsc{nl}}}$ can depend critically on the assumed *b**ϕ*(*b*1) relation, which motivates further theoretical studies on it. Such an interesting set of studies can involve determining the values of *p* predicted from current state-of-the-art galaxy formation simulations (as Ref. did recently with IllustrisTNG) and use the mean and scatter of the predictions to inform priors on *p*. This is akin to the case of baryonic effects on the small-scale total matter power spectrum that is a major source of uncertainty in weak-lensing data analysis, and whose priors are also often informed by hydrodynamical simulations.\nIt is important to note also that *b**ϕ* = 2*δ**c*(*b*1 − *p*) is a purely phenomenological variant of the universality relation that was shown to describe well the stellar-mass selected objects simulated with IllustrisTNG for *p* = 0.55. It could well be the case that the same functional form is not a good fit to the results from other galaxy formation simulations, nor to galaxy samples selected by properties other than stellar-mass. In fact, the results of Ref. show that *b**ϕ* = 2*δ**c*(*b*1 − *p*) ceases to be an adequate description for objects selected by color or black hole accretion rate. These are particularly relevant considerations for multitracer analyses with samples selected by different criteria, in which case it would be likely inadequate to assume a fixed form of *b**ϕ*(*b*1) for all samples (cf. discussion in Sec. [sec:numerics1]). These remarks all motivate further work to determine better functional forms for the *b**ϕ*(*b*1) relation and/or design alternative ways to place priors on *b**ϕ*. Likewise, it would be interesting to test also the validity of the universality relation Eq. ([eq:univbphidelta]) for the second-order bias parameter *b**ϕ**δ*.\nWe would like to thank Giovanni Cabass, Elisabeth Krause and Fabian Schmidt for useful comments and discussions. The author acknowledges support from the Starting Grant (ERC-2015-STG 678652) “GrInflaGal” from the European Research Council.\nAspects of the derivation of the galaxy bispectrum of Eq. ([eq:bgggmodel])\n==========================================================================\nThe steps to derive Eq. ([eq:bgggmodel]) involve plugging the Fourier transform of Eq. ([eq:biasexp]) into the expectation value of $\\big<\\delta\\_g(\\vk\\_a)\\delta\\_g(\\vk\\_b)\\delta\\_g(\\vk\\_c)\\big>$ (which appears in the expectation value of Eq. ([eq:bgggestimator])), and then simply retain leading-order terms in perturbation theory, as well as terms $\\propto {f\\_\\textnormal{\\textsc{nl}}}$. This is largely a tedious straightforward exercise, although there are a few steps that are more subtle than others. We display a few of these here to illustrate some typical derivation steps and refer the interested reader to Refs. for other recent detailed calculations of galaxy bispectra in ${f\\_\\textnormal{\\textsc{nl}}}$ cosmologies.\nLet us consider as a first example a term involving the tidal field $K\\_{ij}(\\vx)$. Specifically, the term $\\propto b\\_1 b\\_{K^2} b\\_\\phi {f\\_\\textnormal{\\textsc{nl}}}$ in Eq. ([eq:bgggmodelNG]) follows from [eq:examplea1] b1bK2bfnl+ (2 perm.), where $\\int\\_{\\vp} \\equiv \\int {\\rm d}^3\\vp / (2\\pi)^3$ and we have used already that $\\phi(\\vk) = \\delta\\_m^{(1)}(\\vk)/\\mathcal{M}(k)$. In Fourier space, $K\\_{ij}(\\vk) = \\left(k\\_ik\\_j/k^2 - \\delta\\_{ij}/3\\right)\\delta\\_m(\\vk)$, and the derivation proceeds as [eq:examplea2] &&b1bK2bfnl+ (2 perm.) \n&& = b1bK2bfnl+ (2 perm.) \n&& = 2b1bK2bfnl(2)3 D(abc) + (2 perm.) \n&& = 2b1bK2bfnl(2)3 (ab2 - 13) Pmm(ka)Pmm(kb) D(abc) + (2 perm.), where in the first equality we have used Wick’s theorem to write the four-point function in terms of the product of two two-point functions and the connected four-point function; in the second equality we have integrated over $\\vp$ using the Dirac delta functions and dropped the connected four-point function contribution, which is negligible on the scales of interest; finally, in the third equality we have reshuffled the permutations to write down the expression in a more economic manner. Plugging this expression into the expectation value of Eq. ([eq:bgggestimator]) (and further assuming sufficiently narrow bins that it becomes a good approximation to skip the angular and bin-averages) yields the corresponding term in Eq. ([eq:bgggmodelNG]).\nAs another example, consider the term $\\propto b\\_1^3{f\\_\\textnormal{\\textsc{nl}}}$ in Eq. ([eq:bgggmodelNG]), which comes from the contribution of ${f\\_\\textnormal{\\textsc{nl}}}$ to the linear matter density field [eq:exampleb1] m(1)() = (k). This term then follows from $b\\_1^3\\langle\\delta\\_m(\\vk\\_a)\\delta\\_m(\\vk\\_b)\\delta\\_m(\\vk\\_c)\\rangle$ as [eq:exampleb2] && b13 (kc)<m(a)m(b)(G(c) + fnl G() G(c - ))> + (2 perm.) \n&=& b13 fnl(ka)(kb)(kc) <G(a)G(b) G() G(c - )> + (2 perm.) \n&=& b13 fnl(ka)(kb)(kc) (2)6 P(ka)P(kb) \n&+& (2 perm.) \n&=& 2 b13 fnl(2)3 (kc) D(abc) + (2 perm.), where in the first equality we have used that $\\delta\\_m^{(1)}(\\vk) = \\mathcal{M}(k) \\phi(\\vk)$ and discarded the bispectrum of $\\phi\\_{\\rm G}$ (which is zero for a Gaussian field) and next-to-leading-order contributions; in the second equality we have used Wick’s theorem to write the four-point function as the product of two two-point functions (and discarded the connected four-point function, which is zero); finally, in the third equality we have integrated over $\\vp$ using the Dirac delta functions and have written the auto power spectrum of the primordial potential as *P**ϕ**ϕ*(*k*) = *P**m**m*(*k*)/M(*k*)2. In keeping with the narrow bin approximation mentioned above, plugging this expression in the expectation value of Eq. ([eq:bgggestimator]) yields the desired term in Eq. ([eq:bgggmodelNG]).\nAnother perhaps more subtle term is that $\\propto b\\_1^2 b\\_\\phi {f\\_\\textnormal{\\textsc{nl}}}$ in Eq. ([eq:bgggmodelNG]). For this term, it matters explicitly to leading order that the primordial potential *ϕ* in Eq. ([eq:biasexp]) is evaluated at the Lagrangian position $\\vq$, which is related to the potential evaluated at the Eulerian position $\\vx$ via a displacement term as [eq:examplec1] Lag() = Eul() - si() iEul(), where the displacement field is given by $s^i(\\vx) = -(\\partial\\_i/\\nabla^2) \\delta^{(1)}(\\vx)$. In Fourier space, we have[9](#fn9) [eq:examplec2] Lag() = Eul() + m(1)() Eul(- ). In these equations we have marked when *ϕ* is evaluated in Lagrangian or Eulerian space; in the following, *ϕ* is always the Eulerian one and we drop the subscripts to ease the notation. The corresponding term $\\propto b\\_1^2 b\\_\\phi {f\\_\\textnormal{\\textsc{nl}}}$ in Eq. ([eq:bgggmodelNG]) is then given by two contributions [eq:examplec3] && b12 bfnl \n&+& b12 bfnl. \nThe first of these can be worked out analogously to as in Eq. ([eq:examplea2]). The derivation of the second contribution is in all analogous to the derivation of the tree-level matter bispectrum in perturbation theory, just with the caveat that one of the modes in the three-point function is always a linear one, i.e., only two modes are expanded as $\\delta\\_m(\\vk) = \\delta\\_m^{(1)}(\\vk) + \\delta\\_m^{(2)}(\\vk) + \\cdots$, with $\\delta\\_m^{(2)}(\\vk) = \\int\\_{\\vr} F\\_2(\\vr, \\vk - \\vr)\\delta\\_m^{(1)}(\\vr)\\delta\\_m^{(1)}(\\vk-\\vr)$. From this point onwards, the derivation follows straightforwardly and yields the corresponding term in Eq. ([eq:bgggmodelNG]) (always in keeping with the assumption of sufficiently narrow bins in Eq. ([eq:bgggestimator]) to justify skipping performing the bin averages explicitly).\nThe impact of covariance composition\n====================================\nIn this appendix, we outline the derivation of the covariance matrix of the combined power spectrum and bispectrum data vector. We also show the impact that varying levels of completion of the covariance calculation have on the resulting ${f\\_\\textnormal{\\textsc{nl}}}$ bounds.\nThe covariance of the galaxy auto-power spectrum is defined as [eq:covPPderivation1] PAAPAA(k1, k2) &=& <ggAA(k1)ggAA(k2)> - PggAA(k1)PggAA(k2) \n&=& k1 d3a k2 d3b <gA(a)gA(-a)gA(b)gA(-b)> - PggAA(k1)PggAA(k2) \n&=& k1 d3a k2 d3b (2)6[PggAA(ka)]2 \n&=& k1k2k1 d3a [PggAA(ka)]2 \n&=& k1k2 [PggAA(k1)]2, where in the third equality we have used Wick’s theorem and discarded the contribution from the connected four-point function, which is negligible on the large scales where the ${f\\_\\textnormal{\\textsc{nl}}}$ contribution is important; in the fourth equality we have integrated using the Dirac delta functions, which imposes the constrain *δ**k*1*k*2 that the two wavenumbers must belong to the same bin for the result to be non-zero (we have used also that *δ**D*(0) ≡ *V**s*/(2*π*)3); in the last equality we have assumed sufficiently narrow bins to take the power spectrum out of the integral. The steps above are for the derivation of the covariance of the auto-power spectrum of galaxy sample A, which gives the upper left entry in Eq. ([eq:covPP]); the remainder of the entries of the multitracer power spectrum covariance follow analogously.\nThe covariance of the galaxy bispectrum is defined as (recall, in the main body of the paper, we considered the bispectrum of the subsample A) [eq:covBBderivation1] && BAAABAAA(k1, k2, k3,k1’, k2’, k3’) = <AAAggg(k1, k2, k3)AAAggg(k1’, k2’, k3’)> \n&-& BAAAggg(k1, k2, k3)BAAAggg(k1’, k2’, k3’) \n&=& k1 d3a k2 d3b k3 d3c k1’ d3a’ k2’ d3b’ k3’ d3c’ \n&& <Ag(a)Ag(b)Ag(c)Ag(a’)Ag(b’)Ag(c’)> D(abc) D(abc’) - BAAAggg(k1, k2, k3)BAAAggg(k1’, k2’, k3’). \nUsing Wick’s theorem, the six-point function contribution can be split into terms proportional to the product of three two-point functions, called *P**P**P* term; terms proportional to the product of two three-point functions, called the *B**B* term; terms proportional to the product of a two-point function and a four-point function, called the *T**P* term; and finally, the connected six-point function term. In the main body of the paper, we considered only the contribution from the *P**P**P* and the *B**B* term, which are the most straightforward to evaluate and yield Eqs. ([eq:covBBPPP]) and ([eq:covBBBB]), respectively (see e.g. Refs. for detailed derivations/discussions). As argued recently in Refs. , the contribution from the connected six-point function is expected to be negligible on the large scales and squeezed configurations relevant for ${f\\_\\textnormal{\\textsc{nl}}}$ constraints. The *T**P* term has the same structure of ratios *U*/*V*123*V*1ʹ2ʹ3ʹ and the same powers of power spectra as the *B**B* term, so they display normally the same order of magnitude (especially in the squeezed limit ). Below, we measure roughly the importance of the *T**P* term by doubling the contribution from the *B**B* term.\nFinally, the cross-covariance of the power spectrum and matter bispectrum is defined as [eq:covBPderivation1] && BAAAPAB(k1, k2, k3,k1’) = <AAAggg(k1, k2, k3)ABgg(k1’)> - BAAAggg(k1, k2, k3)PABgg(k1’) \n&=& k1 d3a k2 d3b k3 d3c k1’ d3a’ <gA(a)gA(b)gA(c)gA(a’)gB(-a’)> D(abc) \n&& - BAAAggg(k1, k2, k3)PABgg(k1’) \n&=&, where the third equality follows from Wick’s theorem (dropping the contribution from the connected five-point function) plus derivation steps analogous to those already sketched above. This expression, as written, corresponds to the cross-covariance of the bispectrum with the *P̂**A**B* part of the data vector, but the steps for the *P̂**A**A* and *P̂**B**B* are analogous. Further, $B\\_{ggg}^{\\rm AAB}$ denotes a cross-bispectrum defined as $\\big<\\delta\\_g^{\\rm A}(\\vk\\_a)\\delta\\_g^{\\rm A}(\\vk\\_b)\\delta\\_g^{\\rm B}(\\vk\\_c)\\big> = (2\\pi)^3B\\_{ggg}^{\\rm AAB}(\\vk\\_a, \\vk\\_b, \\vk\\_c)\\delta\\_D(\\vk\\_{abc})$.\nIt is important to mention also that we do not account for any super-sample covariance (SSC) contributions that arise from modes with wavelengths larger than the size of the surveys (see also Refs. ). This is justified for the power spectrum part, for which on the large scales of interest, the SSC contributions are negligible compared to the dominant $\\operatorname{\\rm \\bf Cov}^{PP}$. Similarly, for the bispectrum covariance, Refs. conclude that SSC contributions are only a negligible contribution compared to the *P**P**P*, *B**B* and *T**P* terms. The SSC contributions to the cross-covariance $\\operatorname{\\rm \\bf Cov}^{BP}$ have not been studied with as much detail, although Ref. finds that it could be more important than for $\\operatorname{\\rm \\bf Cov}^{BB}$ for the matter bispectrum; Ref. further suggests that although it can have a small impact on signal-to-noise ratios, its presence may help regularize the covariance matrix and keep it positive-definite (see below).\n![Marginalized 1\\sigma constraints on {f_\\textnormal{\\textsc{nl}}} from combined P_{gg} + B_{ggg} analysis with parametrization 1, b_\\phi = 2\\delta_c(b_1-p). The different colors correspond to the different priors on p indicated in the x-axis. The different symbols show the constraints obtained with varying levels of composition of the covariance matrix, as labeled (see text for more details). The PPP+BB is the default case used in the main body of the paper. The orange band marks the constraints with PPP+BB and p = p_{\\rm fidu}=0.55.](fig_paper_variantscov_sigmafNL_vs_pprior_parametrization_0.png \"fig:\") [fig:sigmascov]\nFigure [fig:sigmascov] helps understand the relative importance of the various contributions to the covariance matrix. The figure shows the marginalized 1*σ* constraints on ${f\\_\\textnormal{\\textsc{nl}}}$ obtained from combined *P**g**g* + *B**g**g**g* analyses with parametrization 1, *b**ϕ* = 2*δ**c*(*b*1 − *p*). The different symbols show the result for varying levels of composition of the covariance matrix. The one labeled *P**P**P* + *B**B* is our default calculation with $\\operatorname{\\rm \\bf Cov}^{BB} = \\operatorname{\\rm \\bf Cov}^{BB}\\_{PPP} + \\operatorname{\\rm \\bf Cov}^{BB}\\_{BB}$ and $\\operatorname{\\rm \\bf Cov}^{BP}=0$. The one labeled *P**P**P* drops the *B**B* term from the bispectrum covariance, which as expected, results in an overestimate of the constraining power of the data, i.e., smaller uncertainties $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$. For example, for the $p = p\\_{\\rm fidu} = 0.55$ case, dropping the *B**B* term shrinks $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$ by roughly 60%. Note that many studies in the literature include only the *P**P**P* contribution, and may therefore represent too optimistic scenarios. Further, the impact of doubling the size of the *B**B* term (labeled *P**P**P* + 2*B**B*), results in roughly 15% increase in $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$ for the cases $p \\in p\\_{\\rm fidu} \\pm 0.5$ and $p = p\\_{\\rm fidu} = 0.55$; recall, this represents a very approximate way to account for the missing *T**P* term. Finally, further adding the power spectrum-bispectrum cross-covariance can increase $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$ by about 15% to 30% for the same two $p \\in p\\_{\\rm fidu} \\pm 0.5$ and $p = p\\_{\\rm fidu} = 0.55$ cases, respectively.\nThe result depicted in Fig. [fig:sigmascov] motivates future forecast studies on ${f\\_\\textnormal{\\textsc{nl}}}$ to begin incorporating covariance contributions beyond the *P**P**P* (as it is commonly done), since they may significantly degrade the constraining power on ${f\\_\\textnormal{\\textsc{nl}}}$. On the other hand, robust forecasts on ${f\\_\\textnormal{\\textsc{nl}}}$ should also include modelling of systematic errors, which add to the total error budget, and therefore reduce the importance of the statistical covariance matrix. A main message here is that future robust survey-specific forecasts on ${f\\_\\textnormal{\\textsc{nl}}}$ should perform convergence tests similar to that depicted in Fig. [fig:sigmascov] to guarantee that the constraining power of the data is not being overestimated.\nAs a final remark, we would like to mention that, at least in the context of our idealized forecast setup, the covariance matrix ceases to be positive definite if the cross-covariance term is taken into account without doubling the contribution from the *B**B* term. We did not investigate this issue further, but speculate that it could be either due to the missing contribution from the connected five-point function (which could include SSC terms) in the cross-covariance part, and/or that the bispectrum covariance matrix must meet some level of completeness (i.e., contain the *T**P* terms) before the cross-covariance part is added without spoiling positive-definiteness. We have further found that this is an issue that develops only if $k\\_{\\rm max} \\gtrsim 0.14 h / \\rm {Mpc}$, which gives some support to the explanation that this could be due to missing contributions that grow important on smaller scales.\n---\n1. Note that the incorporation of more galaxy samples into the analysis does not break this degeneracy since more galaxy bias parameters would need to be added as well.[↩](#fnref1)\n2. In the forecast analysis on PNG presented in Ref. , the authors include also the contribution from a 1-loop term $\\propto {f\\_\\textnormal{\\textsc{nl}}}$ given by an integral of a certain trispectrum configuration (Fourier transform of the four-point function). This term was first derived and discussed in Refs. , although in the context of a galaxy bias expansion that did not include the *ϕ* field in Eq. ([eq:biasexp]). In this paper, we work at tree level in the galaxy bispectrum, and importantly, with a set of operators in the galaxy bias expansion that is complete and closed under renormalization. Contributions from 1-loop terms on the large scales of interest are therefore absorbed by renormalized bias parameters at tree level.[↩](#fnref2)\n3. The labels *B* here should not be confused with the labels ${\\rm B}$ that refer to one of the multitracer subsamples.[↩](#fnref3)\n4. We have also ran constraints using the bispectrum of the full combined sample and found the same conclusions.[↩](#fnref4)\n5. Explicit bounds on ${f\\_\\textnormal{\\textsc{nl}}}$ can also always be obtained with parametrization 2 by imposing priors on *b**ϕ* *a posteriori*.[↩](#fnref5)\n6. In our “wide *p* prior” results, we still enforce *p* ∈ [ − 100, + 100] to prevent the chains from indefinitely exploring the degeneracy.[↩](#fnref6)\n7. For such large values of ${f\\_\\textnormal{\\textsc{nl}}}$, terms $\\propto {f\\_\\textnormal{\\textsc{nl}}}^2$ that we neglected in the galaxy bispectrum become more important and can impact the constraints we obtained with parametrization 2. The point still stands however that the galaxy bispectrum has reduced importance with parametrization 2 because of the strong degeneracy between ${f\\_\\textnormal{\\textsc{nl}}}$ and ${f\\_\\textnormal{\\textsc{nl}}}b\\_{\\phi\\delta}$.[↩](#fnref7)\n8. Makes it impossible, in fact, for power spectrum only analysis.[↩](#fnref8)\n9. Here, we used the convolution theorem and that $i(k\\_i/k^2)\\delta\\_m^{(1)}(\\vk)$ and $ik\\_i\\phi\\_{\\rm Eul}(\\vk)$ are the Fourier transforms of $s^i(\\vx)$ and $\\partial\\_i\\phi\\_{\\rm Eul}(\\vx)$, respectively; the imaginary unit *i* should not be confused with the subscripts *i*.[↩](#fnref9)\n\n On the impact of galaxy bias uncertainties on primordial non-Gaussianity constraints\n=====================================================================================\nIntroduction\n============\nOne of the main goals of modern cosmology is to determine the statistical properties of the density fluctuations of the primordial universe and gain insights into the physics of the mechanisms that generated them during the epoch of inflation. Current observational constraints are compatible with the simplest models of inflation that involve a single scalar field slowly rolling down its potential. A key prediction of these models is that the resulting fluctuations should be Gaussian distributed, and consequently, studies of primordial non-Gaussianity (PNG) have become a major focus in theoretical and observational cosmology given its power to discriminate between single-field models and more elaborate models involving multiple fields (see Ref. for an overview). A popular characterization of PNG is that of the so-called *local* type, in which the primordial gravitational (Bardeen) potential $\\phi(\\vx)$ is expanded as [eq:fnl] () = G() + fnl, where $\\phi\\_{\\rm G}$ is a Gaussian distributed random field, $\\left<\\cdots\\right>$ denotes ensemble average and the parameter ${f\\_\\textnormal{\\textsc{nl}}}$ quantifies the departures from non-Gaussianity. The current tightest bounds on ${f\\_\\textnormal{\\textsc{nl}}}$ come from the analysis of three-point statistics of the cosmic microwave background (CMB) by the Planck satellite, which constrain ${f\\_\\textnormal{\\textsc{nl}}}= -0.9 \\pm 5.1\\ (1\\sigma)$.\nThe next major improvements on the precision of ${f\\_\\textnormal{\\textsc{nl}}}$, $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$, are expected to come from analyses of the statistics of the galaxy distribution. For Gaussian distributed primordial fluctuations, on sufficiently large scales, the galaxy number density contrast *δ**g* can be written as $\\delta\\_g(\\vx) = b\\_1 \\delta\\_m(\\vx) + \\eps(\\vx)$, where *δ**m* denotes matter density contrast fluctuations, $\\eps$ is a noise field and *b*1 is the linear local-in-matter-density galaxy bias parameter that describes the response of galaxy number counts to long-wavelength total matter fluctuations (see Ref. for a review on galaxy bias). In the presence of local PNG, the galaxy distribution gets another contribution $\\delta\\_g(\\vx) \\supset b\\_\\phi {f\\_\\textnormal{\\textsc{nl}}}\\phi(\\vq)$, where *b**ϕ* is the galaxy bias parameter that describes the response of galaxy number counts to long-wavelength primordial gravitational potential perturbations with local PNG ($\\vq$ is the Lagrangian coordinate associated with the Eulerian coordinate $\\vx$). This ${f\\_\\textnormal{\\textsc{nl}}}$ contribution can therefore be used to place bounds on local PNG using galaxy observations. For example, Ref. showed that the galaxy power spectrum *P**g**g*(*k*) (the Fourier transform of the two-point correlation function; *k* denotes wavenumber) acquires a specific scale-dependent signature $\\propto b\\_1b\\_\\phi {f\\_\\textnormal{\\textsc{nl}}}/ k^2$ that becomes important on the largest observable scales. The galaxy bispectrum *B**g**g**g*(*k*1, *k*2, *k*3) (the Fourier transform of the three-point correlation function) is also a notoriously good probe of ${f\\_\\textnormal{\\textsc{nl}}}$, and it is sensitive to it also via the non-zero primordial matter bispectrum that it induces (we discuss the ${f\\_\\textnormal{\\textsc{nl}}}$ contributions to the power spectrum and bispectrum in more detail in Sec. [sec:model]). Current constraints on local PNG from galaxy surveys are of order $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}} \\sim 50$, but a number of forecast studies have been suggesting that $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}} \\sim \\mathcal{O}(1)$ or below could be achieved with upcoming surveys.\nA main theoretical uncertainty in searches for local PNG using the galaxy distribution concerns the galaxy bias parameters. They are functions of redshift and of galaxy properties such as their total mass, stellar mass, luminosity, etc. They formally describe the response of galaxy formation to the long-wavelength environment, and as a result, they are extremely challenging to predict given the many astrophysical processes involved in galaxy formation and evolution. In observational searches for local PNG, one should therefore fit for the bias parameters simultaneously with ${f\\_\\textnormal{\\textsc{nl}}}$, but naturally, the wider the priors adopted for the bias parameters, the weaker the constraining power on ${f\\_\\textnormal{\\textsc{nl}}}$. This motivates work on theoretical predictions of galaxy bias in order to determine the range of values a given observed galaxy sample is expected to take and/or determine relations between the various bias parameters to reduce the dimensionality of the parameter spaces explored in constraint analyses. A widely popular example of such relations is the so-called universality relation between *b**ϕ* and *b*1. Concretely, considering gravity-only dynamics in the formation of dark matter haloes and further assuming that their mass function is universal, it can be shown that *b**ϕ* = 2*δ**c*(*b*1 − 1), where *δ**c* = 1.686 is the threshold overdensity for spherical collapse. The adoption of such a relation is crucial to constrain ${f\\_\\textnormal{\\textsc{nl}}}$. For example, in the contribution to the galaxy power spectrum $\\propto b\\_1b\\_\\phi {f\\_\\textnormal{\\textsc{nl}}}/ k^2$, the value of *b*1 can be constrained by the smaller-scale part of the power spectrum, but then, if *b**ϕ* is allowed complete freedom, it becomes impossible to constrain ${f\\_\\textnormal{\\textsc{nl}}}$ (cf. upper left panel of Fig. [fig:contributions] below).\nEffectively all recent constraint and forecast studies on ${f\\_\\textnormal{\\textsc{nl}}}$ using the galaxy distribution adopt some form for the relation *b**ϕ*(*b*1). The most popular case is the use of the universality relation mentioned above, but this relation has been shown to represent only an approximation of the bias values measured for haloes in *N*-body simulations. More specifically, the simulation results are better described by a variant of the relation *b**ϕ* = *q*2*δ**c*(*b*1 − 1), where *q* ∈ [0.5, 0.9] (the exact value depends on redshift, halo mass and halo finding criterion). Another variant of the universality relation, *b**ϕ* = 2*δ**c*(*b*1 − *p*) with *p* = 1.6 was put forward by Ref. , who argued that it was a more appropriate description for recently formed haloes, which could host quasars (see also Ref. ). More recently, Ref. found using hydrodynamical simulations of galaxy formation with the IllustrisTNG model that *b**ϕ* = 2*δ**c*(*b*1 − *p*) with *p* ≈ 0.55 describes well the bias relation for galaxies selected by their stellar mass. Naturally, uncertainties on galaxy bias in general, and on the *b**ϕ*(*b*1) relation in particular, will propagate to the resulting inferred values of ${f\\_\\textnormal{\\textsc{nl}}}$ and $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$. This is a fact that has been acknowledged is past literature, but that has not been the subject of detailed and dedicated work, despite its utmost importance for PNG constraints. Our main goal in this paper is to take a few steps forward in understanding how such uncertainties can impact the inferred ${f\\_\\textnormal{\\textsc{nl}}}$ and how to mitigate them.\nSpecifically in this paper, we work with an idealized forecast setup to illustrate how galaxy bias uncertainties affect the constraints on ${f\\_\\textnormal{\\textsc{nl}}}$ obtained with a combination of multitracer galaxy power spectrum and bispectrum data. We explore two treatments of the galaxy bias *b**ϕ*: (i) one called parametrization 1, in which we assume *b**ϕ* = 2*δ**c*(*b*1 − *p*) and treat *p* as a free parameter, and (ii) another called parametrization 2, in which instead of constraining ${f\\_\\textnormal{\\textsc{nl}}}$, we constrain directly the product of $b\\_\\phi{f\\_\\textnormal{\\textsc{nl}}}$. In the latter approach, one is less interested in the actual value of ${f\\_\\textnormal{\\textsc{nl}}}$, but more on distinguishing it from zero as $b\\_\\phi{f\\_\\textnormal{\\textsc{nl}}}\\neq 0$ implies ${f\\_\\textnormal{\\textsc{nl}}}\\neq 0$, which is sufficient to distinguish single-field from multifield inflation. With parametrization 1, we will see that the shape of the likelihood becomes appreciably bimodal if *p* is allowed complete freedom. This complicates the interpretation of marginalized constraints on ${f\\_\\textnormal{\\textsc{nl}}}$, but we will see that priors $\\Delta p \\lesssim 0.5$ for fiducial values of *p* = 0.55 are sufficient to return unbiased constraints. Regarding parametrization 2, which is completely independent of any assumptions on the PNG bias parameters, we will find that the significance of the detection of local PNG (i.e. ${f\\_\\textnormal{\\textsc{nl}}}\\neq 0$) is similar to that from parametrization 1, when using the galaxy power spectrum. However, with parametrization 2, the addition of the galaxy bispectrum contributes much less significantly to improving the constraints, compared to the case with parametrization 1.\nOur goal in the remainder of this paper is not to draw precise quantitative estimates of $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$, but rather to outline and discuss some strategies to deal with galaxy bias uncertainties. These can and should be straightforwardly implemented in more complete forecast pipelines to derive more robust and survey-specific bounds on ${f\\_\\textnormal{\\textsc{nl}}}$. The outline of this paper is as follows. In Sec. [sec:model], we display the expressions of the multitracer galaxy power spectrum, galaxy bispectrum and corresponding covariance matrix that we use in our analysis. In Sec. [sec:results], we describe our forecast methodology, and present and discuss our numerical constraint results. Finally, we summarize and conclude in Sec. [sec:summary]. In App. [app:bgggderivation], we comment on a few aspects of the derivation of the galaxy bispectrum, and in App. [app:cov], we analyse the impact that varying levels of completeness of the calculation of the covariance matrix have on ${f\\_\\textnormal{\\textsc{nl}}}$ constraints.\nThe galaxy power spectrum and bispectrum with local PNG\n=======================================================\nIn this section we display the expressions of the galaxy power spectrum, galaxy bispectrum and corresponding covariances that we consider in our analysis.\nWe work with the following expression for the rest-frame galaxy number density constrast $\\delta\\_g(\\vx, z)$, [eq:biasexp] g(, z) &=& b1(z)m(, z) + b2(z) [m(, z)]2 + bK2(z)[Kij(, z)]2 + () + ()m(, z) \n&+& fnl, where *δ**m* is a long-wavelength total matter density fluctuation, *K**i**j* = [∂*i*∂*j*/∇2 − *δ**i**j*/3]*δ**m* is a long-wavelength tidal field and *ϕ* is the long-wavelength primordial gravitational potential (these terms and their products should be understood as *renormalized operators* in this galaxy bias expansion ). The density and tidal fields are evaluated at the evolved Eulerian position $\\vx$, whereas the primordial potential *ϕ* is evaluated at the initial Lagrangian position $\\vq$. The parameters *b*1, *b*2, *b**K*2, *b**ϕ*, *b**ϕ**δ* are the *galaxy bias parameters* that describe the response of the galaxy number density to the presence of the corresponding long-wavelength perturbations that each multiplies (see Ref. for a review); the bias parameters are a function of redshift, as well as of galaxy properties like stellar mass or luminosity. The stochastic terms $\\eps, \\eps\\_\\delta, \\eps\\_\\phi$ encapsulate the dependence of galaxy number counts on the smaller-wavelength properties of the environment where the galaxies form.\nThe terms in Eq. ([eq:biasexp]) are all that are needed to self-consistently derive the leading-order galaxy power spectrum and bispectrum (see also Sec. 7 of Ref. for a review of the contribution of PNG to the galaxy bias expansion). For simplicity, we skip modelling redshift space distortions (RSD) and always work in real space. Further, we skip considering so-called relativistic effects that can affect galaxy statistics with the same scale dependence as ${f\\_\\textnormal{\\textsc{nl}}}$. The form of their contribution is however known and it depends on the magnification and time-evolution bias parameters, which can both be estimated from the data. We note that these and other simplifications (such as ignoring observational systematics) can have an impact on the resulting ${f\\_\\textnormal{\\textsc{nl}}}$ and $\\sigma\\_{{f\\_\\textnormal{\\textsc{nl}}}}$ values, but here we are more interested in the relative impact of galaxy bias uncertainties on the constraints, which depends less sensitively on the level of completion of the rest of the analysis. We retain also only terms that are linear in ${f\\_\\textnormal{\\textsc{nl}}}$, which are the most relevant given current observational bounds, $|{f\\_\\textnormal{\\textsc{nl}}}| \\lesssim 5$.\nMultitracer galaxy power spectrum\n---------------------------------\nThe galaxy power spectrum *P**g**g*(*k*) is defined as $(2\\pi)^3P\\_{gg}(k)\\delta\\_D(\\vk + \\vk') = \\big<\\delta\\_g({\\vk})\\delta\\_g({\\vk'})\\big>$, where $\\delta\\_g(\\vk)$ is the Fourier transform of $\\delta\\_g(\\vx)$ and *δ**D* is the Dirac delta function (from hereon we skip writing explicitly the redshift dependence in the arguments to lighten the expressions). We make use of the multitracer framework for the galaxy power spectrum, in which a given galaxy sample in a given redshift bin is split into two subsamples with different bias parameters. The multitracer technique helps to reduce sample variance errors on the largest observed scales as both galaxy samples share the same large-scale modes; this allows to efficiently constrain ${f\\_\\textnormal{\\textsc{nl}}}$ via the relative amplitude of the clustering of the two samples. We will consider the case of two subsamples for simplicity, sample A and sample B, and consider the following estimators for their auto- and cross-power spectra [eq:pggestimators] ggAA(k) &=& k d3’ gA(’)gA(-’), \nggAB(k) &=& k d3’ gA(’)gB(-’), \nggBB(k) &=& k d3’ gB(’)gB(-’), where *V**s* is the survey volume and $\\int\\_k {\\rm d}^3\\vk'$ represents integrating over a spherical shell in Fourier space with radius *k*, width Δ*k* and volume *V**k* = 4*π**k*2Δ*k*; the superscripts $^{\\rm A,B}$ indicate the galaxy subsample, i.e., $\\hat{P}\\_{gg}^{\\rm AB}$ represents the estimator of the cross-power spectrum. Plugging the Fourier transform of Eq. ([eq:biasexp]) in the expectation value of the above equations, and retaining only terms to leading order in perturbation theory and ${f\\_\\textnormal{\\textsc{nl}}}$ yields [eq:pggmodelAA]<ggAA(k)> PAAgg(k) &=& 2 Pmm(k) + 2b1AbA fnlPm(k) + PAA, \n[eq:pggmodelAB]<ggAB(k)> PABgg(k) &=& b1Ab1B Pmm(k) + fnlPm(k), \n[eq:pggmodelBB]<ggBB(k)> PBBgg(k) &=& 2 Pmm(k) + 2b1BbB fnlPm(k) + PBB, where *P**m**m*(*k*) denotes the linear matter power spectrum and *P**m**ϕ* the cross matter-potential power spectrum. The linear matter density perturbations are related to the primordial potential as $\\delta\\_m^{(1)}(\\vk) = \\mathcal{M}(k) \\phi(\\vk)$, where the superscript (1) indicates the linear perturbation theory contribution to the total density field *δ**m* = *δ**m*(1) + *δ**m*(2) + ⋯, and M(*k*) = (2/3)*k*2*T**m*(*k*)/(Ω*m*0*H*02), with *T**m* being the matter transfer function, Ω*m*0 the present-day fractional cosmic mean matter density and *H*0 the present-day Hubble expansion rate. Hence, *P**m**ϕ* = *P**m**m*(*k*)/M(*k*). Finally, $P\\_{\\eps\\eps}$ is the power spectrum of the noise, which we assume to be Poissonian $P\\_{\\eps\\eps} = 1/\\bar{n}\\_g$, where *n̄**g* is the mean observed galaxy number density; note that we ignore the cross-power spectrum of the noise fields $P^{\\rm AB}\\_{\\eps\\eps}$.\nThe galaxy auto power spectrum is shown in the upper left panel of Fig. [fig:contributions]. On scales $k \\lesssim 0.01\\ h/{\\rm Mpc}$, *T**m* tends to a constant and the ${f\\_\\textnormal{\\textsc{nl}}}$ contribution becomes ∝ 1/*k*2, as shown by the brown line. The ratio of the total galaxy power spectrum (black line) and the contributions assuming Gaussian primordial fluctuations (grey) is therefore scale-dependent on large scales, which is why this signature from ${f\\_\\textnormal{\\textsc{nl}}}$ is popularly referred to as the *scale-dependent bias* effect.\nThe upper left panel of Fig. [fig:contributions] illustrates also neatly the critical importance of galaxy bias uncertainties. The local PNG signature is proportional to $b\\_1b\\_\\phi{f\\_\\textnormal{\\textsc{nl}}}$, and hence, in order to constrain ${f\\_\\textnormal{\\textsc{nl}}}$, one needs either additional data and/or prior information on the bias parameters. If prior information on the amplitude of *P**m**m* is available, then the value of *b*1 can be constrained from the smaller-scale part of the power spectrum where the ${f\\_\\textnormal{\\textsc{nl}}}$ contribution is negligible. Nonetheless, even if *b*1 is perfectly determined, one is still left with a perfect degeneracy between *b**ϕ* and ${f\\_\\textnormal{\\textsc{nl}}}$. The way of breaking this degeneracy that is most popular in the literature[1](#fn1) consists in relating *b**ϕ* and *b*1 via the universality relation mentioned in the previous section, *b**ϕ* = 2*δ**c*(*b*1 − 1). There is however no guarantee that observed galaxies obey this relation. Indeed, as mentioned already in Sec. [sec:introduction], studies have shown that the universality relation is not exactly satisfied even for haloes in gravity-only simulations. Further, *b**ϕ* = 2*δ**c*(*b*1 − 1.6) has been argued to be a better description of haloes that have recently undergone a major merger, and *b**ϕ* = 2*δ**c*(*b*1 − 0.55) has been put forward as a more adequate description of simulated stellar-mass selected galaxies. These results suggest that the exact *b**ϕ*(*b*1) relation is very likely tracer-dependent (i.e., different for different galaxy types) and currently fairly uncertain. This makes it a pressing matter to study and develop ways to incorporate and mitigate uncertainties around the *b**ϕ*(*b*1) relation in ${f\\_\\textnormal{\\textsc{nl}}}$ constraints; that is the subject of Sec. [sec:results] below.\n![The different panels show, as labeled, the galaxy auto-power spectrum and the galaxy bispectrum in a squeezed configuration (k_1=k_2 > k_3, k_3 = 4k_{\\rm F} \\approx 0.0027h/{\\rm Mpc}; k_{\\rm F} is the fundamental mode of a survey with 100 {\\rm Gpc^3}/h^3), equilateral configurations (k_1=k_2=k_3) and folded configurations (k_1=2k_2=2k_3). The result shown is the leading-order one in perturbation theory and {f_\\textnormal{\\textsc{nl}}} at z=1, and for the parameters of sample A in Tab. [table:samples]. In each panel, the black line shows the total contribution and the grey line shows the result expected for primordial Gaussian fluctuations ({f_\\textnormal{\\textsc{nl}}}= 0). The remaining curves show the various contributions \\propto {f_\\textnormal{\\textsc{nl}}}, as labeled (the legend in the lower right panel applies to all bispectrum panels).](fig_paper_contributions_PggBggg.png \"fig:\") [fig:contributions]\nGalaxy bispectrum\n-----------------\nThe galaxy bispectrum is defined as $(2\\pi)^3 B\\_{ggg}(k\\_1, k\\_2, k\\_3)\\delta\\_D({\\vk\\_{123}}) = \\big<\\delta\\_g({\\vk\\_1})\\delta\\_g({\\vk\\_2})\\delta\\_g({\\vk\\_3})\\big>$ (in our notation, $\\vk\\_{123} = \\vk\\_1 + \\vk\\_2 + \\vk\\_3$), and we consider the following estimator, [eq:bgggestimator] ggg(k1,k2,k3) = k1 d3a k2 d3b k3 d3c g(a)g(b)g(c)D(abc), where *V*123 = 8*π*2*k*1*k*2*k*3Δ*k*1Δ*k*2Δ*k*3. In our analysis we work with the bispectrum of only one of the samples used in the multitracer power spectrum part for simplicity. We use the bispectrum of one of the samples rather than the two samples combined to reduce the number of free bias parameters. In App. [app:bgggderivation] we comment on a few aspects of the derivation of the galaxy bispectrum (see also Refs. for a number of past works on the galaxy bispectrum in local PNG cosmologies). The final result can be written as [eq:bgggmodel] <ggg(k1,k2,k3)> Bggg(k1,k2,k3) = BgggG(k1,k2,k3) + BgggNG(k1,k2,k3), where [eq:bgggmodelG] &&BgggG(k1,k2,k3) = b13Bmmm(k1, k2, k3) + + B \n&+&+ \nis the contribution due to nonlinear gravitational clustering that is present even if ${f\\_\\textnormal{\\textsc{nl}}}= 0$, and [eq:bgggmodelNG] &&BgggNG(k1,k2,k3) = [ 2b13fnl(k3) + 2bfnlP \n&+& b12bfnlPmm(k1)Pmm(k2) (12( + ) + 2F2(k1, k2, 12)( + )) \n&+& (b12b + b1b2b + 2b1bK2b (122 - 13))fnlPmm(k1)Pmm(k2) ( + ) + (2 perm.) ] \nis the contribution due to local PNG that is $\\propto {f\\_\\textnormal{\\textsc{nl}}}$.[2](#fn2) In the equations above, *μ**a**b* is the cosine of the angle between the *k**a* and *k**b* legs of the triangle, and likewise to the case of the power spectrum, we work to leading order in the bispectrum (i.e., second order in perturbation theory) and in ${f\\_\\textnormal{\\textsc{nl}}}$. Further, $B\\_{\\eps\\eps\\eps}$ and $P\\_{\\eps\\eps\\_\\delta}$ denote, respectively, the bispectrum of the noise field $\\eps$ and the cross-power spectrum of the fields $\\eps$ and $\\eps\\_\\delta$. Assuming Poissonian noise these are given by $B\\_{\\eps\\eps\\eps} = 1/\\bar{n}\\_g^2$ and $P\\_{\\eps\\eps\\_\\delta} = b\\_1/(2\\bar{n}\\_g)$. To derive the second term on the right-hand side of Eq. ([eq:bgggmodelNG]), we have also assumed Poissonian statistics to relate $P\\_{\\eps\\eps\\_\\phi} = (b\\_\\phi/b\\_1)P\\_{\\eps\\eps\\_\\delta}$. Further, *B**m**m**m* denotes the tree-level matter bispectrum given by [eq:bmmm] Bmmm(k1, k2, k3) = 2F2(1, 2) Pmm(k1)Pmm(k2) + (2 perm.), where [eq:f2] F2(1, 2) F2(k1, k2, 12) = 57 + + 27 122 is the second-order perturbation theory mode-coupling kernel.\nIn Fig. [fig:contributions], the upper right, lower left and lower right panels show, respectively, the galaxy bispectrum for squeezed ($k\\_1=k\\_2 > k\\_3\\approx0.0027h/{\\rm Mpc}$), equilateral (*k*1 = *k*2 = *k*3) and folded (*k*1 = 2*k*2 = 2*k*3) configurations. A first noteworthy and well-known aspect is that the local PNG contributions are most prominent in the squeezed-limit. This is shown by the larger difference between the black and grey lines in the upper right panel. Note that the local PNG contribution is sizeable in the squeezed bispectrum for all modes *k*1 = *k*2 probed, which is why increasing the maximum wavenumber analysed $k\\_{\\rm max}$ can help improve significantly ${f\\_\\textnormal{\\textsc{nl}}}$ constraints with the bispectrum, but less so with the power spectrum. Further, contrary to the power spectrum case, in the galaxy bispectrum the ${f\\_\\textnormal{\\textsc{nl}}}$ contribution is not perfectly degenerate with *b**ϕ*. This degeneracy is broken by the terms $\\propto b\\_1^3 {f\\_\\textnormal{\\textsc{nl}}}$ (blue lines in Fig. [fig:contributions]) and $\\propto b\\_1^2b\\_{\\phi\\delta}{f\\_\\textnormal{\\textsc{nl}}}$ (red lines in Fig. [fig:contributions]). In the latter case, *b**ϕ**δ* can also be calculated assuming the universality of the halo mass function: [eq:univbphidelta] b = b- b1 + 1 + c[b2 - (8/21)(b1-1)]. The same caveats about the validity of the universality relation for *b**ϕ* apply likewise to *b**ϕ**δ*. To the best of our knowledge, however, this *b**ϕ**δ* expression has never been tested neither for halos in gravity-only simulations nor simulated galaxies in hydrodynamical simulations. In our forecasts below, when we study deviations from the universality relation for *b**ϕ*, we shall still assume the universality relation for *b**ϕ**δ*. The latter should be eventually tested with *N*-body simulations, but we note that this term contributes a smaller amount compared to terms ∝ *b**ϕ* to the constraining power on ${f\\_\\textnormal{\\textsc{nl}}}$, and hence, uncertainties on *b**ϕ**δ* are not as critical.\nCovariance of the multitracer galaxy power spectrum and bispectrum\n------------------------------------------------------------------\nIn our forecast analysis we work with the following Gaussian likelihood function [eq:likelihood] (), where $\\vD$ is the data vector, ${\\operatorname{\\rm \\bf Cov}}$ its covariance matrix and $\\vM$ is a theoretical prediction for the data vector that depends on parameters $\\vtheta$. The data vector consists of hypothetical measurements of the real-space multitracer power spectrum and bispectrum [eq:datavector] = { ggAA, ggAB, ggBB, gggAAA}; we will work with the bispectrum of subsample A, which is the higher number density one (cf. Sec. [sec:methodology]). Our theoretical predictions $\\vM$ are evaluated using Eqs. ([eq:pggmodelAA]), ([eq:pggmodelAB]), ([eq:pggmodelBB]), ([eq:bgggmodel]), ([eq:bgggmodelG]) and ([eq:bgggmodelNG]); we construct the data vector using the same equations evaluated at a fiducial cosmology. The covariance matrix of the data vector contains three main blocks:\n \n$$\\label{eq:cov\\_def}\n{\\operatorname{\\rm \\bf Cov}} = \n\\begin{pmatrix}\n\\operatorname{\\rm \\bf Cov}^{PP} & \\operatorname{\\rm \\bf Cov}^{BP} \\\\[1.5ex]\n\\cdots & \\operatorname{\\rm \\bf Cov}^{BB} \\\\[1.5ex]\n\\end{pmatrix}\n\\,\\,,$$ \nwhere $\\operatorname{\\rm \\bf Cov}^{PP}$, $\\operatorname{\\rm \\bf Cov}^{BP}$ and $\\operatorname{\\rm \\bf Cov}^{BB}$ denote, respectively, the covariance of the power spectrum part of the data vector, the cross-covariance of the bispectrum and power spectrum, and the covariance of the bispectrum part; the dots indicate the block is equal to the corresponding symmetric block. In App. [app:cov], we outline the derivation of the covariance matrix, and comment on the relative impact of the various contributions to it (see also Refs. for works on bispectrum covariances). In this section, we limit ourselves to displaying only the final expressions. For $\\operatorname{\\rm \\bf Cov}^{PP}$, we consider only the contribution from the disconnected part of the four-point function (the so-called Gaussian term); for the case of the multitracer power spectrum, the result is a 3 × 3 block diagonal matrix:\n \n$$\\label{eq:covPP}\n{\\operatorname{\\rm \\bf Cov}}^{PP}(k\\_1, k\\_2) = 2 \\frac{(2\\pi)^3\\delta\\_{k\\_1k\\_2}}{V\\_sV\\_{k\\_1}}\n\\begin{pmatrix}\n[P\\_{gg}^{AA}(k\\_1)]^2 & P\\_{gg}^{AB}(k\\_1)P\\_{gg}^{AA}(k\\_1) & [P\\_{gg}^{AB}(k\\_1)]^2 \\\\[1.5ex]\n\\cdots & \\frac12\\left[P\\_{gg}^{AA}(k\\_1)P\\_{gg}^{BB}(k\\_1) + [P\\_{gg}^{AB}(k\\_1)]^2\\right] & P\\_{gg}^{BB}(k\\_1)P\\_{gg}^{AB}(k\\_1) \\\\[1.5ex]\n\\cdots & \\cdots & [P\\_{gg}^{BB}(k\\_1)]^2\n\\end{pmatrix}\n\\,\\,.$$ \nWe evaluate the bispectrum covariance block as $\\operatorname{\\rm \\bf Cov}^{BB} = \\operatorname{\\rm \\bf Cov}^{BB}\\_{PPP} + \\operatorname{\\rm \\bf Cov}^{BB}\\_{BB}$, with (in our notation, the superscripts in $\\operatorname{\\rm \\bf Cov}$ indicate which estimators we are taking the covariance of, whereas the subscripts label the various contributions[3](#fn3)) [eq:covBBPPP] BBPPP(k1, k2, k3,k1’, k2’, k3’) = TT’ Pgg(k1)Pgg(k2)Pgg(k3) and [eq:covBBBB] BBBB(k1, k2, k3,k1’, k2’, k3’) &=& Bggg(k1’, k2, k3)Bggg(k1, k2’, k3’) k1k1’ + (8 perm.), \nwhere *U*(*k*1, *k*1ʹ) = 16*π*3*k*2*k*3*k*2ʹ*k*3ʹΔ*k*2Δ*k*3Δ*k*2ʹΔ*k*3ʹΔ*k*1, *δ**T**T*ʹ is a Kronecker delta function that is non-zero only if both triangles *T* = {*k*1, *k*2, *k*3} and *T*ʹ = {*k*1ʹ, *k*2ʹ, *k*3ʹ} are the same, and $S\\_{\\rm shape} = 6,2,1$ for equilateral, isosceles and scalene triangles, respectively; in Eq. ([eq:covBBBB]), the permutations are all 9 that link each side of one triangle to each side of the other. Finally, in our main results we will neglect the cross-covariance term, $\\operatorname{\\rm \\bf Cov}^{BP} = 0$; its contribution is analysed and discussed in App. [app:cov].\nResults\n=======\nIn this section we present our main results on the impact of local PNG galaxy bias uncertainties on ${f\\_\\textnormal{\\textsc{nl}}}$ constraints. We begin in Sec. [sec:methodology] by describing our idealized forecast setup and the two galaxy bias parametrizations that we explore in this paper. The numerical forecast results are then shown and discussed in Secs. [sec:numerics1] and [sec:numerics2].\nMethodology and forecast setup\n------------------------------\nlcccccccccccccc & Sample & *n̄**g* $\\left[{h^3/{\\rm Mpc^3}}\\right]$ & *b*1 & *b*2 & *b**K*2 & *b**ϕ* & *b**ϕ**δ* & \n& A & 1.74 × 10− 3 & 1.58 & − 0.62 & − 0.17 & 3.49 & 1.48 & \n& B & 1.07 × 10− 4 & 2.37 & 0.66 & − 0.39 & 6.15 & 5.01 & \n[table:samples]\nIn our forecast study we consider a galaxy sample at redshift *z* = 1. For the multitracer power spectrum part, we split this galaxy sample into two subsamples A and B as follows. We assume the galaxy sample follows a stellar mass function described by a Schechter function [eq:smf] (M\\*) = \\* ()- exp, with $\\phi\\_\\* = 0.0074\\ h^3/{\\rm Mpc^3}/{\\rm dex}$, *M*\\*ʹ = 1011 *M*⊙/*h* and *α* = 0.38; we have verified this is sufficiently realistic in that it matches well the stellar mass function of the IllustrisTNG simulations at *z* = 1. We then define the two subsamples via a cut in stellar mass *M*\\*: subsample A with *M*\\* ∈ [5 × 1010; 2 × 1011]*M*⊙/*h* and subsample B with *M*\\* > 2 × 1011*M*⊙/*h*. The mean number density *n̄**g* and linear bias parameter *b*1 of each sample are obtained by integrating, respectively, [eq:ng] |ng = M\\* , b1 = M\\* b1T(M\\*) over the corresponding mass ranges. In the equation for *b*1, $b\\_1^{\\rm T}(M\\_\\*)$ denotes the linear halo bias fitting formula of Ref. evaluated in terms of stellar mass according to the prescription described in Ref. . We evaluate the higher-order bias terms *b*2, *b**K*2 using, respectively, the polynomial fit of Ref. *b*2 = 0.412 − 2.143*b*1 + 0.929*b*12 + 0.008*b*13, and the Lagrangian linear-in-matter-density prediction *b**K*2 = − (2/7)(*b*1 − 1). The fiducial values of *b**ϕ* are obtained with the variant of the universality relation *b**ϕ* = 2*δ**c*(*b*1 − *p*) with *p* = 0.55, which is roughly the value preferred by the stellar-mass selected objects in the IllustrisTNG model. We use Eq. ([eq:univbphidelta]) to evaluate *b**ϕ**δ*. These specifications of the two subsamples are summarized in Tab. [table:samples]. For the bispectrum part of the analysis, we consider the bispectrum of subsample A, which has higher number density.[4](#fn4)\nFor the fiducial cosmology we assume a spatially flat $\\Lambda{\\rm CDM}$ model with present-day baryon density Ω*b*0 = 0.0486, cold dark matter density Ω*c*0 = 0.2603, dimensionless Hubble parameter *h* = 0.6774, scalar spectral index *n**s* = 0.967 and primordial scalar amplitude parameter A*s* = 2.068 × 10− 9 at a pivot scale $k\\_{\\rm pivot} = 0.05\\ /{\\rm Mpc}$. We neglect the impact of neutrino masses, Ω*m*0 = Ω*b*0 + Ω*c*0. We further consider a fiducial value of ${f\\_\\textnormal{\\textsc{nl}}}=3$, which is within the range currently allowed by CMB observations ${f\\_\\textnormal{\\textsc{nl}}}= -0.9 \\pm 5.1\\ (1\\sigma)$ ; we choose a non-zero value for ${f\\_\\textnormal{\\textsc{nl}}}$ to entertain the possibility of non-zero detections in our analysis, which is the most interesting scenario. For simplicity and to remain focused on the impact of galaxy bias uncertainties, we will vary only the cosmological parameters ${f\\_\\textnormal{\\textsc{nl}}}$ and A*s*. In real data analysis, one would always adopt priors on the remaining cosmological parameters from other data sets anyway. We evaluate all the linear power spectra and transfer functions with the CAMB code.\nOur assumed survey volume is $V\\_s = 100{\\rm Gpc^3}/h^3$, which is roughly the comoving volume of a full-sky survey spanning *z* = 1 ± 0.5. We consider 30 *k* bins between $k\\_{\\rm min} = k\\_{\\rm F} = \\pi/V\\_s^{1/3}$ and $k\\_{\\rm max} = 0.2\\ h/{\\rm Mpc}$ uniformly distributed in log-space. This value of $k\\_{\\rm max}$ is approximately that above which our leading-order treatment of the galaxy power spectrum and bispectrum becomes inadequate. In power spectrum only analysis (denoted by *P**g**g*-only from hereon), the value of $k\\_{\\rm max}$ is not as important as in combined analysis with the bispectrum (denoted by *P**g**g* + *B**g**g**g* from hereon); this is because the latter can still be appreciably sensitive to ${f\\_\\textnormal{\\textsc{nl}}}$ on small scales for squeezed configurations with large *k*1, *k*2 and small *k*3 (cf. upper right panel of Fig. [fig:contributions]). We have checked that with the more conservative option $k\\_{\\rm max} = 0.1\\ h/{\\rm Mpc}$, the *P**g**g* + *B**g**g**g* analysis had effectively the same constraining power on ${f\\_\\textnormal{\\textsc{nl}}}$ as the *P**g**g*-only analysis. We therefore opted for a higher $k\\_{\\rm max}$ value to study the consequences of galaxy bias uncertainties when the constraining power of the data is stronger. With this *k*-binning, we evaluate the bispectrum at all possible triangles with $k\\_{\\rm min} \\leq k\\_1, k\\_2, k\\_3 \\leq k\\_{\\rm max}$, which yields a total of 873 triangles.\nIn our numerical results, we discuss the impact of galaxy bias uncertainties on local PNG constraints under the following two characterizations of the parameter space:\n* In this case, *b**ϕ* is related to *b*1 via the variant of the universality relation *b**ϕ* = 2*δ**c*(*b*1 − *p*), with *p* being treated as a free parameter. We will be specifically interested on the impact of different assumed priors on *p*. The parameter space that we wish to constrain is 11 dimensional and given by [eq:params1] = {fnl, b1A, b1B, PA, PB, p, s, b2A, bK2A, PA, BA }, where *δ*A*s* is defined as $\\mathcal{A}\\_s = \\mathcal{A}\\_{s, \\rm fidu}\\left[1 + \\delta\\_{\\mathcal{A}\\_s}\\right]$, with $\\mathcal{A}\\_{s, \\rm fidu}$ being the fiducial value. For *P**g**g*-only analyses we vary just the first six of these parameters; we keep *δ*A*s* fixed, which would otherwise be perfectly degenerate with $b\\_1^{\\rm A}, b\\_1^{\\rm B}$. Note that we assume the value of *p* to be the same for both subsamples in the multitracer analysis. Important to recall is also the fact that we still assume the universality prediction of Eq. ([eq:univbphidelta]) to derive *b**ϕ**δ*, although this term contributes subdominantly compared to others.\n* In this case, instead of attempting to make assumptions on *b**ϕ* to constrain ${f\\_\\textnormal{\\textsc{nl}}}$, one fits instead directly for the products of ${f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi$ and ${f\\_\\textnormal{\\textsc{nl}}}b\\_{\\phi\\delta}$ that contribute to the galaxy power spectrum and bispectrum. The parameter space then becomes 13 dimensional and given by [eq:params2] = {[fnlbA], [fnlbB], b1A, b1B, PA, PB, fnl, s, [fnlbA], b2A, bK2A, PA, BA }. For *P**g**g*-only analyses we vary just the first six of these parameters; this parametrization is completely independent of any assumptions on *b**ϕ*, but one loses the ability to constraint the absolute value of ${f\\_\\textnormal{\\textsc{nl}}}$. The value of this approach lies instead on the possibility to *model-independently* detect non-zero $[{f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi^{\\rm A}]$ or $[{f\\_\\textnormal{\\textsc{nl}}}b\\_\\phi^{\\rm B}]$, which would imply ${f\\_\\textnormal{\\textsc{nl}}}\\neq 0$, and consequently, rule out single-field models of inflation. For *P**g**g* + *B**g**g**g* analysis, the absolute value of ${f\\_\\textnormal{\\textsc{nl}}}$ can still be formally constrained because of the term $\\propto b\\_1^3{f\\_\\textnormal{\\textsc{nl}}}$ in Eq. ([eq:bgggmodelNG]) (although with much reduced constraining power, as we will see in Sec. [sec:numerics2]).[5](#fn5)\nFinally, we sample the parameter space with the EMCEE Python implementation of the affine-invariant Markov Chain Monte Carlo (MCMC) sampler proposed in Ref. . We assume wide uninformative linear priors on all parameters and use 32 *walkers* with a nominal chain convergence criterion that the chain size must be 100 times the autocorrelation time and the latter having varied less than 1% the last time it was calculated (every few thousand samples, in our case). We have also visually inspected the contour plots and found them to be stable enough for our conclusions well before nominal convergence was achieved. We have further rerun the constraints with the walkers initialized at different starting points, which returned the same constraints. As another cross-check, we have further compared the outcome of EMCEE with that of a two-dimensional grid search for the parameters ${f\\_\\textnormal{\\textsc{nl}}}$ and *p* in parametrization 1 (which as we will see next displays a non-trivial bimodal shape of the likelihood), which we found to be effectively indistinguishable.\nResults from parametrization 1: *b**ϕ* = 2*δ**c*(*b*1 − *p*)\n------------------------------------------------------------\nTo build intuition for the shape of the constraints on ${f\\_\\textnormal{\\textsc{nl}}}$ with the *b**ϕ* = 2*δ**c*(*b*1 − *p*) parametrization, we study first the results from constraints in which only ${f\\_\\textnormal{\\textsc{nl}}}$ and *p* are varied and the remaining parameters are kept fixed at their fiducial values. In Fig. [fig:intuition0Pgg], this is shown for the *P**g**g*-only constraints. The left panel shows that the two-dimensional distribution is appreciably bimodal with one peak at ${f\\_\\textnormal{\\textsc{nl}}}> 0$ and *p* < *b*1*A*, *b*2*A*, and another at ${f\\_\\textnormal{\\textsc{nl}}}< 0$ and *p* > *b*1*A*, *b*2*A*. This is as expected since the multitracer galaxy power spectrum is weakly sensitive to a simultaneous change of sign of ${f\\_\\textnormal{\\textsc{nl}}}$ and *b*1 − *p*; for a single galaxy sample, the power spectrum would be completely unchanged under this change of sign. This shape of the distribution can be understood by inspecting the degeneracy directions from the three spectra in our multitracer power spectrum analysis. Concretely, Eqs. ([eq:pggmodelAA]), ([eq:pggmodelAB]) and ([eq:pggmodelBB]) remain unchanged if ${f\\_\\textnormal{\\textsc{nl}}}$ and *p* vary, respectively, according to the relations [eq:degdirs] Degeneracy from PggAA :: p &=& b1, fiduA -, \nDegeneracy from PggAB :: p &= &"},"data_id":{"kind":"string","value":"arxiv_0002839"}}},{"rowIdx":693,"cells":{"content":{"kind":"string","value":"Ultra-Fast Converging Path-Integral Approach for RotatingIdeal Bose-Einstein Condensates\n========================================================================================\nA recently developed efficient recursive approach for analytically calculating the short-time evolution of the one-particle propagator to extremely high orders is applied here for numerically studying the thermodynamical and dynamical properties of a rotating ideal Bose gas of 87Rb atoms in an anharmonic trap. At first, the one-particle energy spectrum of the system is obtained by diagonalizing the discretized short-time propagator. Using this, many-boson properties such as the condensation temperature, the ground-state occupancy, density profiles, and time-of-flight absorption pictures are calculated for varying rotation frequencies. The obtained results improve previous semiclassical calculations, in particular for smaller particle numbers. Furthermore, we find that typical time scales for a free expansion are increased by an order of magnitude for the delicate regime of both critical and overcritical rotation.\nBose-Einstein condensation Effective action Exact diagonalization Time-of-flight 03.75.Hh 31.15.xk 51.30.+i\nIntroduction\n============\nBose-Einstein condensation (BEC) represents a macroscopic quantum phenomenon of recent broad research interest. Since its first experimental realization in 1995, it has been extensively studied experimentally, analytically, and numerically. The two main research directions are weakly-interacting dilute gases in magneto-optical traps and strongly-interacting quantum gases in optical lattices. The behavior of a Bose-Einstein condensate under rotation is essential for understanding many fundamental BEC phenomena. For instance, its response to rotation represents one of the seminal hallmarks of superfluidity. However, once a harmonically trapped Bose-Einstein condensate is rotated critically, i.e. the rotation frequency becomes so large that it compensates the radially confining harmonic frequency, the system turns out to be radially no longer trapped. In the absence of additional potential terms the condensate would start to expand perpendicular to the rotation axis. For an overcritical rotation, this expansion would even be accelerated by the presence of a residual centrifugal force. In order to reach experimentally this delicate regime of critical or overcritical rotation, Fetter suggested in Ref. to add an additional quartic term to the harmonic trap potential. Using a Gaussian laser beam propagating in the *z*-direction, this has been realized experimentally in Paris by Dalibard and co-workers for a BEC of 3 ⋅ 105 atoms of 87Rb. The resulting axially-symmetric trap with a small quartic anharmonicity in the *x**y*-plane, seen by individual atoms, has the form\n \n$$\\label{eq:trap}\n\\hspace\\*{-1mm}\nV\\_{\\rm BEC}=\\frac{M\\_\\mathrm{Rb}}{2}(\\omega\\_\\perp^2-\\Omega^2) r\\_\\perp^2+\\frac{M\\_\\mathrm{Rb}}{2}\\omega\\_z^2 z^2+\\frac{k}{4}r\\_\\perp^4\\,,$$ \nwith the perpendicular radius $r\\_\\perp =\\sqrt{x^2+y^2}$, as well as the trap frequencies *ω*⊥ = 2*π* × 64.8 Hz, *ω**z* = 2*π* × 11.0 Hz, and the trap anharmonicity *k* = *k*BEC = 2.6 × 10− 11 Jm− 4. Furthermore, the rotation frequency Ω, which is measured in units of *ω*⊥, i.e. it is expressed by the ratio *r* = Ω/*ω*⊥, represents the tunable control parameter which could be experimentally increased up to *r* = 1.04. This highest possible rotation frequency seems to coincide with an instability which follows from a Thomas-Fermi solution of the Gross-Pitaevskii equation.\nAs long as we can ignore the presence of two-particle interactions and approximately describe the system with the ideal Bose gas, its many-particle properties in the grand-canonical ensemble are exclusively derivable from one-particle states. When considering the thermodynamic limit, usually the semiclassical approximation is applied, where the one-particle ground state *E*0 is retained and treated quantum mechanically, while all one-particles states above *E*0 are approximately treated as a continuum. This semiclassical approximation remains reasonable good irrespective of the rotation frequency Ω once the total particle number *N* is large enough and the trap anharmonicity *k* small enough. The latter condition implies that the underlying one-particle potential ([eq:trap]) has a small curvature around its minimum. However, in this context the question arises for which system parameters such a semiclassical approximation is not sufficient for a precise description of BEC phenomena, as well as when it finally breaks down, requiring a full quantum mechanical treatment of the system.\nIn order to analyze this fundamental problem more quantitatively, it is mandatory to determine the one-particle energy eigenvalues and eigenfunctions fully quantum mechanically. To this end we apply a recently developed ultra-fast converging path-integral approach for calculating the imaginary-time evolution amplitude of general non-relativistic many-body systems. In this approach a hierarchy of discretized effective actions is introduced where higher terms, compared to the naive action, substantially reduce errors in calculations of transition amplitudes. In particular, this allows a systematic derivation of discretized effective actions of level *p*, which lead to a 1/*N**p*-convergence of discretized time-sliced transition amplitudes within the continuum limit *N* → ∞ of infinitely-many time slices. In addition, the improved convergence of transition amplitudes, calculated with the level *p* effective action, can be used to construct higher-order analytic approximations for short-time propagators. Furthermore, the *N* = 1 time-slice approximation turns out to be valid for short imaginary times and useful for small values of the inverse temperature in applications in quantum statistical physics. The recent Refs. demonstrate that this path-integral approach allows to determine a huge number of both eigenvalues and eigenfunctions of quantum systems with very high accuracy using exact numerical diagonalization.\nIn this Letter we show how this path-integral approach is applied for studying both global and local properties of fast-rotating Bose-Einstein condensates. To this end we proceed as follows: Sec. [sec:grand] briefly reviews the main ingredients of the path-integral approach in the general context of ideal Bose-Einstein condensates. Then we calculate in Sec. [sec:energy] a large number of energy eigenvalues and eigenfunctions for the anharmonic one-particle potential ([eq:trap]). Afterwards, Sec. [sec:finite] discusses how a finite number of numerically available energy eigenvalues affects the results and how they can be improved by introducing systematic semiclassical corrections. On the basis of this precise numerical one-particle information, Sec. [sec:global] studies global properties of a rotating condensate, for instance the condensation temperature *T*c as a function of rotation frequency Ω and the ground-state occupancy *N*0/*N* as a function of temperature *T*. Finally, Sec. [sec:local] is devoted to the calculation of local properties of the condensate, such as density profiles and time-of-flight absorption pictures. Sec. [sec:con] briefly summarizes the main results presented in this Letter.\nMany- Versus One-Particle Physics\n=================================\nAt first we demonstrate that a precise numerical access to one-particle eigenstates allows the calculation of the condensation temperature and other thermodynamic properties of an ideal Bose gas. Afterwards, we briefly review details of the path-integral effective action approach for a numerical study of a 87Rb BEC in the anharmonic trap ([eq:trap]).\nIdeal Bose Gas\n--------------\nFor high temperatures, the grand-canonical partition function of an ideal Bose gas is given by\n \n$$\\label{eq:Z}\n{\\cal Z}=\\sum\\_\\nu e^{-\\beta(E\\_\\nu-\\mu N\\_\\nu)}\\,,$$ \nwhere *ν* enumerates all possible configurations of the system, *β* = 1/*k**B**T* represents the inverse temperature, and *μ* denotes the chemical potential. As the ideal bosons do not interact, the system energy *E**ν* can be expressed in terms of single-particle energy eigenvalues\n \n*E**ν* = ∑*n**N**ν*(*n*) *E**n* , \nwhere *n* counts single-particle energy states, while *N**ν*(*n*) = 0, 1, 2, … and *E**n* stand for the occupancy and the energy eigenvalue of level *n*, respectively. Correspondingly, the number of particles in the system reads\n \n*N**ν* = ∑*n**N**ν*(*n*) . \nThus, the grand-canonical free energy ${\\cal F}=-(\\ln {\\cal Z})/\\beta$ results to be\n \n$${\\cal F}=\\frac{1}{\\beta}\\sum\\_n\\ln \\left[ 1-e^{-\\beta(E\\_n-\\mu)} \\right]\\,,$$ \nwhere a subsequent Taylor expansion of the logarithm yields\n \n$$\\label{eq:F}\n{\\cal F}=-\\frac{1}{\\beta}\\sum\\_{m=1}^\\infty\\frac{e^{m\\beta\\mu}}{m} Z\\_1(m\\beta) \\,,$$ \nusually denoted as the cumulant expansion. Thus, the many-body thermodynamic potential ([eq:F]) of an ideal Bose gas is exclusively determined by single-particle states via the one-particle partition function:\n \n*Z*1(*β*) = ∑*n**e*− *β**E**n* . \nIn principle, the above outlined exact calculation of the many-body free energy allows a full numerical description of ideal Bose gases, and can be also applied for studies of dilute Bose gases in the case when interactions are negligible. However, it becomes numerically very involved even for simple trapping potentials at low temperatures. In addition to this, the BEC phase transition is achieved only in the thermodynamic limit of an infinite number of atoms, thus making numerical studies of the condensation increasingly difficult. Usually, this problem is solved by fixing the chemical potential *μ* at the low temperatures of the condensate phase to the ground-state energy, i.e. by setting *μ* = *E*0, and to treat the ground state separately, by explicitly taking into account its macroscopic occupation *N*0. Thus, for low enough temperatures the grand-canonical free energy ([eq:F]) is modified to\n \n$$\\begin{aligned}\n{\\cal F}&=&-\\frac{1}{\\beta}\\sum\\_{m=1}^\\infty\\frac{e^{m\\beta\\mu}}{m} \\left[ Z\\_1(m\\beta) - e^{- m \\beta E\\_0} \\right] \\nonumber \\\\ \n&&+ N\\_0 ( E\\_0 - \\mu)\\,.\\end{aligned}$$ \nIn order to avoid any double-counting, we have subtracted in the first line the contribution of the ground state within the one-particle partition function, whereas the second line takes into account a possible macroscopic occupation of the ground state. The resulting total number of particles $N = - \\partial {\\cal F} / \\partial \\mu$ follows to be\n \n*N* = *N*0 + ∑*m* = 1∞*e**m**β**μ*[*Z*1(*m**β*) − *e*− *m**β**E*0] . \nThis particle number equation serves different purposes in the respective phases. Within the gas phase, where the macroscopic occupation of the ground state vanishes, i.e. we have *N*0 = 0, Eq. ([eq:N]) determines the temperature dependence of the chemical potential *μ*. On the other hand, within the BEC phase the chemical potential *μ* coincides with its minimal value, i.e. the ground-state energy *E*0, so Eq. ([eq:N]) yields the temperature dependence of *N*0. Therefore, the value of *β*c = 1/*k**B**T*c, which characterizes the boundary between both phases, follows from Eq. ([eq:N]) by setting *N*0 = 0 and *μ* = *E*0:\n \n*N* = ∑*m* = 1∞[*e**m**β*c*E*0*Z*1(*m**β*c) − 1] . \nWe conclude that, for a given number *N* of ideal bosons, the condensation temperature can be exactly calculated only if both the single-particle ground-state energy *E*0 and the full temperature dependence of the one-particle partition function ([eq:Z1]) are known.\nPath-Integral Approach\n----------------------\nThe recently developed path-integral approach allows an efficient and fast-converging numerical calculation of all one-particle properties of quantum systems. Note that this general numerical approach is suitable to treat arbitrarily-shaped trap potentials and is also applicable for exact studies of many-body problems. In this Letter we consider BEC experiments performed by Jean Dalibard’s group, where a trapping potential with a quartic anharmonicity was realized. For this reason, we will specialize our numerical approach in the following to such potentials in order to quantitatively analyze this seminal experiment.\nThe quantum statistical imaginary-time transition amplitude $A(\\mathbf a, \\mathbf b;t)=\\langle \\mathbf b|e^{- t\\hat{H}/\\hbar} |\\mathbf a \\rangle $ follows in the continuum limit *N* → ∞ of infinitely-many time slices\n \n$$A(\\mathbf a,\\mathbf b;t)=\\lim\\_{N \\to \\infty} A\\_N (\\mathbf a,\\mathbf b;t)\\,.$$ \nHere the time-sliced amplitude $A\\_N (\\mathbf a, \\mathbf b;t)$ is a product of short-time amplitudes corresponding to the introduced time steps, and is expressed as a multiple integral of the function *e*− *S**N*/ℏ, where *S**N* is usually called discretized (Euclidean) action. For the simplest one-particle theory with the Euclidean Lagrangian $L=\\dot{\\mathbf q}^2 / 2 +V(\\mathbf q)$, where the coordinates are rescaled so that the mass is equal to unity, the naive discretized action is given by\n \n$$S\\_N=\\sum\\_{n=0}^{N-1}\\left[ \\frac{\\boldsymbol\\delta\\_n^2}{2\\varepsilon}+\\varepsilon V(\\bar{\\mathbf q}\\_n)\\right]\\,,$$ \nwith the abbreviations *ɛ* = *t*/*N*, $\\boldsymbol\\delta\\_n=\\mathbf q\\_{n+1}-\\mathbf q\\_n$, $\\bar{\\mathbf q}\\_n=(\\mathbf q\\_{n+1}+\\mathbf q\\_n)/2$ and boundary conditions $\\mathbf q\\_0=\\mathbf a$, $\\mathbf q\\_N=\\mathbf b$. Using the naive discretized action leads to the convergence of the transition amplitudes to the continuum as slow as 1/*N*.\nStarting from the one-particle Schr\" odinger equation in *d* spatial dimensions, we have derived a hierarchy of discretized effective actions *S**N*(*p*) with a systematically improved convergence to the continuum result. The general short-time amplitude is first written in the form\n \n$$\\begin{aligned}\n&&\\hspace\\*{-10mm}\nA(\\mathbf q\\_n,\\mathbf q\\_{n+1};\\varepsilon)\\nonumber\\\\\n&&=\\frac{1}{(2\\pi \\hbar\\varepsilon)^{d/2}}\\, e^{-\\frac{\\boldsymbol\\delta\\_n^2}{2 \\hbar\\varepsilon}-\\frac{\\varepsilon}{\\hbar}W(\\bar{\\mathbf q}\\_n,\\boldsymbol\\delta\\_n;\\varepsilon)}\\,,\n\\label{eq:Aqnqn1}\\end{aligned}$$ \nwere $W(\\bar{\\mathbf q}\\_n,\\boldsymbol\\delta\\_n;\\varepsilon)$ stands for the ideal discretized effective action, giving exact transition amplitudes in any discretization. In fact, the above equation is valid for any propagation time *t*, not just for short times, and if we were able to calculate exactly the ideal effective potential *W*, we could use it directly to calculate the long-time transition amplitude $A(\\mathbf a, \\mathbf b;t)$. This is practically possible only for a limited number of exactly solvable models. However, we have derived a recursive approach which allows for an efficient analytical calculation of a high-order short-time expansion of the effective potential. If we insert the expansion *W*(*p* − 1) calculated to order *ɛ**p* − 1 into Eq. ([eq:Aqnqn1]), we get the short-time amplitude\n \n$$\\begin{aligned}\n&&\\hspace\\*{-10mm}A^{(p)}(\\mathbf q\\_n,\\mathbf q\\_{n+1};\\varepsilon)\\nonumber\\\\\n\\label{eq:Ap}\n&&= \\frac{1}{(2\\pi \\hbar \\varepsilon)^{d/2}}\\, e^{-\\frac{\\boldsymbol\\delta\\_n^2}{2 \\hbar \\varepsilon}-\\frac{\\varepsilon}{\\hbar} W^{(p-1)}(\\bar{\\mathbf q}\\_n,\\boldsymbol\\delta\\_n;\\varepsilon)}\\,,\\end{aligned}$$ \nwhich is designated by *p* due to the fact that *W*(*p* − 1) is multiplied by an additional factor of *ɛ* in the exponent. This yields a result for the exponent which is correct up to order *ɛ**p*, i.e. its error is proportional to *ɛ**p* + 1. If we take into account the pre-factor 1/(2*π*ℏ*ɛ*)*d*/2, the above expression for the short time amplitude *A*(*p*) gives overall errors proportional to *ɛ**p* + 1/2 in *d* = 1. Consequently, in *d* = 2 spatial dimensions, which is the case relevant here, the errors of short-time transition amplitudes would be of the order *ɛ**p*. However, if we consider the *N* dependence of the above expression, the errors are always proportional to 1/*N**p* + 1, since the pre-factor 1/(2*π*ℏ*ɛ*)*d*/2 will be absorbed in the normalization of the amplitude $A(\\mathbf a, \\mathbf b;t)$. When used to calculate a long-time transition amplitude, the product of *N* short-time amplitudes will subsequently lead to errors proportional to 1/*N**p*, as demonstrated conclusively earlier in Refs. .\nTypical values of the inverse temperature *β* in BEC experiments are quite small compared to the typical energy scale which is defined by the harmonic trap frequencies. For example, in the Paris experiment the dimensionless value of ℏ*β**ω*⊥ ranges between 10− 3 and 10− 1. Therefore, one can immediately use the above formula for the amplitude *A*(*p*) and calculate the corresponding one-particle partition function by numerically integrating the diagonal amplitude $A(\\mathbf a, \\mathbf a, \\beta)$ over the coordinate $\\mathbf a$. For small enough *β* the above formula converges rapidly, and the amplitudes can be calculated exactly for all practical purposes. We refer to this approximation as the *N* = 1 approximation, since it corresponds to having only one time-slice in the standard definition of the transition amplitude in the path-integral formalism.\nHowever, the direct use of this approach has several disadvantages. First of all, one still has to perform an integral over the diagonal coordinate $\\mathbf a$ in order to calculate the partition function. Second, this has to be done repeatedly for each value of the inverse temperature *β*. And most importantly, one has also to extract the value of the ground-state energy in view of the particle number equation ([eq:N]). In principle, this is done by studying the high-*β* regime, where the short-time expansion ([eq:Ap]) is not valid. Although this procedure works also for lower values of *β*, it requires the numerical calculation of the one-particle partition function and a detailed study of its dependence on the inverse temperature in order to obtain the ground-state energy with sufficient precision. For this reason, the algorithm becomes numerically complex and difficult to use, especially in cases where the ground state is degenerate.\nThe approach based on a numerical diagonalization of the space-discretized evolution operator effectively resolves all of the above issues. It allows the precise numerical calculation of a large number of energy eigenvalues and eigenstates of the single-particle Hamiltonian. Once calculated in the low-*β* regime, these data can be used to obtain the one-particle partition function for any value of the inverse temperature *β* in a numerically inexpensive way. Calculations based on this approach do not have restrictions in the high-*β* regime, where, in fact, they turn out to yield results with even better precision. In addition, the one-particle energy eigenfunctions obtained by the exact diagonalization will allow us to calculate local properties of Bose-Einstein condensates with very high accuracy.\nNumerical Calculation of Energy Eigenvalues and Eigenstates\n===========================================================\nThe most efficient method for calculating properties of few-body quantum systems is the direct diagonalization of the space-discretized propagator in imaginary time $\\hat{U}(t)=\\exp(-t \\hat H / \\hbar)$. Here *t* represents the appropriately chosen propagation time, for which we can calculate the matrix elements of the evolution operator analytically within our path-integral approach in the *N* = 1 approximation.\n[!ht]\n|c|c|c| & \n*n* & *r* = 0&*r* = 1 \n 0 & 1.0009731351803 & 0.1162667164134 \n1 & 2.0029165834022 & 0.2674689968905 \n2 & 2.0029165834022 & 0.2674689968905 \n3 & 3.0058275442161 & 0.4426927375269 \n4 & 3.0058275442161 & 0.4426927375270 \n5 & 3.0067964582067 & 0.4725275724941 \n6 & 4.0097032385903 & 0.6368178804983 \n7 & 4.0097032385903 & 0.6368178804984 \n8 & 4.0116368851078 & 0.6848142470356 \n9 & 4.0116368851078 & 0.6848142470357 \n|c|c|c| & \n*n* & *r* = 0&*r* = 1 \n 0 & 1.468486725893 & 1.162667164134 \n1 & 3.213056378201 & 2.674689968905 \n2 & 3.213056378201 & 2.674689968905 \n3 & 5.163819069871 & 4.426927375269 \n4 & 5.163819069871 & 4.426927375270 \n5 & 5.406908088225 & 4.725275724941 \n6 & 7.282930987460 & 6.368178804982 \n7 & 7.282930987460 & 6.368178804982 \n8 & 7.690584058915 & 6.848142470357 \n9 & 7.690584058915 & 6.848142470357 \n[tab:underandcrit]\n[!t]\n|c|c|c| & \n*n* & *k*BEC&103 *k*BEC \n 0 & -0.6617041825660 & 1.135693826206 \n1 & -0.6465857464220 & 2.628129903790 \n2 & -0.6465857464220 & 2.628129903790 \n3 & -0.6032113415949 & 4.363876633929 \n4 & -0.6032113415948 & 4.363876633929 \n5 & -0.5349860004310 & 4.667653582963 \n6 & -0.5349860004309 & 6.290444734007 \n7 & -0.4451224795419 & 6.290444734007 \n8 & -0.4451224795419 & 6.777210773169 \n9 & -0.3362724309903 & 6.777210773169 \n[tab:over]\nOn a given real-space grid defined by a set of coordinates $\\mathbf x\\_{\\mathbf j}=\\mathbf j\\Delta$, where Δ denotes the spacing, $\\mathbf j \\in [-L/\\Delta, L/\\Delta]^d$ is a *d*-dimensional vector of integers, and *L* denotes the spatial cutoff, matrix elements of the propagator are just short-time transition amplitudes\n \n$$\\label{eq:U}\nU\\_{\\mathbf i \\mathbf j}(t)=\\langle \\mathbf i \\Delta |\\hat{U}(t)|\\mathbf j\\Delta\\rangle \\,.$$ \nNote that throughout the paper we use dimensionless units, in which any energy is expressed in terms of ℏ*ω*⊥, while the length unit is the corresponding harmonic oscillator length $\\sqrt{\\hbar/M\\_\\mathrm{Rb} \\omega\\_\\perp}$.\nThe eigenvectors $\\psi\\_n(\\mathbf j\\Delta)$ of such a discretized evolution operator matrix correspond to the space-discretized eigenfunctions of the original Hamiltonian, and the corresponding eigenvalues *E**n* are related to the eigenvalues of the single-particle Hamiltonian by *e*− *t**E**n*/ℏ. A detailed analysis of discretization errors of this method is given in Ref. . It provides a practical algorithm for choosing optimal discretization parameters which minimizes the associated errors.\nAnother source of errors is related to the fact that for non-trivial potentials it is not possible to calculate exactly evolution operator matrix elements. Higher-order effective actions minimize the errors for a given propagation time *t* and allow an optimal choice of this parameter as well by taking into account other discretization errors, as was shown in Ref. . This approach is able to give very accurate energy eigenvalues even for moderate values of the propagation time *t* of the order 0.1. Table [tab:underandcrit] presents the first several energy eigenvalues for the two-dimensional (*x**y*-) part of the BEC potential ([eq:trap]) for the non-rotating case (*r* = 0), as well as for the critically-rotating condensate (*r* = 1). The top table gives the energy spectrum of the potential with the anharmonicity *k* = *k*BEC used in the experiment, while the bottom table shows the spectrum for the much larger anharmonicity *k* = 103 *k*BEC. The degeneracies of numerically obtained eigenstates in all cases correspond to the expected structure of the spectrum, which can be deduced from the symmetry of the problem. In addition to this, the interesting case of critical rotation (*r* = 1) allows a further verification of the numerical results. To this end we recall that the energy eigenvalues of a pure quartic oscillator, to which *V*BEC reduces in this case, are proportional to *k*1/3 due to a spatial rescaling in the underlying Schrödinger equation. Therefore, we expect that the energy eigenvalues for *k* = 103 *k*BEC are precisely 10 times larger than the corresponding eigenvalues for *k* = *k*BEC. Comparing the rightmost columns in Table [tab:underandcrit] we see exactly this scaling. This demonstrates conclusively that the presented method can be successfully applied also in this deeply non-perturbative parameter regime. Furthermore, Table [tab:over] gives the energy spectrum of an over-critically rotating (*r* = 1.04) condensate, illustrating that the same approach can be used in this delicate regime as well.\n![Numerically calculated density of states for xy-part of the BEC potential with k=k_\\mathrm{BEC} for a critically rotating condensate, obtained by using level p=21 effective action. The discretization parameters are L=22.3, L/\\Delta=100, and t=0.2. The dashed line is the corresponding semiclassical approximation for the density of states.](fig1 \"fig:\") [fig:DOS]\nWith these results single-particle partition functions *Z*1(*β*) can now be calculated according to Eq. ([eq:Z1]). This is especially suitable for the low-temperature regime, when higher energy levels give a negligible contribution. Although the above described approach is able to accurately give several thousands of energy eigenvalues, their number is always necessarily limited. This is easily seen from Fig. [fig:DOS], where we compare the density of states for a critically rotating condensate with the corresponding semiclassical approximation for the density of states. This comparison allows to estimate the maximal reliable two-dimensional energy eigenvalue *E*max which can be obtained numerically for a given set of discretization parameters. For example, from Fig. [fig:DOS] we can estimate *E*max ≈ 90 for *r* = 1 with the discretization parameters *L* = 22.3, *L*/Δ = 100, and *t* = 0.2. Table [tab:Emax] gives estimates for the maximal reliable energy eigenvalue for the anharmonicites *k* = *k*BEC and *k* = 103 *k*BEC for several values of rotation frequencies. These results are obtained from numerical calculations using the SPEEDUP codes. This table gives also an overview over those discretization parameters which were used for a numerical diagonalization of the BEC potential ([eq:trap]) in order to calculate both global and local properties of the condensate throughout the whole paper.\n[!t]\n|c|c|c|c|c| & & \n*r*&*E*max/ℏ*ω*⊥ &*L* &*E*max/ℏ*ω*⊥ &*L* \n 0.0 & 140 & 14.2 &190 & 3.90 \n 0.2 & 140 & 14.4 & 190 & 3.90 \n 0.4 & 140 & 15.0 & 180 & 3.91 \n 0.6 & 140 & 16.3 & 180 & 3.92 \n 0.8 & 130 & 18.6 & 180 & 3.94 \n 1.0 & 90 & 22.3 & 170 & 3.96 \n 1.04 & 90 & 23.2 & 170 & 3.96 \n[tab:Emax]\nIn the low-temperature limit the finiteness of the number of known energy eigenstates does not present a problem. In fact, a precise knowledge of a large number of energy eigenvalues makes this approach a preferred method for a numerically exact treatment of low-temperature phenomena. On the other hand, the high-temperature regime, where thermal contributions of higher energy states play a significant role, is not treatable in the same way. This regime is usually not relevant for studies of BEC experiments, but we consider it for the sake of completeness. Furthermore, we want to demonstrate that the effective action approach can even be successfully used for studying this parameter range. In the case that the temperature is high enough, so that effects of higher energy eigenstates cannot be neglected, the inverse temperature *β* becomes a small parameter. Thus, it becomes possible to calculate numerically the single-particle partition function as a sum of diagonal amplitudes, i.e.\n \n$$\\label{eq:Z1trace}\nZ\\_1(\\beta)=\\sum\\_{\\mathbf j} U\\_{\\mathbf j \\mathbf j}(\\beta)\\Delta^d\\,,$$ \nwhere Δ represents the spacing and, as before, the values of $\\mathbf j$ are defined by $\\mathbf j \\in [-L/\\Delta, L/\\Delta]^d$, with the spatial cutoff *L* chosen in such a way to ensure the localization of the evolution matrix within the interval [ − *L*, *L*]*d*. Note that this can be verified, since the evolution operator matrix elements are calculated using the analytic approximation ([eq:Ap]) with level *p* effective potential\n \n$$\\label{eq:Ukk}\nU\\_{\\mathbf j \\mathbf j}^{(p)}(\\beta)=\\frac{1}{(2\\pi\\hbar^2\\beta)^{d/2}}\\, e^{-\\beta W^{(p-1)}(\\mathbf j\\Delta, 0; \\hbar\\beta)}\\,,$$ \nwhere ℏ*β* plays now the role of the imaginary time, and the influence of the kinetic term is not present since $U\\_{\\mathbf j \\mathbf j}$ is a diagonal transition amplitude ($\\boldsymbol\\delta=0$).\nFinite Number of Energy Eigenvalues and Semiclassical Corrections\n=================================================================\nIn the previous section we have described a numerical approach which is capable of providing a large number of accurate energy eigenvalues for a general quantum system. Its major application is for few-body systems, where the complexity of the algorithm is very low. For instance, we are able to calculate typically 104 energy eigenvalues for the considered BEC potential ([eq:trap]). In this section we discuss in more detail how the finiteness of numerically available energy eigenstates affects the calculation of thermodynamic properties of Bose-Einstein condensates.\nAs outlined in Sec. [sec:grand], the information on single-particle eigenvalues is sufficient for calculating the condensation temperature according to Eq. ([eq:Tc]). Below the condensation temperature, the ground-state occupancy follows from solving the equation\n \n*N* = *N*0 + ∑*m* = 1∞[*e**m**β**E*0 *Z*1(*m**β*) − 1] . \nIn practical calculations, however, one is inevitably forced to restrict the sum over *m* in the cumulant expansion ([eq:NN0]) to some finite cutoff *M*, resulting in the following approximation for the number of thermal atoms\n \n*N* − *N*0 = ∑*m* = 1*M*∑*n* = 1∞*e*− *m**β*(*E**n* − *E*0) . \nThus the ground-state occupancy *N*0 depends for each particle number *N* and temperature *T* also on this cutoff *M*. In particular, solving ([eq:NN0M]) for the (inverse) condensation temperature, obtained by demanding *N*0 = 0, will yield *β*c(*M*) with an explicit dependence on *M*. The exact condensation temperature *β*c is only obtained in the limit *M* → ∞.\n![Number of thermally excited atoms N-N_0 as a function of the cutoff M in the cumulant expansion ([eq:NN0M]). The results are given for a non-rotating condensate at T=105.18 nK and for a critically rotating condensate at T=63.30 nK. The results are obtained by level p=21 effective action, and all available numerical eigenstates are used to calculate N-N_0. The discretization parameters were L=14.2 for r=0 and L=22.3 for r=1. In both cases the spacing was chosen according to L/\\Delta=100, and the propagation time was t=0.2.](fig2 \"fig:\") [fig:N-T-M-noSC]\nFig. [fig:N-T-M-noSC] illustrates the *M*-dependence resulting from Eq. ([eq:NN0M]) for both a non-rotating and a critically rotating condensate. As expected, the sum saturates for high values of *M* to some finite number *N* − *N*0. By tuning the temperature in such a way that the sum saturates at the desired value of the total number of atoms *N* in the system, which implies *N*0 = 0, one is, in principle, able to extract the condensation temperature *T*c.\nAlthough the results in Fig. [fig:N-T-M-noSC] suggest that this approach can be applied straightforwardly, a closer look at the results for numerically calculated values of *N* − *N*0 reveals several problems that have to be addressed. At first we have to investigate how the results depend on the number of energy eigenstates used in the numerical calculation. Fig. [fig:N-T-Emax-noSC] gives this dependence for a critically rotating condensate at its critical temperature *T*c = 63.30 nK. We can see that the dependence on the maximal available two-dimensional energy eigenvalue *E*max is quite significant. The inset of this figure reveals another problem: the value to which number *N*0 − *N* saturates depends in addition on the cumulant cutoff *M*, as explained earlier. While the *M*-dependence can be dealt with by using a very large value of the cumulant cutoff in numerical calculations, the dependence on the maximal energy eigenvalue *E*max must be eliminated by taking into account a proper semiclassical correction to the single-particle partition functions.\n![Number of thermally excited atoms N-N_0 calculated as a function of the maximal available two-dimensional energy eigenvalue E_\\mathrm{max} at T=63.30 nK. The results are given for two different values of the cumulant cutoff M for a critically rotating condensate, with the same parameters as in Fig. [fig:N-T-M-noSC]. The horizontal line corresponds to the number of atoms N=3\\cdot 10^5 in the experiment.](fig3 \"fig:\") [fig:N-T-Emax-noSC]\nNamely, the finite number of energy eigenstates implies that the single-particle partition functions are only estimated by\n \n*Z*1(*β*) ≈ ∑*n* = 0*n*max*e*− *β**E**n* , \nwhere *n*max corresponds to the value *E*max of the numerically available maximal energy eigenvalue. A semiclassical correction to this value, can be calculated according to Ref. as\n \n$$\\begin{aligned}\n&&\\hspace\\*{-1.2cm}\\Delta Z\\_1(\\beta, E\\_\\mathrm{max})=\\nonumber\\\\\n&&\\hspace\\*{-0.9cm}\\int\\frac{d^3\\mathbf{x}\\, d^3\\mathbf{p}}{(2\\pi\\hbar)^3}\\, e^{-\\beta H(\\mathbf{x}, \\mathbf{p})}\\, \\Theta(H(\\mathbf{x}, \\mathbf{p})-E\\_\\mathrm{max})\\,,\n\\label{semikling}\\end{aligned}$$ \nwhere *H*(**x**, **p**) represents the classical Hamiltonian of the system, while Θ denotes the Heaviside step-function.\nFor the trap potential ([eq:trap]) we have in *z*-direction a pure harmonic potential which can be treated exactly. Therefore, we focus only on the two-dimensional problem in the *x**y*-plane. In this case, the semiclassical correction for the single-particle partition function ([semikling]) can be expressed in terms of the complementary error function:\n \n$$\\begin{aligned}\n&&\\hspace\\*{-1cm}\\Delta Z\\_1^{(2)}(\\beta, E\\_\\mathrm{max})=\\frac{1}{2\\beta}\\left\\{ \\frac{e^{-\\beta E\\_\\mathrm{max}}}{k} \\left[-(1-r^2) \\right. \\right. \\nonumber\\\\\n&&\\hspace\\*{-0.5cm}\\left. +\\sqrt{(1-r^2)^2+4kE\\_\\mathrm{max}} \\,\\right]\\, +\\sqrt{\\frac{\\pi}{k\\beta}} e^{\\frac{\\beta (1-r^2)^2}{4k}}\\nonumber\\\\\n&&\\hspace\\*{-0.5cm}\\left. \\times \\mathrm{Erfc}\\left(\\sqrt{\\beta E\\_\\mathrm{max}+\\frac{\\beta (1-r^2)^2}{4k}}\\right)\\right\\}\\,.\n\\label{eq:2DSC}\\end{aligned}$$ \n![Number of thermally excited atoms N-N_0 calculated as a function of E_\\mathrm{max} with and without semiclassical corrections, calculated with any large cumulant cutoff M=10^4 to eliminate the M-dependence. The results correspond to a critically rotating condensate with the same parameters as in Fig. [fig:N-T-M-noSC]. The horizontal line corresponds to N=301 834 which represents the exact value at T_c=63.30 nK.](fig4 \"fig:\") [fig:N-T-Emax-SC-M1e4]\nWhen this semiclassical correction is taken into account, the numerical results show almost no dependence on *E*max, as can be seen from Fig. [fig:N-T-Emax-SC-M1e4]. Here we have used an excessively large value of the cumulant cutoff *M* = 104 in order to completely eliminate any *M* dependence. From the inset in this graph we also see that *E*max must be chosen in accordance with the value estimated in the previous section for the maximal reliable energy eigenvalue obtained by numerical diagonalization. If we use a value *E*max larger than this, we will be underestimating the higher part of the energy spectra, and obtain incorrect results. For a critically rotating condensate with the anharmonicity *k* = *k*BEC the estimated value of *E*max from Table [tab:Emax] is around 90 ℏ*ω*⊥, which agrees with the results from the inset of Fig. [fig:N-T-Emax-SC-M1e4]. If we use this value for *E*max and calculate properties of the condensate using numerically obtained eigenstates below *E*max with semiclassical corrections according to Eq. ([eq:2DSC]), we will obtain the exact results with very high accuracy.\nGlobal Properties of Rotating Bose-Einstein Condensates\n=======================================================\nIn this section we will apply this approach to calculate different global properties of rotating Bose-Einstein condensates.\nCondensation Temperature\n------------------------\nIf we take into account semiclassical corrections as explained in the previous section, we can calculate, for instance, the condensation temperature of the condensate for different rotation frequencies. This implies that we have to find the temperature for which the number of thermal atoms *N* − *N*0 saturates precisely at the total number of atoms *N*. In practice, this works the other way around: for a given condensation temperature *T*c we numerically calculate the particle number in the system using Eq. ([eq:NN0M]), which gives the number of atoms in the system required for a condensation temperature to be equal to *T*c. This procedure is implemented in Fig. [fig:N-T-T-knfac1] for several values of the rotation frequency Ω in units of *r* = Ω/*ω*⊥. For example, for *T*c = 63.14 nK we see that the corresponding number of particles is *N* = 3 ⋅ 105, which coincides with the value for a critically rotating condensate in the experiment of Dalibard and co-workers.\nIn principle, such a procedure is only applicable for low-accuracy calculations of the critical temperature, since otherwise one has to use very large values of the cutoff *M* which would practically slow-down numerical calculations. If one is interested in more precise results, a suitable *M*-dependence must be properly taken into account. In order to be able to efficiently extract the correct value of *β*c, we will derive an analytical estimate of the asymptotic error Δ*β*c = *β*c − *β*c(*M*) which is introduced by the cutoff *M*. Note that always Δ*β*c > 0, since *β*c(*M*) < *β*c compensates the missing terms in the sum ([eq:NN0M]).\n![Number of thermally excited atoms N-N_0 as a function of the temperature T for different values of the rotation frequency and the quartic anharmonicity k=k_\\mathrm{BEC}. The discretization parameters are given in Table [tab:Emax], and the results are calculated by taking into account semiclassical corrections. The dashed line corresponds to the number of atoms N=3\\cdot 10^5 in the experiment. For comparison, the full lines depict the semiclassical results from Ref. .](fig5 \"fig:\") [fig:N-T-T-knfac1]\nIf we insert *β*c = *β*c(*M*) + Δ*β*c into Eq. ([eq:NN0M]), the error Δ*β*c can considered to be small for sufficiently large value of the cutoff *M*. By comparing Eq. ([eq:NN0M]) with the exact expression ([eq:Tc]) we obtain\n \n$$\\begin{aligned}\n&&\\hspace\\*{-11mm}\\sum\\_{m=M+1}^\\infty \\sum\\_{n=1}^\\infty e^{-m\\beta\\_\\mathrm{c} (E\\_n-E\\_0)}\\approx\\nonumber\\\\\n&&\\hspace\\*{-8mm}\\Delta\\beta\\_\\mathrm{c} \\sum\\_{m=1}^M\\sum\\_{n=1}^\\infty m(E\\_n-E\\_0)\\,e^{-m\\beta\\_\\mathrm{c} (E\\_n-E\\_0)} \\,.\\end{aligned}$$ \nThe term *m*(*E**n* − *E*0) within the sum can be obtained by setting *N*0 = 0 and applying the partial derivative ∂/∂*β*c to Eq. ([eq:NN0M]):\n \n$$\\begin{aligned}\n&&\\hspace\\*{-11mm}-\\Delta\\beta\\_\\mathrm{c}\\frac{\\partial N}{\\partial\\beta\\_\\mathrm{c}} \\approx\\nonumber \\\\\n&&\\hspace\\*{-8mm}\\left[1-\\Delta\\beta\\_\\mathrm{c} \\frac{\\partial}{\\partial\\beta\\_\\mathrm{c}}\\right]\\sum\\_{m=M+1}^\\infty\\sum\\_{n=1}^\\infty e^{-m\\beta\\_\\mathrm{c} (E\\_n-E\\_0)}\\,.\n\\label{eq:Deltaeq}\\end{aligned}$$ \nNote that the derivative of the particle number *N* with respect to *β*c is not equal to zero, since *N* is here effectively defined by the sum ([eq:Tc]). Therefore, we have instead\n \n$$\\label{eq:Delta}\n\\hspace\\*{-1mm}\\frac{\\partial N}{\\partial\\beta\\_\\mathrm{c}}\n= -\\sum\\_{m=1}^\\infty\\sum\\_{n=1}^\\infty m\\, (E\\_n-E\\_0)\\, e^{-m\\beta\\_\\mathrm{c} (E\\_n-E\\_0)}\\,.$$ \nClearly, the right-hand side is a negative quantity which does not depend on *M*. However, it does depend on *β*c and the energy spectrum of the system.\nIf the system is close to a *d*-dimensional harmonic oscillator, which is the case for the potential ([eq:trap]) with the small anharmonicity relevant for the experiment, for large values of *M* we have approximately\n \n$$\\hspace\\*{-1mm}\n\\sum\\_{m=M+1}^\\infty\\sum\\_{n=1}^\\infty e^{-m\\beta\\_\\mathrm{c} (E\\_n-E\\_0)}\\approx\nd\\, \\frac{e^{-(M+1)\\beta\\_\\mathrm{c}\\hbar\\omega}}{1-e^{-\\beta\\_\\mathrm{c}\\hbar\\omega}}\\,,$$ \nwhere *ω* denotes an effective harmonic frequency. For the case of a large anharmonicity, the effective frequency *ω* would depend on *k*, representing the harmonic expansion of the potential around its minimum. With such an estimate, Eq. ([eq:Deltaeq]) reduces to\n \n$$\\begin{aligned}\n\\label{eq:E1}\n&&\\hspace\\*{-12mm}\n\\Delta\\beta\\_\\mathrm{c}\\approx -\\frac{d}{1-e^{-\\beta\\_\\mathrm{c}\\hbar\\omega}}\\times\\nonumber\\\\\n&&\\hspace{-8mm} \\frac{e^{-(M+1)\\beta\\_\\mathrm{c}\\hbar\\omega}}{\\partial N / \\partial \\beta\\_\\mathrm{c} \n+(M+1)\\, e^{-(M+1)\\beta\\_\\mathrm{c}\\hbar\\omega}\\frac{d\\hbar\\omega}{1-e^{-\\beta\\_\\mathrm{c}\\hbar\\omega}}}\\,.\\end{aligned}$$ \nThe term (*M* + 1) *e*− (*M* + 1)*β*cℏ*ω* in the denominator of the second factor can be neglected for large enough values of the cutoff *M*, yielding as a simplified version of the above expression:\n \n$$\\begin{aligned}\n\\label{eq:E2}\n\\Delta\\beta\\_\\mathrm{c}\\approx - \\frac{d\\, e^{-(M+1)\\beta\\_\\mathrm{c}\\hbar\\omega}}{\\partial N / \\partial \\beta\\_\\mathrm{c} \\, (1-e^{-\\beta\\_\\mathrm{c}\\hbar\\omega})}\\,.\\end{aligned}$$ \n![Dependence of \\beta_\\mathrm{c} on the cumulant cutoff M for an over-critically (r=1.04) rotating condensate of N=3\\cdot 10^5 atoms of ^{87}Rb with the quartic anharmonicity of the trap k=k_\\mathrm{BEC}. The discretization parameters are given in Table [tab:Emax]. The dashed line corresponds to a value of \\beta_\\mathrm{c} obtained by fitting the numerical results to the function ([eq:fit]), while the full line gives the fitted function f(M).](fig6 \"fig:\") [fig:betac-E-SC-M]\nIn order to use the derived estimates for Δ*β*c, apparently one would already have to know the sought-after value of *β*c as well as the difficult derivative ∂*N*/∂*β*c. However, in practical applications this obstacle can be circumvented as follows. The expressions ([eq:E1]) and ([eq:E2]) can be used for fitting the numerical data for *β*c(*M*) = *β*c − Δ*β*c, as is illustrated in Fig. [fig:betac-E-SC-M]. In this standard approach, all unknown values are fit parameters, obtained numerically by the least-square method. Note that not only *β*c is obtained by such a fitting procedure, but also other parameters, such as ∂*N*/∂*β*c, or the effective harmonic frequency *ω*. The important point here is to capture the correct *M*-dependence, while all other parameters do not depend on it, so that they can be extracted by fitting. For example, in Fig. [fig:betac-E-SC-M] we have used the fitting function\n \n$$\\begin{aligned}\nf(M)=\\beta\\_\\mathrm{c}-\\frac{c\\_1e^{-c\\_2 (M+1)}}{1+c\\_3(M+1)\\, e^{-c\\_4 (M+1)}}\\,,\n\\label{eq:fit}\\end{aligned}$$ \nwhich reproduces the numerical data quite accurately and gives high-precision results for the condensation temperature *T*c. The virtue of the derived estimates lies in the fact that they can be used to extract the information on the condensation temperature even for moderate values of *M*, when a saturation is not yet achieved. This substantially speeds up the numerical calculation of condensation temperatures, especially when it has to be done for different values of potential parameters, such as the frequency ratio *r* = Ω/*ω*⊥.\n![The condensation temperature as a function of the rotation frequency for the condensate of N=3\\cdot 10^5 and N=1\\cdot 10^4 atoms of ^{87}Rb, with the quartic anharmonicity of the trap k=k_\\mathrm{BEC}. The discretization parameters are given in Table [tab:Emax]. The full lines correspond to the semiclassical approximation for T_\\mathrm{c} from Ref. .](fig7 \"fig:\") [fig:Tc-r]\nFig. [fig:Tc-r] summarizes the numerical results for the condensation temperature *T*c for the anharmonicity *k* = *k*BEC as well as the particle numbers *N* = 3 ⋅ 105 and *N* = 1 ⋅ 104. If we compare the obtained numerical results with the semiclassical approximation from Ref. , we see that the agreement turns out to be relatively good for the undercritical regime, but it becomes worse for an overcritical rotation of the condensate. After presenting results for the ground-state occupancy, which were obtained from this approach in the next section, we will compare our numerically exact results with the semiclassical approximation in more detail, and identify the parameter ranges where a full numerical treatment becomes necessary.\nGround-State Occupancy\n----------------------\nThe ground-state occupancy is the next important global property of Bose-Einstein condensates we will look into. Below the condensation temperature a non-trivial fraction of atoms is in the ground state, thus yielding a macroscopic value of the occupancy ratio *N*0/*N*.\n![Ground-state occupancy N_0/N as a function of the temperature T for non-rotating and critically rotating condensate for different values of the total number of atoms. The quartic anharmonicity is k=10^3~k_\\mathrm{BEC}, and the discretization parameters are given in Table [tab:Emax]. The full lines depict the semiclassical results from Ref. .](fig8 \"fig:\") [fig:N0N-T-knfac1e3]\nUsing the same approach as above, we can calculate the ground-state occupancy from Eq. ([eq:NN0]). After determining the ground-state energy *E*0 from an exact diagonalization of the evolution operator, we obtain\n \n$$\\label{eq:Nnew}\n\\frac{N\\_0}{N}=1-\\frac{1}{N}\\sum\\_{m=1}^\\infty \\left[ e^{m\\beta E\\_0}{\\cal Z}\\_1 (m\\beta)-1 \\right ]\\,.$$ \nIn order to calculate *N*0/*N*, we need the full single-particle energy spectrum. For low temperatures, the large number of energy eigenstates obtained within the exact diagonalization is sufficient. In the sum of Eq. ([eq:Nnew]) we have again to introduce a cutoff *M* and to eliminate it by applying the methods described in previous sections. To this end one uses either a very large value for the cutoff or one derives the appropriate finite correction term, and fits the results to the derived function.\nFig. [fig:N0N-T-knfac1e3] presents numerical results for the ground-state occupancy of the condensate. The quartic anharmonicity of the trap is chosen to be *k* = 103 *k*BEC, and the resuts are given for the nonrotating case with the total number of atoms *N* = 1 ⋅ 104 and for critically rotating codensate with *N* = 5 ⋅ 104 atoms. A comparison with the semiclassical results derived in Ref. shows that the deviations increase with larger temperatures and smaller particle numbers.\nComparison with Semiclassical Approximation\n-------------------------------------------\n![Relative error of semiclassical results for the condensation temperature as a function of the rotation frequency \\Omega in units of r=\\Omega/\\omega_\\perp for N=3\\cdot 10^5 (top) and N=1\\cdot 10^4 (bottom). The quartic anharmonicity is k=k_\\mathrm{BEC}, and the discretization parameters are given in Table [tab:Emax]. The insets in both plots give the corresponding results for the large anharmonicity k=10^3~k_\\mathrm{BEC}.](fig9b \"fig:\") [fig:Tc-E-SC-SC]\nFig. [fig:Tc-E-SC-SC] depicts the errors of the semiclassically calculated condensation temperature, where the exact values are obtained by using the presented numerical approach. As we can see, the agreement is relatively good for large particle numbers and small anharmonicity if the condensate rotates under-critically. The error in this case is of the order of 1 % to 1.5 %, and turns out to be minimal for a critical rotation. However, the error significantly increases for an overcritical rotation up to almost 3.5 % for *r* = 1.1. Therefore, while the semiclassical approximation is acceptable for undercritical rotation, in the overcritical regime a numerical treatment becomes necessary. This is even more pronounced if we decrease the particle number to 104, which is quite typical for many BEC experiments. In that case, semiclassical results already have an error of the order of 20 %. For large anharmonicity the rotation effect is not so important, as we can see from insets on both graphs in Fig. [fig:Tc-E-SC-SC]. However, increasing the particle number to 104 makes a numerical treatment indispensable, since the errors of the semiclassical results amount up to 5 %.\nLocal Properties of Rotating Bose-Einstein Condensates\n======================================================\nLocal properties of ultra-cold quantum gases are ubiquitously used to observe and study the phenomenon of Bose-Einstein condensation. The prominent peak in time-of-flight absorption pictures, which appears suddenly when the temperature is decreased below *T*c, is a clear signature for the occurrence of a BEC phase transition. It is experimentally used to measure the thermodynamic properties of the condensate. In this section we will show how the presented numerical approach can be applied to calculate both the density profiles and the time-of-flight absorption imaging profiles.\nDensity Profiles\n----------------\nThe two-point propagator $\\rho(\\mathbf x\\_1,\\mathbf x\\_2)=\\langle\\hat\\Psi^\\dagger(\\mathbf x\\_1) \\hat\\Psi(\\mathbf x\\_2)\\rangle$ defines via its diagonal element, i.e. $n(\\mathbf x)=\\rho(\\mathbf x,\\mathbf x)$, the density profile of atoms in a trap. For the ideal Bose gas, the density profile can be written as\n \n$$n(\\mathbf x)=N\\_0|\\psi\\_0(\\mathbf x)|^2+\\sum\\_{n\\geq1}N\\_n |\\psi\\_n(\\mathbf x)|^2\\,,\n\\label{eq:npsi}$$ \nwhere the second term represents the thermal contribution to the density profile. Furthermore, the quantities *ψ**n* represent single-particle eigenstates, while the occupancies *N**n* with *n* ≥ 1 are given by the Bose-Einstein distribution\n \n$$N\\_n=\\frac{1}{e^{\\beta (E\\_n-E\\_0)}-1}\\,.$$ \n![Density profile in xy-plane for a non-rotating (top) and a critically rotating (bottom) condensate of N=3\\cdot 10^5 atoms of ^{87}Rb with the anharmonicity k=k_\\mathrm{BEC} at T=30 nK. The dimensionless unit length on both graphs corresponds to 1.34 \\mum, i.e. the linear size of the profile is approximately 16.1 \\mum (top) and 32.2 \\mum (bottom). The discretization parameters are given in Table [tab:Emax].](fig10b \"fig:\") [fig:n-knfac1-T30]\nHaving at our disposal numerically calculated energy eigenvalues and eigenfunctions, we can calculate the density profile of the condensate. In order to do so, we first have to obtain the ground-state occupancy number *N*0 using the approach described in the previous section. Once this is done, Eq. ([eq:npsi]) allows to calculate the density profile. In view of a comparison with absorption imaging, which always produces two-dimensional profiles, we have to integrate our numerically determined three-dimensional particle density $n(\\mathbf x)$ along the imaging axis. Fig. [fig:n-knfac1-T30] presents typical results for the resulting density profiles of Bose-Einstein condensates for both the non-rotating and the critically-rotating case. Obviously, a rotation of the condensate leads to an effective spreading due to the appearance of a centrifugal potential.\nAlthough this approach is sufficient for treating the low-temperature regime, where the condensate is present, we emphasize that the same method can also be used to deal with the thermal regime, when the temperature is increased beyond *T*c. For even higher temperatures, when the number of energy eigenstates, that need to be taken into account, exceeds the number of numerically accessible eigenstates, the presented approach can be extended in a similar way as the partition function was calculated previously as a sum of diagonal transition amplitudes. Using the cumulant expansion of occupancies and the spectral decomposition of thermal transition amplitudes, the density profile can be written for high enough temperatures as\n \n$$\\begin{aligned}\n&&\\hspace\\*{-12mm}n(\\mathbf x)=N\\_0|\\psi\\_0(\\mathbf x)|^2\\nonumber\\\\\n&&\\hspace\\*{-7mm}+ \\sum\\_{m\\geq1}\\left[e^{m\\beta E\\_0}A(\\mathbf x, 0; \\mathbf x, m\\beta\\hbar) - |\\psi\\_0(\\mathbf x)|^2\\right]\\,.\n\\label{eq:nA}\\end{aligned}$$ \nHere $A(\\mathbf x, 0; \\mathbf x, m\\beta\\hbar)$ represents the imaginary-time amplitude for a single-particle transition from the position $\\mathbf x$ in the initial imaginary time *t* = 0 to the position $\\mathbf x$ in the final imaginary time *t* = *m**β*ℏ.\nWhile both definitions ([eq:npsi]) and ([eq:nA]) are mathematically equivalent in the case when one is able to calculate infinitely many energy eigenstates and amplitudes for an arbitrary propagation time, the first one is more suitable for low temperatures, when the number of relevant energy eigenstates is moderate, and the second one is suitable for high temperatures, when the imaginary propagation time ℏ*β* is small, and the short-time expansion can be successfully applied.\nTime-of-Flight Graphs for BECs\n------------------------------\nIn typical BEC experiments, a trapping potential is switched off and the gas is allowed to expand freely during a short flight time *t* which is of the order of several tens of miliseconds. Afterwards an absorption picture is taken which maps the density profile to the plane perpendicular to the laser beam. For the ideal Bose condensate, the density profile after time *t* is given by\n \n$$n(\\mathbf x, t)=N\\_0|\\psi\\_0(\\mathbf x, t)|^2+\\sum\\_{n\\geq1}N\\_n |\\psi\\_n(\\mathbf x, t)|^2\\,,\n\\label{eq:ntpsi}$$ \nwhere the density profile has to be integrated along the imaging axis, and the eigenstates $\\psi\\_n(\\mathbf x, t)$ are propagated according to the free Hamiltonian, containing only the kinetic term, since the trapping potential is switched off. If the energy eigenstates are available exactly, either analytically or numerically, their propagation time can be calculated by performing two consecutive Fourier transformations:\n \n$$\\hspace\\*{-2mm}\n\\psi\\_n(\\mathbf x, t)=\n\\int \\frac{d^3 \\mathbf k\\, d^3 \\mathbf R}{(2\\pi)^3}\\,\ne^{i[\\mathbf k\\cdot(\\mathbf r-\\mathbf R)-\\omega\\_{\\mathbf k}t]}\\,\\psi\\_n(\\mathbf R)\\,,\n\\label{eq:psit}$$ \nwhere the term $e^{-i\\omega\\_{\\mathbf k}t}$ accounts for a free-particle propagation in $\\mathbf k$-space. In practical applications, when the energy eigenstates are calculated by a numerical diagonalization of space-discretized transition amplitudes, the natural way to calculate the above free-particle time evolution is to use Fast Fourier Transform (FFT) numerical libraries.\n![Time-of-flight absorption density profiles in xy-plane for an over-critically rotating (r=1.04) condensate of N=3\\cdot 10^5 atoms of ^{87}Rb with the anharmonicity k=k_\\mathrm{BEC} at T=30 nK. The flight time, designated as TOF, is given at each plot. The dimensionless unit length on all graphs corresponds to 1.34 \\mum and the linear size of profiles is approximately 53.6 \\mum. The discretization parameters are given in Table [tab:Emax].](fig11f \"fig:\") [fig:tofovercrit]\nFor high temperatures we can use a mathematically equivalent definition of the density profile which is derived again from using the cumulant expansion of occupancy numbers and the spectral decomposition of transition amplitudes:\n \n$$\\begin{aligned}\n&&\\hspace\\*{-12mm}n(\\mathbf x, t)=N\\_0|\\psi\\_0(\\mathbf x, t)|^2\\nonumber\\\\\n&&\\hspace\\*{-8mm}+\\sum\\_{m\\geq1}\\left[e^{m\\beta E\\_0}\\int \\frac{d^3 \\mathbf k\\_1\\, d^3 \\mathbf k\\_2\\, d^3 \\mathbf X\\_1\\, d^3 \\mathbf X\\_2}{(2\\pi)^6}\\, \\right.\\nonumber\\\\\n&&\\hspace\\*{-4mm}\\times e^{i[(\\mathbf k\\_1-\\mathbf k\\_2)\\cdot\\mathbf x-\\mathbf k\\_1\\cdot\\mathbf X\\_1 + \\mathbf k\\_2\\cdot\\mathbf X\\_2-(\\omega\\_{\\mathbf k\\_1}-\\omega\\_{\\mathbf k\\_2})t] }\\nonumber\\\\\n&&\\hspace\\*{-4mm}\\times A(\\mathbf X\\_1, 0; \\mathbf X\\_2, m\\beta\\hbar) - |\\psi\\_0(\\mathbf x, t)|^2\\Bigg]\\,.\n\\label{eq:ntA}\\end{aligned}$$ \nIn both approaches it is first necessary to calculate the ground-state energy *E*0 and the eigenfunction $\\psi\\_0(\\mathbf x)$, as well as the ground-state occupancy *N*0. If we rely on Eqs. ([eq:ntpsi]) and ([eq:psit]) to calculate time-of-flight graphs, we have to calculate as many eigenstates as possible by numerical diagonalization. Conversely, if it is possible to use directly Eq. ([eq:ntA]), we can apply the effective action short-time expansion of thermal transition amplitudes. In both cases FFT is ideally suited for calculating time-of-flight graphs.\nOvercritical Rotation\n---------------------\nThe case of critical and overcritical rotation *r* ≥ 1 is realized in the Paris experiment by introducing the anharmonic part of the potential ([eq:trap]), so that the condensate is confined even when the harmonic part of the trapping potential is completely compensated or overcompensated by the rotation. The experimental realization of this delicate balance was difficult to achieve, but nevertheless when the condensate was successfully confined while rotating over-critically, the measurements of its properties can be done using the standard techniques, including absorption imaging. Within the semiclassical approach one has to carefully consider this situation, since the chemical potential is defined by the minimum of the potential, and now cannot be simply set to zero anymore. In our numerical approach, however, the implementation of the methods described in previous sections is straightforward even for overcritical rotation. First one calculates energy eigenvalues and eigenstates using exact diagonalization, yielding negative values for the first several eigenstates. Table [tab:over] shows the resulting energy spectrum of an over-critically rotating condensate (*r* = 1.04) for the experimental value of the anharmonicity *k*BEC, as well as for the case of large anharmonicity 103 *k*BEC.\n![Condensate density at the origin of xy-plane as a function of the time of flight (TOF) for the condensate of N=3\\cdot 10^5 atoms of ^{87}Rb at T=30 nK for several rotation frequencies \\Omega in units of r=\\Omega/\\omega_\\perp. The quartic anharmonicity is k=k_\\mathrm{BEC} and the discretization parameters are given in Table [tab:Emax].](fig12 \"fig:\") [fig:dorigin]\nCondensation temperature and other global properties as well as local properties of overcritically rotating condensates can also be calculated as before. Fig. [fig:tofovercrit] gives the time-of-flight absorption imaging sequence in the *x**y*-plane for an overcritically (*r* = 1.04) rotating Bose-Einstein condensate with the anharmonicity *k* = *k*BEC and the particle number *N* = 3 ⋅ 105 at *T* = 30 nK, exhibiting an interesting behavior. The initial density profile has a minimum at the origin, due to the shape of the anharmonic potential. The free expansion of the condensate leads to an increase in the particle density at the origin, and only afterwards the condensate density profile expands monotonically. Fig. [fig:dorigin] presents the time dependence of the particle density at the origin for varying rotation frequencies, parametrized by the ratio *r* = Ω/*ω*⊥. We read off that approaching the critical rotation slows down the expansion of the condensate. For overcritical rotation this is even more pronounced, due to the appearance of the peak in the particle density for the expansion time *t* > 0. This leads to an expansion which is typically an order of magnitude slower for the rotation with *r* > 1.\nConclusions\n===========\nIn this Letter a new method for numerically calculating short-time transition amplitudes based on the effective action approach is applied to the study of ideal Bose gases. Earlier derived higher-order discretized effective actions are used for an efficient numerical calculation of both global and local properties of fast-rotating Bose-Einstein condensates. To this end we have calculated large numbers of single-particle eigenvalues and eigenstates using an exact numerical diagonalization of the space-discretized evolution operator matrix. Using this information, we have calculated the condensation temperature and the ground-state occupancy of the condensate, as well as density profiles and time-of-flight absorption graphs. We have also shown that a critical and an overcritical rotation can be studied using the presented numerical approach, and that it leads to a substantial increase in the time scale for the free expansion of the condensate after the trapping potential is switched off. Finally, we note that our approach can also be used for numerical studies of properties of rotating ultra-cold Fermi gases.\nAcknowledgments\n===============\nThis work was supported in part by the Ministry of Science and Technological Development of the Republic of Serbia, under project No. OI141035 and bilateral project PI-BEC funded jointly with the German Academic Exchange Service (DAAD), and the European Commission under EU Centre of Excellence grant CX-CMCS. Numerical simulations were run on the AEGIS e-Infrastructure, supported in part by FP7 projects EGEE-III and SEE-GRID-SCI.\n00\nL. Pitaevskii and S. Stringari, *Bose-Einstein Condensation* (Oxford University Press, Oxford, 2003).\nC. Pethick and H. 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Pelster: Europ. Phys. J. D **54**, 667 (2009).\n\nUltra-Fast Converging Path-Integral Approach for RotatingIdeal Bose-Einstein Condensates\n========================================================================================\nA recently developed efficient recursive approach for analytically calculating the short-time evolution of the one-particle propagator to extremely high orders is applied here for numerically studying the thermodynamical and dynamical properties of a rotating ideal Bose gas of 87Rb atoms in an anharmonic trap. At first, the one-particle energy spectrum of the system is obtained by diagonalizing the discretized short-time propagator. Using this, many-boson properties such as the condensation temperature, the ground-state occupancy, density profiles, and time-of-flight absorption pictures are calculated for varying rotation frequencies. The obtained results improve previous semiclassical calculations, in particular for smaller particle numbers. Furthermore, we find that typical time scales for a free expansion are increased by an order of magnitude for the delicate regime of both critical and overcritical rotation.\nBose-Einstein condensation Effective action Exact diagonalization Time-of-flight 03.75.Hh 31.15.xk 51.30.+i\nIntroduction\n============\nBose-Einstein condensation (BEC) represents a macroscopic quantum phenomenon of recent broad research interest. Since its first experimental realization in 1995, it has been extensively studied experimentally, analytically, and numerically. The two main research directions are weakly-interacting dilute gases in magneto-optical traps and strongly-interacting quantum gases in optical lattices. The behavior of a Bose-Einstein condensate under rotation is essential for understanding many fundamental BEC phenomena. For instance, its response to rotation represents one of the seminal hallmarks of superfluidity. However, once a harmonically trapped Bose-Einstein condensate is rotated critically, i.e. the rotation frequency becomes so large that it compensates the radially confining harmonic frequency, the system turns out to be radially no longer trapped. In the absence of additional potential terms the condensate would start to expand perpendicular to the rotation axis. For an overcritical rotation, this expansion would even be accelerated by the presence of a residual centrifugal force. In order to reach experimentally this delicate regime of critical or overcritical rotation, Fetter suggested in Ref. to add an additional quartic term to the harmonic trap potential. Using a Gaussian laser beam propagating in the *z*-direction, this has been realized experimentally in Paris by Dalibard and co-workers for a BEC of 3 ⋅ 105 atoms of 87Rb. The resulting axially-symmetric trap with a small quartic anharmonicity in the *x**y*-plane, seen by individual atoms, has the form\n \n$$\\label{eq:trap}\n\\hspace\\*{-1mm}\nV\\_{\\rm BEC}=\\frac{M\\_\\mathrm{Rb}}{2}(\\omega\\_\\perp^2-\\Omega^2) r\\_\\perp^2+\\frac{M\\_\\mathrm{Rb}}{2}\\omega\\_z^2 z^2+\\frac{k}{4}r\\_\\perp^4\\,,$$ \nwith the perpendicular radius $r\\_\\perp =\\sqrt{x^2+y^2}$, as well as the trap frequencies *ω*⊥ = 2*π* × 64.8 Hz, *ω**z* = 2*π* × 11.0 Hz, and the trap anharmonicity *k* = *k*BEC = 2.6 × 10− 11 Jm− 4. Furthermore, the rotation frequency Ω, which is measured in units of *ω*⊥, i.e. it is expressed by the ratio *r* = Ω/*ω*⊥, represents the tunable control parameter which could be experimentally increased up to *r* = 1.04. This highest possible rotation frequency seems to coincide with an instability which follows from a Thomas-Fermi solution of the Gross-Pitaevskii equation.\nAs long as we can ignore the presence of two-particle interactions and approximately describe the system with the ideal Bose gas, its many-particle properties in the grand-canonical ensemble are exclusively derivable from one-particle states. When considering the thermodynamic limit, usually the semiclassical approximation is applied, where the one-particle ground state *E*0 is retained and treated quantum mechanically, while all one-particles states above *E*0 are approximately treated as a continuum. This semiclassical approximation remains reasonable good irrespective of the rotation frequency Ω once the total particle number *N* is large enough and the trap anharmonicity *k* small enough. The latter condition implies that the underlying one-particle potential ([eq:trap]) has a small curvature around its minimum. However, in this context the question arises for which system parameters such a semiclassical approximation is not sufficient for a precise description of BEC phenomena, as well as when it finally breaks down, requiring a full quantum mechanical treatment of the system.\nIn order to analyze this fundamental problem more quantitatively, it is mandatory to determine the one-particle energy eigenvalues and eigenfunctions fully quantum mechanically. To this end we apply a recently developed ultra-fast converging path-integral approach for calculating the imaginary-time evolution amplitude of general non-relativistic many-body systems. In this approach a hierarchy of discretized effective actions is introduced where higher terms, compared to the naive action, substantially reduce errors in calculations of transition amplitudes. In particular, this allows a systematic derivation of discretized effective actions of level *p*, which lead to a 1/*N**p*-convergence of discretized time-sliced transition amplitudes within the continuum limit *N* → ∞ of infinitely-many time slices. In addition, the improved convergence of transition amplitudes, calculated with the level *p* effective action, can be used to construct higher-order analytic approximations for short-time propagators. Furthermore, the *N* = 1 time-slice approximation turns out to be valid for short imaginary times and useful for small values of the inverse temperature in applications in quantum statistical physics. The recent Refs. demonstrate that this path-integral approach allows to determine a huge number of both eigenvalues and eigenfunctions of quantum systems with very high accuracy using exact numerical diagonalization.\nIn this Letter we show how this path-integral approach is applied for studying both global and local properties of fast-rotating Bose-Einstein condensates. To this end we proceed as follows: Sec. [sec:grand] briefly reviews the main ingredients of the path-integral approach in the general context of ideal Bose-Einstein condensates. Then we calculate in Sec. [sec:energy] a large number of energy eigenvalues and eigenfunctions for the anharmonic one-particle potential ([eq:trap]). Afterwards, Sec. [sec:finite] discusses how a finite number of numerically available energy eigenvalues affects the results and how they can be improved by introducing systematic semiclassical corrections. On the basis of this precise numerical one-particle information, Sec. [sec:global] studies global properties of a rotating condensate, for instance the condensation temperature *T*c as a function of rotation frequency Ω and the ground-state occupancy *N*0/*N* as a function of temperature *T*. Finally, Sec. [sec:local] is devoted to the calculation of local properties of the condensate, such as density profiles and time-of-flight absorption pictures. Sec. [sec:con] briefly summarizes the main results presented in this Letter.\nMany- Versus One-Particle Physics\n=================================\nAt first we demonstrate that a precise numerical access to one-particle eigenstates allows the calculation of the condensation temperature and other thermodynamic properties of an ideal Bose gas. Afterwards, we briefly review details of the path-integral effective action approach for a numerical study of a 87Rb BEC in the anharmonic trap ([eq:trap]).\nIdeal Bose Gas\n--------------\nFor high temperatures, the grand-canonical partition function of an ideal Bose gas is given by\n \n$$\\label{eq:Z}\n{\\cal Z}=\\sum\\_\\nu e^{-\\beta(E\\_\\nu-\\mu N\\_\\nu)}\\,,$$ \nwhere *ν* enumerates all possible configurations of the system, *β* = 1/*k**B**T* represents the inverse temperature, and *μ* denotes the chemical potential. As the ideal bosons do not interact, the system energy *E**ν* can be expressed in terms of single-particle energy eigenvalues\n \n*E**ν* = ∑*n**N**ν*(*n*) *E**n* , \nwhere *n* counts single-particle energy states, while *N**ν*(*n*) = 0, 1, 2, … and *E**n* stand for the occupancy and the energy eigenvalue of level *n*, respectively. Correspondingly, the number of particles in the system reads\n \n*N**ν* = ∑*n**N**ν*(*n*) . \nThus, the grand-canonical free energy ${\\cal F}=-(\\ln {\\cal Z})/\\beta$ results to be\n \n$${\\cal F}=\\frac{1}{\\beta}\\sum\\_n\\ln \\left[ 1-e^{-\\beta(E\\_n-\\mu)} \\right]\\,,$$ \nwhere a subsequent Taylor expansion of the logarithm yields\n \n$$\\label{eq:F}\n{\\cal F}=-\\frac{1}{\\beta}\\sum\\_{m=1}^\\infty\\frac{e^{m\\beta\\mu}}{m} Z\\_1(m\\beta) \\,,$$ \nusually denoted as the cumulant expansion. Thus, the many-body thermodynamic potential ([eq:F]) of an ideal Bose gas is exclusively determined by single-particle states via the one-particle partition function:\n \n*Z*1(*β*) = ∑*n**e*− *β**E**n* . \nIn principle, the above outlined exact calculation of the many-body free energy allows a full numerical description of ideal Bose gases, and can be also applied for studies of dilute Bose gases in the case when interactions are negligible. However, it becomes numerically very involved even for simple trapping potentials at low temperatures. In addition to this, the BEC phase transition is achieved only in the thermodynamic limit of an infinite number of atoms, thus making numerical studies of the condensation increasingly difficult. Usually, this problem is solved by fixing the chemical potential *μ* at the low temperatures of the condensate phase to the ground-state energy, i.e. by setting *μ* = *E*0, and to treat the ground state separately, by explicitly taking into account its macroscopic occupation *N*0. Thus, for low enough temperatures the grand-canonical free energy ([eq:F]) is modified to\n \n$$\\begin{aligned}\n{\\cal F}&=&-\\frac{1}{\\beta}\\sum\\_{m=1}^\\infty\\frac{e^{m\\beta\\mu}}{m} \\left[ Z\\_1(m\\beta) - e^{- m \\beta E\\_0} \\right] \\nonumber \\\\ \n&&+ N\\_0 ( E\\_0 - \\mu)\\,.\\end{aligned}$$ \nIn order to avoid any double-counting, we have subtracted in the first line the contribution of the ground state within the one-particle partition function, whereas the second line takes into account a possible macroscopic occupation of the ground state. The resulting total number of particles $N = - \\partial {\\cal F} / \\partial \\mu$ follows to be\n \n*N* = *N*0 + ∑*m* = 1∞*e**m**β**μ*[*Z*1(*m**β*) − *e*− *m**β**E*0] . \nThis particle number equation serves different purposes in the respective phases. Within the gas phase, where the macroscopic occupation of the ground state vanishes, i.e. we have *N*0 = 0, Eq. ([eq:N]) determines the temperature dependence of the chemical potential *μ*. On the other hand, within the BEC phase the chemical potential *μ* coincides with its minimal value, i.e. the ground-state energy *E*0, so Eq. ([eq:N]) yields the temperature dependence of *N*0. Therefore, the value of *β*c = 1/*k**B**T*c, which characterizes the boundary between both phases, follows from Eq. ([eq:N]) by setting *N*0 = 0 and *μ* = *E*0:\n \n*N* = ∑*m* = 1∞[*e**m**β*c*E*0*Z*1(*m**β*c) − 1] . \nWe conclude that, for a given number *N* of ideal bosons, the condensation temperature can be exactly calculated only if both the single-particle ground-state energy *E*0 and the full temperature dependence of the one-particle partition function ([eq:Z1]) are known.\nPath-Integral Approach\n----------------------\nThe recently developed path-integral approach allows an efficient and fast-converging numerical calculation of all one-particle properties of quantum systems. Note that this general numerical approach is suitable to treat arbitrarily-shaped trap potentials and is also applicable for exact studies of many-body problems. In this Letter we consider BEC experiments performed by Jean Dalibard’s group, where a trapping potential with a quartic anharmonicity was realized. For this reason, we will specialize our numerical approach in the following to such potentials in order to quantitatively analyze this seminal experiment.\nThe quantum statistical imaginary-time transition amplitude $A(\\mathbf a, \\mathbf b;t)=\\langle \\mathbf b|e^{- t\\hat{H}/\\hbar} |\\mathbf a \\rangle $ follows in the continuum limit *N* → ∞ of infinitely-many time slices\n \n$$A(\\mathbf a,\\mathbf b;t)=\\lim\\_{N \\to \\infty} A\\_N (\\mathbf a,\\mathbf b;t)\\,.$$ \nHere the time-sliced amplitude $A\\_N (\\mathbf a, \\mathbf b;t)$ is a product of short-time amplitudes corresponding to the introduced time steps, and is expressed as a multiple integral of the function *e*− *S**N*/ℏ, where *S**N* is usually called discretized (Euclidean) action. For the simplest one-particle theory with the Euclidean Lagrangian $L=\\dot{\\mathbf q}^2 / 2 +V(\\mathbf q)$, where the coordinates are rescaled so that the mass is equal to unity, the naive discretized action is given by\n \n$$S\\_N=\\sum\\_{n=0}^{N-1}\\left[ \\frac{\\boldsymbol\\delta\\_n^2}{2\\varepsilon}+\\varepsilon V(\\bar{\\mathbf q}\\_n)\\right]\\,,$$ \nwith the abbreviations *ɛ* = *t*/*N*, $\\boldsymbol\\delta\\_n=\\mathbf q\\_{n+1}-\\mathbf q\\_n$, $\\bar{\\mathbf q}\\_n=(\\mathbf q\\_{n+1}+\\mathbf q\\_n)/2$ and boundary conditions $\\mathbf q\\_0=\\mathbf a$, $\\mathbf q\\_N=\\mathbf b$. Using the naive discretized action leads to the convergence of the transition amplitudes to the continuum as slow as 1/*N*.\nStarting from the one-particle Schr\" odinger equation in *d* spatial dimensions, we have derived a hierarchy of discretized effective actions *S**N*(*p*) with a systematically improved convergence to the continuum result. The general short-time amplitude is first written in the form\n \n$$\\begin{aligned}\n&&\\hspace\\*{-10mm}\nA(\\mathbf q\\_n,\\mathbf q\\_{n+1};\\varepsilon)\\nonumber\\\\\n&&=\\frac{1}{(2\\pi \\hbar\\varepsilon)^{d/2}}\\, e^{-\\frac{\\boldsymbol\\delta\\_n^2}{2 \\hbar\\varepsilon}-\\frac{\\varepsilon}{\\hbar}W(\\bar{\\mathbf q}\\_n,\\boldsymbol\\delta\\_n;\\varepsilon)}\\,,\n\\label{eq:Aqnqn1}\\end{aligned}$$ \nwere $W(\\bar{\\mathbf q}\\_n,\\boldsymbol\\delta\\_n;\\varepsilon)$ stands for the ideal discretized effective action, giving exact transition amplitudes in any discretization. In fact, the above equation is valid for any propagation time *t*, not just for short times, and if we were able to calculate exactly the ideal effective potential *W*, we could use it directly to calculate the long-time transition amplitude $A(\\mathbf a, \\mathbf b;t)$. This is practically possible only for a limited number of exactly solvable models. However, we have derived a recursive approach which allows for an efficient analytical calculation of a high-order short-time expansion of the effective potential. If we insert the expansion *W*(*p* − 1) calculated to order *ɛ**p* − 1 into Eq. ([eq:Aqnqn1]), we get the short-time amplitude\n \n$$\\begin{aligned}\n&&\\hspace\\*{-10mm}A^{(p)}(\\mathbf q\\_n,\\mathbf q\\_{n+1};\\varepsilon)\\nonumber\\\\\n\\label{eq:Ap}\n&&= \\frac{1}{(2\\pi \\hbar \\varepsilon)^{d/2}}\\, e^{-\\frac{\\boldsymbol\\delta\\_n^2}{2 \\hbar \\varepsilon}-\\frac{\\varepsilon}{\\hbar} W^{(p-1)}(\\bar{\\mathbf q}\\_n,\\boldsymbol\\delta\\_n;\\varepsilon)}\\,,\\end{aligned}$$ \nwhich is designated by *p* due to the fact that *W*(*p* − 1) is multiplied by an additional factor of *ɛ* in the exponent. This yields a result for the exponent which is correct up to order *ɛ**p*, i.e. its error is proportional to *ɛ**p* + 1. If we take into account the pre-factor 1/(2*π*ℏ*ɛ*)*d*/2, the above expression for the short time amplitude *A*(*p*) gives overall errors proportional to *ɛ**p* + 1/2 in *d* = 1. Consequently, in *d* = 2 spatial dimensions, which is the case relevant here, the errors of short-time transition amplitudes would be of the order *ɛ**p*. However, if we consider the *N* dependence of the above expression, the errors are always proportional to 1/*N**p* + 1, since the pre-factor 1/(2*π*ℏ*ɛ*)*d*/2 will be absorbed in the normalization of the amplitude $A(\\mathbf a, \\mathbf b;t)$. When used to calculate a long-time transition amplitude, the product of *N* short-time amplitudes will subsequently lead to errors proportional to 1/*N**p*, as demonstrated conclusively earlier in Refs. .\nTypical values of the inverse temperature *β* in BEC experiments are quite small compared to the typical energy scale which is defined by the harmonic trap frequencies. For example, in the Paris experiment the dimensionless value of ℏ*β**ω*⊥ ranges between 10− 3 and 10− 1. Therefore, one can immediately use the above formula for the amplitude *A*(*p*) and calculate the corresponding one-particle partition function by numerically integrating the diagonal amplitude $A(\\mathbf a, \\mathbf a, \\beta)$ over the coordinate $\\mathbf a$. For small enough *β* the above formula converges rapidly, and the amplitudes can be calculated exactly for all practical purposes. We refer to this approximation as the *N* = 1 approximation, since it corresponds to having only one time-slice in the standard definition of the transition amplitude in the path-integral formalism.\nHowever, the direct use of this approach has several disadvantages. First of all, one still has to perform an integral over the diagonal coordinate $\\mathbf a$ in order to calculate the partition function. Second, this has to be done repeatedly for each value of the inverse temperature *β*. And most importantly, one has also to extract the value of the ground-state energy in view of the particle number equation ([eq:N]). In principle, this is done by studying the high-*β* regime, where the short-time expansion ([eq:Ap]) is not valid. Although this procedure works also for lower values of *β*, it requires the numerical calculation of the one-particle partition function and a detailed study of its dependence on the inverse temperature in order to obtain the ground-state energy with sufficient precision. For this reason, the algorithm becomes numerically complex and difficult to use, especially in cases where the ground state is degenerate.\nThe approach based on a numerical diagonalization of the space-discretized evolution operator effectively resolves all of the above issues. It allows the precise numerical calculation of a large number of energy eigenvalues and eigenstates of the single-particle Hamiltonian. Once calculated in the low-*β* regime, these data can be used to obtain the one-particle partition function for any value of the inverse temperature *β* in a numerically inexpensive way. Calculations based on this approach do not have restrictions in the high-*β* regime, where, in fact, they turn out to yield results with even better precision. In addition, the one-particle energy eigenfunctions obtained by the exact diagonalization will allow us to calculate local properties of Bose-Einstein condensates with very high accuracy.\nNumerical Calculation of Energy Eigenvalues and Eigenstates\n===========================================================\nThe most efficient method for calculating properties of few-body quantum systems is the direct diagonalization of the space-discretized propagator in imaginary time $\\hat{U}(t)=\\exp(-t \\hat H / \\hbar)$. Here *t* represents the appropriately chosen propagation time, for which we can calculate the matrix elements of the evolution operator analytically within our path-integral approach in the *N* = 1 approximation.\n[!ht]\n|c|c|c| & \n*n* & *r* = 0&*r* = 1 \n 0 & 1.0009731351803 & 0.1162667164134 \n1 & 2.0029165834022 & 0.2674689968905 \n2 & 2.0029165834022 & 0.2674689968905 \n3 & 3.0058275442161 & 0.4426927375269 \n4 & 3.0058275442161 & 0.4426927375270 \n5 & 3.0067964582067 & 0.4725275724941 \n6 & 4.0097032385903 & 0.6368178804983 \n7 & 4.0097032385903 & 0.6368178804984 \n8 & 4.0116368851078 & 0.6848142470356 \n9 & 4.0116368851078 & 0.6848142470357 \n|c|c|c| & \n*n* & *r* = 0&*r* = 1 \n 0 & 1.468486725893 & 1.162667164134 \n1 & 3.213056378201 & 2.674689968905 \n2 & 3.213056378201 & 2.674689968905 \n3 & 5.163819069871 & 4.426927375269 \n4 & 5.163819069871 & 4.426927375270 \n5 & 5.406908088225 & 4.725275724941 \n6 & 7.282930987460 & 6.368178804982 \n7 & 7.282930987460 & 6.368178804982 \n8 & 7.690584058915 & 6.848142470357 \n9 & 7.690584058915 & 6.848142470357 \n[tab:underandcrit]\n[!t]\n|c|c|c| & \n*n* & *k*BEC&103 *k*BEC \n 0 & -0.6617041825660 & 1.135693826206 \n1 & -0.6465857464220 & 2.628129903790 \n2 & -0.6465857464220 & 2.628129903790 \n3 & -0.6032113415949 & 4.363876633929 \n4 & -0.6032113415948 & 4.363876633929 \n5 & -0.5349860004310 & 4.667653582963 \n6 & -0.5349860004309 & 6.290444734007 \n7 & -0.4451224795419 & 6.290444734007 \n8 & -0.4451224795419 & 6.777210773169 \n9 & -0.3362724309903 & 6.777210773169 \n[tab:over]\nOn a given real-space grid defined by a set of coordinates $\\mathbf x\\_{\\mathbf j}=\\mathbf j\\Delta$, where Δ denotes the spacing, $\\mathbf j \\in [-L/\\Delta, L/\\Delta]^d$ is a *d*-dimensional vector of integers, and *L* denotes the spatial cutoff, matrix elements of the propagator are just short-time transition amplitudes\n \n$$\\label{eq:U}\nU\\_{\\mathbf i \\mathbf j}(t)=\\langle \\mathbf i \\Delta |\\hat{U}(t)|\\mathbf j\\Delta\\rangle \\,.$$ \nNote that throughout the paper we use dimensionless units, in which any energy is expressed in terms of ℏ*ω*⊥, while the length unit is the corresponding harmonic oscillator length $\\sqrt{\\hbar/M\\_\\mathrm{Rb} \\omega\\_\\perp}$.\nThe eigenvectors $\\psi\\_n(\\mathbf j\\Delta)$ of such a discretized evolution operator matrix correspond to the space-discretized eigenfunctions of the original Hamiltonian, and the corresponding eigenvalues *E**n* are related to the eigenvalues of the single-particle Hamiltonian by *e*− *t**E**n*/ℏ. A detailed analysis of discretization errors of this method is given in Ref. . It provides a practical algorithm for choosing optimal discretization parameters which minimizes the associated errors.\nAnother source of errors is related to the fact that for non-trivial potentials it is not possible to calculate exactly evolution operator matrix elements. Higher-order effective actions minimize the errors for a given propagation time *t* and allow an optimal choice of this parameter as well by taking into account other discretization errors, as was shown in Ref. . This approach is able to give very accurate energy eigenvalues even for moderate values of the propagation time *t* of the order 0.1. Table [tab:underandcrit] presents the first several energy eigenvalues for the two-dimensional (*x**y*-) part of the BEC potential ([eq:trap]) for the non-rotating case (*r* = 0), as well as for the critically-rotating condensate (*r* = 1). The top table gives the energy spectrum of the potential with the anharmonicity *k* = *k*BEC used in the experiment, while the bottom table shows the spectrum for the much larger anharmonicity *k* = 103 *k*BEC. The degeneracies of numerically obtained eigenstates in all cases correspond to the expected structure of the spectrum, which can be deduced from the symmetry of the problem. In addition to this, the interesting case of critical rotation (*r* = 1) allows a further verification of the numerical results. To this end we recall that the energy eigenvalues of a pure quartic oscillator, to which *V*BEC reduces in this case, are proportional to *k*1/3 due to a spatial rescaling in the underlying Schrödinger equation. Therefore, we expect that the energy eigenvalues for *k* = 103 *k*BEC are precisely 10 times larger than the corresponding eigenvalues for *k* = *k*BEC. Comparing the rightmost columns in Table [tab:underandcrit] we see exactly this scaling. This demonstrates conclusively that the presented method can be successfully applied also in this deeply non-perturbative parameter regime. Furthermore, Table [tab:over] gives the energy spectrum of an over-critically rotating (*r* = 1.04) condensate, illustrating that the same approach can be used in this delicate regime as well.\n![Numerically calculated density of states for xy-part of the BEC potential with k=k_\\mathrm{BEC} for a critically rotating condensate, obtained by using level p=21 effective action. The discretization parameters are L=22.3, L/\\Delta=100, and t=0.2. The dashed line is the corresponding semiclassical approximation for the density of states.](fig1 \"fig:\") [fig:DOS]\nWith these results single-particle partition functions *Z*1(*β*) can now be calculated according to Eq. ([eq:Z1]). This is especially suitable for the low-temperature regime, when higher energy levels give a negligible contribution. Although the above described approach is able to accurately give several thousands of energy eigenvalues, their number is always necessarily limited. This is easily seen from Fig. [fig:DOS], where we compare the density of states for a critically rotating condensate with the corresponding semiclassical approximation for the density of states. This comparison allows to estimate the maximal reliable two-dimensional energy eigenvalue *E*max which can be obtained numerically for a given set of discretization parameters. For example, from Fig. [fig:DOS] we can estimate *E*max ≈ 90 for *r* = 1 with the discretization parameters *L* = 22.3, *L*/Δ = 100, and *t* = 0.2. Table [tab:Emax] gives estimates for the maximal reliable energy eigenvalue for the anharmonicites *k* = *k*BEC and *k* = 103 *k*BEC for several values of rotation frequencies. These results are obtained from numerical calculations using the SPEEDUP codes. This table gives also an overview over those discretization parameters which were used for a numerical diagonalization of the BEC potential ([eq:trap]) in order to calculate both global and local properties of the condensate throughout the whole paper.\n[!t]\n|c|c|c|c|c| & & \n*r*&*E*max/ℏ*ω*⊥ &*L* &*E*max/ℏ*ω*⊥ &*L* \n 0.0 & 140 & 14.2 &190 & 3.90 \n 0.2 & 140 & 14.4 & 190 & 3.90 \n 0.4 & 140 & 15.0 & 180 & 3.91 \n 0.6 & 140 & 16.3 & 180 & 3.92 \n 0.8 & 130 & 18.6 & 180 & 3.94 \n 1.0 & 90 & 22.3 & 170 & 3.96 \n 1.04 & 90 & 23.2 & 170 & 3.96 \n[tab:Emax]\nIn the low-temperature limit the finiteness of the number of known energy eigenstates does not present a problem. In fact, a precise knowledge of a large number of energy eigenvalues makes this approach a preferred method for a numerically exact treatment of low-temperature phenomena. On the other hand, the high-temperature regime, where thermal contributions of higher energy states play a significant role, is not treatable in the same way. This regime is usually not relevant for studies of BEC experiments, but we consider it for the sake of completeness. Furthermore, we want to demonstrate that the effective action approach can even be successfully used for studying this parameter range. In the case that the temperature is high enough, so that effects of higher energy eigenstates cannot be neglected, the inverse temperature *β* becomes a small parameter. Thus, it becomes possible to calculate numerically the single-particle partition function as a sum of diagonal amplitudes, i.e.\n \n$$\\label{eq:Z1trace}\nZ\\_1(\\beta)=\\sum\\_{\\mathbf j} U\\_{\\mathbf j \\mathbf j}(\\beta)\\Delta^d\\,,$$ \nwhere Δ represents the spacing and, as before, the values of $\\mathbf j$ are defined by $\\mathbf j \\in [-L/\\Delta, L/\\Delta]^d$, with the spatial cutoff *L* chosen in such a way to ensure the localization of the evolution matrix within the interval [ − *L*, *L*]*d*. Note that this can be verified, since the evolution operator matrix elements are calculated using the analytic approximation ([eq:Ap]) with level *p* effective potential\n \n$$\\label{eq:Ukk}\nU\\_{\\mathbf j \\mathbf j}^{(p)}(\\beta)=\\frac{1}{(2\\pi\\hbar^2\\beta)^{d/2}}\\, e^{-\\beta W^{(p-1)}(\\mathbf j\\Delta, 0; \\hbar\\beta)}\\,,$$ \nwhere ℏ*β* plays now the role of the imaginary time, and the influence of the kinetic term is not present since $U\\_{\\mathbf j \\mathbf j}$ is a diagonal transition amplitude ($\\boldsymbol\\delta=0$).\nFinite Number of Energy Eigenvalues and Semiclassical Corrections\n=================================================================\nIn the previous section we have described a numerical approach which is capable of providing a large number of accurate energy eigenvalues for a general quantum system. Its major application is for few-body systems, where the complexity of the algorithm is very low. For instance, we are able to calculate typically 104 energy eigenvalues for the considered BEC potential ([eq:trap]). In this section we discuss in more detail how the finiteness of numerically available energy eigenstates affects the calculation of thermodynamic properties of Bose-Einstein condensates.\nAs outlined in Sec. [sec:grand], the information on single-particle eigenvalues is sufficient for calculating the condensation temperature according to Eq. ([eq:Tc]). Below the condensation temperature, the ground-state occupancy follows from solving the equation\n \n*N* = *N*0 + ∑*m* = 1∞[*e**m**β**E*0 *Z*1(*m**β*) − 1] . \nIn practical calculations, however, one is inevitably forced to restrict the sum over *m* in the cumulant expansion ([eq:NN0]) to some finite cutoff *M*, resulting in the following approximation for the number of thermal atoms\n \n*N* − *N*0 = ∑*m* = 1*M*∑*n* = 1∞*e*− *m**β*(*E**n* − *E*0) . \nThus the ground-state occupancy *N*0 depends for each particle number *N* and temperature *T* also on this cutoff *M*. In particular, solving ([eq:NN0M]) for the (inverse) condensation temperature, obtained by demanding *N*0 = 0, will yield *β*c(*M*) with an explicit dependence on *M*. The exact condensation temperature *β*c is only obtained in the limit *M* → ∞.\n![Number of thermally excited atoms N-N_0 as a function of the cutoff M in the cumulant expansion ([eq:NN0M]). The results are given for a non-rotating condensate at T=105.18 nK and for a critically rotating condensate at T=63.30 nK. The results are obtained by level p=21 effective action, and all available numerical eigenstates are used to calculate N-N_0. The discretization parameters were L=14.2 for r=0 and L=22.3 for r=1. In both cases the spacing was chosen according to L/\\Delta=100, and the propagation time was t=0.2.](fig2 \"fig:\") [fig:N-T-M-noSC]\nFig. [fig:N-T-M-noSC] illustrates the *M*-dependence resulting from Eq. ([eq:NN0M]) for both a non-rotating and a critically rotating condensate. As expected, the sum saturates for high values of *M* to some finite number *N* − *N*0. By tuning the temperature in such a way that the sum saturates at the desired value of the total number of atoms *N* in the system, which implies *N*0 = 0, one is, in principle, able to extract the condensation temperature *T*c.\nAlthough the results in Fig. [fig:N-T-M-noSC] suggest that this approach can be applied straightforwardly, a closer look at the results for numerically calculated values of *N* − *N*0 reveals several problems that have to be addressed. At first we have to investigate how the results depend on the number of energy eigenstates used in the numerical calculation. Fig. [fig:N-T-Emax-noSC] gives this dependence for a critically rotating condensate at its critical temperature *T*c = 63.30 nK. We can see that the dependence on the maximal available two-dimensional energy eigenvalue *E*max is quite significant. The inset of this figure reveals another problem: the value to which number *N*0 − *N* saturates depends in addition on the cumulant cutoff *M*, as explained earlier. While the *M*-dependence can be dealt with by using a very large value of the cumulant cutoff in numerical calculations, the dependence on the maximal energy eigenvalue *E*max must be eliminated by taking into account a proper semiclassical correction to the single-particle partition functions.\n![Number of thermally excited atoms N-N_0 calculated as a function of the maximal available two-dimensional energy eigenvalue E_\\mathrm{max} at T=63.30 nK. The results are given for two different values of the cumulant cutoff M for a critically rotating condensate, with the same parameters as in Fig. [fig:N-T-M-noSC]. The horizontal line corresponds to the number of atoms N=3\\cdot 10^5 in the experiment.](fig3 \"fig:\") [fig:N-T-Emax-noSC]\nNamely, the finite number of energy eigenstates implies that the single-particle partition functions are only estimated by\n \n*Z*1(*β*) ≈ ∑*n* = 0*n*max*e*− *β**E**n* , \nwhere *n*max corresponds to the value *E*max of the numerically available maximal energy eigenvalue. A semiclassical correction to this value, can be calculated according to Ref. as\n \n$$\\begin{aligned}\n&&\\hspace\\*{-1.2cm}\\Delta Z\\_1(\\beta, E\\_\\mathrm{max})=\\nonumber\\\\\n&&\\hspace\\*{-0.9cm}\\int\\frac{d^3\\mathbf{x}\\, d^3\\mathbf{p}}{(2\\pi\\hbar)^3}\\, e^{-\\beta H(\\mathbf{x}, \\mathbf{p})}\\, \\Theta(H(\\mathbf{x}, \\mathbf{p})-E\\_\\mathrm{max})\\,,\n\\label{semikling}\\end{aligned}$$ \nwhere *H*(**x**, **p**) represents the classical Hamiltonian of the system, while Θ denotes the Heaviside step-function.\nFor the trap potential ([eq:trap]) we have in *z*-direction a pure harmonic potential which can be treated exactly. Therefore, we focus only on the two-dimensional problem in the *x**y*-plane. In this case, the semiclassical correction for the single-particle partition function ([semikling]) can be expressed in terms of the complementary error function:\n \n$$\\begin{aligned}\n&&\\hspace\\*{-1cm}\\Delta Z\\_1^{(2)}(\\beta, E\\_\\mathrm{max})=\\frac{1}{2\\beta}\\left\\{ \\frac{e^{-\\beta E\\_\\mathrm{max}}}{k} \\left[-(1-r^2) \\right. \\right. \\nonumber\\\\\n&&\\hspace\\*{-0.5cm}\\left. +\\sqrt{(1-r^2)^2+4kE\\_\\mathrm{max}} \\,\\right]\\, +\\sqrt{\\frac{\\pi}{k\\beta}} e^{\\frac{\\beta (1-r^2)^2}{4k}}\\nonumber\\\\\n&&\\hspace\\*{-0.5cm}\\left. \\times \\mathrm{Erfc}\\left(\\sqrt{\\beta E\\_\\mathrm{max}+\\frac{\\beta (1-r^2)^2}{4k}}\\right)\\right\\}\\,.\n\\label{eq:2DSC}\\end{aligned}$$ \n![Number of thermally excited atoms N-N_0 calculated as a function of E_\\mathrm{max} with and without semiclassical corrections, calculated with any large cumulant cutoff M=10^4 to eliminate the M-dependence. The results correspond to a critically rotating condensate with the same parameters as in Fig. [fig:N-T-M-noSC]. The horizontal line corresponds to N=301 834 which represents the exact value at T_c=63.30 nK.](fig4 \"fig:\") [fig:N-T-Emax-SC-M1e4]\nWhen this semiclassical correction is taken into account, the numerical results show almost no dependence on *E*max, as can be seen from Fig. [fig:N-T-Emax-SC-M1e4]. Here we have used an excessively large value of the cumulant cutoff *M* = 104 in order to completely eliminate any *M* dependence. From the inset in this graph we also see that *E*max must be chosen in accordance with the value estimated in the previous section for the maximal reliable energy eigenvalue obtained by numerical diagonalization. If we use a value *E*max larger than this, we will be underestimating the higher part of the energy spectra, and obtain incorrect results. For a critically rotating condensate with the anharmonicity *k* = *k*BEC the estimated value of *E*max from Table [tab:Emax] is around 90 ℏ*ω*⊥, which agrees with the results from the inset of Fig. [fig:N-T-Emax-SC-M1e4]. If we use this value for *E*max and calculate properties of the condensate using numerically obtained eigenstates below *E*max with semiclassical corrections according to Eq. ([eq:2DSC]), we will obtain the exact results with very high accuracy.\nGlobal Properties of Rotating Bose-Einstein Condensates\n=======================================================\nIn this section we will apply this approach to calculate different global properties of rotating Bose-Einstein condensates.\nCondensation Temperature\n------------------------\nIf we take into account semiclassical corrections as explained in the previous section, we can calculate, for instance, the condensation temperature of the condensate for different rotation frequencies. This implies that we have to find the temperature for which the number of thermal atoms *N* − *N*0 saturates precisely at the total number of atoms *N*. In practice, this works the other way around: for a given condensation temperature *T*c we numerically calculate the particle number in the system using Eq. ([eq:NN0M]), which gives the number of atoms in the system required for a condensation temperature to be equal to *T*c. This procedure is implemented in Fig. [fig:N-T-T-knfac1] for several values of the rotation frequency Ω in units of *r* = Ω/*ω*⊥. For example, for *T*c = 63.14 nK we see that the corresponding number of particles is *N* = 3 ⋅ 105, which coincides with the value for a critically rotating condensate in the experiment of Dalibard and co-workers.\nIn principle, such a procedure is only applicable for low-accuracy calculations of the critical temperature, since otherwise one has to use very large values of the cutoff *M* which would practically slow-down numerical calculations. If one is interested in more precise results, a suitable *M*-dependence must be properly taken into account. In order to be able to efficiently extract the correct value of *β*c, we will derive an analytical estimate of the asymptotic error Δ*β*c = *β*c − *β*c(*M*) which is introduced by the cutoff *M*. Note that always Δ*β*c > 0, since *β*c(*M*) < *β*c compensates the missing terms in the sum ([eq:NN0M]).\n![Number of thermally excited atoms N-N_0 as a function of the temperature T for different values of the rotation frequency and the quartic anharmonicity k=k_\\mathrm{BEC}. The discretization parameters are given in Table [tab:Emax], and the results are calculated by taking into account semiclassical corrections. The dashed line corresponds to the number of atoms N=3\\cdot 10^5 in the experiment. For comparison, the full lines depict the semiclassical results from Ref. .](fig5 \"fig:\") [fig:N-T-T-knfac1]\nIf we insert *β*c = *β*c(*M*) + Δ*β*c into Eq. ([eq:NN0M]), the error Δ*β*c can considered to be small for sufficiently large value of the cutoff *M*. By comparing Eq. ([eq:NN0M]) with the exact expression ([eq:Tc]) we obtain\n \n$$\\begin{aligned}\n&&\\hspace\\*{-11mm}\\sum\\_{m=M+1}^\\infty \\sum\\_{n=1}^\\infty e^{-m\\beta\\_\\mathrm{c} (E\\_n-E\\_0)}\\approx\\nonumber\\\\\n&&\\hspace\\*{-8mm}\\Delta\\beta\\_\\mathrm{c} \\sum\\_{m=1}^M\\sum\\_{n=1}^\\infty m(E\\_n-E\\_0)\\,e^{-m\\beta\\_\\mathrm{c} (E\\_n-E\\_0)} \\,.\\end{aligned}$$ \nThe term *m*(*E**n* − *E*0) within the sum can be obtained by setting *N*0 = 0 and applying the partial derivative ∂/∂*β*c to Eq. ([eq:NN0M]):\n \n$$\\begin{aligned}\n&&\\hspace\\*{-11mm}-\\Delta\\beta\\_\\mathrm{c}\\frac{\\partial N}{\\partial\\beta\\_\\mathrm{c}} \\approx\\nonumber \\\\\n&&\\hspace\\*{-8mm}\\left[1-\\Delta\\beta\\_\\mathrm{c} \\frac{\\partial}{\\partial\\beta\\_\\mathrm{c}}\\right]\\sum\\_{m=M+1}^\\infty\\sum\\_{n=1}^\\infty e^{-m\\beta\\_\\mathrm{c} (E\\_n-E\\_0)}\\,.\n\\label{eq:Deltaeq}\\end{aligned}$$ \nNote that the derivative of the particle number *N* with respect to *β*c is not equal to zero, since *N* is here effectively defined by the sum ([eq:Tc]). Therefore, we have instead\n \n$$\\label{eq:Delta}\n\\hspace\\*{-1mm}\\frac{\\partial N}{\\partial\\beta\\_\\mathrm{c}}\n= -\\sum\\_{m=1}^\\infty\\sum\\_{n=1}^\\infty m\\, (E\\_n-E\\_0)\\, e^{-m\\beta\\_\\mathrm{c} (E\\_n-E\\_0)}\\,.$$ \nClearly, the right-hand side is a negative quantity which does not depend on *M*. However, it does depend on *β*c and the energy spectrum of the system.\nIf the system is close to a *d*-dimensional harmonic oscillator, which is the case for the potential ([eq:trap]) with the small anharmonicity relevant for the experiment, for large values of *M* we have approximately\n \n$$\\hspace\\*{-1mm}\n\\sum\\_{m=M+1}^\\infty\\sum\\_{n=1}^\\infty e^{-m\\beta\\_\\mathrm{c} (E\\_n-E\\_0)}\\approx\nd\\, \\frac{e^{-(M+1)\\beta\\_\\mathrm{c}\\hbar\\omega}}{1-e^{-\\beta\\_\\mathrm{c}\\hbar\\omega}}\\,,$$ \nwhere *ω* denotes an effective harmonic frequency. For the case of a large anharmonicity, the effective frequency *ω* would depend on *k*, representing the harmonic expansion of the potential around its minimum. With such an estimate, Eq. ([eq:Deltaeq]) reduces to\n \n$$\\begin{aligned}\n\\label{eq:E1}\n&&\\hspace\\*{-12mm}\n\\Delta\\beta\\_\\mathrm{c}\\approx -\\frac{d}{1-e^{-\\beta\\_\\mathrm{c}\\hbar\\omega}}\\times\\nonumber\\\\\n&&\\hspace{-8mm} \\frac{e^{-(M+1)\\beta\\_\\mathrm{c}\\hbar\\omega}}{\\partial N / \\partial \\beta\\_\\mathrm{c} \n+(M+1)\\, e^{-(M+1)\\beta\\_\\mathrm{c}\\hbar\\omega}\\frac{d\\hbar\\omega}{1-e^{-\\beta\\_\\mathrm{c}\\hbar\\omega}}}\\,.\\end{aligned}$$ \nThe term (*M* + 1) *e*− (*M* + 1)*β*cℏ*ω* in the denominator of the second factor can be neglected for large enough values of the cutoff *M*, yielding as a simplified version of the above expression:\n \n$$\\begin{aligned}\n\\label{eq:E2}\n\\Delta\\beta\\_\\mathrm{c}\\approx - \\frac{d\\, e^{-(M+1)\\beta\\_\\mathrm{c}\\hbar\\omega}}{\\partial N / \\partial \\beta\\_\\mathrm{c} \\, (1-e^{-\\beta\\_\\mathrm{c}\\hbar\\omega})}\\,.\\end{aligned}$$ \n![Dependence of \\beta_\\mathrm{c} on the cumulant cutoff M for an over-critically (r=1.04) rotating condensate of N=3\\cdot 10^5 atoms of ^{87}Rb with the quartic anharmonicity of the trap k=k_\\mathrm{BEC}. The discretization parameters are given in Table [tab:Emax]. The dashed line corresponds to a value of \\beta_\\mathrm{c} obtained by fitting the numerical results to the function ([eq:fit]), while the full line gives the fitted function f(M).](fig6 \"fig:\") [fig:betac-E-SC-M]\nIn order to use the derived estimates for Δ*β*c, apparently one would already have to know the sought-after value of *β*c as well as the difficult derivative ∂*N*/∂*β*c. However, in practical applications this obstacle can be circumvented as follows. The expressions ([eq:E1]) and ([eq:E2]) can be used for fitting the numerical data for *β*c(*M*) = *β*c − Δ*β*c, as is illustrated in Fig. [fig:betac-E-SC-M]. In this standard approach, all unknown values are fit parameters, obtained numerically by the least-square method. Note that not only *β*c is obtained by such a fitting procedure, but also other parameters, such as ∂*N*/∂*β*c, or the effective harmonic frequency *ω*. The important point here is to capture the correct *M*-dependence, while all other parameters do not depend on it, so that they can be extracted by fitting. For example, in Fig. [fig:betac-E-SC-M] we have used the fitting function\n \n$$\\begin{aligned}\nf(M)=\\beta\\_\\mathrm{c}-\\frac{c\\_1e^{-c\\_2 (M+1)}}{1+c\\_3(M+1)\\, e^{-c\\_4 (M+1)}}\\,,\n\\label{eq:fit}\\end{aligned}$$ \nwhich reproduces the numerical data quite accurately and gives high-precision results for the condensation temperature *T*c. The virtue of the derived estimates lies in the fact that they can be used to extract the information on the condensation temperature even for moderate values of *M*, when a saturation is not yet achieved. This substantially speeds up the numerical calculation of condensation temperatures, especially when it has to be done for different values of potential parameters, such as the frequency ratio *r* = Ω/*ω*⊥.\n![The condensation temperature as a function of the rotation frequency for the condensate of N=3\\cdot 10^5 and N=1\\cdot 10^4 atoms of ^{87}Rb, with the quartic anharmonicity of the trap k=k_\\mathrm{BEC}. The discretization parameters are given in Table [tab:Emax]. The full lines correspond to the semiclassical approximation for T_\\mathrm{c} from Ref. .](fig7 \"fig:\") [fig:Tc-r]\nFig. [fig:Tc-r] summarizes the numerical results for the condensation temperature *T*c for the anharmonicity *k* = *k*BEC as well as the particle numbers *N* = 3 ⋅ 105 and *N* = 1 ⋅ 104. If we compare the obtained numerical results with the semiclassical approximation from Ref. , we see that the agreement turns out to be relatively good for the undercritical regime, but it becomes worse for an overcritical rotation of the condensate. After presenting results for the ground-state occupancy, which were obtained from this approach in the next section, we will compare our numerically exact results with the semiclassical approximation in more detail, and identify the parameter ranges where a full numerical treatment becomes necessary.\nGround-State Occupancy\n----------------------\nThe ground-state occupancy is the next important global property of Bose-Einstein condensates we will look into. Below the condensation temperature a non-trivial fraction of atoms is in the ground state, thus yielding a macroscopic value of the occupancy ratio *N*0/*N*.\n![Ground-state occupancy N_0/N as a function of the temperature T for non-rotating and critically rotating condensate for different values of the total number of atoms. The quartic anharmonicity is k=10^3~k_\\mathrm{BEC}, and the discretization parameters are given in Table [tab:Emax]. The full lines depict the semiclassical results from Ref. .](fig8 \"fig:\") [fig:N0N-T-knfac1e3]\nUsing the same approach as above, we can calculate the ground-state occupancy from Eq. ([eq:NN0]). After determining the ground-state energy *E*0 from an exact diagonalization of the evolution operator, we obtain\n \n$$\\label{eq:Nnew}\n\\frac{N\\_0}{N}=1-\\frac{1}{N}\\sum\\_{m=1}^\\infty \\left[ e^{m\\beta E\\_0}{\\cal Z}\\_1 (m\\beta)-1 \\right ]\\,.$$ \nIn order to calculate *N*0/*N*, we need the full single-particle energy spectrum. For low temperatures, the large number of energy eigenstates obtained within the exact diagonalization is sufficient. In the sum of Eq. ([eq:Nnew]) we have again to introduce a cutoff *M* and to eliminate it by applying the methods described in previous sections. To this end one uses either a very large value for the cutoff or one derives the appropriate finite correction term, and fits the results to the derived function.\nFig. [fig:N0N-T-knfac1e3] presents numerical results for the ground-state occupancy of the condensate. The quartic anharmonicity of the trap is chosen to be *k* = 103 *k*BEC, and the resuts are given for the nonrotating case with the total number of atoms *N* = 1 ⋅ 104 and for critically rotating codensate with *N* = 5 ⋅ 104 atoms. A comparison with the semiclassical results derived in Ref. shows that the deviations increase with larger temperatures and smaller particle numbers.\nComparison with Semiclassical Approximation\n-------------------------------------------\n![Relative error of semiclassical results for the condensation temperature as a function of the rotation frequency \\Omega in units of r=\\Omega/\\omega_\\perp for N=3\\cdot 10^5 (top) and N=1\\cdot 10^4 (bottom). The quartic anharmonicity is k=k_\\mathrm{BEC}, and the discretization parameters are given in Table [tab:Emax]. The insets in both plots give the corresponding results for the large anharmonicity k=10^3~k_\\mathrm{BEC}.](fig9b \"fig:\") [fig:Tc-E-SC-SC]\nFig. [fig:Tc-E-SC-SC] depicts the errors of the semiclassically calculated condensation temperature, where the exact values are obtained by using the presented numerical approach. As we can see, the agreement is relatively good for large particle numbers and small anharmonicity if the condensate rotates under-critically. The error in this case is of the order of 1 % to 1.5 %, and turns out to be minimal for a critical rotation. However, the error significantly increases for an overcritical rotation up to almost 3.5 % for *r* = 1.1. Therefore, while the semiclassical approximation is acceptable for undercritical rotation, in the overcritical regime a numerical treatment becomes necessary. This is even more pronounced if we decrease the particle number to 104, which is quite typical for many BEC experiments. In that case, semiclassical results already have an error of the order of 20 %. For large anharmonicity the rotation effect is not so important, as we can see from insets on both graphs in Fig. [fig:Tc-E-SC-SC]. However, increasing the particle number to 104 makes a numerical treatment indispensable, since the errors of the semiclassical results amount up to 5 %.\nLocal Properties of Rotating Bose-Einstein Condensates\n======================================================\nLocal properties of ultra-cold quantum gases are ubiquitously used to observe and study the phenomenon of Bose-Einstein condensation. The prominent peak in time-of-flight absorption pictures, which appears suddenly when the temperature is decreased below *T*c, is a clear signature for the occurrence of a BEC phase transition. It is experimentally used to measure the thermodynamic properties of the condensate. In this section we will show how the presented numerical approach can be applied to calculate both the density profiles and the time-of-flight absorption imaging profiles.\nDensity Profiles\n----------------\nThe two-point propagator $\\rho(\\mathbf x\\_1,\\mathbf x\\_2)=\\langle\\hat\\Psi^\\dagger(\\mathbf x\\_1) \\hat\\Psi(\\mathbf x\\_2)\\rangle$ defines via its diagonal element, i.e. $n(\\mathbf x)=\\rho(\\mathbf x,\\mathbf x)$, the density profile of atoms in a trap. For the ideal Bose gas, the density profile can be written as\n \n$$n(\\mathbf x)=N\\_0|\\psi\\_0(\\mathbf x)|^2+\\sum\\_{n\\geq1}N\\_n |\\psi\\_n(\\mathbf x)|^2\\,,\n\\label{eq:npsi}$$ \nwhere the second term represents the thermal contribution to the density profile. Furthermore, the quantities *ψ**n* represent single-particle eigenstates, while the occupancies *N**n* with *n* ≥ 1 are given by the Bose-Einstein distribution\n \n$$N\\_n=\\frac{1}{e^{\\beta (E\\_n-E\\_0)}-1}\\,.$$ \n![Density profile in xy-plane for a non-rotating (top) and a critically rotating (bottom) condensate of N=3\\cdot 10^5 atoms of ^{87}Rb with the anharmonicity k=k_\\mathrm{BEC} at T=30 nK. The dimensionless unit length on both graphs corresponds to 1.34 \\mum, i.e. the linear size of the profile is approximately 16.1 \\mum (top) and 32.2 \\mum (bottom). The discretization parameters are given in Table [tab:Emax].](fig10b \"fig:\") [fig:n-knfac1-T30]\nHaving at our disposal numerically calculated energy eigenvalues and eigenfunctions, we can calculate the density profile of the condensate. In order to do so, we first have to obtain the ground-state occupancy number *N*0 using the approach described in the previous section. Once this is done, Eq. ([eq:npsi]) allows to calculate the density profile. In view of a comparison with absorption imaging, which always produces two-dimensional profiles, we have to integrate our numerically determined three-dimensional particle density $n(\\mathbf x)$ along the imaging axis. Fig. [fig:n-knfac1-T30] presents typical results for the resulting density profiles of Bose-Einstein condensates for both the non-rotating and the critically-rotating case. Obviously, a rotation of the condensate leads to an effective spreading due to the appearance of a centrifugal potential.\nAlthough this approach is sufficient for treating the low-temperature regime, where the condensate is present, we emphasize that the same method can also be used to deal with the thermal regime, when the temperature is increased beyond *T*c. For even higher temperatures, when the number of energy eigenstates, that need to be taken into account, exceeds the number of numerically accessible eigenstates, the presented approach can be extended in a similar way as the partition function was calculated previously as a sum of diagonal transition amplitudes. Using the cumulant expansion of occupancies and the spectral decomposition of thermal transition amplitudes, the density profile can be written for high enough temperatures as\n \n$$\\begin{aligned}\n&&\\hspace\\*{-12mm}n(\\mathbf x)=N\\_0|\\psi\\_0(\\mathbf x)|^2\\nonumber\\\\\n&&\\hspace\\*{-7mm}+ \\sum\\_{m\\geq1}\\left[e^{m\\beta E\\_0}A(\\mathbf x, 0; \\mathbf x, m\\beta\\hbar) - |\\psi\\_0(\\mathbf x)|^2\\right]\\,.\n\\label{eq:nA}\\end{aligned}$$ \nHere $A(\\mathbf x, 0; \\mathbf x, m\\beta\\hbar)$ represents the imaginary-time amplitude for a single-particle transition from the position $\\mathbf x$ in the initial imaginary time *t* = 0 to the position $\\mathbf x$ in the final imaginary time *t* = *m**β*ℏ.\nWhile both definitions ([eq:npsi]) and ([eq:nA]) are mathematically equivalent in the case when one is able to calculate infinitely many energy eigenstates and amplitudes for an arbitrary propagation time, the first one is more suitable for low temperatures, when the number of relevant energy eigenstates is moderate, and the second one is suitable for high temperatures, when the imaginary propagation time ℏ*β* is small, and the short-time expansion can be successfully applied.\nTime-of-Flight Graphs for BECs\n------------------------------\nIn typical BEC experiments, a trapping potential is switched off and the gas is allowed to expand freely during a short flight time *t* which is of the order of several tens of miliseconds. Afterwards an absorption picture is taken which maps the density profile to the plane perpendicular to the laser beam. For the ideal Bose condensate, the density profile after time *t* is given by\n \n$$n(\\mathbf x, t)=N\\_0|\\psi\\_0(\\mathbf x, t)|^2+\\sum\\_{n\\geq1}N\\_n |\\psi\\_n(\\mathbf x, t)|^2\\,,\n\\label{eq:ntpsi}$$ \nwhere the density profile has to be integrated along the imaging axis, and the eigenstates $\\psi\\_n(\\mathbf x, t)$ are propagated according to the free Hamiltonian, containing only the kinetic term, since the trapping potential is switched off. If the energy eigenstates are available exactly, either analytically or numerically, their propagation time can be calculated by performing two consecutive Fourier transformations:\n \n$$\\hspace\\*{-2mm}\n\\psi\\_n(\\mathbf x, t)=\n\\int \\frac{d^3 \\mathbf k\\, d^3 \\mathbf R}{(2\\pi)^3}\\,\ne^{i[\\mathbf k\\cdot(\\mathbf r-\\mathbf R)-\\omega\\_{\\mathbf k}t]}\\,\\psi\\_n(\\mathbf R)\\,,\n\\label{eq:psit}$$ \nwhere the term $e^{-i\\omega\\_{\\mathbf k}t}$ accounts for a free-particle propagation in $\\mathbf k$-space. In practical applications, when the energy eigenstates are calculated by a numerical diagonalization of space-discretized transition amplitudes, the natural way to calculate the above free-particle time evolution is to use Fast Fourier Transform (FFT) numerical libraries.\n![Time-of-flight absorption density profiles in xy-plane for an over-critically rotating (r=1.04) condensate of N=3\\cdot 10^5 atoms of ^{87}Rb with the anharmonicity k=k_\\mathrm{BEC} at T=30 nK. The flight time, designated as TOF, is given at each plot. The dimensionless unit length on all graphs corresponds to 1.34 \\mum and the linear size of profiles is approximately 53.6 \\mum. The discretization parameters are given in Table [tab:Emax].](fig11f \"fig:\") [fig:tofovercrit]\nFor high temperatures we can use a mathematically equivalent definition of the density profile which is derived again from using the cumulant expansion of occupancy numbers and the spectral decomposition of transition amplitudes:\n \n$$\\begin{aligned}\n&&\\hspace\\*{-12mm}n(\\mathbf x, t)=N\\_0|\\psi\\_0(\\mathbf x, t)|^2\\nonumber\\\\\n&&\\hspace\\*{-8mm}+\\sum\\_{m\\geq1}\\left[e^{m\\beta E\\_0}\\int \\frac{d^3 \\mathbf k\\_1\\, d^3 \\mathbf k\\_2\\, d^3 \\mathbf X\\_1\\, d^3 \\mathbf X\\_2}{(2\\pi)^6}\\, \\right.\\nonumber\\\\\n&&\\hspace\\*{-4mm}\\times e^{i[(\\mathbf k\\_1-\\mathbf k\\_2)\\cdot\\mathbf x-\\mathbf k\\_1\\cdot\\mathbf X\\_1 + \\mathbf k\\_2\\cdot\\mathbf X\\_2-(\\omega\\_{\\mathbf k\\_1}-\\omega\\_{\\mathbf k\\_2})t] }\\nonumber\\\\\n&&\\hspace\\*{-4mm}\\times A(\\mathbf X\\_1, 0; \\mathbf X\\_2, m\\beta\\hbar) - |\\psi\\_0(\\mathbf x, t)|^2\\Bigg]\\,.\n\\label{eq:ntA}\\end{aligned}$$ \nIn both approaches it is first necessary to calculate the ground-state energy *E*0 and the eigenfunction $\\psi\\_0(\\mathbf x)$, as well as the ground-state occupancy *N*0. If we rely on Eqs. ([eq:ntpsi]) and ([eq:psit]) to calculate time-of-flight graphs, we have to calculate as many eigenstates as possible by numerical diagonalization. Conversely, if it is possible to use directly Eq. ([eq:ntA]), we can apply the effective action short-time expansion of thermal transition amplitudes. In both cases FFT is ideally suited for calculating time-of-flight graphs.\nOvercritical Rotation\n---------------------\nThe case of critical and overcritical rotation *r* ≥ 1 is realized in the Paris experiment by introducing the anharmonic part of the potential ([eq:trap]), so that the condensate is confined even when the harmonic part of the trapping potential is completely compensated or overcompensated by the rotation. The experimental realization of this delicate balance was difficult to achieve, but nevertheless when the condensate was successfully confined while rotating over-critically, the measurements of its properties can be done using the standard techniques, including absorption imaging. Within the semiclassical approach one has to carefully consider this situation, since the chemical potential is defined by the minimum of the potential, and now cannot be simply set to zero anymore. In our numerical approach, however, the implementation of the methods described in previous sections is straightforward even for overcritical rotation. First one calculates energy eigenvalues and eigenstates using exact diagonalization, yielding negative values for"},"data_id":{"kind":"string","value":"arxiv_0002840"}}},{"rowIdx":694,"cells":{"content":{"kind":"string","value":"The Magellanic Edges Survey I. Description and First Results\n============================================================\n### Accepted XXX. Received YYY; in original form ZZZ\n[firstpage]\nWe present an overview of, and first science results from, the Magellanic Edges Survey (MagES), an ongoing spectroscopic survey mapping the kinematics of red clump and red giant branch stars in the highly substructured periphery of the Magellanic Clouds. In conjunction with *Gaia* astrometry, MagES yields a sample of ∼ 7000 stars with individual 3D velocities that probes larger galactocentric radii than most previous studies. We outline our target selection, observation strategy, data reduction and analysis procedures, and present results for two fields in the northern outskirts (>10∘ on-sky from the centre) of the Large Magellanic Cloud (LMC). One field, located in the vicinity of an arm-like overdensity, displays apparent signatures of perturbation away from an equilibrium disk model. This includes a large radial velocity dispersion in the LMC disk plane, and an asymmetric line-of-sight velocity distribution indicative of motions vertically out of the disk plane for some stars. The second field reveals 3D kinematics consistent with an equilibrium disk, and yields *V*circ = 87.7 ± 8.0km s− 1 at a radial distance of ∼ 10.5kpc from the LMC centre. This leads to an enclosed mass estimate for the LMC at this radius of (1.8 ± 0.3) × 1010M⊙.\nMagellanic Clouds – galaxies:kinematics and dynamics – stars: kinematics and dynamics – galaxies: stellar content – galaxies: structure\nIntroduction\n============\nThe Large and Small Magellanic Clouds (LMC/SMC) are of fundamental importance in numerous areas of astronomy. The LMC, as the most massive Milky Way (MW) satellite – with recent estimates of its total mass exceeding 1011M⊙ – has significant effects on our Galaxy. For example, it can induce warps in the MW disk, generate overdensities in the MW dark matter halo, perturb the orbits of smaller satellites and stellar streams, and has likely brought with it several dwarf satellites of its own. The Clouds are also the closest pair of interacting dwarf galaxies, at distances of 50kpc and 60kpc for the LMC and SMC respectively. This makes them ideally situated for a detailed study of the influence of interactions on galaxy evolution. The SMC in particular is significantly distorted, with a line of sight depth of up to 20kpc and an asymmetric, irregular morphology, both of which encode valuable information about its extensive interaction history.\nIt is evident that having precise information on the masses and orbits of the Clouds, as well as their interaction and star formation histories, is important for our understanding of both the local and more distant universe. In order to obtain information on these topics, the Clouds have been the focus of numerous surveys, with efforts intensifying as the availability of instruments able to survey quickly the large on-sky area of the Clouds increases. One example is the Dark Energy Camera, which is situated on the 4m Blanco Telescope at the Cerro Tololo Inter-American Observatory in Chile, and has a 3 square degree field of view. Several surveys including the Survey of the Magellanic Stellar History, and the Magellanic SatelLites Survey have utilised DECam to obtain deep multi-band photometry across the Magellanic system. In combination with DECam photometry from the Dark Energy Survey, and additional imaging from, this has provided an almost complete photometric picture of the Clouds and their surrounds (Mackey et al. in prep).\nA key result from these surveys is the discovery of a wealth of low-surface-brightness substructure across the periphery of the Clouds ; clear evidence of tidal interaction between the two Clouds, and/or the Clouds and the Milky Way. However, in order to piece together precise details of the interactions forming these features, kinematic information for stars in the substructures and across the Clouds, which is not provided by photometric surveys, is needed.\nSpectroscopic surveys have long been used to characterise line-of-sight kinematics in the Clouds, though these have predominantly targeted stars (or star clusters) in the interior, rather than the outskirts, of the Clouds. Studies of the LMC have largely focussed on its internal rotation, with older tracer populations such as carbon stars, red giant branch stars, and star clusters found to have larger velocity dispersions compared to younger populations such as red supergiants. Even within the RGB population, metal-poor (and, by extension, older) stars are found to have increased dispersions relative to more metal-rich stars. Some studies have also found potential evidence for a high-dispersion halo population around the LMC.\nIn contrast to the relatively ordered motion within the LMC, studies of the SMC reveal more complex, disturbed kinematics. Both younger and older populations have large velocity dispersions and spatial velocity gradients indicative of the SMC being disrupted by the LMC (though note also find some evidence for coherent rotation within the SMC). SMC debris has been found in not only the bridge region between the Clouds, but also at large distances from the Clouds, and even within the LMC itself.\nIn addition to kinematic studies, spectroscopic measurements of the CaII triplet equivalent width (pioneered by and ) have often been used to obtain metallicity estimates for RGB stars in the Clouds. Metallicity gradients as a function of galactocentric radius are found in both Clouds, with median [Fe/H] abundances decreasing from around − 0.5 in the central ( ≤ 6kpc) LMC disk to around − 1 further out. In the more metal-poor SMC, [Fe/H] abundances decrease from − 1 in the central ( ≤ 2∘) regions, to approx. − 1.5 further out (e.g. ; but see also ).\nWhile spectroscopic studies are useful, measuring line-of-sight kinematics alone is insufficient to constrain the full 3D velocity field of the Clouds. This is particularly relevant when considering distant substructures, as full 3D kinematic information is required in order to distinguish between different formation mechanisms for the observed stellar substructures. In this respect, *Gaia* DR2 has been a boon, providing proper motion measurements down to red clump magnitudes ($G\\lesssim19$) in both Clouds. This has allowed substructures to be kinematically traced out to 25∘ from the centre of the Clouds, and detailed analyses of internal LMC dynamics based on proper motions to be performed. However, the Clouds are sufficiently distant that the *Gaia* spectrograph does not reach the faint magnitudes required to provide line-of-sight velocities for the old stellar populations in the Clouds that comprise the peripheral substructures.\nAs such, to date, there have been no large scale studies of 3D kinematics in the outskirts of the Clouds. The Magellanic Edges Survey (MagES) is designed to fill this gap. The core aim of the survey is to obtain spectra for large numbers of red clump and red giant branch stars that trace substructures across the Magellanic periphery, in order to derive line-of-sight velocities that can be used in conjunction with *Gaia* data to obtain the full 3D kinematic information necessary to unravel the interaction history of the Clouds. To do so, it utilises observations with the 2dF fibre positioner coupled with the dual-arm AAOmega spectrograph at the 3.9m Anglo–Australian Telescope (AAT) to obtain simultaneous spectra for ∼ 370 stars across each ∼ 2∘ diameter field. The survey began in 2015, with observations taken for several nights per year to date (details provided in Table [tab:fieldloc]).\nIn this paper, we present the detailed methodology of MagES, and our first science results. §[sec:design] presents the survey fields and target selection procedure. §[sec:data] describes the reduction and data validation processes. §[sec:isolating] discusses the method used to isolate Magellanic stars and extract aggregate field kinematics. We report our first science results, a determination of the LMC disk motion using distant tracers, in §[sec:results], followed by our conclusions and future plans for MagES in §[sec:concs].\nSurvey Design and Target Selection\n==================================\nAs MagES is intended to shed light on interactions between the Clouds – a major signature of which is the formation tidal disturbances in the periphery – MagES fields largely target overdense regions and substructures in the outskirts of the Clouds. Over time, as the photometric coverage of the Magellanic periphery has increased, the positioning of MagES fields has evolved to continually target the most conspicuous features. To date, twenty-six 2dF fields have been observed; these are detailed in Table [tab:fieldloc], with Fig. [fig:map] presenting a visual representation of the targeted fields overplotted on a stellar density map of red clump stars across the Clouds.\nThe earliest observed fields target a large arm-like feature to the north of the LMC first discussed in. Subsequent runs have focussed on spoke-like features to the south of the LMC disk, and extended red clump features surrounding the SMC. The most recent observations target another apparent tidal feature extending from the SMC which curves around the southern LMC, discussed in, and thought to be a counterpoint to the northern arm feature.\n[tab:fieldloc]\nlrrrr>m3.2cm>m2cm>m3.5cm Field & & & & & & & \n \n11 & 05 19 42.63 & -56 53 06.88 & -1.30 & 12.80 & & 27000 & 12.7 (L) \n13 & 05 35 05.69 & -55 06 03.11 & 0.90 & 14.70 & 19 Aug 20151, 1 Feb 2016 & 18380& 14.6 (L) \n15 & 06 00 07.40 & -54 17 53.14 & 4.70 & 15.30 & 20 Aug 2015 & 8700& 16.0 (L) \n16 & 06 12 13.07 & -53 52 32.45 & 6.60 & 15.50 & l 3 Feb 2016, 4 Feb 2016, \n5 Feb 2016 & 16200& 16.8 (L) \n \n6 & 03 22 33.00 & -80 40 55.00 & -5.00 & -12.75 & 14 Dec 2017, 1 Oct 2018 & 12600& 13.1 (L) \n7 & 03 26 04.00 & -77 26 18.00 & -6.50 & -9.75 & 1 Dec 2017 & 10800& 11.0 (L) \n8 & 03 39 15.00 & -73 43 48.00 & -7.50 & -6.00 & 15 Dec 2017, 16 Dec 2017 & 10800& 8.8 (L) \n10 & 04 36 23.00 & -79 07 17.00 & -2.50 & -10.00 & 12 Dec 2017 & 9000& 9.9 (L) \n14 & 05 50 22.00 & -79 21 18.00 & 1.00 & -10.00 & 13 Dec 2017 & 10800& 10.0 (L) \n17 & 06 32 16.00 & -80 59 36.00 & 2.50 & -12.00 & 30 Sep 2018 & 12600& 12.2 (L) \n19 & 06 40 29.00 & -53 29 04.00 & 11.00 & 15.00 & 12 Dec 2017 & 10800& 18.6 (L) \n20 & 07 04 01.00 & -53 37 01.00 & 14.50 & 13.75 & 14 Dec 2017 & 12600& 19.9 (L) \n \n1 & 00 56 26.00 & -67 43 32.00 & -22.00 & -12.00 & 30 Sep 2018, 1 Oct 2018 & 10800& 5.4 (S) \n2 & 00 59 30.00 & -79 10 57.00 & -10.50 & -16.75 & 2 Oct 2018 & 10800& 6.1 (S) \n3 & 01 20 00.00 & -82 30 00.00 & -6.97 & -17.51 && 11600& 9.5 (S) \n4 & 01 45 11.00 & -79 15 22.00 & -9.25 & -14.75 & 30 Sep 2018 & 12600& 6.9 (S) \n5 & 02 06 32.00 & -76 29 09.00 & -10.75 & -12.00 & 1 Oct 2018, 2 Oct 2018 & 16200& 6.0 (S) \n9 & 03 40 00.00 & -86 17 13.12 & -1.78 & -17.73 && 14410& 17.1 (L) \n12 & 05 20 00.00 & -59 18 00.00 & -1.17 & 10.29 & 27 Feb 2019, 28 Feb 2019 & 10800& 10.3 (L) \n18 & 06 40 00.00 & -62 30 00.00 & 8.19 & 5.89 & 27 Feb 2019, 28 Feb 2019 & 12200& 10.7 (L) \n21 & 07 17 12.00 & -76 36 00.00 & 6.14 & -8.58 & 2 Mar 2019, 5 Mar 2019 & 14600& 10.9 (L) \n22 & 07 25 34.00 & -52 04 52.00 & 18.50 & 14.00 & 1 Oct 2018, 2 Oct 2018 & 9000& 22.8 (L) \n23 & 07 36 00.00 & -71 00 00.00 & 9.99 & -4.19 & 6 Mar 2019, 7 Mar 2019 & 14400& 11.4 (L) \n24 & 07 58 48.00 & -84 12 00.00 & 3.67 & -16.31 & 4 Mar 2019 & 9300& 16.4 (L) \n25 & 08 32 00.00 & -67 00 00.00 & 16.74 & -4.01 & 3 Mar 2019 & 11300& 17.5 (L) \n26 & 08 48 00.00 & -79 00 00.00 & 8.67 & -13.59 & 1 Mar 2019 & 9000& 16.1 (L) \non 19 Aug 2015, these pilot fields were observed with only the red arm of AAOmega; subsequent observations were taken in the typical setup with both arms of the spectrograph, as discussed in §[sec:reductions].\n![Location of observed 2dF fields across the Magellanic system; fields are predominantly located on substructures or overdensities in the periphery of the Clouds. Green circles indicate D fields, blue circles indicate M fields, and purple circles indicate G fields. Fields 12 and 18 are discussed in detail in this paper. The background image shows the log density of red clump and red giant stars per square degree. These were selected from Gaia DR2 according to the process outlined by ; repeated here in §[sec:gfields]. On this map, north is up and east is to the left; (\\eta, \\xi) are coordinates in a tangent-plane projection centred on the LMC (\\alpha_0=82.25^{\\circ}, \\delta_0=-69.5^{\\circ}). Orange dashed circles mark angular separations of 8^{\\circ}, 12^{\\circ}, 16^{\\circ}, and 20^{\\circ} from the LMC centre, as well as 4^{\\circ} and 8^{\\circ} from the SMC centre. Within 8^{\\circ} of the LMC and 4^{\\circ} of the SMC, wide-field optical images are displayed. The red x-signs mark the locations of Canopus (the second brightest star in the sky, which limits field placement to avoid spectral contamination from scattered light) and the south celestial pole (which limits field placement due to telescope pointing limits).](fig1_small \"fig:\") [fig:map]\nTarget Selection\n----------------\nMagES primarily targets red clump stars, as this region in colour-magnitude space has high contrast for Magellanic stars relative to background contaminants (see Fig. [fig:cmds]). Whilst even stronger contrast exists for the Magellanic main sequence turn-off population, these stars are ∼ 2.5 magnitudes fainter than the red clump, and as such would require prohibitively long integration times to reach sufficient S/N.\nccc & [-4.5cm] & [-4.5cm] \n\n[fig:cmds]\nIn addition to the field placement evolving, the target selection procedure has also changed as new data have become available. Consequently, there are three distinct target selection procedures that have been applied during different phases of the survey:\n1. Fields within the DES footprint, observed prior to the release of *Gaia* DR2 (2015–2016). These are denoted as **D** fields;\n2. Fields outside the DES footprint, observed prior to the release of *Gaia* DR2 (2017); these are denoted as **M** fields; and\n3. Fields observed post-*Gaia* DR2 release (2018+); these are denoted as **G** fields.\nThe three procedures are detailed in the following sections; Table [tab:fieldloc] provides classification of each field into one of these three groups, in addition to the location of the field’s centre, dates observed and total exposure time. **D** fields are listed first, followed by **M** and **G** fields; this is in approximately chronological order of observations. Within each grouping, fields are listed in order of increasing right ascension.\nOnce a list of possible targets is compiled for each field, they are assigned various priorities between 1 and 9 (with 9 being the highest). Higher priorities are given to stars most likely to be of Magellanic origin, though how this is defined varies based on the specific selection procedure, and is discussed in the following sections. The 2dF allocation software configure uses the priorities to inform fibre allocation: higher ranked targets are more likely to be observed. By design, the selection procedure is such that there are almost always more possible targets than available 2dF fibres – as such, prioritisation strongly influences which stars are observed in any given field.\nAll three procedures involve target selection and prioritisation based on cuts in extinction-corrected colour-magnitude space. Where DECam photometry is used, the de-reddening is done using dust maps and updated coefficients from. Where *Gaia* DR2 photometry is used, the correction uses the procedure described in : the first two terms of Eq. 1 from are used in conjunction with the dust maps. No correction is made for reddening internal to the Clouds as this is not expected to be significant in the low-density peripheral regions targeted by MagES.\n### D Fields\nInitial fields observed by MagES were located entirely within the photometric footprint of DES year 1 ; target selection is thus based on (*r*0, (*g* − *r*)0) and (*i*0, (*g* − *i*)0) colour magnitude diagrams. Within each colour magnitude diagram, an inner and outer box are defined, centred on the red clump, as in panels a and b of Fig. [fig:cmds]. Priorities for each star are defined based on their location on the two diagrams: stars in the inner box in both diagrams are given the highest priority, with decreasing priority given to stars located in the outer boxes, or located within box boundaries on only one of the diagrams.\nNote that the boundaries of selection boxes for **D** fields are defined based on photometry of the northern disk of the LMC, where the position of the red clump in colour-magnitude space is well defined; the same box is then applied to fields covering fainter substructures. Selection boxes are designed to be sufficiently generous that small changes in CMD position of the red clump (due to, for example, field-to-field differences in line-of-sight distance), do not affect target selection.\n### M fields\nFields designated **M** are located outside of the DES survey footprint, and are selected based on g- and r-band DECam photometry obtained by.We refer interested readers to for details of the data reduction and photometric analysis. Three boxes are defined on the (*r*0, (*g* − *r*)0) colour magnitude diagram, as in panel c of Fig. [fig:cmds]: an inner and outer box surrounding the red clump, similar to those used for the **D** fields, as well as a lower box designed to capture any faint red clump extension. As with **D** fields, these CMD boxes are defined based on photometry of the northern LMC disk. Highest priority is assigned to stars in the inner box; followed by the outer box. Stars in the third box are assigned lowest priority as, while useful, this region of the CMD has higher Milky Way contamination than the canonical red clump.\n### G Fields\nFields observed after the release of *Gaia* DR2 utilise these data exclusively in the selection procedure; even in regions where DECam photometry exists. Unlike previous selections, a combination of photometry and astrometry are used. Highest priority is given to stars that pass the selection procedure presented in. This uses *Gaia* photometry to select red clump and RGB stars (see panel d of Fig. [fig:cmds]). **G** fields are the only fields to contain RGB stars, though these are few in number compared to red clump stars. In addition, parallax ($\\varpi<0.2$) and proper motion ( − 0.6 < *μ**B* (mas yr− 1) < 1.4, 0.9 < *μ**L* (mas yr− 1) < 2.8)[2](#fn2) cuts are applied to isolate Magellanic stars. Lower priority is given to stars within a slightly offset selection box surrounding the red clump, with the same parallax cut and more generous proper motion cuts ( − 1.0 < *μ**α*[3](#fn3) (mas yr− 1) < 4.0 and − 4.0 < *μ**δ* (mas yr− 1) < 4.0) which increase the selection area in proper motion space by a factor of 25. This lower-priority selection is used when the number of target stars passing the initial, higher-priority selection criteria is significantly lower than the number of 2dF fibres available – while less efficient, we have confirmed additional Magellanic stars are captured through this second, less restrictive selection.\nData Acquisition\n================\nObservations and Data Reduction\n-------------------------------\nAll observations were taken using the 2dF/AAOmega instrument on the AAT at Siding Spring Observatory. 2df is a multi-object fibre positioner which allows for target placement within a two-degree field on the sky. It has a total of 400 fibres, of which ∼ 365 are available for science targets (the remainder being dedicated to guide stars and sky observations, detailed later in this section). AAOmega is a dual beam optical spectrograph; for these observations, the light was split using the 580V dichroic (i.e. at 5800Å). On the blue arm, the 1500V grating was utilised, obtaining a spectral resolution of R ∼ 3700, and wavelength coverage of 4910–5615Å[4](#fn4)\n. This is designed to cover the 5167Å, 5172Å and 5183Å MgIb lines to provide precise line-of-sight (LOS) velocity estimates.\nOn the red arm, the 1700D grating was used, providing a resolution of R ∼ 10000 and wavelength coverage 8370–8790Å. This is designed to cover the 8498Å, 8542Å and 8662Å CaII triplet at sufficiently high resolution to both allow for an estimation of metallicity, as well as provide a second LOS velocity estimate complementary to that obtained from the blue arm of the spectrograph.\nIn general, our survey strategy was to observe fields for between 10800–12600s, split into 1800s exposures to avoid skyline saturation and mitigate cosmic ray contamination. This results in typical signal to noise (S/N) values of ∼ 10 per pixel in both the red and blue data (at least in spectral regions not heavily contaminated by night sky emission; poor sky subtraction during the data reduction process degrades the S/N in some regions of the red spectra). In practice, total exposure times vary in accordance with observing conditions, with shorter exposures acceptable in very good conditions, but additional repeated exposures required when conditions were poor.\nFig. [fig:sndists] shows histograms of ‘quality measure’ (QM: an empirical S/N indicator covering spectral regions of interest, described further in §[sec:losvels]) for both red and blue arms of the spectrograph in representative **M** and **G** fields. **D** fields have distributions comparable to **M** fields. Red clump stars have similar QM values in both red and blue spectra. In contrast, RGB stars (found only in **G** fields) have significantly higher QM values in the red spectra: this is the source of the second ‘bump’ in the QM distribution in panel d of Fig. [fig:sndists]. As RGB stars are much redder than clump stars, this increase in QM is not prominent in the blue spectra.\n| | |\n| --- | --- |\n| image | image |\n[fig:sndists]\nData are reduced using the pipeline, which undertakes the subsequent steps. First, all observations are debiased using bias frames taken at the start of each night. Next, spectral traces are located with a fibre-flat field, taken immediately prior to each set of science exposures. These traces are used to extract the data for each fibre. The extracted spectra are then divided by corresponding normalised trace from the fibre-flat to correct for pixel-to-pixel variations along the CCD for each fibre. Wavelength calibration is performed using traces from an arc frame also obtained immediately prior to each set of science exposures, via a least-squares polynomial fit. A secondary wavelength calibration tweak, based on night sky emission features and utilising a lower-order polynomial fit, is also performed after the initial calibration.\nBecause the target stars are faint, the subtraction of signal from the night sky – both continuum and line emission – is a crucial part of the reductions. To facilitate this, within each 2dF field, 25 dedicated fibres are used to measure the night-sky flux across the observed spectral range. Sky fibre locations are selected by configure from a list of 150 possible locations in each field, which were cross-checked against the photometric catalogues available at the time of observation (DES or *Gaia*) to ensure no sources are located within a radius of 10 arcsec from the fibre position. During the reduction process, we discard any sky fibres where there are indications of non-sky signal present.\nThe sky-subtraction process must take into account fibre-to-fibre throughput variations; in, the relative throughputs of each fibre are determined using night sky emission features. Several features are identified within each fibre, and the total flux within each feature measured. The median flux of the feature is taken across all dedicated sky fibres in the field; the ratio between this median, and the total flux of the feature measured in each target fibre, gives the relative throughput of the fibre. This procedure is repeated for several night sky emission features; the median throughput is used as the final value. In the blue spectra, as there is only a single strong night sky emission feature (at 5577Å), the throughput derived from this feature is used directly. The median sky spectrum, obtained from all sky fibres in each field, is then normalised by the relative fibre throughputs and subtracted from each fibre.\nFinally, all science exposures for a given field on a given night are combined. However, in order to account for variations in data quality (caused by, for example, variable seeing) or exposure time differences between observations, the relative weight each exposure will contribute to the final combined frame must first be determined. This is calculated using the frames flux weighting algorithm in ; which compares the total flux summed across each object spectrum to that expected (calculated by based on the supplied object magnitude and the total exposure time across all exposures). The median offset between the observed and expected flux for all objects in a given exposure is calculated, and subsequently inverted and scaled such that the ‘best’ exposure (i.e. with the smallest offset) is given a weight of unity, with shorter or poorer-quality exposures given commensurately reduced weights.\nOnce the relative weight of each exposure is determined, all exposures are combined into a single frame according to the following process in. The weighted median value of each pixel is taken across all exposures to create an initial estimate of the combined frame. Each individual exposure is compared to this median estimate; if the value of any pixel in an individual frame exceeds the corresponding median pixel value by 10*σ*, that pixel is flagged as contaminated by a cosmic ray in the individual exposure. The final combined frame is calculated by taking the weighted mean of each pixel in each exposure, excluding those flagged as contaminated by cosmic rays. In this way, pixel values where exposures are flagged as contaminated are effectively ‘filled-in’ by the equivalent pixels in exposures which are not flagged as contaminated by cosmic rays. When a given field was observed over multiple nights, frames for each night were reduced separately, with LOS velocity estimates combined later (see §[sec:losvels]).\nExamples of typical reduced spectra are presented in Fig. [fig:spectra]. The faint target magnitudes, combined with the relatively low S/N of the spectra, preclude the determination of detailed abundance estimates for individual stars, with the exception of [Fe/H] as based on the CaII triplet (described in §[sec:metallicity]). However, the quality of the spectra is sufficient to derive LOS velocity estimates as described in §[sec:losvels].\ncc \n \n[fig:spectra]\nLOS Velocity Determination\n--------------------------\nLOS velocity estimates for each star were obtained by cross-correlation of the spectra against velocity templates using the iraf fxcor routine. A synthetic template from the library, using stellar parameters appropriate for LMC red clump stars[5](#fn5) and rebinned to the same dispersion as the observed spectra, was used for cross-correlation of the blue spectra. For the red spectra, observations of the star HD 160043 (a standard star observed as part of the program described in, which used an identical setup to our observations) was used for cross-correlation. Only portions[6](#fn6)\nof the entire observed spectrum were used for cross-correlation; these were selected to avoid regions with substantial night-sky residuals. The rvcorrect routine was used to convert the obtained velocities to the heliocentric frame.\nA number of quality cuts are subsequently performed to identify and eliminate any targets with poor or untrustworthy velocity measurements. Plots combining a bespoke ‘quality measure’ (QM: defined as the ratio of median signal to standard deviation in a relatively flat region of the spectrum, after performing a single 3*σ* clip to remove any remaining night-sky residuals), velocity uncertainty and cross-correlation peak height (both as reported by fxcor) are inspected to determine field-by-field thresholds on each of these parameters, for both red and blue spectra; non-static thresholds are required to account for variation in data quality over the course of the survey. The QM we describe is effectively an empirical signal-to-noise measurement across a truncated region of each spectrum – and is therefore different to analytical S/N ratios calculated for the spectra on a per-pixel or per-Angstron basis. Fig. [fig:qualcuts] demonstrates an example of the cuts made for the spectra observed in field 20; a similar number of stars are retained for both red and blue spectra, but the cut values applied are substantially different. Stars where at least one spectrum passes the quality cuts are retained for further analysis. Representative QM distributions for stars passing quality cuts are presented in Fig. [fig:sndists]: stars with low QM are preferentially removed by all quality cuts applied.\n![Quality control plots for field 20, showing quality measure vs. line of sight velocity uncertainty (top row) and quality measure vs. FXCOR cross-correlation peak height (bottom row), for blue (left column) and red (right column) spectra. Note that FXCOR peak height values are discrete, as this is reported by the software to only two decimal places. Stars passing quality cuts are marked in black. Stars that fail quality cuts are coloured according to the cut that is failed; where multiple cuts are failed, stars are coloured by the criterion which is failed by the largest value. Dashed lines indicate the values of cuts applied.](fig5 \"fig:\") [fig:qualcuts]\nFor stars where both red and blue spectra pass the quality cuts, LOS velocities obtained from the two spectra are compared. We confirm for each field that no systematic offsets are observed, with the median velocity difference between the two associated velocities consistent with zero for all fields. Additionally, we calculate the standard deviation of the velocity difference distribution, and compare this to the median uncertainty in the difference (calculated as the quadrature sum of the uncertainties in both associated velocities). We find these are comparable for all fields, indicating the fxcor velocity uncertainties are reflective of the true velocity uncertainty.\nThe average of the two derived velocities, weighted by the inverse of the velocity uncertainty (taken directly as the velocity error reported by fxcor), is taken as the final LOS velocity, provided that the difference between the two individual velocities is less than 100km s− 1. Stars where the difference in the two associated velocities exceeds 100km s− 1 are excluded from further analysis, as such large differences indicate a failure in the cross-correlation process. Typically < 5 stars per field are excluded under this condition. Stars where the difference in the two associated velocities exceeds 50 km s− 1 (which are also very few in number) are manually inspected; in every case, these stars have LOS velocity estimates that preclude them from being Magellanic, and are subsequently de-weighted in §[sec:isolating] such that they do not contribute to field aggregate properties. These large velocity differences are typically associated with either:\n1. Unusually low signal, likely associated with small fibre misalignments which, for the relatively red stars targeted in this survey, predominantly affects the blue spectrum; or, more commonly,\n2. Poor skyline subtraction in the red spectrum, when prominent skylines overlap the 8498Å and 8542Å CaII lines and result in an incorrect velocity determination. Magellanic stars have LOS velocities that shift these CaII lines sufficiently far from the problematic skylines that this overlap occurs only for non-members.\nFor stars where only one associated spectrum passes the quality cuts, the LOS velocity derived from this spectrum is used directly as the final LOS velocity.\nAn additional step was applied to fields observed over multiple nights. When stars pass the aforementioned quality cuts on more than one night, the LOS velocities determined from each night are compared. Again, no systematic offsets are observed. The average of each derived velocity, weighted by the inverse of the velocity uncertainty, is taken as the final LOS velocity, provided that the difference between each of the individual velocities is less than 100km s− 1. Stars where the difference in the two associated velocities exceeds 100km s− 1 are excluded from further analysis. Again, any stars where the difference in the two associated velocities exceeds 50 km s− 1 are manually inspected; typically 10–20 stars per field meet this condition. In every case, these stars have LOS velocity estimates that preclude them from being Magellanic, and are subsequently de-weighted in §[sec:isolating] such that they do not contribute to field aggregate properties. We calculate the median velocity differences between stars observed on multiple nights, and find this is on the order of the median velocity uncertainty on each individual night, indicating the fxcor velocity uncertainties are reflective of the true velocity uncertainty. When stars are observed over multiple nights, but only satisfy quality requirements on a single night, the LOS velocity derived from that night is used as the final LOS velocity.\nThe above process results in typical LOS velocity uncertainties of 5–10km s− 1 per star in all observed fields. Velocity uncertainty distributions in each field do have tails to higher values, which result from stars where only single observations or spectra are analysed. The largest velocity uncertainty retained is 30km s− 1; although, by design, such stars contribute very little information to the field aggregate properties described in §[sec:aggregates].\n*Gaia* cross-matching\n---------------------\nIn addition to LOS velocities, proper motions are required to obtain full 3D kinematics. To obtain these, we cross-match all MagES stars with heliocentric velocities against the *Gaia* DR2 catalogue. A match radius of 1 arcsec is used; every star returns a single *Gaia* match under this condition.\nThe resulting sample is further filtered by requiring *Gaia* parameters <1.5 and (AEN)<1.0. These criteria act to remove any blended or extended sources, and unresolved binaries, which may have erroneous proper motions or LOS velocities. While the AEN cut is more lenient than that used to select Magellanic stars in e.g., MagES fields are located in diffuse regions where blending/crowding is not expected to be significant, and most non-stellar sources or unresolved binaries are expected to be removed through the quality cuts already applied to the LOS velocities. This is supported by the fact that very few stars are removed by applying these criteria. In addition, we test alternate quality criteria ; doing so leaves our results essentially unchanged.\nThe median proper motion uncertainty, per component, is ∼ 0.5 mas yr− 1, across all observed fields. As no cuts are applied to the sample based on proper motion uncertainties, some individual stars have significantly higher uncertainties (in the worst cases, up to 2 mas yr− 1). However, such stars are few in number, and contribute very little information to the aggregate field measurements described in §[sec:aggregates].\nThe outcome of this overall process is a sample of ∼ 7000 stars across 26 fields that have both line-of-sight velocities and proper motions. These include both true Magellanic stars, and some foreground contaminants (which are removed as described in §[sec:isolating]). Note that no explicit parallax cuts are applied to remove contaminants at this stage. Any foreground stars with large parallaxes that survive the reduction process (for **D** and **M** fields; target selection in **G** fields precludes any stars with parallax > 0.2 mas yr− 1) have sufficiently different LOS and proper motions compared to other Magellanic stars that they are removed in §[sec:isolating].\nIsolating Magellanic Stars\n==========================\nThough the target selection procedures outlined in Section [sec:design] are designed to isolate candidate Magellanic stars, there remains some level of contamination from the Milky Way. This particularly affects **D** and **M** fields, which were observed prior to the release of *Gaia* DR2. An example is shown in Fig. [fig:rawveldists], which shows LOS velocities and proper motions for stars in fields 11 (a typical **D** field, located in a low-surface-brightness substructure to the north of the LMC) and 12 (a typical **G** field, located in the northern LMC disk). In panel a, showing the LOS velocity distribution of stars in field 11, there is a strong kinematic peak in the LOS velocities at ∼ 280 km s− 1 associated with the LMC, but also a large population of contaminants at lower LOS velocities which are foreground Milky Way stars. In contrast, the LOS velocity distribution of stars in field 12 (shown in panel c) lacks Milky Way contaminants almost entirely. In proper motion space, there is a clear clustering of proper motions between 0<*μ**α*(mas yr− 1)<3 and -2<*μ**δ*(mas yr− 1)<2, corresponding to stars with LOS velocities ∼ 300 km s− 1 in both fields. However, in field 11 (panel b), this is embedded in a broader proper motion distribution associated with the Milky Way. This component is missing in panel d for field 12, as **G** fields have proper motion cuts applied during target selection.\nThese two fields sit on opposite ends of a contamination spectrum: field 11 was observed with less efficient target selection criteria, and is also located in a low-surface-brightness substructure where the density of true Magellanic stars is low. In contrast, field 12 is located in the LMC disk, where the density of Magellanic stars is high, and was observed using the strictest target selection criteria. Most MagES fields have levels of MW contamination between these two extremes. Consequently, in order to reliably determine kinematics for the Magellanic system, we need to remove the Milky Way contamination to generate a sample of stars that are likely genuinely associated with the Clouds. We utilise a probabilistic method to do this, rather than applying hard cuts – although the LOS kinematic peak associated with the LMC is well-separated from the Milky Way contamination in Fig. [fig:rawveldists], this is not the case for all observed fields. A probabilistic method is thus better suited for those fields where Magellanic and contaminant populations more closely overlap, while still allowing a homogeneous algorithm to be applied across the entire sample. We now discuss the processes used to select stars that have a high probability of Magellanic Cloud membership, and how this information is used to determine aggregate kinematics for each observed field.\n \n [fig:rawveldists]\nContamination model\n-------------------\nIn order to differentiate between Magellanic stars and contaminants, an empirical representation of the observed Milky Way contaminant profile in each field is required. As the observed contaminant profile varies across the large footprint of MagES, it is generated on a field-by-field basis using the Besançon Model of the Galaxy (described in, and accessed as version 1603 through the web service[7](#fn7)). The process used to generate the empirical model for each field is as follows.\nThe Besançon model is used to generate mock stars located within a 1∘ radius surrounding the field centre (the same field-of-view size as each observed 2dF field). The appropriate selection cuts for the field (as described in §[sec:design], including photometric cuts for **D** and **M** fields, and both photometric and kinematic cuts applied to **G** fields), are subsequently applied in order to obtain lists of mock Milky Way stars, within each priority band, that could conceivably have been observed in the field. This process is repeated for 10 unique iterations of the Besançon model, and the results aggregated, to ensure the kinematic parameter space is sufficiently sampled.\nWe rescale the number of stars in each priority list by repeating each priority list *n* times, where *n* is the number of stars actually observed in that priority bin in the field. This effectively weights the contribution of each priority bin to the combined kinematic distribution of all priority bins, by the fraction of stars actually observed in that bin in the field. This process is equivalent to repeated sampling of each bin *n* times, and accounts for the preferential selection implemented by configure used to generate the observed target lists.\nThe final list of model stars, of all priorities, is subsequently split into separate lists based on the population (disk[8](#fn8) or halo) each star belongs to, as defined by the Besançon model itself. These have significantly different kinematic distributions, and we therefore found it easiest to treat these separately when generating empirical models. Once each population is defined, we generate a simple representation of the distribution of each population in velocity space using a Gaussian mixture model. We later use these analytical descriptions to inform the probability of individual stars being associated with either a contaminant population, or the Magellanic Clouds.\nThe log-likelihood of the mixture model for each population is described by Eq. [eq:modlikelihood], where *P*(**x***j*∣MWpop) is the likelihood of each individual star in the population belonging to any Gaussian within the mixture model. Each of the *J* individual stars in the population has kinematics **x***j* (comprising a LOS velocity *v**j*, and proper motions *μ**α*, *j* and *μ**δ*, *j*). Note that *μ**α* always refers to proper motion in the *α*cos(*δ*) direction, such that it is perpendicular to *μ**δ*.\n \n$$\\label{eq:modlikelihood}\n \\begin{split}\n \\log\\left(\\mathcal{L}\\right) &= \\sum^J\\_{j=1}\\log\\left(P({\\mathbf{x}\\_j}|\\text{MW}\\_{\\text{pop}},\\phi)\\right) \\\\ &=\\sum^J\\_{j=1}\\log\\left[\\sum^\\kappa\\_{k=1}\\left(\\eta\\_k\\mathcal{N}(\\mathbf{x}\\_{j}|\\mathbf{m}\\_{k}, \\mathbf{C}\\_{k}) \\right) \\right]\n \\end{split}$$ \nHere, N(**x***j*∣**m***k*, **C***k*) is the probability density function of each Gaussian comprising the mixture model: each of which has means **m***k* and a covariance matrix **C***k*. The probability density function of each component is given in Eq. [eq:modprobdist].\n \n$$\\begin{gathered}\n\\label{eq:modprobdist}\n \\mathcal{N}(\\mathbf{x}\\_j|\\mathbf{m}\\_k, \\mathbf{C}\\_k)=(2\\pi)^{-d/2}\\det(\\mathbf{C}\\_k)^{-1} \\\\ \\times \\exp\\left[-\\frac{1}{2} (\\mathbf{x}\\_j-\\mathbf{m}\\_k)^{\\intercal} \\mathbf{C}^{-1} (\\mathbf{x}\\_j-\\mathbf{m}\\_k)\\right] \\end{gathered}$$ \nHere, *d* is the dimensionality of the Gaussians comprising the mixture model. Whilst it is possible to fit LOS velocities and proper motions simultaneously (implying a dimensionality of 3), we choose to fit these separately, as this allows us to fit a varying number of Gaussians to each kinematic component in order to best describe the overall population. For example, if disk stars have an asymmetric LOS velocity distribution, we parameterise this using two Gaussians; however in proper motion space, these stars may be sufficiently described by a single Gaussian. The total number *κ* of Gaussians fit to each kinematic component and population is given in Table [tab:MWmods]. We note that **G** fields require fewer Gaussian components compared to **D** and **M** fields as fewer Milky Way stars survive the stricter target selection criteria applied to **G** fields. For populations where multiple Gaussians are fit, the parameter *η**k* is used to describe the relative fraction of stars in each Gaussian: ∑*k**κ**η**k* = 1.\n[tab:MWmods]\nlcc|cc && \n& Disk &Halo &Disk & Halo \nLOS velocities & 2 & 1 & 1 & 1 \nProper motions & 1 & 2 & 1 & 1 \nThe mean and covariance matrices for each kinematic component are given in Eqs. [eq:modms]-[eq:modcs]. LOS velocity Gaussians have systematic velocities of *v**k* and dispersions of *σ**k*, while proper motion Gaussians have systematic velocities of *μ**k* and *μ**δ*, *k*, dispersions of *σ**α*, *k* and *σ**δ*, *k*, and covariance parameters of *ρ**k*. As we fit the LOS velocity and proper motion distributions separately, an underlying assumption of our method is that there is no correlation between the LOS velocity, and either of the proper motion components.\n \n$$\\begin{aligned}\n\\label{eq:modms}\n \\mathbf{m}\\_{k, \\text{PM}} &= \\begin{pmatrix}\n \\mu\\_{\\alpha,k} \\\\ \\mu\\_{\\delta,k}\\\\ \\end{pmatrix} &\n \\mathbf{m}\\_{k, \\text{LOS}} &= \\begin{pmatrix} v\\_{k} \\end{pmatrix}\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\label{eq:modcs}\n \\mathbf{C}\\_{k, \\text{PM}} &= \\begin{pmatrix}\n \\sigma\\_{\\alpha,k}^2 & \\rho\\_k\\sigma\\_{\\alpha,k}\\sigma\\_{\\delta,k}\\\\\n \\rho\\_k\\sigma\\_{\\alpha,k}\\sigma\\_{\\delta,k} & \\sigma\\_{\\delta,k}^2 \\\\ \\end{pmatrix} &\n \\mathbf{C}\\_{k, \\text{LOS}} &= \\begin{pmatrix} \\sigma\\_{v,k}^2 \\end{pmatrix}\\end{aligned}$$ \nTo determine the best-fitting parameters for each of the *κ* Gaussians within the mixture model for each population, we sample the posterior distribution of the model parameters – which we abbreviate as *ϕ* = (**m***k*, **C***k*, *η**k*) – using the Markov Chain Monte Carlo ensemble sampler emcee in order to maximise the log-likelihood given in Eq. [eq:modlikelihood]. In this process, 50 walkers each take 2000 steps, with the burn-in phase of the first 1000 steps discarded when computing the final parameter values and their associated uncertainties. Uniform priors are applied to all parameters. Note that in subsequent analysis, we always use only the best-fitting parameter estimates ($\\hat\\phi$), as the effect of drawing from the confidence intervals calculated by emcee is negligible, as demonstrated in Appendix [sec:appendix].\nOnce the best-fitting parameters for each population are known, the likelihood functions for both disk and halo populations are summed as per Eq. [eq:modtot] to give the overall likelihood function for a given model star to belong to any of *M* Milky Way components within a given field. By definition, *M* = *κ*disk + *κ*halo. Here, *γ* refers to the relative fractions of disk and halo stars per field in the Besançon model: *γ*disk + *γ*halo = 1. Unlike each *η**k*, which are fit using emcee for each Gaussian within each population, *γ*disk and *γ*halo are calculated explicitly. The overall relative weighting of each Milky Way component is *η**m*. **m***m* and **C***m* are identical in form to Eqs. [eq:modms]-[eq:modcs].\n \n$$\\label{eq:modtot}\n \\begin{split}\n P(\\mathbf{x}\\_j|\\text{MW},\\hat\\phi) &= \\gamma\\_{\\text{disk}} P(\\mathbf{x}\\_j|\\text{MW}\\_{\\text{disk}},\\hat\\phi) + \\gamma\\_{\\text{halo}}P(\\mathbf{x}\\_j|\\text{MW}\\_{\\text{halo}},\\hat\\phi) \\\\\n &= \\gamma\\_{\\text{disk}}\\sum^{\\kappa\\_{\\text{disk}}}\\_{k\\_{\\text{disk}}=1} \\left[ \\eta\\_{k, \\text{disk}}\\mathcal{N}\\left(\\mathbf{x}\\_j|\\mathbf{m}\\_{k, \\text{disk}},\\mathbf{C}\\_{k, \\text{disk}}\\right)\\right]\\\\ &+ \\gamma\\_{\\text{halo}}\\sum^{\\kappa\\_{\\text{halo}}}\\_{k\\_{\\text{halo}}=1} \\left[ \\eta\\_{k, \\text{halo}}\\mathcal{N}\\left(\\mathbf{x}\\_j|\\mathbf{m}\\_{k, \\text{halo}},\\mathbf{C}\\_{k, \\text{halo}}\\right)\\right] \\\\\n &=\\sum^M\\_{m=1}\\left[\\eta\\_m\\mathcal{N}\\left(\\mathbf{x}\\_{j}|\\mathbf{m}\\_{m}, \\mathbf{C}\\_{m}\\right)\\right]\n \\end{split}$$ \nGenerating membership probabilities\n-----------------------------------\nThe outcome of the above process is a list of fitted parameter values, with uncertainties, which specify an approximate analytic form for the predicted Milky Way contamination within a given field. This is used in conjunction with the observed data, in a procedure similar to that outlined in, to assign probabilistic Magellanic membership to each observed star in a given field.\nAs evident in Fig. [fig:rawveldists], stars associated with the Clouds are concentrated in relatively cold (narrow) kinematic peaks, that are distinguishable from the profiles associated with Milky Way contaminants. As such, we generate probability density functions that describe the likelihood a given observed star belongs to either the Clouds, or one of the Milky Way contaminant profiles: under our parameterisation, if a star does not belong to the Milky Way, it must belong to a separate kinematic peak, which we associate with the Magellanic Clouds.\nUnlike stars generated using the Besançon models, observed stars have associated uncertainties in their kinematics, with LOS velocities *v**i* ± *u**v*, *i* and proper motions *μ**α*, *i* ± *u**α*, *i* and *μ**δ*, *i* ± *u**δ*, *i*. In addition, the uncertainties in the two proper motion directions are correlated: *ρ**i* (as obtained directly from the *Gaia* source catalogue using the column ) describes this correlation. These uncertainties must be included in the calculation of probability density functions, in order to separate the intrinsic dispersion of the fitted Gaussians from observational broadening due to measurement error. The kinematics of each observed star **x***i* and its uncertainties **C***i* are described by Eqs. [eq:obsxs]-[eq:obscs]. As we calculate the probability density functions for the LOS velocities and proper motions of the stars separately, we inherently assume the LOS velocity uncertainties of the stars are uncorrelated with the uncertainties in either proper motion component.\n \n$$\\begin{aligned}\n\\label{eq:obsxs}\n \\mathbf{x}\\_{i,\\text{PM}} &= \\begin{pmatrix}\n \\mu\\_{\\alpha,i} \\\\ \\mu\\_{\\delta,i}\\\\ \\end{pmatrix} &\n \\mathbf{x}\\_{i,\\text{LOS}} &= \\begin{pmatrix} v\\_{i} \\end{pmatrix}\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\label{eq:obscs}\n \\mathbf{C}\\_{i,\\text{PM}} &= \\begin{pmatrix}\n \\sigma\\_{\\alpha,k}^2 & \\rho\\_i\\sigma\\_{\\alpha,i}\\sigma\\_{\\delta,i}\\\\\n \\rho\\_i\\sigma\\_{\\alpha,i}\\sigma\\_{\\delta,i} & \\sigma\\_{\\delta,i}^2 \\\\ \\end{pmatrix} &\n \\mathbf{C}\\_{i,\\text{LOS}} &= \\begin{pmatrix} \\sigma\\_{v,i}^2 \\end{pmatrix}\\end{aligned}$$ \nThe likelihood for a given observed star to be a Milky Way contaminant is defined in Eq. [eq:mwprob]. Here, the total likelihood is the sum of the probabilities of the star being associated with any of the *M* Milky Way components used to fit the Besançon models. **m***m* and **C***m* are as described in Eq. [eq:modtot], and use the best-fitting parameters derived for each component fit to the Besançon model.\n \n$$\\label{eq:mwprob}\n P({\\mathbf{x}\\_i|\\text{MW}},\\hat\\phi)=\\sum^M\\_{m=1}\\left[\\eta\\_m\\mathcal{N}\\left(\\mathbf{x}\\_{i}|\\mathbf{m}\\_{m}, [\\mathbf{C}\\_{m}+\\mathbf{C}\\_{i}]\\right)\\right]$$ \nIf a star does not belong to the Milky Way, then under our parameterisation it must belong to a separate kinematic peak, which we associate with the Magellanic Clouds, and assume to be Gaussian in nature. The likelihood for a given observed star to be associated with such a peak is given by Eq. [eq:mcprob]. Note that in this parameterisation, only a single peak associated with the Clouds is fitted; however, particularly for fields located between the two Clouds, it is possible multiple separate populations associated with the Clouds are present. In such cases, the procedure can be generalised to allow the fitting of multiple Gaussians associated with Magellanic peaks, as necessary.\n \n*P*(**x***i*∣MC, *φ*) = N(**x***i*∣**m**MC, [**C**MC + **C***i*]) \n**m**MC and **C**MC describe the properties of the means and covariances of the Magellanic peak respectively, and are given in Eqs. [eq:mpk]-[eq:cpk]. Here, *v*MC is the systemic LOS velocity of the peak; *μ**α*, MC and *μ**δ*, MC are the systemic proper motions of the peak; *σ**v*, MC is the velocity dispersion of the peak; *σ**α*, MC and *σ**δ*, MC are the proper motion dispersions of the peak; and *ρ*MC describes the covariance of the proper motion dispersions. We assume there is no correlation between the LOS velocity dispersion and the proper motion dispersions of the peak.\n \n$$\\begin{aligned}\n \\label{eq:mpk}\n \\mathbf{m}\\_{\\text{MC, PM}} &= \\begin{pmatrix}\n \\mu\\_{\\alpha,\\text{MC}} \\\\ \\mu\\_{\\delta,\\text{c}}\\\\ \\end{pmatrix} &\n \\mathbf{m}\\_{\\text{MC, LOS}} &= \\begin{pmatrix} v\\_{\\text{MC}} \\end{pmatrix}\\end{aligned}$$ \n \n$$\\label{eq:cpk}\n \\begin{split}\n &\\mathbf{C}\\_{\\text{MC, PM}} = \\begin{pmatrix}\n \\sigma\\_{\\alpha,\\text{c}}^2 & \\rho\\_{\\text{MC}}\\sigma\\_{\\alpha,\\text{MC}}\\sigma\\_{\\delta,\\text{MC}}\\\\\n \\rho\\_{\\text{MC}}\\sigma\\_{\\alpha,\\text{MC}}\\sigma\\_{\\delta,\\text{MC}} & \\sigma\\_{\\delta,\\text{MC}}^2 \\\\ \\end{pmatrix} \\\\\n &\\mathbf{C}\\_{\\text{MC, LOS}} = \\begin{pmatrix} \\sigma\\_{v,\\text{MC}}^2 \\end{pmatrix}\n \\end{split}$$ \nIn order to identify the characteristics of the Magellanic kinematic peak, we use emcee to sample the posterior distribution of each of the peak parameters – which we abbreviate as *φ* = (*γ*MC, **m**MC, **C**MC) – in order to maximise the log-likelihood function given in Eq. [eq:log1pass]. Here, *N* is the total number of observed stars, *γ*MW describes the fraction of observed stars in a given field that are associated with the Milky Way (as opposed to being Magellanic in origin), and *γ*MC describes the fraction of observed stars in a given field that are associated with the Magellanic Clouds (as opposed to being associated with any component of the Milky Way). By definition, *γ*MW + *γ*MC = 1. Note that the value of the kinematic peak parameters derived in this process are not the final kinematic properties of the Clouds at this location: they simply indicate a region in velocity space, roughly consistent with the expected motions of the Clouds, where an excess of stars above the Milky Way contamination baseline exists.\n \n$$\\label{eq:log1pass}\n \\log\\left(\\mathcal{L}\\right) = \\sum^N\\_{i=1}\\log\\left[\\gamma\\_{\\text{MC}} P(\\mathbf{x}\\_i|\\text{MC},\\varphi) + \\gamma\\_{\\text{MW}} P({\\mathbf{x}\\_i|\\text{MW}},\\hat\\phi)\\right]$$ \nOnce the initial properties of the Magellanic kinematic peak ($\\hat\\varphi$) are known, these are used in Eq. [eq:pfrac] to calculate the individual probability that a given observed star belongs to the peak, and is therefore associated with the Clouds. Separate independent probabilities are generated based on (1) the LOS velocity distribution $P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)\\_{\\text{LOS}}$ and (2) the 2D proper motion distribution $P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)\\_{\\text{PM}}$. These are multiplicatively combined as per Eq. [eq:ptot] to determine an overall probability $P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)$ that each observed star is associated with the Clouds.\n \n$$\\label{eq:pfrac}\n P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)\\_{\\text{LOS/PM}} = \\frac{\\gamma\\_{\\text{MC}} P(\\mathbf{x}\\_i|\\text{MC},\\hat\\varphi)}{\\gamma\\_{\\text{MC}} P(\\mathbf{x}\\_i|\\text{MC},\\hat\\varphi)+\\gamma\\_{\\text{MW}} P({\\mathbf{x}\\_i|\\text{MW}},\\hat\\phi)} \\\\$$ \n \n$$\\label{eq:ptot}\n P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi) = P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)\\_{\\text{LOS}} \\times P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)\\_{\\text{PM}}$$ \nDetermining field aggregate properties\n--------------------------------------\nOnce each star in a field has been assigned an aggregate association probability $P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)$, these are used to calculate the aggregate 3D motion of the Clouds, and the dispersion in each of the three velocity components, across the field. A single Gaussian with mean **m**MC and covariance **C**MC, taking identical form to those given in Eqs. [eq:mpk] and [eq:cpk], is used to describe the field kinematics. emcee is used to sample the posterior distribution of each of these parameters to maximise the log-likelihood function given in Eq. [eq:finlikelihood]; each term of which is weighted by $P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)$. In this way, stars that are very unlikely to be associated with the Clouds contribute minimally to the calculated field aggregate properties. The resulting parameters describe the field aggregate properties of the Clouds at each location. We report the 68 per cent confidence interval as the 1*σ* uncertainty in each parameter.\n \n$$\\begin{gathered}\n\\label{eq:finlikelihood}\n \\log\\left(\\mathcal{L}\\right) = \\sum^N\\_{i=1}\\log\\Biggl( P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)\\gamma\\_{\\text{MC}}\\mathcal{N}(\\mathbf{x}\\_{i}|\\mathbf{m}\\_{\\text{MC}}, \\left[\\mathbf{C}\\_{\\text{MC}}+\\mathbf{C}\\_i\\right] )\\\\+\\left[1-P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)\\right]\\gamma\\_{\\text{MW}} P({\\mathbf{x}\\_i|\\text{MW}},\\hat\\phi) \\Biggr)\\end{gathered}$$ \nMetallicity determination\n-------------------------\nIn addition to field kinematics, [Fe/H] estimates are also determined for stars with high probability of being associated with the Clouds (defined here as having $P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)>50\\%$). The procedure used broadly follows that outlined in, although with some modifications. In Da Costa’s method, the equivalent widths of the 8542Å and 8662Å CaII lines, present in the red-arm spectra of each star, are first measured by fitting a combined Gaussian plus Lorentzian function, and summed (see for further details of the measurement technique). Next, the reduced equivalent width, *W*ʹ, is calculated as per Eq. [eq:ewidth].\n[eq:ewidth] W’ = EW-(-0.6600.016)(V0-V,0)\nHere, − 0.660 ± 0.016 is the slope of the *E**W* − *W*ʹ relation derived in. *V*0 is the de-reddened *V*-band magnitude of the star; this is calculated from the *Gaia* photometry of the star using the transformations given in. *V*HB, 0 is the horizontal branch magnitude, which we take as equal to the median red clump magnitude in the surrounding field. This median is calculated by taking the median *Gaia* *G*0 magnitude for stars in a selection box surrounding the Magellanic red clump on the *Gaia* (*G*0, (*G*BP − *G*RP)0) CMD. The boundaries of the selection box are drawn on a field-by-field basis, but in all cases covering only a narrow (*G*BP − *G*RP)0 range to minimise contamination from Milky Way stars, many of which are located near to the Magellanic red clump (as seen in Fig. [fig:cmds]). The median *G*0 magnitude is then converted to a *V*-band magnitude using the relations given in. Finally, the reduced equivalent width is transformed into an [Fe/H] estimate using Eq. 2 in, reproduced here as Eq. [eq:fehcalc]. This equation is valid in the range $-2.4\\lesssim\\text{[Fe/H]}\\lesssim0.1$ dex.\n \n[Fe/H] = (0.528 ± 0.017)*W*ʹ − (3.420 ± 0.077) \nHowever, the 8662Å line used in the above calculation is within a region of the spectrum relatively heavily contaminated by night-sky emission, which is often poorly-subtracted during the data reduction process. This, in combination with the relatively faint magnitudes of the observed red clump stars, can result in inaccurate measurements of the line’s equivalent width. The 8542Å line is not as strongly affected, but is still difficult to accurately measure in lower-S/N spectra. In order to mitigate this effect, and prevent biasing of the derived metallicities, we implement two modifications to Da Costa’s method.\nThe first of these is that spectra for red clump stars, after being shifted into the rest frame using their observed LOS velocities, are stacked in groups of at least 10. This increases the contrast of the two CaII absorption features relative to the residual night-sky emission (which is stochastically either over- or under-subtracted, and is therefore suppressed when multiple spectra are stacked). This allows for more accurate determination of the equivalent widths of the lines. Note that as red clump stars only occupy a small magnitude range (and relatively small ranges in other stellar parameters), stacking spectra is not expected to bias the resulting equivalent widths. It will, however, result in metallicity estimates that tend toward the mean metallicity of the field. As such, we only use stacked spectra when analysing aggregate metallicity properties across an entire field, and do not include results from stacked spectra when analysing the metallicity distribution within a given field.\nUnfortunately, even when considering stacked spectra, it remains impossible to determine accurate equivalent widths for the 8662Å line for ∼ 50 per cent of spectra. In order to derive metallicities for these spectra, we implement a similar process as described above, but which does not utilise the equivalent width of the 8662Å line. Instead, the slope of the *E**W* − *W*ʹ relation, and the coefficients in Eq. 2 of, are recalculated using only the equivalent width of the 8542Å line. The resulting relations are provided in Eqs. [eq:nuw] and [eq:nufeh].\n \n*W*ʹ = *E**W* − ( − 0.366 ± 0.036) × (*V*0 − *V*HB, 0) \n \n[Fe/H] = (0.884 ± 0.001)*W*ʹ − ( − 3.336 ± 0.004) \nThe propagated uncertainty in each individual metallicity value is dominated by systematic and photometric uncertainties in the *W*ʹ − *E**W* relation. Whilst the uncertainty in the equivalent width of each line decreases as the S/N of the spectrum increases, brighter stars – which have higher S/N spectra – have a correspondingly larger value of (*V*0 − *V*HB, 0), which results in a larger uncertainty in this term of the *W*ʹ − *E**W* relation than that contributed by the equivalent width itself. As a result, the overall metallicity uncertainty does not correlate strongly with either spectrum S/N, or [Fe/H] value.\nFor stars where both CaII lines can be measured accurately (which are typically the brightest stars in any given field) we compare the [Fe/H] values derived using the single and double-line methods. We find the [Fe/H] values derived have an ∼ 0.2dex scatter around the 1:1 relation, with no systematic offset between the derived values. This scatter is significantly larger than the propagated uncertainty in each individual metallicity value. We therefore take 0.2dex as the total uncertainty on the metallicity value of each individual star, regardless of which method is used.\nResults\n=======\nThe result of MagES data processing is a set of six kinematic parameters for each 2dF field, describing the apparent systemic velocity and dispersion of the Clouds in 3D at that location, and a set of metallicity estimates for each location. Detailed analysis of these data, covering various substructures in the Magellanic periphery, will be presented in forthcoming papers. Here, we focus on initial results from two fields (12 and 18) in the northern outer disk of the LMC: both to verify our approach, and to provide a basis for future comparison with more distant fields. Table [tab:obsresults] provides the observed kinematic properties of these two fields (LOS velocity and dispersion, and the two components of proper motion and their dispersions), their median metallicities, and the standard deviation of their [Fe/H] distributions. The reported uncertainty on the median metallicity is the standard error of the mean, equal to the standard deviation of the distribution divided by the square root of the number of stars with metallicity determinations.\n[tab:obsresults]\nl>m2cmcccccccc Field & Distance (∘) from LMC COM & & & & & & & Median [Fe/H] & *σ*[Fe/H] \n18 & 10.7 & 324.8 ± 1.1& 19.8 ± 0.8 & 1.45 ± 0.02 & 0.13 ± 0.02 & 0.96 ± 0.01 & 0.11 ± 0.02 & − 1.0 ± 0.1 & 0.3 \n12 & 10.3 & 287.5 ± 1.5& 24.3 ± 1.0 & 1.77 ± 0.02 & 0.13 ± 0.02 & 0.19 ± 0.02 & 0.21 ± 0.02 & − 1.1 ± 0.1 & 0.5 \nWhilst Table [tab:obsresults] reports 3D kinematics in observable units, it is more informative to consider these in the reference frame of the LMC disk itself. As such, the framework presented in and is used to describe the LMC disk velocity field, and transform the observed components into velocities in a cylindrical coordinate system. This coordinate system is aligned with the LMC disk, and has its origin at the LMC centre of mass (COM). This transformation includes the subtraction of the systemic motion of the LMC COM, as projected at each field location.\nHowever, various studies of the Clouds have reported COM positions which differ by up to a degree on the sky, depending on the chosen tracer. Given that our sample is primarily red clump stars, for consistency we adopt the COM position reported by, for their ‘PMs+Old *v*LOS Sample’: i.e. 79.88∘ ± 0.83∘, − 69.59∘ ± 0.25∘. This is a kinematic centre, derived from a simultaneous fit of HST field-aggregate proper motions, combined with LOS velocities for an ‘old’[9](#fn9) stellar sample. This is as similar as possible to the data used in the present work. We further adopt the associated bulk motion reported by applicable for this choice of centre: i.e. *μ**δ*, 0 = 0.287 ± 0.054 mas yr− 1, *μ**α*, 0 = 1.895 ± 0.024 mas yr− 1, and *v*LOS, 0 = 261.1 ± 2.2km s− 1. The bulk proper motions reported are, within uncertainty, consistent with those reported by.\nThe geometry of the LMC disk must also be assumed during this coordinate transform. When considering estimates derived using relatively old tracers (similar to the population observed with MagES) the inclination of the LMC disk has traditionally been reported as ∼ 35∘ ; though some more recent studies suggest ∼ 25∘. However, all such measurements have been derived using stars at much smaller radial distances from the LMC COM than even the innermost of our fields. Moreover, warps and a twisting of the position angle of the line of nodes (LON; represented as Θ)[10](#fn10) have been found in the LMC disk. Given this, the behaviour of the LMC disk at radii commensurate with our fields is largely unconstrained; so the most appropriate choice of geometry for these fields is not obvious.\nIn this work, we therefore decided to test two different LMC disk geometries, spanning the range of recent measurements reported in the literature. The first is taken from the same field-aggregate proper motion and old stellar LOS measurements as used for the LMC COM properties (with *i* = 34.0∘, Θ = 139.1∘). The second is taken as the best-fitting model from (*i* = 25.86∘, Θ = 149.23∘), which is derived solely from photometric data[11](#fn11). Future work (e.g. Mackey et al. in prep) should provide direct disk geometry measurements at the locations of several MagES fields, which can be used to validate the assumptions made here. For simplicity, in what follows we assume no precession or nutation of the LMC disk, consistent with the measurements of.\nFor each of the assumed geometries, the observed kinematic parameters for our two fields are transformed into physical velocities and dispersion in the LMC disk frame. We calculate *V**ϕ*, the azimuthal streaming or rotation velocity; *V**R*, the radial velocity in the disk plane; and *V**Z*, the vertical velocity perpendicular to the disk plane, as well as dispersions in each of these components. Fig. [fig:diskkin] displays these velocities for the two northern LMC disk fields. Error bars on each point are obtained by using Monte Carlo error propagation to simultaneously propagate uncertainty in the observed velocity components, the LMC disk geometry, and the bulk motion of the LMC COM. Uncertainty in the location of the LMC COM is not propagated as this is found to be negligible compared to the other uncertainty sources.\n![LMC disk velocities and dispersions in fields 18 and 12, calculated using both and disk geometries. Top panels show the azimuthal velocity component (panel a) and its dispersion (panel d); positive values indicate clockwise rotation. Middle panels show the radial velocity component (panel b) and its dispersion (panel e); positive values indicate movement outward from the LMC COM in the LMC disk plane. Bottom panels show the vertical velocity component (panel c) and its dispersion (panel f); positive values indicate movement perpendicular to the disk plane, in a direction primarily towards the observer. For each velocity component, the values within a given field are the same within uncertainty, regardless of the assumed geometry.](fig7 \"fig:\") [fig:diskkin]\nAs is apparent from Fig. [fig:diskkin], the calculated velocities and dispersions are, within uncertainty, the same for both tested disk geometries. This is partly due to the relatively large uncertainties in the disk geometry parameters themselves. For example, the model has a large uncertainty in the inclination ( ± 7∘), while the et al. model has a large uncertainty in the position angle of the line of nodes ( ± 8.35∘). Nevertheless, the lack of substantial sensitivity to the parameters of the tested disk geometries indicates that the conclusions drawn in the following analysis are robust to differences between the actual LMC disk geometry at these locations, and the values assumed in this paper. Consequently, in subsequent discussion, we adopt disk velocities and dispersions assuming the geometry of, for consistency with our adopted COM position and bulk velocity. These disk measurements are reported in Table [tab:diskresults], which presents the azimuthal, radial, and vertical velocity components, and their dispersions.\n[tab:diskresults]\n| Field | *V**ϕ* (km s− 1) | *σ**ϕ* (km s− 1) | *V**R* (km s− 1) | *σ**R* (km s− 1) | *V**Z* (km s− 1) | *σ**Z* (km s− 1) |\n| --- | --- | --- | --- | --- | --- | --- |\n| Field 18 | 70.9 ± 14.0 | 25.8 ± 3.0 | − 1.3 ± 11.4 | 24.4 ± 2.7 | 5.4 ± 9.6 | 20.6 ± 1.0 |\n| Field 12 | 58.6 ± 10.8 | 29.5 ± 3.7 | 15.5 ± 14.5 | 44.8 ± 5.1 | 2.8 ± 10.3 | 26.8 ± 1.7 |\nThe [Fe/H] distributions for the two fields are presented in Fig. [fig:metdist]. The median metallicity in both fields ([Fe/H] = − 1.0 ± 0.1 for field 18, and [Fe/H] = − 1.1 ± 0.1 for field 12) is consistent with literature spectroscopic metallicity determinations for stars at similar distances from the LMC COM. Both distributions have tails to lower metallicities, with this tail being most pronounced in field 12; this inflates the standard deviation of the distribution. We look for evidence that any stars we observe may form part of a halo-like component by comparing the kinematics of stars in the metal-poor tails of the [Fe/H] distributions (defined here as having [Fe/H]<–1.5) to those stars with higher [Fe/H] values. While there are only few ‘metal-poor’ stars, simple Kolmogorov–Smirnov (K–S) tests indicate no significant differences in the kinematics of lower- and higher-metallicity stars in either field.\n \n [fig:metdist]\nLMC Disk motions\n----------------\nIn this section, we discuss the derived velocities and dispersions of two fields observed in the northern LMC disk. We remind readers that these values are derived assuming the geometry, and associated bulk velocity, of ; uncertainties in these values, and in the distance to the LMC, are propagated through and contribute to the uncertainty in the values reported here.\nFig. [fig:rotcurve] shows the azimuthal velocity (*V**ϕ*) for the two MagES disk fields, relative to similar measurements obtained by. It should be noted that the proper motions are based on Hubble Space Telescope (HST) astrometry, and as such, represent the mean proper motion of all stellar populations in each given field. It is known from LOS velocity measurements that younger stellar populations in the Magellanic Clouds rotate more quickly than older populations. As such, the rotation velocity derived from HST proper motions (which combine both populations) is higher than that derived using just LOS velocities for older stars. Also plotted are rotation velocities derived from,, and. These are derived using the proper motions of large samples of individual Magellanic RGB and carbon stars.\n![Azimuthal velocities in the LMC disk as a function of distance from the LMC COM. Orange points indicate measurements for the two MagES fields in the northern LMC disk; error bars include propagated uncertainties from all parameters. Dark blue points show values derived from HST proper motion measurements of mixed young and old populations, while light blue/aqua points show values derived from line-of-sight observations for their old stellar population. Error bars on these points only include uncertainty in the observed motions, and not disk geometry or COM location, and are thus smaller than those for the MagES fields. The solid lines reflect the best‐fitting rotation models derived from (labelled Vasiliev18), (labelled Wan20), (labelled Helmi18) and (labelled vdM14). Surrounding shaded regions indicate 1\\sigma uncertainty propagated from all parameters; these are thus comparable to the errorbars of the MagES fields. Dashed continuations of the solid lines indicate where these models have been extrapolated outwards in order to facilitate comparison with the two MagES points: the observations used to derive the velocities shown are generally located much closer to the LMC COM than the MagES fields.](fig9 \"fig:\") [fig:rotcurve]\nThe azimuthal velocities for the two MagES fields (70.9 ± 14.0km s− 1, and 58.6 ± 10.8km s− 1 for fields 18 and 12 respectively) are both consistent with one another within uncertainty, and consistent with all other sets of measurements in Fig. [fig:rotcurve]. This is unsurprising; the old stellar populations used to derive each literature rotation curve are similar to the population observed by MagES; and therefore should have similar kinematics, as is observed.\nIt is worth noting these northern-LMC MagES fields provide an estimate of the LMC rotation at radii more distant from the LMC COM (>10∘ on-sky) than all previous estimates (which typically have data confined to <10∘ of the LMC COM). As these measurements are consistent with measurements derived at more central locations, this indicates the LMC rotation curve remains flat even at very large distances from the LMC COM, where external perturbations (e.g. due to the SMC) might be expected to disturb the disk motion. For example, at comparable radii on the southern side of the LMC disk, clear substructures are seen.\nThe azimuthal velocity dispersion (*σ**ϕ*) within the two MagES fields (25.8 ± 3.0km s− 1 and 29.5 ± 3.7km s− 1 for fields 18 and 12 respectively) are moderately lower than that measured by (37.1 ± 0.7km s− 1). This difference can at least partially be attributed to the fact that assume a constant velocity dispersion at all radii. As their data are relatively centrally concentrated (with data at radii predominantly within 6∘), the recovered dispersion is predominantly reflective of the large dispersion in the inner LMC. However, there is evidence that the azimuthal velocity dispersion decreases with radius in disk galaxies. As the MagES fields are situated at substantially larger galactocentric radii ( ∼ 10.5∘ from the LMC COM) than the data, it is reasonable that the azimuthal velocity dispersion in the MagES fields is correspondingly smaller.\nBy this reasoning, it might also be expected that the azimuthal velocity dispersion in the MagES fields should be smaller than that measured by ( ∼ 20 km s− 1 at ∼ 8∘ from the LMC COM). However, the aforementioned decrease in azimuthal velocity dispersion with radius is strongest in the inner regions of the disk, and levels off (implying a relatively constant dispersion as a function with radius) in the disk outskirts. Accordingly, consistency between the dispersions measured in the MagES fields, and that measured by, is not surprising. This is true for field 18, although the dispersion in field 12 is somewhat higher than that measured by. We note, however, that field 12 is located only a small distance radially inward from the base of the arm-like feature discussed in. We hypothesise that this increased dispersion may be due to the same perturbation which formed the feature. Further evidence of perturbation in field 12 is discussed below.\nFor both MagES fields, the vertical motion (*V**Z*) perpendicular to the LMC disk plane is, within uncertainties, consistent with zero. This is as expected; in an equilibrium system, a roughly equivalent number of stars will, at any one time, be moving vertically in both directions, resulting in a mean motion of zero across the field. The vertical velocity dispersion for the two fields (20.6 ± 1.0km s− 1 and 26.8 ± 1.7km s− 1 for fields 18 and 12 respectively) are slightly higher, but not significantly different from, that measured by ( ∼ 15 km s− 1 at ∼ 8∘ from the LMC COM). As is the case for the vertical velocity, in an equilibrium system, the mean radial velocity (*V**R*) across a field is expected to be zero, with a roughly equivalent number of stars moving in both directions. This is true for field 18; however for field 12, the radial velocity (15.5 ± 14.5km s− 1) does not overlap zero within ∼ 1*σ*. The (small) positive value suggests a mild net motion radially outward for stars in this field.\nThe source of the net outward motion in this field is not obvious. As noted above, field 12 is located nearby the base of an arm-like structure in the outer LMC. It is possible this radial motion is a signature of the perturbation which formed the feature. Interestingly, also find positive radial velocities for some stars between ∼ 4∘–8∘, which they suggest may be due to non-equilibrium effects induced by interactions between the Clouds. A future paper (Cullinane et al. in prep) will investigate the hypothesis that interactions can cause such positive radial velocities in further detail.\nThe radial velocity dispersion (*σ**R*) in field 18 (24.4 ± 2.7km s− 1) is, within uncertainty, equal to the azimuthal velocity dispersion. This is consistent with the behaviour reported in and. The magnitude of the radial dispersion measured here is again somewhat smaller than that reported in ; the difference can be attributed to the same reasons outlined above for the azimuthal velocity dispersion. However, the radial dispersion is approximately consistent with the ∼ 20kms reported by at his most distant point. In this field, the radial and vertical velocity dispersions are also consistent with each other, within uncertainties. This is similar to the behaviour of the Milky Way thick disk.\nIn contrast, field 12 has a radial velocity dispersion (44.8 ± 5.1km s− 1) almost double that of field 18. This is significantly higher than either the azimuthal or vertical velocity dispersions measured in the field, and, by coincidence, is closer to that measured in the inner LMC disk by and. As noted above, field 12 is located nearby the base of an arm-like structure in the outer LMC. Further, use a N-body model of the interaction between the LMC and SMC to demonstrate that such events can cause increased radial velocity dispersions, particularly in the outer regions of the LMC disk. Consequently, it seems plausible that the same perturbation which formed the nearby interaction feature, might also have increased the radial velocity dispersion in the outer disk as measured here. This idea will be explored in greater detail in a forthcoming paper (Cullinane et al. in prep).\nAsymmetric LOS Velocity Distributions\n-------------------------------------\nWhen the LOS velocity distribution of stars in the two northern disk fields are plotted, as in Fig. [fig:losdists], it is apparent that the distributions are asymmetric: there are clear tails to lower LOS velocities. We quantify this asymmetry by calculating the ‘excess’ fraction of stars in the low-velocity tail. To do this, we first fit a half-normal distribution to stars with LOS velocities exceeding the peak velocity of the field, using a least-squares fitting algorithm. The centre of the half-Gaussian is fixed to the peak velocity reported in Table [tab:obsresults]; only the dispersion of the half-Gaussian is fit. This ‘reduced dispersion’ reflects the dispersion value that would be calculated if the LOS velocity distribution were truly Gaussian in nature. Using this ‘reduced dispersion’, we then calculate the fraction of stars with LOS velocities greater than 1*σ* below the peak value. If the distribution were perfectly Gaussian, 15.865 per cent of stars would have velocities further than 1*σ* from each side of the peak value.\n| |\n| --- |\n| image |\n| image |\n| |\n[fig:losdists]\nIf we perform this test for stars with LOS velocities exceeding the peak velocity, this is approximately true: field 18 has 16.2 per cent, and field 12 has 13.3 per cent, of stars greater than 1*σ* above the peak value. Given the finite size of the sample, 1–2 per cent difference between the calculated values is expected. In contrast, if we perform the same test for stars with LOS velocities under the peak velocity, substantially different results are observed. In field 18, 21.2 per cent of stars are greater than 1*σ* below the peak value, while for field 18, this increases to 30.2 per cent. This is significantly more than expected for a perfectly Gaussian distribution.\nThis asymmetry was not accounted for when fitting Gaussians to these distributions as described in §[sec:isolating]. Consequently, it is possible that the field kinematics discussed above are slightly biased. To demonstrate this is not the case, new estimates of the aggregate field kinematics are determined by repeating the process described in §[sec:aggregates], but including in this calculation only stars with LOS velocities exceeding a particular velocity threshold, so as to effectively ‘exclude’ the low-LOS-velocity tail from the calculation. If doing so does not change the aggregate field properties derived, we can be satisfied the analysis in §[sec:diskvels] remains unaffected by the asymmetry in the LOS velocity distribution.\nThe velocity threshold imposed does not take a fixed value; instead, it is varied in 5 km s− 1 steps for both fields. The most stringent threshold is equal to *V*LOS − *σ*LOS, as reported in Table [tab:obsresults]: this corresponds to 1*σ* below the aggregate LOS velocity of the field. The weakest threshold imposed passes all stars with Magellanic membership probabilities $P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)>30\\%$.\nIn both fields, imposing a LOS velocity threshold introduces small changes to the LOS kinematic properties: as the LOS threshold becomes more stringent, the field aggregate LOS velocity increases, and the LOS velocity dispersion decreases. In field 18, these both change by ∼ 5 km s− 1; in field 12, slightly larger shifts ( ∼ 8 km s− 1 each) are observed. This is not surprising: excluding LOS velocities below a threshold naturally increases the median LOS velocity of the remaining population; and, by reducing the range of LOS values in the surviving population, naturally decreases its dispersion.\nOf greater interest is any effect on the proper motions of the population. The most stringent threshold applied to field 12 (265 km s− 1) results in reductions to both proper motion dispersions by ∼ 0.03 mas yr− 1 (corresponding to differences of ∼ 7 km s− 1 at the distance of the Clouds). However, as the proper motion components have larger uncertainties than the LOS velocity component, these shifts remain within the 1*σ* uncertainty of the value obtained when no threshold is applied. In field 18, observed differences in proper motions are even smaller – on the order of ∼ 2 km s− 1 at the distance of the Clouds – and therefore not significant.\nOf more import is whether these small shifts in observed kinematic properties engender differences when transformed into the LMC disk frame. The same transformation as described in §[sec:results] is performed to generate LMC disk velocities for each set of observed velocities, with uncertainties in both the observed kinematics, and the LMC disk geometry, propagated.\nIn field 18, the only effect of imposing a LOS velocity threshold is a reduction in the vertical velocity dispersion (*σ**Z*), which drops by ∼ 4 km s− 1 at the most stringent velocity threshold of 305 km s− 1. Considering the dispersion derived without any threshold imposed is 20.6 ± 1.0 km s− 1, this represents a ∼ 4*σ* reduction in the dispersion. All other disk velocities are well within the 1*σ* uncertainty of the original values. The same pattern is observed in field 12, with a reduction in the vertical velocity dispersion of ∼ 6 km s− 1 at the most stringent velocity threshold of 265 km s− 1. This represents a 3.5*σ* reduction in the dispersion. While the azimuthal and radial velocity dispersions in this field also drop by a few km s− 1, due primarily to the reduced dispersion in the observed proper motions, these also remain within the 1*σ* uncertainty of the original derived values.\nIt is no surprise that only the vertical velocity dispersions differ by any substantive amount; the relatively low inclination of the LMC means the LOS velocity dispersion (which is most significantly affected by imposing a LOS velocity threshold) is translated almost directly into the vertical velocity dispersion. Further, despite these reductions, the vertical velocity dispersions calculated remain consistent with the most distant estimates derived by. Thus the conclusions drawn in §[sec:diskvels] are unaffected by the asymmetry in the LOS velocity distribution of the stars.\nHaving satisfied ourselves that the results in §[sec:diskvels] remain valid, we now turn to analysing the asymmetry itself, and its possible origins. We first check for possible correlations between LOS velocity, and other properties of individual stars, testing proper motions, *Gaia* *G*0 magnitude, on-sky position, and metallicity where available. Unfortunately, the relatively large uncertainties on individual measurements of these quantities are sufficient to mask any such correlations if they exist. Consequently, we instead analyse aggregate properties of stars with lower and higher LOS velocities – as these aggregate properties have smaller associated uncertainties, any significant differences in the overall kinematics of the two groups should be more clearly apparent. To do this, the same range of LOS velocity thresholds discussed above are used to divide the stars in each field into two subgroups; a ‘low-velocity’ sample containing stars with LOS velocities below the threshold, and a ‘high-velocity’ sample containing stars with LOS velocities that exceed the threshold.\nHowever, each individual star has an uncertainty in its LOS velocity, and this could change how each star is classified between the two subsamples. This would consequently affect the aggregate properties of the two groups. In order to account for this, the observed kinematics of each star are drawn randomly from a Gaussian distribution with width equal to the 1*σ* uncertainty in its velocity. This process is repeated in order to generate a set of 500 ‘low-velocity’ and ‘high-velocity’ groups for any given threshold, the aggregate properties of which can be compared to one another.\nK–S tests are used to determine whether the properties of the high- and low-LOS velocity groups are statistically similar. Tests are performed on the median proper motions, (*G*BP − *G*RP)0 colour, *Gaia* *G*0 magnitude, on-sky position, metallicity, and fibre number (to confirm no systematic differences linked to the observational setup are present). Two-dimensional K–S tests are used to compare the positions and proper motions of the groups, as these properties are correlated; all other tests are one-dimensional. The dispersions of the two groups are not compared as there are always significantly fewer stars in the low-velocity group; the dispersion calculated is therefore not likely to be representative of the true dispersion of the population. For the properties which are tested, each of the 500 distribution sets is compared, and the median of the resulting p-value distribution assessed. In all cases, this p-value is >0.05, indicating there is no significant difference in the properties of the stars comprising the two subgroups (apart from, by definition, their mean LOS velocities).\nIn order to better understand the implications of the LOS velocity asymmetry, we transform the aggregate properties of the two groups into the LMC disk frame using the procedure outlined in §[sec:results]. We find that differences exist between the vertical and azimuthal velocity components of the two groups, but the radial velocity of the two groups remains consistent within uncertainty, regardless of the threshold used to separate the groups. This is true of both fields analysed.\nBy far the most significant difference is in the vertical velocity component (*V**Z*); in both fields, the low-LOS-velocity group has *V**Z* values of ∼ 40 km s− 1, indicating motion perpendicular to the disk plane, in a direction roughly towards the Earth. This is primarily a consequence of the relatively low inclination of the LMC disk, such that differences in LOS velocity naturally correspond to differences in the vertical velocity. Compared to the behaviour of the high-LOS-velocity sample (which has median vertical velocities consistent with 0 km s− 1, as expected for an equilibrium stellar disk), the large vertical velocity of the low-velocity sample is indicative of mean motion away from the disk for these stars.\nThere are also differences in the azimuthal velocity of the two groups. In both fields, the low-velocity group rotates ∼ 25 km s− 1 more slowly than the high-velocity group, though this difference is barely significant at the 1*σ* level. The large uncertainties in the azimuthal velocities, which may mask the significance of this difference, are a direct consequence of the large uncertainties in the proper motions of the stars from which the azimuthal velocity is derived. Future *Gaia* releases, with reduced proper motion uncertainties, will likely clarify whether this small difference is genuinely significant.\nThe difference in azimuthal velocity of the two groups bears similarities to the signature of a kinematically distinct population of stars discussed in, which they attribute to infalling SMC stars either moving counter to LMC disk rotation, or located in a plane strongly inclined relative to the LMC disk. However, it is unlikely our low-velocity group is part of the same population. At the large radii of our fields, we would expect any difference in distance associated with the stars being located in very different planes to result in a detectable difference in red clump magnitude, which is not observed. Further, the difference in azimuthal velocity between the two groups is identical in both fields, despite these being located more than 10∘ apart, suggesting that both groups are likely linked to the LMC disk. At the large galactocentric radii of our fields, the median LMC [Fe/H] abundance of approx. − 1 is less easily distinguishable from typical SMC metallicities.\nWe speculate that the low-velocity tail of the LOS velocity distribution may be the result of an external perturbation. This is consistent with the fact that there is a higher relative fraction of stars in the low-velocity group in field 12 (which, as discussed above, shows other indications of being perturbed) compared to field 18. While there are other possibilities, it is certainly plausible that an interaction, with either or both of the SMC or Milky Way, could begin to pull stars out of the LMC outer disk in one direction preferentially, generating the non-zero vertical velocity observed for these stars. Numerical models of interactions in the Magellanic system are required to test the veracity of this signature, and its possible links with the northern arm. The MagES collaboration is actively working to follow up this avenue of investigation.\nLMC Mass estimate\n-----------------\nUnder the assumption that stars in the outer LMC disk are following equilibrium or near equilibrium motions[12](#fn12), it is possible to calculate an estimate for the dynamical mass of the LMC using the azimuthal rotational velocities derived in the preceding analysis. This assumption is likely valid for field 18; but, as discussed above, there are indications of possible non-equilibrium behaviour in field 12. As such, despite the fact that azimuthal velocities for both MagES disk fields are consistent within uncertainty, only information derived from field 18 is used in the following analysis. To determine the dynamical mass, Eq. [eq:mass] is used.\n[eq:mass] M=\nHere, *M*enc is the enclosed mass of the LMC within *R* kpc of the LMC COM; *G* = 4.3007 × 10− 6 kpc M⊙− 1(km s− 1)2; and *V*circ is the circular velocity (in km s− 1) at distance *R* from the LMC COM. Note that the azimuthal rotation velocity is not *V*circ, the velocity of a tracer on a circular orbit in the equatorial plane; to determine this first requires correction for asymmetric drift. To make this correction, we use Eq. [eq:vcirc], taken from, which relates azimuthal velocity *V**ϕ* to *V*circ.\n[eq:vcirc] V2 = V2+2\nHere, *R**d* is the disk scale length (which we take as 1.5kpc from ) and *σ*LOS is the line of sight velocity dispersion of stars in the field. Note that Eq. [eq:vcirc] only applies to the simplified case of an axisymmetric exponential disk system embedded within an isothermal dark halo. Although, as is apparent from Fig. [fig:map], axisymmetry breaks down at large distances from the LMC COM in the south, at the location of field 18 in the northern LMC disk, this remains a reasonable assumption.\nThe circular velocity calculated using the above procedure is 87.7 ± 8.0km s− 1. This is consistent with values reported by (91.7 ± 18.8km s− 1) and ( ∼ 90 km s− 1), but moderately lower than the circular velocity reported by (123.6 ± 1.9km s− 1). However, as discussed in §[sec:diskvels], the radial velocity dispersion measured by is 10–15 km s− 1 larger than those measured by MagES, and more closely reflects inner disk kinematics. By extension, when this is used in the asymmetric drift correction, it results in a significantly larger circular velocity than that derived from the MagES data.\nUsing the MagES circular velocity in Eq. [eq:mass] results in a total enclosed LMC mass, within 10kpc, of (1.8 ± 0.3) × 1010M⊙. To compare this mass to that derived in, we project their enclosed mass estimate (determined within a radius of 8.7kpc) out to a distance of 10kpc. The resulting mass of (2.1 ± 0.7) × 1010M⊙ is consistent with our estimate. Assuming that this radius is sufficient to encompass the majority of light from the LMC, a mass-to-light (M/L) ratio for the LMC can be calculated. We calculate the V-band luminosity of the LMC using its absolute magnitude relative to the absolute magnitude of the Sun. Using this with our enclosed mass estimate implies a M/L ratio of 12.5 ± 2.3 M⊙/L⊙.\nThe derived mass is low compared to mass measurements derived using more indirect methods, such as perturbations to stellar streams, the timing argument, or cosmological simulations of similar systems. This difference is to be expected, as each of the above methods provides the total infall mass of the LMC, including its dark halo. In contrast, the MagES field considered here, despite being at a greater distance from the LMC COM than most previous kinematic estimates, is still located well within the LMC dark halo: studies such as or have found likely LMC-associated stars at distances almost three times greater than field 18. As such, the enclosed mass derived simply does not capture a significant fraction of the total LMC mass. If, however, the assumption is made that the LMC rotation curve remains flat out to 29kpc (the furthest distance LMC-associated stars have been found to date as per ), and that the LMC is embedded in a typical dark matter halo, the inferred LMC enclosed mass would be (1.1 ± 0.2) × 1011M⊙, which is more in line with total infall mass estimates, and the mass calculated under similar assumptions in. In this scenario, the implied M/L ratio of the LMC rises to 58.2 ± 6.8 M⊙/L⊙.\nSummary\n=======\nIn this paper, we have described the Magellanic Edges Survey (MagES), a spectroscopic survey that, in conjunction with *Gaia* astrometry, is designed to obtain and interpret 3D stellar kinematics across the Magellanic periphery. Conducted using 2dF+AAOmega at the AAT, it primarily targets red clump stars and will ultimately yield 3D velocities for ∼ 7000 stars in 26 two-degree diameter fields in the outskirts of the Clouds, and metallicities for a limited subset with sufficiently high S/N spectra. It will constitute the largest sample of Magellanic stars with homogeneous 3D velocity information to date, in fields at larger galactocentric radii than most previous studies. In combination, this will provide significant insight into the evolution and interaction history of the Magellanic system.\nAs an early science demonstration, we present results for two MagES fields in the outer northern disk of the LMC. One field is located near the base of an arm-like feature to the north of the LMC first discovered by, and has 3D kinematics indicative of perturbation from an equilibrium disk. It has a non-zero radial velocity outwards in the LMC disk plane, in the direction towards the substructure, and an elevated azimuthal velocity dispersion. Further, it has a significant ( ∼ 44 km s− 1) radial velocity dispersion; which, as illustrated by, can be caused by LMC/MW/SMC interactions. The other field, located ∼ 10∘ from any known photometric substructures, behaves as expected for an equilibrium disk. Its kinematics are consistent with literature values derived from similar populations closer to the LMC centre, indicating the rotation curve of the LMC remains flat even at very large radii. The kinematics derived for both fields are robust against moderate changes to the assumed geometry of the LMC disk.\nBoth fields display an asymmetric LOS velocity distribution, with tails to low LOS velocities, though this is more pronounced in field 12. The asymmetry does not affect the field-aggregate properties discussed in §[sec:diskvels], and K–S tests confirm no statistically significant differences exist between stars with lower and higher LOS velocities. However, when transformed into the LMC disk frame, stars with low LOS velocities are found to have vertical velocities of ∼ 40 km s− 1, indicative of a subset of stars being perturbed away from the assumed LMC disk plane. As the asymmetry is strongest in the field nearest the arm-like substructure, we hypothesise that it is a signature of interaction. Further analysis in conjunction with dynamical models is required to fully understand this behaviour.\nThe kinematics of the ‘undisturbed’ field are used to estimate the LMC mass; one of the most distant estimates derived using stellar kinematics. The derived circular velocity of the stars is 87.7 ± 8.0km s− 1, with a resulting enclosed mass of (1.8 ± 0.3) × 1010M⊙ within ∼ 10kpc. This is consistent with other enclosed mass values derived using stellar kinematics ; but, as is typical for such estimates, is lower than masses derived using more indirect methods, for example perturbations to orbits of MW stellar streams, which are sensitive to the total halo mass.\nAcknowledgements\n================\nThis work has made use of data from the European Space Agency (ESA) mission **Gaia** (<https://www.cosmos.esa.int/gaia>), processed by the **Gaia** Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the *Gaia* Multilateral Agreement. Based on data acquired at the Anglo-Australian Observatory. We acknowledge the traditional owners of the land on which the AAT stands, the Gamilaraay people, and pay our respects to elders past, present and emerging. This research has been supported in part by the Australian Research Council (ARC) Discovery Projects grant DP150103294. ADM is supported by an ARC Future Fellowship (FT160100206). SK is partially supported by NSF grants AST-1813881, AST-1909584 and Heising-Simons foundation grant 2018-1030. DMN acknowledges support from NASA under award Number 80NSSC19K0589. AK gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 138713538 – SFB 881 (“The Milky Way System”), subprojects A03, A05, A11.\nEffect of uncertainties on maximum likelihood results\n=====================================================\nIn Section [sec:isolating], several maximum likelihood steps are used to determine fit parameters; each of which, in addition to returning parameter values that maximize the given likelihood function, also provide 1*σ* confidence intervals for the fit parameters. In the main analysis, we always utilise the best-fit values for each parameter in subsequent steps, with the inherent assumption that the effect of these uncertainties is negligible. Here, we confirm this assumption is reasonable.\nEffect of uncertainties in the contamination model\n--------------------------------------------------\nThe calculation in §[sec:memberprob], to determine initial estimates for the properties of Magellanic kinematic peaks, requires the use of parameters that describe the expected Milky Way foreground contamination, derived from the Besançon models in §[sec:besancon]. However, each of these Milky Way contamination parameters – i.e. those within $\\hat\\phi$, comprised of *v**m*, *μ**δ*, *m*, *μ**α*, *m*, *σ**v*, *m*, *σ**δ*, *m*, *σ**α*, *m*, *ρ**m*, and *η**m* – has an associated 1*σ* uncertainty. The effect of varying these parameters within their uncertainties on the initial estimate of the parameters defining the Magellanic peak is tested to ensure it is negligible.\nWe do this by calculating the Magellanic peak parameters 500 times, each time using Milky Way contamination parameters drawn randomly from Gaussian distributions centred on the best-fitting parameter values, with width equal to the 1*σ* equivalent uncertainty in the parameter. The resulting distributions of each Magellanic peak parameter are inspected, and the standard deviation calculated as an estimate of the uncertainty introduced by varying the Milky Way contamination parameters.\nIn every case, we find the distributions of Magellanic peak parameters introduced by varying the contamination model input parameters, are much narrower than the 1*σ* uncertainties in the Magellanic peak parameters when determined using the best-fitting contamination model as input. In other words, the dominant source of uncertainty in the Magellanic peak parameters is that driven by observational uncertainties in the stellar kinematics, and not uncertainties associated with the parameters of the model contaminant population; validating the assumption made in the text.\nEffect of uncertainties in Magellanic kinematic peak properties\n---------------------------------------------------------------\nThe initial estimates for the Magellanic peak properties are used to calculate the probability of each star being associated with the Clouds; which is subsequently used to calculate the aggregate kinematics of each field in §[sec:aggregates]. As discussed in §[sec:modeluncs], each of these parameters has associated uncertainty. We test the effect of varying these parameters within their uncertainties on the membership probability of each star, and the field aggregate properties, to ensure this is negligible.\nTo begin, the membership probability $P(\\text{MC}|i,\\varphi,\\hat\\phi)$ of each star is calculated 500 times, each time using Magellanic peak parameters within *φ* – that is, *v*MC, *μ**δ*, MC, *μ**α*, MC, *σ**v*, MC, *σ**δ*, MC, *σ**α*, pk, and *ρ*MC – drawn randomly from Gaussian distributions centred on the best-fitting parameters, with width equal to the 1*σ* uncertainties on the parameters. As $P(\\text{MC}|i,\\varphi,\\hat\\phi)$ requires information from both LOS velocity and proper motion, these values are varied simultaneously. The resulting $P(\\text{MC}|i,\\varphi,\\hat\\phi)$ distributions are all relatively narrow; we characterise the width of these distributions as half the difference between the minimum and maximum $P(\\text{MC}|i,\\varphi,\\hat\\phi)$ values calculated for each star.\nWe then calculate the field aggregate properties as per Eq. [eq:finlikelihood] 500 times. Each time, membership probabilities for all stars are drawn randomly from a Gaussian distribution centred on the original $P(\\text{MC}|i,\\hat\\varphi,\\hat\\phi)$ value assigned to each star, with width equal to the characteristic width of the $P(\\text{MC}|i,\\varphi,\\hat\\phi)$ distributions.\nIn every case, we find that the distributions of each field aggregate property introduced by varying the membership probability, are much narrower than the 1*σ* uncertainties in the aggregate properties when determined using the best-fitting membership probabilities as input. In other words, the dominant source of uncertainty in the field aggregate parameters is that driven by observational uncertainties in the stellar kinematics, and not uncertainties associated with the membership probabilities of each individual star, or initial peak parameter estimates.\n[lastpage]\n---\n1. E-mail: lara.cullinane@anu.edu.au (LRC)[↩](#fnref1)\n2. *L* and *B* are Magellanic longitude and latitude respectively, as defined in. *μ**L* is the proper motion in the *L*cos(*B*) direction, such that it is perpendicular to *μ**B*.[↩](#fnref2)\n3. *μ**α* refers to proper motion in the *α*cos(*δ*) direction, as obtained directly from the *Gaia* source catalogue using the column.[↩](#fnref3)\n4. The design of AAOmega is such that the wavelength coverage varies between individual fibres; the quoted range includes only those wavelengths that are accessible in every fibre.[↩](#fnref4)\n5. T=5000K, log(*g*) = 2.5, [Fe/H] = − 0.5, [*α*/Fe] = 0.[↩](#fnref5)\n6. 5100–5400Å for the blue spectra, and 8470–8740Å for the red spectra.[↩](#fnref6)\n7. <https://model.obs-besancon.fr/>[↩](#fnref7)\n8. No thin disk stars survive the selection criteria applied to the Model; thus all ‘disk’ stars are defined as being associated with the thick disk of the Milky Way.[↩](#fnref8)\n9. Comprised of carbon stars, AGB and RGB stars that are predominantly older than 1–2 Gyr and therefore similar in age to the red clump population.[↩](#fnref9)\n10. The axis along which the plane of the inclined LMC disk intersects the plane of the sky.[↩](#fnref10)\n11. We do not use the model parameters derived using only the outermost radial bin in the analysis as that data comes only from a small portion of the southern LMC disk; at this stage, it is not clear if the reported warping of the disk in the south has a counterpart in the northern LMC.[↩](#fnref11)\n12. i.e. that the mean *V**R* and *V**Z* in a field are identically zero.[↩](#fnref12)\n\nThe Magellanic Edges Survey I. Description and First Results\n============================================================\n### Accepted XXX. Received YYY; in original form ZZZ\n[firstpage]\nWe present an overview of, and first science results from, the Magellanic Edges Survey (MagES), an ongoing spectroscopic survey mapping the kinematics of red clump and red giant branch stars in the highly substructured periphery of the Magellanic Clouds. In conjunction with *Gaia* astrometry, MagES yields a sample of ∼ 7000 stars with individual 3D velocities that probes larger galactocentric radii than most previous studies. We outline our target selection, observation strategy, data reduction and analysis procedures, and present results for two fields in the northern outskirts (>10∘ on-sky from the centre) of the Large Magellanic Cloud (LMC). One field, located in the vicinity of an arm-like overdensity, displays apparent signatures of perturbation away from an equilibrium disk model. This includes a large radial velocity dispersion in the LMC disk plane, and an asymmetric line-of-sight velocity distribution indicative of motions vertically out of the disk plane for some stars. The second field reveals 3D kinematics consistent with an equilibrium disk, and yields *V*circ = 87.7 ± 8.0km s− 1 at a radial distance of ∼ 10.5kpc from the LMC centre. This leads to an enclosed mass estimate for the LMC at this radius of (1.8 ± 0.3) × 1010M⊙.\nMagellanic Clouds – galaxies:kinematics and dynamics – stars: kinematics and dynamics – galaxies: stellar content – galaxies: structure\nIntroduction\n============\nThe Large and Small Magellanic Clouds (LMC/SMC) are of fundamental importance in numerous areas of astronomy. The LMC, as the most massive Milky Way (MW) satellite – with recent estimates of its total mass exceeding 1011M⊙ – has significant effects on our Galaxy. For example, it can induce warps in the MW disk, generate overdensities in the MW dark matter halo, perturb the orbits of smaller satellites and stellar streams, and has likely brought with it several dwarf satellites of its own. The Clouds are also the closest pair of interacting dwarf galaxies, at distances of 50kpc and 60kpc for the LMC and SMC respectively. This makes them ideally situated for a detailed study of the influence of interactions on galaxy evolution. The SMC in particular is significantly distorted, with a line of sight depth of up to 20kpc and an asymmetric, irregular morphology, both of which encode valuable information about its extensive interaction history.\nIt is evident that having precise information on the masses and orbits of the Clouds, as well as their interaction and star formation histories, is important for our understanding of both the local and more distant universe. In order to obtain information on these topics, the Clouds have been the focus of numerous surveys, with efforts intensifying as the availability of instruments able to survey quickly the large on-sky area of the Clouds increases. One example is the Dark Energy Camera, which is situated on the 4m Blanco Telescope at the Cerro Tololo Inter-American Observatory in Chile, and has a 3 square degree field of view. Several surveys including the Survey of the Magellanic Stellar History, and the Magellanic SatelLites Survey have utilised DECam to obtain deep multi-band photometry across the Magellanic system. In combination with DECam photometry from the Dark Energy Survey, and additional imaging from, this has provided an almost complete photometric picture of the Clouds and their surrounds (Mackey et al. in prep).\nA key result from these surveys is the discovery of a wealth of low-surface-brightness substructure across the periphery of the Clouds ; clear evidence of tidal interaction between the two Clouds, and/or the Clouds and the Milky Way. However, in order to piece together precise details of the interactions forming these features, kinematic information for stars in the substructures and across the Clouds, which is not provided by photometric surveys, is needed.\nSpectroscopic surveys have long been used to characterise line-of-sight kinematics in the Clouds, though these have predominantly targeted stars (or star clusters) in the interior, rather than the outskirts, of the Clouds. Studies of the LMC have largely focussed on its internal rotation, with older tracer populations such as carbon stars, red giant branch stars, and star clusters found to have larger velocity dispersions compared to younger populations such as red supergiants. Even within the RGB population, metal-poor (and, by extension, older) stars are found to have increased dispersions relative to more metal-rich stars. Some studies have also found potential evidence for a high-dispersion halo population around the LMC.\nIn contrast to the relatively ordered motion within the LMC, studies of the SMC reveal more complex, disturbed kinematics. Both younger and older populations have large velocity dispersions and spatial velocity gradients indicative of the SMC being disrupted by the LMC (though note also find some evidence for coherent rotation within the SMC). SMC debris has been found in not only the bridge region between the Clouds, but also at large distances from the Clouds, and even within the LMC itself.\nIn addition to kinematic studies, spectroscopic measurements of the CaII triplet equivalent width (pioneered by and ) have often been used to obtain metallicity estimates for RGB stars in the Clouds. Metallicity gradients as a function of galactocentric radius are found in both Clouds, with median [Fe/H] abundances decreasing from around − 0.5 in the central ( ≤ 6kpc) LMC disk to around − 1 further out. In the more metal-poor SMC, [Fe/H] abundances decrease from − 1 in the central ( ≤ 2∘) regions, to approx. − 1.5 further out (e.g. ; but see also ).\nWhile spectroscopic studies are useful, measuring line-of-sight kinematics alone is insufficient to constrain the full 3D velocity field of the Clouds. This is particularly relevant when considering distant substructures, as full 3D kinematic information is required in order to distinguish between different formation mechanisms for the observed stellar substructures. In this respect, *Gaia* DR2 has been a boon, providing proper motion measurements down to red clump magnitudes ($G\\lesssim19$) in both Clouds. This has allowed substructures to be kinematically traced out to 25∘ from the centre of the Clouds, and detailed analyses of internal LMC dynamics based on proper motions to be performed. However, the Clouds are sufficiently distant that the *Gaia* spectrograph does not reach the faint magnitudes required to provide line-of-sight velocities for the old stellar populations in the Clouds that comprise the peripheral substructures.\nAs such, to date, there have been no large scale studies of 3D kinematics in the outskirts of the Clouds. The Magellanic Edges Survey (MagES) is designed to fill this gap. The core aim of the survey is to obtain spectra for large numbers of red clump and red giant branch stars that trace substructures across the Magellanic periphery, in order to derive line-of-sight velocities that can be used in conjunction with *Gaia* data to obtain the full 3D kinematic information necessary to unravel the interaction history of the Clouds. To do so, it utilises observations with the 2dF fibre positioner coupled with the dual-arm AAOmega spectrograph at the 3.9m Anglo–Australian Telescope (AAT) to obtain simultaneous spectra for ∼ 370 stars across each ∼ 2∘ diameter field. The survey began in 2015, with observations taken for several nights per year to date (details provided in Table [tab:fieldloc]).\nIn this paper, we present the detailed methodology of MagES, and our first science results. §[sec:design] presents the survey fields and target selection procedure. §[sec:data] describes the reduction and data validation processes. §[sec:isolating] discusses the method used to isolate Magellanic stars and extract aggregate field kinematics. We report our first science results, a determination of the LMC disk motion using distant tracers, in §[sec:results], followed by our conclusions and future plans for MagES in §[sec:concs].\nSurvey Design and Target Selection\n==================================\nAs MagES is intended to shed light on interactions between the Clouds – a major signature of which is the formation tidal disturbances in the periphery – MagES fields largely target overdense regions and substructures in the outskirts of the Clouds. Over time, as the photometric coverage of the Magellanic periphery has increased, the positioning of MagES fields has evolved to continually target the most conspicuous features. To date, twenty-six 2dF fields have been observed; these are detailed in Table [tab:fieldloc], with Fig. [fig:map] presenting a visual representation of the targeted fields overplotted on a stellar density map of red clump stars across the Clouds.\nThe earliest observed fields target a large arm-like feature to the north of the LMC first discussed in. Subsequent runs have focussed on spoke-like features to the south of the LMC disk, and extended red clump features surrounding the SMC. The most recent observations target another apparent tidal feature extending from the SMC which curves around the southern LMC, discussed in, and thought to be a counterpoint to the northern arm feature.\n[tab:fieldloc]\nlrrrr>m3.2cm>m2cm>m3.5cm Field & & & & & & & \n \n11 & 05 19 42.63 & -56 53 06.88 & -1.30 & 12.80 & & 27000 & 12.7 (L) \n13 & 05 35 05.69 & -55 06 03.11 & 0.90 & 14.70 & 19 Aug 20151, 1 Feb 2016 & 18380& 14.6 (L) \n15 & 06 00 07.40 & -54 17 53.14 & 4.70 & 15.30 & 20 Aug 2015 & 8700& 16.0 (L) \n16 & 06 12 13.07 & -53 52 32.45 & 6.60 & 15.50 & l 3 Feb 2016, 4 Feb 2016, \n5 Feb 2016 & 16200& 16.8 (L) \n \n6 & 03 22 33.00 & -80 40 55.00 & -5.00 & -12.75 & 14 Dec 2017, 1 Oct 2018 & 12600& 13.1 (L) \n7 & 03 26 04.00 & -77 26 18.00 & -6.50 & -9.75 & 1 Dec 2017 & 10800& 11.0 (L) \n8 & 03 39 15.00 & -73 43 48.00 & -7.50 & -6.00 & 15 Dec 2017, 16 Dec 2017 & 10800& 8.8 (L) \n10 & 04 36 23.00 & -79 07 17.00 & -2.50 & -10.00 & 12 Dec 2017 & 9000& 9.9 (L) \n14 & 05 50 22.00 & -79 21 18.00 & 1.00 & -10.00 & 13 Dec 2017 & 10800& 10.0 (L) \n17 & 06 32 16.00 & -80 59 36.00 & 2.50 & -12.00 & 30 Sep 2018 & 12600& 12.2 (L) \n19 & 06 40 29.00 & -53 29 04.00 & 11.00 & 15.00 & 12 Dec 2017 & 10800& 18.6 (L) \n20 & 07 04 01.00 & -53 37 01.00 & 14.50 & 13.75 & 14 Dec 2017 & 12600& 19.9 (L) \n \n1 & 00 56 26.00 & -67 43 32.00 & -22.00 & -12.00 & 30 Sep 2018, 1 Oct 2018 & 10800& 5.4 (S) \n2 & 00 59 30.00 & -79 10 57.00 & -10.50 & -16.75 & 2 Oct 2018 & 10800& 6.1 (S) \n3 & 01 20 00.00 & -82 30 00.00 & -6.97 & -17.51 && 11600& 9.5 (S) \n4 & 01 45 11.00 & -79 15 22.00 & -9.25 & -14.75 & 30 Sep 2018 & 12600& 6.9 (S) \n5 & 02 06 32.00 & -76 29 09.00 & -10.75 & -12.00 & 1 Oct 2018, 2 Oct 2018 & 16200& 6.0 (S) \n9 & 03 40 00.00 & -86 17 13.12 & -1.78 & -17.73 && 14410& 17.1 (L) \n12 & 05 20 00.00 & -59 18 00.00 & -1.17 & 10.29 & 27 Feb 2019, 28 Feb 2019 & 10800& 10.3 (L) \n18 & 06 40 00.00 & -62 30 00.00 & 8.19 & 5.89 & 27 Feb 2019, 28 Feb 2019 & 12200& 10.7 (L) \n21 & 07 17 12.00 & -76 36 00.00 & 6.14 & -8.58 & 2 Mar 2019, 5 Mar 2019 & 14600& 10.9 (L) \n22 & 07 25 34.00 & -52 04 52.00 & 18.50 & 14.00 & 1 Oct 2018, 2 Oct 2018 & 9000& 22.8 (L) \n23 & 07 36 00.00 & -71 00 00.00 & 9.99 & -4.19 & 6 Mar 2019, 7 Mar 2019 & 14400& 11.4 (L) \n24 & 07 58 48.00 & -84 12 00.00 & 3.67 & -16.31 & 4 Mar 2019 & 9300& 16.4 (L) \n25 & 08 32 00.00 & -67 00 00.00 & 16.74 & -4.01 & 3 Mar 2019 & 11300& 17.5 (L) \n26 & 08 48 00.00 & -79 00 00.00 & 8.67 & -13.59 & 1 Mar 2019 & 9000& 16.1 (L) \non 19 Aug 2015, these pilot fields were observed with only the red arm of AAOmega; subsequent observations were taken in the typical setup with both arms of the spectrograph, as discussed in §[sec:reductions].\n![Location of observed 2dF fields across the Magellanic system; fields are predominantly located on substructures or overdensities in the periphery of the Clouds. Green circles indicate D fields, blue circles indicate M fields, and purple circles indicate G fields. Fields 12 and 18 are discussed in detail in this paper. The background image shows the log density of red clump and red giant stars per square degree. These were selected from Gaia DR2 according to the process outlined by ; repeated here in §[sec:gfields]. On this map, north is up and east is to the left; (\\eta, \\xi) are coordinates in a tangent-plane projection centred on the LMC (\\alpha_0=82.25^{\\circ}, \\delta_0=-69.5^{\\circ}). Orange dashed circles mark angular separations of 8^{\\circ}, 12^{\\circ}, 16^{\\circ}, and 20^{\\circ} from the LMC centre, as well as 4^{\\circ} and 8^{\\circ} from the SMC centre. Within 8^{\\circ} of the LMC and 4^{\\circ} of the SMC, wide-field optical images are displayed. The red x-signs mark the locations of Canopus (the second brightest star in the sky, which limits field placement to avoid spectral contamination from scattered light) and the south celestial pole (which limits field placement due to telescope pointing limits).](fig1_small \"fig:\") [fig:map]\nTarget Selection\n----------------\nMagES primarily targets red clump stars, as this region in colour-magnitude space has high contrast for Magellanic stars relative to background contaminants (see Fig. [fig:cmds]). Whilst even stronger contrast exists for the Magellanic main sequence turn-off population, these stars are ∼ 2.5 magnitudes fainter than the red clump, and as such would require prohibitively long integration times to reach sufficient S/N.\nccc & [-4.5cm] & [-4.5cm] \n\n[fig:cmds]\nIn addition to the field placement evolving, the target selection procedure has also changed as new data have become available. Consequently, there are three distinct target selection procedures that have been applied during different phases of the survey:\n1. Fields within the DES footprint, observed prior to the release of *Gaia* DR2 (2015–2016). These are denoted as **D** fields;\n2. Fields outside the DES footprint, observed prior to the release of *Gaia* DR2 (2017); these are denoted as **M** fields; and\n3. Fields observed post-*Gaia* DR2 release (2018+); these are denoted as **G** fields.\nThe three procedures are detailed in the following sections; Table [tab:fieldloc] provides classification of each field into one of these three groups, in addition to the location of the field’s centre, dates observed and total exposure time. **D** fields are listed first, followed by **M** and **G** fields; this is in approximately chronological order of observations. Within each grouping, fields are listed in order of increasing right ascension.\nOnce a list of possible targets is compiled for each field, they are assigned various priorities between 1 and 9 (with 9 being the highest). Higher priorities are given to stars most likely to be of Magellanic origin, though how this is defined varies based on the specific selection procedure, and is discussed in the following sections. The 2dF allocation software configure uses the priorities to inform fibre allocation: higher ranked targets are more likely to be observed. By design, the selection procedure is such that there are almost always more possible targets than available 2dF fibres – as such, prioritisation strongly influences which stars are observed in any given field.\nAll three procedures involve target selection and prioritisation based on cuts in extinction-corrected colour-magnitude space. Where DECam photometry is used, the de-reddening is done using dust maps and updated coefficients from. Where *Gaia* DR2 photometry is used, the correction uses the procedure described in : the first two terms of Eq. 1 from are used in conjunction with the dust maps. No correction is made for reddening internal to the Clouds as this is not expected to be significant in the low-density peripheral regions targeted by MagES.\n### D Fields\nInitial fields observed by MagES were located entirely within the photometric footprint of DES year 1 ; target selection is thus based on (*r*0, (*g* − *r*)0) and (*i*0, (*g* − *i*)0) colour magnitude diagrams. Within each colour magnitude diagram, an inner and outer box are defined, centred on the red clump, as in panels a and b of Fig. [fig:cmds]. Priorities for each star are defined based on their location on the two diagrams: stars in the inner box in both diagrams are given the highest priority, with decreasing priority given to stars located in the outer boxes, or located within box boundaries on only one of the diagrams.\nNote that the boundaries of selection boxes for **D** fields are defined based on photometry of the northern disk of the LMC, where the position of the red clump in colour-magnitude space is well defined; the same box is then applied to fields covering fainter substructures. Selection boxes are designed to be sufficiently generous that small changes in CMD position of the red clump (due to, for example, field-to-field differences in line-of-sight distance), do not affect target selection.\n### M fields\nFields designated **M** are located outside of the DES survey footprint, and are selected based on g- and r-band DECam photometry obtained by.We refer interested readers to for details of the data reduction and photometric analysis. Three boxes are defined on the (*r*0, (*g* − *r*)0) colour magnitude diagram, as in panel c of Fig. [fig:cmds]: an inner and outer box surrounding the red clump, similar to those used for the **D** fields, as well as a lower box designed to capture any faint red clump extension. As with **D** fields, these CMD boxes are defined based on photometry of the northern LMC disk. Highest priority is assigned to stars in the inner box; followed by the outer box. Stars in the third box are assigned lowest priority as, while useful, this region of the CMD has higher Milky Way contamination than the canonical red clump.\n### G Fields\nFields observed after the release of *Gaia* DR2 utilise these data exclusively in the selection procedure; even in regions where DECam photometry exists. Unlike previous selections, a combination of photometry and astrometry are used. Highest priority is given to stars that pass the selection procedure presented in. This uses *Gaia* photometry to select red clump and RGB stars (see panel d of Fig. [fig:cmds]). **G** fields are the only fields to contain RGB stars, though these are few in number compared to red clump stars. In addition, parallax ($\\varpi<0.2$) and proper motion ( − 0.6 < *μ**B* (mas yr− 1) < 1.4, 0.9 < *μ**L* (mas yr− 1) < 2.8)[2](#fn2) cuts are applied to isolate Magellanic stars. Lower priority is given to stars within a slightly offset selection box surrounding the red clump, with the same parallax cut and more generous proper motion cuts ( − 1.0 < *μ**α*[3](#fn3) (mas yr− 1) < 4.0 and − 4.0 < *μ**δ* (mas yr− 1) < 4.0) which increase the selection area in proper motion space by a factor of 25. This lower-priority selection is used when the number of target stars passing the initial, higher-priority selection criteria is significantly lower than the number of 2dF fibres available – while less efficient, we have confirmed additional Magellanic stars are captured through this second, less restrictive selection.\nData Acquisition\n================\nObservations and Data Reduction\n-------------------------------\nAll observations were taken using the 2dF/AAOmega instrument on the AAT at Siding Spring Observatory. 2df is a multi-object fibre positioner which allows for target placement within a two-degree field on the sky. It has a total of 400 fibres, of which ∼ 365 are available for science targets (the remainder being dedicated to guide stars and sky observations, detailed later in this section). AAOmega is a dual beam optical spectrograph; for these observations, the light was split using the 580V dichroic (i.e. at 5800Å). On the blue arm, the 1500V grating was utilised, obtaining a spectral resolution of R ∼ 3700, and wavelength coverage of 4910–5615Å[4](#fn4)\n. This is designed to cover the 5167Å, 5172Å and 5183Å MgIb lines to provide precise line-of-sight (LOS) velocity estimates.\nOn the red arm, the 1700D grating was used, providing a resolution of R ∼ 10000 and wavelength coverage 8370–8790Å. This is designed to cover the 8498Å, 8542Å and 8662Å CaII triplet at sufficiently high resolution to both allow for an estimation of metallicity, as well as provide a second LOS velocity estimate complementary to that obtained from the blue arm of the spectrograph.\nIn general, our survey strategy was to observe fields for between 10800–12600s, split into 1800s exposures to avoid skyline saturation and mitigate cosmic ray contamination. This results in typical signal to noise (S/N) values of ∼ 10 per pixel in both the red and blue data (at least in spectral regions not heavily contaminated by night sky emission; poor sky subtraction during the data reduction process degrades the S/N in some regions of the red spectra). In practice, total exposure times vary in accordance with observing conditions, with shorter exposures acceptable in very good conditions, but additional repeated exposures required when conditions were poor.\nFig. [fig:sndists] shows histograms of ‘quality measure’ (QM: an empirical S/N indicator covering spectral regions of interest, described further in §[sec:losvels]) for both red and blue arms of the spectrograph in representative **M** and **G** fields. **D** fields have distributions comparable to **M** fields. Red clump stars have similar QM values in both red and blue spectra. In contrast, RGB stars (found only in **G** fields) have significantly higher QM values in the red spectra: this is the source of the second ‘bump’ in the QM distribution in panel d of Fig. [fig:sndists]. As RGB stars are much redder than clump stars, this increase in QM is not prominent in the blue spectra.\n| | |\n| --- | --- |\n| image | image |\n[fig:sndists]\nData are reduced using the pipeline, which undertakes the subsequent steps. First, all observations are debiased using bias frames taken at the start of each night. Next, spectral traces are located with a fibre-flat field, taken immediately prior to each set of science exposures. These traces are used to extract the data for each fibre. The extracted spectra are then divided by corresponding normalised trace from the fibre-flat to correct for pixel-to-pixel variations along the CCD for each fibre. Wavelength calibration is performed using traces from an arc frame also obtained immediately prior to each set of science exposures, via a least-squares polynomial fit. A secondary wavelength calibration tweak, based on night sky emission features and utilising a lower-order polynomial fit, is also performed after the initial calibration.\nBecause the target stars are faint, the subtraction of signal from the night sky – both continuum and line emission – is a crucial part of the reductions. To facilitate this, within each 2dF field, 25 dedicated fibres are used to measure the night-sky flux across the observed spectral range. Sky fibre locations are selected by configure from a list of 150 possible locations in each field, which were cross-checked against the photometric catalogues available at the time of observation (DES or *Gaia*) to ensure no sources are located within a radius of 10 arcsec from the fibre position. During the reduction process, we discard any sky fibres where there are indications of non-sky signal present.\nThe sky-subtraction process must take into account fibre-to-fibre throughput variations; in, the relative throughputs of each fibre are determined using night sky emission features."},"data_id":{"kind":"string","value":"arxiv_0002841"}}},{"rowIdx":695,"cells":{"content":{"kind":"string","value":"STATISTICAL INFERENCE FOR THE PENALIZED EM ALGORITHM TO TEST DIFFERENTIAL ITEM FUNCTIONING\n==========================================================================================\nRecent advancements in testing differential item functioning (DIF) have greatly relaxed restrictions made by the conventional multiple group item response theory (IRT) model with respect to the number of grouping variables and the assumption of predefined DIF-free anchor items. The application of the *L*1 penalty in DIF detection has shown promising results in identifying a DIF item without a priori knowledge on anchor items while allowing the simultaneous investigation of multiple grouping variables. The least absolute shrinkage and selection operator (LASSO) is added directly to the loss function to encourage variable sparsity such that DIF parameters of anchor items are penalized to be zero. Therefore, no predefined anchor items are needed. However, DIF detection using LASSO requires a non-trivial model selection consistency assumption and is difficult to draw statistical inference. Given the importance of identifying DIF items in test development, this study aims to apply the decorrelated score test to test DIF once the penalized method is used. Unlike the existing regularized DIF method which is unable to test the statistical significance of a DIF item selected by LASSO, the decorrelated score test requires weaker assumptions and is able to provide asymptotically valid inference to test DIF. Additionally, the deccorrelated score function can be used to construct asymptotically unbiased normal and efficient DIF parameter estimates via a one-step correction. The performance of the proposed decorrelated score test and the one-step estimator are evaluated via a Monte Carlo simulation study.\nDifferential item functioning, Moderated nonlinear factor analysis, LASSO, *L*1 penalty, LASSO, Score test, Penalized EM algorithm\nIntroduction\n============\nA major goal of educational and psychological assessment is to create interpretable scores on a relevant construct being measured. A well-constructed test score representing a latent trait (e.g., proficiency level), should be reliable and valid. One major threat to the validity of the intended use of scores from an instrument is the lack of measurement invariance or existence of *differential item functioning* (DIF). Sometimes, characteristics of a test may introduce unintended systematic score differences between individual test takers or subgroups of examinees of the same ability level, which can result in invalid score interpretation. Therefore, DIF detection should be conducted for routine operations of psychological and educational assessment to ensure valid interpretations of test scores.\nItem response theory (IRT), a collection of mathematical and statistical models, wasand still remainsthe foundational modeling system in educational assessment. One advantage of an IRT model in conducting a DIF analysis is that it offers a formal statistical framework for assessing DIF. Despite the popularity and advantages of DIF detection using IRT models, the inherent assumptions therein make practical applications of DIF analyses challenging. For example, IRT DIF detection methods usually explicitly or implicitly assume the existence of *predefined* DIF-free items[1](#fn1), also known as anchor items. Typically, DIF-free items are needed to anchor the latent scale so that DIF can be distinguished from the between group difference in latent trait distributions. Correctly specifying anchor items is crucial in correctly identifying a DIF item. Violations to the DIF-free anchor assumption may lead to inflated false detection rates in finding DIF items. A practical obstacle is locating DIF-free anchor items without a priori information. Another challenge is detecting DIF items associated with multiple grouping variables. Conventional IRT DIF detection methods typically investigate DIF with respect to one categorical grouping variable (e.g., gender, ethnicity, SES) at a time. However, it is impossible to investigate the complex nature of DIF due to interconnected background characteristics and thus it is unlikely to truly uncover the source of DIF.\nAs a solution to these practical difficulties encountered by conventional IRT DIF methods, regularization has been applied to the latent regression modeling framework to detect DIF items. In particular, the use of the *L*1 penalty or the LASSO has shown promising results in DIF detection due to its efficiency in variable selection <e.g., >bauer2017more,belzak2020improving,magis2015detection,tutz2015penalty. For example, the *L*1 penalty has been successfully applied to moderated nonlinear factor analysis (MNLFA) model to identify DIF items. Such a modeling framework offers greater flexibility to detect DIF for multiple grouping variables simultaneously. Item parameters and population distribution parameters, upon applying suitable link functions, can be expressed as linear or nonlinear functions of multiple person characteristics. Consequently, DIF detection can be treated as a variable selection problem—non-zero coefficients are only assigned to covariates that cause DIF. The goal of the penalty procedure is to encourage sparseness as much as possible with respect to DIF parameters such that DIF parameters of anchor items are penalized to be zero while those of DIF items are non-zero. Hence, no predefined anchor items are needed as items with DIF parameters penalized to be zero are used as anchors.\nDespite the flexibility of the *L*1 penalty for DIF detection, several concerns need to be addressed carefully. First, the accuracy of DIF detection based on the *L*1 penalty is contingent on non-trivial assumptions. In order for LASSO to select the right covariate for each item, and thus, items exhibiting true DIF, a variable selection consistency condition is needed. In the current case, failing to meet this condition could result in mistakingly flagging non-DIF items or miss real DIF items even in large samples. Second, with a finite sample size especially those encountered in the social sciences, inferential statistics such as confidence interval estimates and p-values are extremely important to differentiate a true DIF item from a sampling error. For example, a common practice in DIF detection in educational and psychological assessments is effect size reporting after DIF is detected. In this case, an uncertainty measure may prove to be helpful in deciding whether a DIF effect size is truly different from zero. Nevertheless, it is difficult to draw statistical inference based on the LASSO-type estimator due to shrinkage resulting from penalization. Lastly, LASSO creates biased parameter estimates and non-normal limiting distribution of the parameter estimates, which makes subsequent analyses difficult. For example, DIF effect sizes, item parameter estimates, and standard error estimates cannot be directly used to quantify uncertainty in latent score estimates.\nPurpose of the Study\n====================\nThe current study aims to fill the gap in the literature to make inferential claims once LASSO is used. Specifically, a decorrelated score test is proposed to detect DIF for binary response data with multiple covariates after the *L*1 penalty is used. The decorrelated score test could potentially be extended to accommodate continuous and discrete response data and multidimensional latent variables. In the current study, the simplest and the most commonly used case was considered to better demonstrate the extension of the decorrelated score test to detect DIF. Unlike the existing regularized DIF method, the decorrelated score test does not require the variable selection consistency assumption and is able to provide valid inference on DIF effects. Specifically, a sparse score vector with respect to the focal parameter is estimated consistently so that the resulting score test statistic has an asymptotically normal reference distribution. Additionally, an asymptotic unbiased estimator can be constructed using a one-step bias correction using the decorrelated score function.\nA Monte Carlo simulation study is conducted to examinee the finite sample behavior of the decorrelated scodre test. Its performance in controlling the Type I error rate, establishing sufficient power, controlling the false detection rate in identifying a DIF item under different DIF-related conditions are compared with three methods: (1) regularization method based on LASSO selection only, (2) a naive model refitting method, and (3) the oracle solution assuming known anchors.\nDecorrelated Score Test to Test DIF\n===================================\nMathematically, DIF can be defined as follows. Considering *n* independent and identically distributed multivariate vectors, $\\mathfrak{Y}=(\\YY\\_1,\\cdots,\\YY\\_n)^\\top$ and $\\YY\\_i=(Y\\_{i1},\\dots, Y\\_{iJ})^\\top$, following a statistical model $\\mathcal{P}=\\{P\\_{\\xxi}:\\xxi \\in \\Xi\\}$, where $\\xxi$ is a *d* − dimensional vector of unknown parameters and $\\Xi \\subset \\real^d$ is the parameter space. The categorical random variable *Y**i**j* representing the item response from person *i*, *i* = 1, ⋯, *n* to item *j* ∈ J = {1, …, *J*}, define the conditional item response function, denoted as $f\\_j(y|\\theta,\\xx)=P(Y\\_{ij}=y|\\theta\\_i=\\theta,\\xx\\_i=\\xx)$, as the probability of endorsing a particular answer conditional on the person’s latent ability level $\\theta\\_i=\\theta\\in\\real$ and person covariate $\\xx\\_i\\in\\real^K$. More broadly, item *j* exhibits DIF if\n \n$$\\begin{aligned}\n \\label{eq: nonpar-dif}\n f\\_j(y|\\theta,\\xx) \\neq f\\_j(y|\\theta, \\tilde{\\xx}),\\end{aligned}$$ \nfor some $\\xx \\neq \\tilde{\\xx}\\in\\real^K$. Equation [eq: nonpar-dif] indicates that DIF exists when the conditional item response function *f**j* differs for any $\\xx \\neq \\tilde{\\xx}$ after controlling for the same level of latent ability. In other words, measurement invariance holds for item *j*, only when $ f\\_j(y|\\theta,\\xx) =\nf\\_j(y|\\theta, \\tilde{\\xx})$ for all $\\xx \\neq \\tilde{\\xx}$.\nModerated Nonlinear Factor Analysis Model\n-----------------------------------------\nThe moderated nonlinear factor analysis model (MNLFA) assumes that the item response for item *j* from person *i*, (i.e., *Y**i**j* ∈ {0, 1}) follows a Bernoulli distribution ($ Y\\_{ij}|\\theta\\_i\n\\sim \\hbox{Bern}(P\\_{ij}(\\theta\\_i))$), where *P**i**j* denotes the probability of endorsing item *j* for person *i*. Incorporating vector of person covariates $\\xx\\_i$ into the conditional item response function (cIRF) to affect both the item response and the latent distribution offers greater flexibility in testing DIF across different levels of both continuous and categorical variables. It also permits testing DIF for multiple grouping variables as well as their interaction effects (e.g., gender × age) at the same time. Using the same notation, the cIRF *f**j* is written as\n \n$$\\begin{aligned}\n\\label{eq:cIRF}\n f\\_j(Y\\_{ij}|\\theta,\\xx)&=P\\_{ij}^{Y\\_{ij}}(1-P\\_{ij})^{1-Y\\_{ij}},\\,\n\\hbox{where}\\nonumber\\\\\n P\\_{ij}=P(Y\\_{ij}&=1|\\theta\\_i=\\theta,\n\\xx\\_i=\\xx)=\\frac{1}{1+\\exp[-\\alpha\\_j(\\xx)-\\beta\\_j(\\xx)\\theta]},\n \\end{aligned}$$ \nwhere $\\alpha\\_j(\\xx)$ and $\\beta\\_j(\\xx)$ are the item intercept and item slope functions, respectively, which can be further expressed as functions of the vector of person covariate $\\xx\\in \\real^{K}$. Latent ability *θ**i* is assumed to be normally distributed: $\\theta\\_i\\sim \\mathcal{N}(\\mu(\\xx),\\sigma^2(\\xx))$, in which the latent population parameters: mean (*μ*) and variance (*σ*2) can also be expressed as functions of the person covariate vector $\\xx$ defined as follows:\n \n$$\\begin{aligned}\n \\label{eq:blatent}\n \\mu(\\xx)=\\ggamma^\\top\\xx \\quad \\sigma^2(\\xx)=\\exp(\\ddelta^\\top\\xx),\\end{aligned}$$ \nwhere *K*-dimensional vectors $\\ggamma$ and $\\ddelta$ represent effects of the person covariate vector $\\xx$ on the latent mean and variance on a logarithmic scale. These two vectors are further referred to as latent population parameters. Also, item intercept and item slope functions are expressed as\n \n$$\\begin{aligned}\n \\alpha\\_j(\\xx)&=d\\_j+\\bbeta\\_{0j}^\\top\\xx, \\label{eq:unif} \\\\\n \\beta\\_j(\\xx)&=a\\_j+\\bbeta\\_{1j}^\\top\\xx \\label{eq:nonunif},\n \\end{aligned}$$ \nwhere *d**j* and *a**j* are the item location and item slope parameters. The *K*-dimensional vectors $\\bbeta\\_{0j}$ and $\\bbeta\\_{1j}$ stand for person covariate effects on the item intercept and item slope parameter, respectively. To detect DIF for item *j*, a composite hypothesis test is typically conducted to test against the null hypothesis\n \n$$\\begin{aligned}\n \\label{eq:null}\n \\hbox{H}\\_0: \\, \\bbeta\\_{0j}=\\bbeta\\_{1j}=\\bf{0}.\n \\end{aligned}$$ \nTypically, a minimal constraint is needed to identify the MNLFA model. This requires that there exists at least one item for each column of $\\bbeta\\_{0j}$ and $\\bbeta\\_{1j}$ is zero/anchor. Without predefined anchor items or constrained parameters, the MNLFA model can be estimated using the Penalized Expectation Maximization (EM) Algorithm and DIF items are those items with remaining DIF parameters.\nRegularized Differential Item Functioning\n-----------------------------------------\nThe marginal likelihood function $f(\\yyi|\\xx\\_i)$ of an observed individual response vector $\\yyi=(y\\_{i1},\\dots,y\\_{1J})^\\top$ assuming conditional independence among item responses given the latent variable and person covariates $\\xx\\_i=(x\\_{i1},\\dots,x\\_{iK})^\\top$ is expressed as\n \n$$\\begin{aligned}\n \\label{eq:marginal}\n f(\\yyi|\\xx\\_i)=\\int\\prod\\_{j=1}^J f\\_j(y\\_{ij}|\\theta\\_i,\\xx\\_i)\\phi(\\theta\\_i|\\xx\\_i)d\\theta\\_i,\n \\end{aligned}$$ \nin which *ϕ*( ⋅ ) is the probably density function of a normal distribution governed by the population parameters $\\ggamma$ and $\\ddelta$. Assuming that the item response vectors Y is independent and identically distributed (i.i.d), the sample likelihood function is written as\n \n$$\\begin{aligned}\n \\label{eq:likelihood}\n f\\_n(\\YY|\\XX)=\\prod\\_{i=1}^nf(\\yyi|\\xx\\_i),\n \\end{aligned}$$ \nwhere $\\YY=(\\yy\\_1,\\cdots,\\yy\\_n)^\\top$ and $\\XX=(\\xx\\_1,\\cdots,\\xx\\_n)^\\top$ represent observed data matrices. Subsequently, the corresponding marginal log-likelihood is expressed as\n \n$$\\begin{aligned}\n \\label{eq:ln}\n \\ell\\_n(\\xxi; \\YY|\\XX)=\\frac{1}{n}\\log f\\_n(\\YY|\\XX)=\\frac{1}{n}\\sum\\_{i=1}^n\\log f(\\yyi|\\xx\\_i),\n \\end{aligned}$$ \nwhere $\\xxi = (a\\_1, d\\_1, \\bbeta\\_{01}^\\top, \\bbeta\\_{11}^\\top, \\dots, a\\_J, d\\_J, \\bbeta\\_{0J}^\\top, \\bbeta\\_{1J}^\\top,\\ggamma^\\top,\\ddelta^\\top)^\\top$ is a collection of model parameters. To estimate the model using the penalized EM algorithm, the *L*1 penalty function is added directly to the loss function\n \n$$\\begin{aligned}\n \\label{eq:pmlikn}\n p\\_n(\\xxi; \\YY, \\lambda|\\XX) =-\\ell\\_n(\\xxi;\\YY|\\XX)+\\lambda\\|\\bbeta\\_j\\|\\_1,\n \\end{aligned}$$ \nwhere $\\bbeta\\_j=(\\bbeta\\_{0j}^\\top,\\bbeta\\_{1j}^\\top)^\\top$ denotes all DIF parameters, $\\ell\\_n(\\xxi;\\YY|\\XX)$ is the sample log-likelihood expressed in Equation [eq:ln] and ∥∥1 indicates the *L*1 norm that sums absolute values of all elements weighted by the penalty weight *λ*. Then the local minimizer of $ p\\_n(\\xxi; \\YY, \\lambda|\\XX)$ with respect to $\\xxi$\n \n$$\\begin{aligned}\n \\hat{\\xxi} &= \\operatorname\\*{\\arg\\min}\\_{\\xxi} p\\_n(\\xxi;\\YY,\\lambda|\\XX)\\end{aligned}$$ \ncan be estimated by the penalized EM algorithm, which is documented in Appendix A.\nIn the upcoming section, the decorrelated score test will be introduced. The intuition about the decorrelated score test resembles the efficient score test when the model is identified. Thus, the efficient score test for an identified model will first be introduced followed by the decorrelated score test when the model is not identified with redundant parameters.\nDecorrelated Score Test\n-----------------------\nPartition model parameters $\\xxi$ into $\\xxi =(\\ppsi,\\eeta^\\top )^\\top$, where $\\ppsi$ is the *d*0 − dimensional vector of focal parameters and $\\eeta^\\top $ stands for *d*1 = (*d* − *d*0) − dimensional vector of nuisance parameters. Given the loss function $\\ell(\\xxi,\\YY)=-\\ell\\_n(\\xxi,\\YY)$, which is the average sample log-likelihood, define $\\II=\\operatorname{\\mathbb{E}}\\_{\\xxi}(\\nabla^2\\ell(\\xxi,\\YY))$. Let $\\xxi^\\*$ be the true value of $\\xxi$. Similarly, denote $\\II^\\*=\\operatorname{\\mathbb{E}}\\_{\\xxi^\\*}(\\nabla^2\\ell(\\xxi^\\*,\\YY))$. Note that for the rest of the study, *d* is fixed. Let’s first consider the case when the model is identified. For example, an MNLFA model with sufficiently large subset of known anchors. The asymptotic normality of the ML estimator [$\\hat{\\xxi}=\\hbox{argmin}\\_{\\xxi \\in \\Xi}\\ell(\\xxi,\\YY)$] follows from a first-order Taylor series expansion of the score function at $\\xxi^\\*$,\n \n$$\\begin{aligned}\n \\bold{0}&=\\nabla\\_{\\xxi}\\ell(\\hat{\\xxi},\\YY)=\\nabla\\_{\\xxi}\\ell({\\xxi^\\*},\\YY)+\\nabla^2\\_{\\xxi,\\xxi}\\ell({\\xxi^\\*},\\YY)(\\hat{\\xxi}-\\xxi^\\*)+R\\_n\\end{aligned}$$ \nwhere the remainder term is of the form $R\\_n=\\frac{1}{2}(\\hat{\\xxi}-\\bar{\\xxi})^T\\nabla\\_{\\xxi,\\xxi}^3\\ell(\\bar{\\xxi},\\YY)(\\hat{\\xxi}-\\bar{\\xxi})$ and $\\bar{\\xxi}$ is in between $\\hat{\\xxi}$ and $\\xxi^\\*$. By rearranging this equation,\n \n$$\\begin{aligned}\n \\sqrt{n}\\,(\\hat{\\xxi}-\\xxi^\\*)&=-\\sqrt{n}\\,\\II^{\\*-1}\\nabla\\_{\\xxi}\\ell({\\xxi^\\*},\\YY)+\\sqrt{n}R\\_n\\\\\n \\Leftrightarrow \\sqrt{n}\\,\\begin{pmatrix}\n \\hat{\\ppsi} -\\ppsi^\\*\\\\ \\hat{\\eeta} -\\eeta^\\*\n \\end{pmatrix}&=-\\sqrt{n}\\,\\begin{pmatrix}\n \\II^\\*\\_{\\ppsi,\\ppsi}& \\II^\\*\\_{\\ppsi,\\eeta}\\\\\n \\II^\\*\\_{\\eeta,\\ppsi}&\\II^\\*\\_{\\eeta,\\eeta}\n \\end{pmatrix}^{-1}\\begin{pmatrix}\n \\nabla\\_{\\ppsi}\\ell({\\xxi^\\*},\\YY)\\\\\n \\nabla\\_{\\eeta}\\ell({\\xxi^\\*},\\YY)\n \\end{pmatrix}-\\sqrt{n}R\\_n,\\end{aligned}$$ \nwhere subscripts of $\\II^\\*$ are corresponding partitions of the matrix. Drop $\\YY$ from the rest of the notations. As $\\II\\_{\\eeta,\\eeta}$ is invertible, using the block inverse formula, it follows that\n \n$$\\begin{aligned}\n \\sqrt{n}\\,(\\hat{\\ppsi}-\\ppsi^\\*)=-\\sqrt{n}\\,\\II^{\\*-1}\\_{\\ppsi|\\eeta}s({\\xxi^\\*})+\\sqrt{n}R\\_n,\\quad R\\_n= o\\_p(1/\\sqrt{n})\n \\end{aligned}$$ \nwhere $\\ss({\\xxi^\\*})=\\nabla\\_{\\ppsi}\\ell(\\xxi^\\*)-\\II^\\*\\_{\\ppsi\\eeta}\\II^{\\*-1}\\_{\\eeta\\eeta}\\nabla\\_{\\eeta}\\ell(\\xxi^\\*)$ and $\\II^\\*\\_{\\ppsi|\\eeta}=\\II^\\*\\_{\\ppsi\\ppsi}-\\II^\\*\\_{\\ppsi\\eeta}\\II^{\\*-1}\\_{\\eeta\\eeta}\\II^\\*\\_{\\eeta\\ppsi}$ are the efficient score and efficient information, respectively. Note that efficient score can be interpreted as the projection of the $\\nabla\\_{\\ppsi}\\ell(\\xxi)$ to the orthogonal complement of the score function with respect to the nuisance parameters. Under $H\\_{0}: \\ppsi^\\*=\\mathbf{0}$, it holds that $n\\nabla\\_{\\ppsi}(\\mathbf{0},\\hat{\\eeta})^{\\top}[\\hat{\\II}\\_{\\ppsi|\\eeta}]^{-1}\\nabla\\_{\\ppsi}(\\mathbf{0},\\hat{\\eeta})\\overset{\\mathcal{D}}{\\rightarrow}\\chi^2\\_{d\\_0}$, where $\\hat{\\eeta}=\\hbox{argmin}\\_{\\eeta}\\ell(\\mathbf{0},\\eeta)$ is constrained parameter estimates. It is straightforward to see that the estimated efficient score ($\\ss(\\mathbf{0},\\hat{\\eeta})$) equals $\\nabla\\_{\\ppsi\n}(\\mathbf{0},\\hat{\\eeta})\n$ due to the fact that $\\nabla\\_{\\eeta}\\ell(\\mathbf{0},\\hat{\\eeta})=\\mathbf{0}$.\nHowever, when the model is not identified, $\\II\\_{\\eeta,\\eeta}$, in general, is not invertible. Intuitively, there exists redundant parameters in the model. A natural extension of the above case when the model is not identified is estimating an “efficient score” so that the influence of entries of the redundant parameters in the nuisance score is minimized. Following the same logic as the efficient score, a sparse score can be estimated by projecting $\\nabla\\_{\\ppsi}\\ell(\\xxi)$ to the orthogonal complement of a low-dimensional subspace spanned by the non-redundant part of the nuisance score vector. Therefore, a sparse vector/matrix is needed to find the best sparse linear combination of $\\nabla\\_{\\eeta}\\ell(\\xxi)$ to approximate $\\nabla\\_{\\ppsi}\\ell(\\xxi)$. Mathematically, the decorrelated score, or the extension of the efficient score when the model is not identified, has no correlation with the score function with respect to the nuisance score (i.e., $\\operatorname{\\mathbb{E}}(\\ss(\\xxi)^\\top\\nabla\\_{\\eeta}\\ell(\\xxi))=\\mathbf{0}$). Geometrically, the decorrelated score has the same interpretation which is the projection of $\\nabla\\_{\\ppsi}\\ell(\\xxi)$ to the orthogonal complement of the linear space spanned by the nuisance score function $\\nabla\\_{\\eeta}\\ell(\\xxi)$. Following this intuition, define the decorrelated score function as\n \n$$\\begin{aligned}\n \\label{eq:dscore}\n\\ss(\\xxi) & = \\nabla\\_{\\ppsi}\\ell(\\xxi)-\\WW^\\top\\nabla\\_{\\eeta}\\ell(\\xxi), \\hbox{ where } \\WW^{\\top}= \\II\\_{\\ppsi\\eeta}\\II^{-1}\\_{\\eeta\\eeta} \\in \\real^{d\\_0\\times d\\_1}.\n \\end{aligned}$$ \nTo find the projection of $\\nabla\\_{\\ppsi}\\ell(\\xxi)$ to the orthogonal complement of the linear space span by the nuisance score function $\\nabla\\_{\\eeta}\\ell(\\xxi)$, the sparse matrix $\\WW$ can be estimated using Algorithm [alg:dscore] (see Table [tab:algorithm1]). As can be seen, the key is to estimate a sparse matrix $\\hat{\\WW}=(\\hat{\\WW}\\_{\\*1},\\cdots,\\hat{\\WW}\\_{\\*d\\_0})$ column by column to construct the decorrelated score ($\\ss(\\ppsi,\\eeta)$) so that the additional and redundant parameters do not influence the decorrelated score. Further the Dscore test statistic can be constructed\n \n$$\\begin{aligned}\n \\hat{T}\\_{Dscore}=n\\hat{\\ss}(\\mathbf{0},\\eeta)^{\\top}[\\hat{\\II}\\_{\\ppsi|\\eeta}]^{-1}\\hat{\\ss}(\\mathbf{0},\\eeta),\\end{aligned}$$ \nwhere $\\hat{\\II}\\_{\\ppsi|\\eeta}=\\nabla^2\\_{\\ppsi,\\ppsi}\\ell(\\hat{\\xxi})-\\hat{\\WW}^{\\top}\\nabla^2\\_{\\eeta,\\ppsi}\\ell(\\hat{\\xxi})$. Under some technical assumptions[2](#fn2), it can be proved that\n \n$$\\begin{aligned}\n\\label{eq:Fdscore}\n \\hat{T}\\_{Dscore}\\xrightarrow{\\cal{D}} \\chi^2\\_{d\\_0}.\\end{aligned}$$ \nMoreover, a one-step asymptotical unbiased estimator $\\tilde{\\ppsi}$ can be constructed using a single Newton step as\n \n$$\\begin{aligned}\n \\label{eq:onestepubiased}\n \\tilde{\\ppsi}=\\hat{\\ppsi}-\\hat{\\II}^{-1}\\_{\\ppsi|\\eeta}\\hat{\\ss}(\\hat{\\xxi})\\end{aligned}$$ \nand with assumptions [asp:1] to [asp:4] documented in Appendix [ch:Appendixb], showed that\n \n$$\\begin{aligned}\n \\label{eq:Funbiased}\n\\sqrt{n}(\\tilde{\\ppsi}-\\ppsi^\\*)\\xrightarrow{\\cal{D}}{{\\cal N}(0,\\II\\_{\\ppsi|\\eeta}^{\\*-1})}.\\end{aligned}$$ \nSubsequently, the (1 − *α*) × 100% confidence interval can be constructed for a linear combination of $\\ppsi^\\*$ (i.e., $\\mathbf{c}^{\\top}\\ppsi^\\*$ where **c**⊤ is a *d*0 − dimensional constant vector) as $[\\tilde{\\ppsi}-n^{-1/2}\\Phi^{-1}(1-\\alpha/2)(\\mathbf{c}^\\top\\hat{\\II}\\_{\\ppsi|\\eeta}\\mathbf{c})^{-1/2},\\tilde{\\ppsi}+n^{-1/2}\\Phi^{-1}(1-\\alpha/2)(\\mathbf{c}^\\top\\hat{\\II}\\_{\\ppsi|\\eeta}\\mathbf{c})^{-1/2}]$.\nMonte Carlo Simulation\n======================\nIn order to investigate the finite sample behavior of the proposed Dscore test in testing DIF under different conditions, a Monte Carlo study is conducted. Performance of the proposed Dscore test is evaluated in comparison to three methods: (1) the Wald test assuming known and correctly specified anchors, (2) the Reg-DIF method (i.e., results based on the LASSO selection only), and (3) and the naive model refitting approach. The simulation study design and evaluation criteria are discussed in Section [ss:studydesign] and Section [ss:evaluationcriterion], respectively. All computations are performed in R. The source code will be made publicly available through the Open Science Framework (OSF) upon acceptance of the manuscript.\nStudy Design\n------------\nBinary response data are generated from a unidimensional MNLFA model under two conditionswith or without DIF items. Two factors are manipulated including (1) the total sample size (*n* = 500, 1, 000, & 2, 500) and (2) the percentage of DIF items (0%, 25%, & 50%). As a result, there are 3 × 3 = 9 fully crossed conditions. These manipulating factors are selected due to their relevance to the DIF detection mechanism of the Dscore test. Moreover, actual values of the manipulated factors were chosen to conform to real-world data analytic scenarios and align with previous methodological studies <e.g.,>bauer2017more,belzak2020improving. Due to the limited DIF research on the topic, model parameters were generated based on a real data analysis using the UK normative sample data of the Revised Eysenck Personality Questionnaire <EPQ-R, >eysenck1985revised[3](#fn3). True data generating values of these DIF items are described in details in the following section.\nThe True Data Generation Model\n------------------------------\nThe binary response for examinee *i* for item *j*, denoted *Y**i**j* ∈ {0, 1}, *i* = 1, …, *n*, *j* = 1, …, *J*, was generated from a MNLFA model (see Equation [eq:cIRF]). The total number of items was fixed at *J* = 12. Model parameters of the data generating model are tabulated in Table [tab:truepar]. Specifically, item discrimination and intercept parameters are the ML estimates of item parameters using the EPQ-R data with five items identified as anchors. Three person covariates are considered in the study which mimic gender, age, and their product. Specially, the dichotomous grouping variable is generated from a Bernoulli distribution with success probability of.5 (i.e., $x \\sim \\hbox{Bern}(0.5)$). Then, for those in category 1, age is generated from N(0.2, 1). Otherwise, age is generated from N(0, 1), which creates a correlation between age and gender of 0.1. An interaction effect is then created by multiplying age and gender. By including an interaction effect, the correlations between the interaction effect and the two variables is large ( > .6). Furthermore, three types of DIF items were generated. Under the 25% DIF condition, items 1 to 3 are DIF items representing items with large, medium, and small DIF effect sizes. When the proportion of DIF items is increased to 50%, items 4 to 6 are added to the DIF set and their DIF effect sizes replicate those of items 1-3.\nTo visualize the generated items, Figure [fig:truepar] displays the true probability of each person endorsing items with the condition of 50% DIF items and *n* = 2, 500. Plots on the left correspond to DIF items whereas those on the right visualize non-DIF items. Gender influence on the item response can be seen from different colors and shapes.\nLastly, the latent variable follows a normal distribution ${\\cal N}(\\ggamma^\\top \\xx,\\ddelta^\\top \\xx)$ conditional on $\\xx$, and similarly true population parameter values are shown in Table [tab:truepar], which are also enlightened by real data. A total of 500 replications for each condition are implemented ensure that the 95% normal approximation confidence band at the nominal level 0.05 to be [0.031, 0.069].\nEstimation\n----------\nFor each replication under each condition, the following four methods are fitted to the binary response data to test DIF at the item level and the parameter level. MNLFA models are estimated using penalized maximum likelihood for the Reg-DIF method and the Dscore method or ML for the refit method and oracle solution. For both estimation methods, integration with respect to the latent variable is approximated by a 49-point Gauss–Hermite quadrature. The four methods are discussed in detail here.\nFirst, each item is tested for DIF based on LASSO selection only (i.e., the Reg-DIF method). The MNLFA model fitted with all three grouping covariates is estimated using the penalized EM algorithm. To select the optimal tuning parameter under each replication, typically BIC is used in latent variable models. However, the current study fixed *λ* at an estimated optimal value (i.e., $c\\sqrt{1/N}$ where the constant *c* is estimated by $\\hat{c} = \\sqrt{n}\\sum^{R}\\_{r=1}(\\hat{\\lambda}\\_r)/R$)[4](#fn4). The conditions with *n* = 500 were used to estimate the constant. Results show that *ĉ* = 0.8291, 0.6883, 0.5727 when the number of DIF items is 0, 3, and 6, respectively. Therefore, *λ* is fixed at the estimated optimal value for each condition. For the reg-DIF method, an item does not exhibit DIF if both $\\bbeta\\_{0j}$ and $\\bbeta\\_{1j}$ are zero vectors. Otherwise, the item is considered as a DIF item.\nNext, the naive model refitting method can be applied by refitting the same MNLFA model with the anchors selected using method 1. This model is estimated using the marginal ML estimation method with the EM algorithm. The marginal likelihood function can be approximated using the same configuration as in the penalized EM algorithm. The convergence tolerance for the log-likelihood change for the penalized EM algorithm and the M-step are set to be 10− 4 and 10− 6, respectively. The maximum number of iterations is set to be 500. Additionally, the Dscore test is conducted after the initial run (method 1) using the penalized EM algorithm. Details of the Dscore test was described in Section [ss:dscoretest] and thus are not repeated here. A critical step to conduct the Dscore test is to estimate $\\hat{\\WW}$ as displayed in Equation [eq:w]. As the theory only requires that *λ* and *λ*′ are of the same rate, *λ*′ will be set to the selected *λ* value in the initial parameter estimation step to speed up the computation and save time. and have found that the decorrelated score test is not sensitive to *λ*′ and both fixed $\\lambda^\\prime=0.5\\sqrt{\\log d/n}$ in their simulation studies. Furthermore, asymptotic unbiased parameter estimates can be estimated using the one-step bias correction (see Equation [eq:onestepubiased]). Note that although the focal parameters can be multidimensional or unidimensional (i.e., the debias step can be conducted at the item level or parameter level), the current study investigates the one-step bias correction by parameter type. For instance, for each item, the bias correction treats $(a\\_j, d\\_j, \\bbeta\\_{0j}, \\bbeta\\_{1j})^\\top$ as the focal parameters and everything else as nuisance parameters. The population parameter estimates are corrected at the same time by treating all item parameters as nuisance parameters.\nLastly, to compare the performance of the Dscore test with the oracle solution assuming anchors are known, Wald tests are performed to test DIF at the item-level. For each replication and each condition, item 11 and item 12 are treated as anchors. In addition, final model parameter estimates and their corresponding standard errors are estimated using anchors selected by the Wald test.\nEvaluation Criteria\n-------------------\nThe comparative inferential performance of the four methods is evaluated in terms of (1) hypothesis testing in testing a DIF item measured by rejection rates at the 0.05 level, (2) parameter recovery including bias and variance of model parameter estimates including a-DIF and d-DIF, and (3) recovery of standard errors. Standard error estimates are evaluated by comparing the square root of the mean of the variance of parameter estimates against the empirical standard errors (i.e., Monte Carlo standard deviation of the parameter estimates).\nResults\n=======\nResults: Hypothesis Testing\n---------------------------\nIn this section, the Type I error rate, false positive rate (FDR), and power of detecting a DIF item are reported. To visualize the comparative performance, the empirical rejection rate at the nominal level 0.05 was plotted under (1) the null condition when there were no DIF items (i.e., Type I error), (2) the alternative condition when there was a mix of DIF and DIF-free items for DIF items only (i.e., Power), and (3) the alternative condition when there was a mix of DIF and DIF-free items for DIF-free items only (i.e., FDR).\n### Type I Error Rate\nFigure [fig:typeierror] shows the Type I error rate of detecting a DIF item under the null condition when there were no DIF items. The horizontal dashed lines show the 0.05 nominal level and the dotted lines represent the 95% normal approximation confidence band at the nominal level 0.05 across 500 replications. If the rejection rate falls within the 95% confidence band (i.e., 95% confidence interval =[0.031, 0.069]), the specific method is considered to have well-controlled Type I error rate of incorrectly detecting a DIF item. Larger values than the upper bound of the confidence band at the nominal level 0.05 represent over-rejecting the null hypothesis whereas smaller values than the lower bound of the confidence band indicate under-rejecting the null hypothesis.\nOverall, the Reg-DIF, model refit method, and decorrelated score test showed controlled Type I error under all manipulated sample size conditions as the rejection rate of all these methods fall within the 95% confidence band.\n### Power\nFigure [fig:power] shows the power of detecting DIF items under different conditions. The power can be negatively impacted by the sample size while more robust to the number of true DIF items. Overall, the decorrelated score test was more powerful than the Reg-DIF and the model refit methods across all conditions. The difference was more obvious under more challenging conditions where the sample size is small. The performance of the Reg-DIF method and the model refit method was nearly identical when the total number of DIF items was small. The difference between the two methods was more obvious when the number of DIF items was large. Specifically, the model refit method was slightly less powerful than Reg-DIF by design. The reason is that once an item is identified as an anchor item, its DIF parameters are excluded in the refitted model. In other words, this item can no longer be identified as a DIF item.\n### False Detection Rate\nFinally, results for incorrectly detecting an anchor item as a DIF item can be visualized using the empirical rejection rate for the true anchor item under alternative conditions. As displayed in Figure [fig:fdr], the performance of the decorrelated score test is outstanding as the FDR of the decorrelated score test falls within the 95% normal-approximation confidence band, indicating controlled FDR. Conversely, the Reg-DIF and model refit method under-rejected the null hypothesis when the number of DIF items is small. This is understandable, as was previously shown that the power of identifying the true DIF item was in general smaller than the decorrelated score test and the oracle solution. However, when the number of DIF items was large, the Reg-DIF method over-rejected the null hypothesis. As was previously illustrated, the model refit method is more conservative in rejecting a null hypothesis as compared to the Reg-DIF method. Intuitively, it makes sense that the FDR is more controlled as compared to the Reg-DIF method. Nevertheless, it is still overly rejecting the null hypothesis when the sample size is not sufficiently large. As the sample size increases, the FDR using the model refit method falls within the 95% confidence band at the nominal 0.05 level.\nResults: Parameter Recovery\n---------------------------\nTo investigate parameter recovery, the bias and variance of the parameter estimates using each method were computed. Discussion is focussed on the DIF parameters, the parameters that inferences will be drawn on. Bias and variance of other model parameter estimates including item slope (*a**j*), item intercept (*d**j*), and population parameters ($\\ggamma,\\ddelta$) using different methods are included in the supplementary materials. For all methods other than the one-step debiased estimator using the decorrelated score function, DIF parameters penalized to be 0 or not included in the final model are treated as 0 to compute bias and variance of an estimator.\n### Bias\nWhen there was no DIF, all methods recover model parameters well with bias generally less than 0.05. This is understandable because when there was no DIF, the Type I error rate, as was previously shown, was well-controlled. The penalized EM performed similarly as the EM since in most cases because *λ* was large enough to penalize all DIF parameters to be 0. Even if this does not happen, the model refit method is always fitting the correct model under the null condition.\nWhen there was a mix of DIF and non-DIF items, the performance of different methods varied. The Reg-DIF method often resulted in biased DIF parameters for the true DIF items (i.e., items 1 to 3 in the 3 DIF item condition and item 1 to 6 in the 6 DIF item condition) due to the shrinkage. The bias for the DIF parameters for the anchor items were relatively small. Compared with the Reg-DIF method, the model refit method reduced bias for the true DIF items by 3% to 92% depending on the item. Note that bias remained relatively large for the a-DIF parameter even sample size was large (see last row of the Figure [fig:biasax]) due to failing to select the right DIF-effect. It was more clear when the sampling distribution of non-zero DIF parameters was investigated (see Figure [fig:samplingdif]). As can be seen, the sampling distribution of the non-zero DIF effect (blue line) using the model refit method was bi-modal with one mode at 0 indicating that the effect was not selected by the LASSO and another non-zero mode.\nAs a comparison, the one-step debiased parameter estimate using the decorrelated score function performed remarkably well. DIF parameters were recovered well as the bias ranged from -0.03 to 0.06 across conditions. Its advantages in reducing bias due to shrinkage were more obvious as compared with the model refit method under smaller sample size or more DIF items conditions. Under the most difficult condition (i.e., small sample size and large number of DIF items), it performed equally well as compared to the oracle solution. More importantly, as the sample size increased, the bias decreased.\n### Variance of DIF Parameter Estimates\nFigure [fig:Varax] and Figure [fig:Vardx] present the variance of the DIF parameter estimates. Large values indicate more uncertainty of the parameter estimates while smaller values indicate less uncertainty. As the sample size increases, variance should, in general, decrease. Among all methods, the Reg-DIF method produced the least variable parameter estimates. The model refit method produced similarly variable a-DIF parameter estimates but notably more variable d-DIF parameter estimates as compared to the Reg-DIF especially for the DIF items. Lastly, the one-step debiased parameter estimate using the decorrelated score function was the most variable estimator for the a-DIF parameter as compared to all other methods especially when the sample size was small. However, it was less variable for the d-DIF parameter for the DIF item as compared with the model refit method.\nResults: Standard Error Estimates\n---------------------------------\nThe recovery of SE was calculated as the ratio between the square root of the mean of the variance of parameter estimates over the empirical standard deviation of the Monte Carlo parameter estimates. If the ratio is closer to 1, the method has a more valid uncertainty measure. One caveat when calculating the SE using the model refit method is that there is no valid SE estimates for DIF parameters penalized to be 0. If the Reg-DIF method does not select a specific DIF effect for a specific replication, the corresponding DIF covariate is excluded from the model refit method. In such cases, SE estimates of the unselected DIF effect is treated as 0. These arbitrary 0 SEs would artificially decrease the average of the estimated SEs across replications and thus should be interpreted with caution. Alternatively, omitting DIF effects penalized to be 0 will often exclude too many DIF effects across replications resulting in instability of the empirical SEs and inaccurate recovery SE measure[5](#fn5).\nThe SE recovery for the a-DIF and d-DIF parameters can be visualized in Figure [fig:SErecoveryax] to [fig:SErecoverydx]. As there are no valid SE estimates from the penalized EM alogrithm, the Reg-DIF method currently presented in the following figures are based on the SE estimates using the model refit method. Note that as some DIF parameters of certain items were never selected by the Reg-DIF method leaving the recovery of SEs unavailable for the model refit method. This is actually one of the disadvantages of using the model refit method that inference can only be drawn on the selected DIF parameters. Again, the model refit method tended to underestimate SEs of DIF parameters. The underestimation of the SEs of DIF effects could contribute to incorrectly detecting an anchor item as a DIF item as was shown previously shown in Figure [fig:fdr]. As a comparison, the decorrelated score method not only can recover SEs of non-zero DIF effects but also zero DIF effects. For a-DIF and d-DIF, SEs ranged from 0.95 to 1.08 and 0.90 to 1.02, respectively. Note that it appears that SEs of a-DIF and d-DIF effects for item 7 were underestimated. However, given that variances of a-DIF and d-DIF parameters for item 7 were strangely large due to the skewness of the sampling distribution, it makes sense that SEs of DIF parameters of item 7 seem to be underestimated.\nDiscussion and Conclusion\n=========================\nSummary\n-------\nOne of the fundamental issues and remaining challenges inherent in DIF detection is finding the correct anchor items. Although recent development in the DIF detection literature (such as using regularization in the MNLFA modeling framework) has shown promising results in detecting DIF items without predefined anchor items, issues such as inflated false detection rate and inability to draw valid inference still remain.\nThe goal of the current study is to apply the decorrelated score test to test DIF items based on the *L*1 − penalized maximum likelihood solution under the MNLFA modeling framework. Unlike DIF detection based on regularization only, the decorrelated score test is valid for all DIF and DIF-free items across all covariates. Additionally, it has been shown that the decorrelated score function can be further used to construct an asymptotically unbiased estimator. Furthermore, the simulation study has shown the comparative performance of the decorrelated score test, DIF detection using regularization only, the model refit method, and the Wald test assuming the correct anchor items in hypothesis testing and parameter recovery.\nOverall, the decorrelated score is advantageous than the other two anchor-free DIF detection methods in three aspects. First, the decorrelated score test shows more statistical power in detecting a DIF item while maintaining controlled false detection rate even compared with the oracle solution with two anchor items available. As summarized and highlighted by previous studies, the Reg-DIF method and the model refit method often result in inflated false detection rate in identifying a true DIF item especially when the sample size is small. Similar to these studies, the current simulation study also shows inflated false detection rate especially under the most difficult yet realistic conditions (i.e., when the sample size is small and the number of DIF items is large). As was previously discussed, the current simulation study also shows that the model refit method has similar performance as the decorrelated score test in correctly identifying a true DIF item while controlling for the Type I error when the sample size is large.\nAnother important finding from the current simulation study is that the one-step debiased estimator using the decorrelated score function yields less biased item and DIF parameter estimates. Specifically, as compared with the model refit method, the one-step debiased estimator recovers the item slopes, item intercepts, a-DIF, and d-DIF parameters well and sometimes even comparable to the oracle solution. As expected, the model refit method reduces the bias for model parameters estimated from the penalized EM algorithm. However, bias still remains. This is expected as LASSO seems to be more sensitive to detecting d-DIF rather than the a-DIF, a point that we return to at the end of this section, it is likely that a-DIF parameters are likely to be mistakenly unselected. Accordingly, the model refit method fits an incorrect model and, therefore, bias remains. Given the less biased item and DIF parameter estimates, it is recommended to use the one-step debiased estimator to produce reliable point estimates without fitting an additional MNFLA model using EM. However, less bias often indicates more parameter variability. This is evidenced by the fact that there are more uncertainty in the item parameter and DIF parameter estimates using the one-step debiased estimator based on the decorrelated score function. Another interesting observation is that consistent with the model refit method, the one-step debiased estimator always recover item intercept parameters better as compared to item slope parameters (see the supplementary materials). This finding is also aligned other studies <see>[for examples]bauer2020simplifying, belzak2021using that d-DIF parameters are in general recovered better than a-DIF parameters, which could potentially explain the superiority of the item intercept parameter recovery. Consequently, a-DIF and item slope parameters have better initial parameter estimates to be plugged into the bias correction function.\nLastly, the decorrelated score test is able to draw reliable statistical inference on a-DIF and d-DIF parameters for zero and non-zero effects, respectively, where the model refit method may fail or is not able to generate one when the DIF effect is zero. As discussed earlier, SEs cannot be computed if DIF effects are penalized to be zero. To this end, the one-step debiased estimator is the only method considered in the present study that can produce reliable statistical inference for the zero DIF effects. As for the remaining DIF effects that are not penalized to be zero, the model refit method often underestimate the SEs of the DIF effects leading to incorrect inferences. This may be of concern, if substantive researchers are interested in testing a-DIF and d-DIF separately. If so, the decorrelated score test offers promising and valid statistical test to identify DIF at both the item and the parameter levels.\nFuture Studies\n==============\nAlthough the proposed decorrelated score test and the one-step debiased estimator have shown promising results in detecting DIF items without predefined anchor items and providing valid uncertainty measures for DIF effects, limitations and issues still remain to be addressed by future studies.\nFirst, the simulation study has suggested that the one-step debiased estimator does not seem to reduce the bias of the penalized ML estimator and that the associated SEs are unreliably estimated using the efficient information. Given that the ill recovery with the uncertainty measure also happens with unpenalized item parameters, it is conjectured that additional penalty weights might be needed for the population parameter or a more careful selection of penalty weights is needed. Some preliminary investigation suggests that if the population parameter is penalized in the initial *L*1 penalization stage, the bias correction using the decorrelated score function shows better performance. Future studies are needed to examine the behavior of the decorrelated score function when population parameters are penalized. Alternatively, finding the optimal *λ*′ for the population parameter for each iteration might also improve the unpredicted behavior of SEs. In the current simulation study, *λ*′ is set to the fixed *λ* based on the estimated rate as the theory suggests $\\lambda \\asymp \\sqrt{1/n}$ and $\\lambda^{\\prime}\\asymp\\sqrt{1/n}$ are approximately the same rate to ensure the *l*2 error bound for initial parameter estimates. It might worth to explore the influence of *λ*′ on the SE recovery of the unpenalized parameter. Although and have reported that the performance of the decorrelated score test is insensitive to the actual *λ*′ value in the high-dimensional regression model and proportional hazard model, it is not clear whether *λ*′ will impact the one-step debiased estimator especially when there are unpenalized parameters in the model.\nSecond, as *λ* is critically important for the initial parameter estimates and the accuracy of DIF detection using the Reg-DIF method and the model refit method, future research is encouraged to investigate different model selection critera in addition to BIC. As mentioned in Section [s:simulation], a pilot study found that BIC calculated based on the penalized EM parameter estimates always selects a relatively smaller *λ* value when the DIF effect size is relatively large and thus results in too many false positives. The current study used a fixed *λ* value estimated from the rate calculated from the *n* = 500 condition based on the ML estimator which potentially avoids the issue of poor performance of BIC. This may also explains the relatively smaller inflated false detection rate and power as compared with other Reg-DIF studies <e.g.,>bauer2020simplifying,belzak2020improving. Given the sparse literature on the model selection accuracy in the regularized DIF framework, future research is needed to find viable and efficient penalty selection approaches.\nThird, future research is needed to investigate the impact of different penalty functions on the performance of the decorrelated score test in DIF detection and the asymptotic behavior of the debiased parameter estimator. The general theory provided in can be directly applied to many penalty functions <e.g., adaptive LASSO,>[]zou2006adaptive. Our simulation study along with the other studies <e.g.,>belzak2020improving showed similar differential performance in recovering different parameters and SEs. In addition, applying different penalty weights for a-DIF and d-DIF may yield better initial parameter estimates and thus improve the performance of the one-step debias estimator.\nLastly, DIF effect size measures can be critically important when deciding to remove, modify, or keep a flagged DIF item. In practice it might be more desirable or more realistic to flag items with large DIF impact, which makes effect size reporting more essential in DIF detection. Latent regression models such as MNLFA models provide a natural DIF effect size measure (i.e., the person covariate effect on the item slope $\\bbeta\\_{1j}$ and item intercept $\\bbeta\\_{0j}$). However, more thorough and careful study on the meaningful cut-off values are needed to facilitate decision making. Alternatively, item level or scale level effects can also be helpful and can be extended to the MNLFA modeling framework. For example, average unsigned difference <see>woods2011dif which calculates weighted differences in the expected response functions between the focal and reference groups can be extended to the MNLFA model for selected values or levels of person covariate to evaluate the magnitude of the DIF effect at the item level. In addition to the item level DIF impact, the differential test function (DTF) index or the expected total test score difference due to DTF can be computed to inform the overall DIF effect on the scale level.\nIn sum, the proposed decorrelated score test and its one-step debiased estimator based on the regularized moderated nonlinear factor analysis model offers a promising solution to the practical obstacles encountered by conventional IRT methods. Valid inference at the DIF parameter level opens doors for more complicated substantive research questions such as investigating the complex nature of DIF due to interconnection of background characteristics.\nAppendix A\n==========\nPenalized Expectation-Maximization Algorithm\n--------------------------------------------\nThe Expectation-Maximization (EM) algorithm is used to find the local minimizer of $ p\\_n(\\xxi; \\YY, \\lambda|\\XX)$ with respect to $\\xxi$. The basic idea of the penalized EM algorithm is to repeatedly approximate the upper bound of the penalized negative sample log-likelihood $p\\_n(\\xxi;\\YY,\\lambda|\\XX) \\leq q\\_n(\\xxi;\\xxi^{(r)}), r=0,1,2,\\cdots$ defined in Equation [eq:q] at each iteration (E-step) and obtain the optimizer (M-step). As a result, the alternation between the E-step and the M-step produces a sequence of parameter updates $\\xxi^{(r)}$. The final parameter estimates $\\hat{\\xxi}$ is referred to as the penalized ML estimator. A critical property of the EM algorithm is that the parameter estimate updates the penalized negative sample log-likelihood in a non-increasing fashion (i.e., $p\\_n(\\xxi^{(r+1)};\\YY,\\lambda|\\XX)-p\\_n(\\xxi^{(r)};\\YY,\\lambda|\\XX)\\leq q\\_n(\\xxi^{(r+1)},\\xxi^{(r)}) - q\\_n(\\xxi^{(r)},\\xxi^{(r)})\\leq 0$ where $q\\_n(\\xxi,\\xxi^{r})$ can be viewed as the upper bound of $p\\_n(\\xxi;\\YY,\\lambda|\\XX)$). Specifically, at iteration *r*, $q\\_n(\\xxi;\\xxi^{(r)})$ is defined as\n \n$$\\begin{aligned}\n \\label{eq:q}\n q\\_n(\\xxi;\\xxi^{(r)}) &= \\mathbb{E}\\_{(\\theta\\_i|\\xxi^{(r)},\\YY,\\XX)}\\left[p\\_n(\\xxi; \\YY;\\theta\\_i|\\XX)\\right]\\\\\n &=-\\frac{1}{n}\\sum\\_{i=1}^n\\mathbb{E}\\_{(\\theta\\_i|\\xxi^{(r)},\\yy\\_i,\\xx\\_i)}\\left[\\log f(\\xxi; \\yy\\_i;\\theta\\_i|\\xx\\_i)\\right] +\\ssum p\\_\\lambda(\\bbeta\\_j)\\\\\n &=-\\frac{1}{n}\\sum\\_i^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl(\\log \\left( f(\\yy\\_i|\\theta\\_i,\\xxi,\\xx\\_i)f(\\theta\\_i|\\xxi,\\xx\\_i) \\right) \\biggl) d \\theta\\_i \\nonumber\\\\\n \\quad &+\\ssum p\\_\\lambda(\\bbeta\\_j)\\end{aligned}$$ \nDropping $\\ssum p\\_\\lambda(\\bbeta\\_j)$ for simplicity, the first term of the right hand side (RHS) equals\n \n$$\\begin{aligned}\n &\\hbox{First term of the RHS}\\\\\n &=-\\frac{1}{n}\\sum\\_{i=1}^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl(\\sum\\_{j=1}^J\\log f\\_j(\\yy\\_{ij}|\\theta\\_i,\\xxi\\_j,\\xx\\_i)+\\log f(\\theta\\_i|\\ggamma,\\ddelta,\\xx\\_i) \\biggl) d \\theta\\_i\\\\\n&=-\\frac{1}{n}\\sum\\_{i=1}^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl(\\sum\\_{j=1}^J\\log f\\_j(\\yy\\_{ij}|\\theta\\_i,\\xxi\\_j,\\xx\\_i) \\biggl) d \\theta\\_i \\\\\n&\\quad -\\frac{1}{n}\\sum\\_{i=1}^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl ( \\log f(\\theta\\_i|\\ggamma,\\ddelta,\\xx\\_i) \\biggl) d \\theta\\_i\\\\\n&=-\\frac{1}{n} \\sum\\_{j=1}^J \\sum\\_{i=1}^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl(\\log f\\_j(\\yy\\_{ij}|\\theta\\_i,\\xxi\\_j,\\xx\\_i) \\biggl) d \\theta\\_i\\\\\n&\\quad -\\frac{1}{n}\\sum\\_{i=1}^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl ( \\log f(\\theta\\_i|\\ggamma,\\ddelta,\\xx\\_i) \\biggl) d \\theta\\_i.\\end{aligned}$$ \nFurther, denote *θ**i**q*, *q* = 1, ⋯, *Q* and *w**i**q* as quadrature nodes and weights for person *i*, respectively. The intractable integral in the $q\\_n(\\xxi;\\xxi^{(r)})$ can be approximated by summations on this quadrature grid as\n \n$$\\begin{aligned}\n q\\_n(\\xxi;\\xxi^{(r)}) &\\approx \n -\\frac{1}{n} \\sum\\_{j=1}^J \\sum\\_{i=1}^{n} \\sum\\_{q=1}^Q e\\_{iq}^{(r)}\\biggl(\\log f\\_j(\\yy\\_{ij}|\\theta\\_{iq},\\xxi\\_j,\\xx\\_i) \\biggl) +\\sum\\_{j}^J p\\_\\lambda(\\bbeta\\_j) \\\\\n&\\quad -\\frac{1}{n} \\sum\\_{i=1}^{n} \\sum\\_{q=1}^Q e\\_{iq}^{(r)}\\biggl ( \\log f(\\theta\\_{iq}|\\ggamma,\\ddelta,\\xx\\_i) \\biggl),\\end{aligned}$$ \nwhere $ e\\_{iq}^{(r)}=f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i, \\xx\\_i)$ is the posterior probability of *θ**i* at iteration *r*, which is approximated by\n \n$$\\begin{aligned}\n \\label{eq:ewt}\n e\\_{iq}^{(r)} = \\frac{\\prod\\_{j=1}^J f\\_j(y\\_{ij}|\\theta\\_{iq},\\xxi\\_j^{(r)},\\xx\\_i)w\\_{iq}}{\\sum\\_{q'=1}^ Q\\prod\\_{j=1}^J f\\_j(y\\_{ij}|\\theta\\_{iq'},\\xxi\\_j^{(r)},\\xx\\_i)w\\_{iq'}}.\\end{aligned}$$ \nFinally, the Bock-Aitkin EM algorithm with the *L*1 penalty can be achieve using the following steps until convergence is reached:\n* E-step. Compute the posterior weights *e**i**q*(*r*), *i* = 1, …, *n*, *q* = 1, …, *Q*;\n* M-step.\n\t+ For latent population parameters, compute\n\t\n\t\n\t \n\t$$\\begin{aligned}\n\t\\label{eq:msteplatent}\n\t (\\ggamma^{(r + 1)},\\ddelta^{(r+1)})^\\prime =\n\t \\arg\\min\\_{\\ggamma,\\ddelta}\\Biggl\\{-\\frac{1}{n}\n\t \\sum\\_{i=1}^{n} \\sum\\_{q=1}^Q e\\_{iq}^{(r)}\\biggl ( \\log f(\\theta\\_{iq}|\\ggamma,\\ddelta,\\xx\\_i) \\biggl)\\Biggr\\}\n\t \\end{aligned}$$\n\t+ For parameters for item *j*, compute\n\t\n\t\n\t \n\t$$\\begin{aligned}\n\t \\label{eq:mstepitem}\n\t \\xxi\\_j^{(r + 1)} =\n\t \\arg\\min\\_{\\xxi\\_j}\\Biggl\\{-\\frac{1}{n}\n\t \\sum\\_{i=1}^{n} \\sum\\_{q=1}^Q e\\_{iq}^{(r)}\\biggl(\\log f\\_j(y\\_{ij}|\\theta\\_{iq},\\xxi\\_j,\\xx\\_i) \\biggl) + p\\_\\lambda(\\bbeta\\_j)\\Biggr\\}.\n\t \\end{aligned}$$\nM-step Optimization\n-------------------\nAs was previously shown, at each iteration *r* + 1, the M-step optimizes $\\xxi^{(r+1)}=\\arg\\min\\_{\\xxi} p\\_n (\\xxi;\\YY,\\lambda|\\XX)$, which can be split into two optimization problems shown in Equations [eq:msteplatent] and [eq:mstepitem]. These two equations update the population parameters (i.e., $\\ggamma$ and $\\ddelta$) and item parameters, respectively. Equation [eq:msteplatent] can be obtained by a Newton-type optimizer. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is used in the current study. In contrast, Equation [eq:mstepitem] for each item *j* needs to be handled separately due to the non-differentiability of the *L*1 penalty. Finding $\\xxi\\_j^{(r+1)}$, *j* = 1, …, *J* in the M-step amounts to a conditional density estimation problem in a sample size *n**Q* with weights *e**i**q*(*r*). Specifically, maximizing the unpenalized conditional density can be achieved by solving the reweighted least square problem. With the *L*1 penalty, the coordinate descent algorithm can be used to solve the penalized weighted least square problem with pseudo data of a sample size *n**Q*.\nAppendix B\n==========\nAs mentioned before, the general theory of the decorrelated score test and its one-step estimator are established under some assumptions. These assumptions need to be verified to guarantee the asymptotic properties of the Dscore test and the asymptotic unbiased estimator as shown in Equations [eq:Fdscore] and [eq:Funbiased]. The following four assumptions are briefly summarized and the verification will be provided in the final dissertation.\n[Consistency conditions for initial parameter estimates][asp:1] For some sequence *a*1(*n*) and *a*2(*n*) converge to 0 as *n* → ∞, it holds\n \n$$\\begin{aligned}\n\\lim\\_{n\\to\\infty} P\\_{\\xxi^\\*}(\\|\\hat{\\xxi}-\\xxi^\\*\\|\\_1 &\\lesssim a\\_1(n)) =1\\\\\n \\lim\\_{n\\to\\infty} P\\_{{\\xxi}^\\*}(\\|\\hat{{\\WW}}-{\\WW}^\\*\\|\\_1 &\\lesssim a\\_2(n)) =1,\\end{aligned}$$ \nwhere ∥∥1 stands for the *L*1 operator norm of a matrix (e.g., ∥**A**∥1 = max1 ≤ *j* ≤ *m*∑*i* = 1*n*∣*a**i**j*∣) and $\\lesssim $ denotes that the left side is less than or equal to the right hand side times some constant *C* > 0. Although estimation consistency of LASSO has been studied in linear and generalized linear models, it has never been studied in the latent variable modeling framework. As the loss function or the negative sample log-likelihood of MNLFA model is not strictly convex, the proof of parameter estimation consistency can be non-trivial. Moreover, as mentioned by, there can be extra difficulty in bounding $\\|\\hat{{\\WW}}-{\\WW}^\\*\\|$, as $\\hat{{\\WW}}$ depends on $\\hat{\\xxi}$.\n[Concentration of the gradient and Hessian][asp:2] Let ${\\bf V}^\\*=({\\bf I}\\_{d\\_0\\times d\\_0}, -{\\WW^{\\*\\top}})^\\top$, then assume\n \n$$\\begin{aligned}\n \\|\\nabla\\ell(\\xxi^\\*)\\|\\_{\\infty}&=\\bigO\\_p(\\sqrt{\\log d/n})\\\\\n \\|\\vv^{\\*\\top}\\nabla^2\\ell(\\xxi^\\*)-\\operatorname{\\mathbb{E}}\\_{\\xxi^\\*}(\\vv^{\\*\\top}\\nabla^2\\ell(\\xxi^\\*))\\|\\_\\infty &=\\bigO\\_p(\\sqrt{\\log d/n}),\\end{aligned}$$ \nwhere ∥∥∞ of a vector *A⃗* for example indicates ∥*A⃗*∥∞ = max1 ≤ *i* ≤ *d*∣*a**i*∣ and of a matrix is the maximum absolute row sum of the matrix (i.e., $\\|{\\mathbf A}\\|\\_\\infty = \\max\\_{1\\leq i \\leq n}\\sum\\_{j=1}^m|a\\_{ij}|$ ). In the low dimensional setting, it might be sufficient to prove under some finite moment assumptions on the gradient and Hessian matrix.\n[Local smoothness on the loss function][asp:3] Let $\\hat{\\xxi}\\_0=({\\bf 0},\\hat{\\eeta}^\\top)^\\top$, $\\hat{\\vv}=({\\bf I}\\_{d\\_0\\times d\\_0},-\\hat{\\WW}^\\top)^\\top$, and ${\\vv}^\\*=({\\bf I}\\_{d\\_0\\times d\\_0},-{\\WW}^{\\*\\top})^\\top$. For both $\\check{\\xxi}=\\hat{\\xxi}\\_0$ and $\\check{\\xxi}=\\hat{\\xxi}$, it holds that\n \n$$\\begin{aligned}\n \\|\\vv^{\\*\\top}\\{\\nabla\\ell(\\check{\\xxi})-\\nabla\\ell({\\xxi}^\\*)-\\nabla^2\\ell({\\xxi}^\\*)(\\check{\\xxi}-\\xxi^\\*)\\}\\|\\_{\\infty}&=o\\_p(n^{-1/2})\\\\\n \\|(\\hat{\\vv}-\\vv^\\*)^\\top(\\nabla\\ell(\\check{\\xxi})-\\nabla\\ell({\\xxi}^\\*))\\|\\_{\\infty}&=o\\_p(n^{-1/2}).\\end{aligned}$$ \nThis assumption implicitly assumes that the $\\ell(\\xxi)$ is second-order differentiable. The verification of the assumption should be straightforward if the loss function is a quadratic form of $\\xxi$.\n[Convergence of the score function][asp:4] Let $\\SSigma^\\*=\\lim\\_{n\\to\\infty} \\hbox{Var}(n^{1/2}\\nabla\\ell(\\xxi^\\*))$. Then the score function holds that\n \n$$\\begin{aligned}\n\\sqrt{n}\\nabla\\ell(\\xxi^\\*)^\\top\\vv^{\\*}{(\\vv^{\\*\\top}\\SSigma^\\*\\vv^\\*)}^{-1}\\vv^{\\*\\top}\\nabla\\ell(\\xxi^\\*)\\xrightarrow{\\mathcal{D}}{\\chi}^2\\_{d\\_0}.\\end{aligned}$$ \nThis assumption can be established by verifying the Lindeberg’s condition, which is a sufficient condition for a sequence of independent random variables.\nBauer, DJ.\n. A more general model for testing measurement invariance and differential item functioning..\nBauer, DJ., Belzak, WC. Cole, VT.\n. Simplifying the assessment of measurement invariance over multiple background variables: Using regularized moderated nonlinear factor analysis to detect differential item functioning..\nBelzak, W.\n. Using regularization to evaluate differential item functioning among multiple covariates: A penalized expectation-maximization algorithm via coordinate descent and soft-thresholding ..\nBelzak, W. Bauer, DJ.\n. Improving the assessment of measurement invariance: Using regularization to select anchor items and identify differential item functioning..\nBock, R. Aitkin, M.\n. Marginal maximum likelihood estimation of item parameters..\nChen, SM., Bauer, DJ., Belzak, WM. Brandt, H.\n. Advantages of spike and slab priors for detecting differential item functioning relative to other bayesian regularizing priors and frequentist lasso..\nEmbretson, SE. Reise, SP.\n. Item response theory for psychologists..\nEysenck, SB., Eysenck, HJ. Barrett, P.\n. A revised version of the psychoticism scale..\nFan, J. Li, R.\n. Variable selection via nonconcave penalized likelihood and its oracle properties..\nFang, EX., Ning, Y. Liu, H.\n. Testing and confidence intervals for high dimensional proportional hazards models..\nFriedman, J., Hastie, T. Tibshirani, R.\n. Regularization paths for generalized linear models via coordinate descent..\n<https://www.jstatsoft.org/v033/i01>\nFu, W. Knight, K.\n. Asymptotics for lasso-type estimators..\nHuang, PH.\n. A penalized likelihood method for multi-group structural equation modelling..\nJacobucci, R., Grimm, KJ. McArdle, JJ.\n. Regularized structural equation modeling..\nKnight, K. Fu, W.\n. Asymptotics for lasso-type estimators..\nLindstrøm, JC. Dahl, FA.\n. Model selection with lasso in multi-group structural equation models..\nLiu, M.\n. Differential item functioning in large-scale mathematics assessments: Comparing the capabilities of the rasch trees model to traditional approaches ..\nLiu, Y. Yang, JS.\n. Bootstrap-calibrated interval estimates for latent variable scores in item response theory..\nMagis, D., Tuerlinckx, F. De Boeck, P.\n. Detection of differential item functioning using the lasso approach..\nNing, Y. Liu, H.\n. A general theory of hypothesis tests and confidence regions for sparse high dimensional models..\nR Core Team.\n. R: A language and environment for statistical computing []..\n<https://www.R-project.org/>\nRoju, NS., Van der Linden, WJ. Fleer, PF.\n. Irt-based internal measures of differential functioning of items and tests..\nShea, CA.\n. Using a mixture IRT model to understand english learner performance on large-scale assessments..\nStark, S., Chernyshenko, OS. Drasgow, F.\n. Examining the effects of differential item (functioning and differential) test functioning on selection decisions: When are statistically significant effects practically important?.\nTutz, G. Schauberger, G.\n. A penalty approach to differential item functioning in rasch models..\nVaart, AWvd.\n. Semiparametric models. Asymptotic statistics ( 358–432)..\nWang, W., Liu, Y. Liu, H.\n. Testing differential item functioning without predefined anchor items using robust regression..\n<https://doi.org/10.3102/10769986221109208>\nWang, WC.\n. Effects of anchor item methods on the detection of differential item functioning within the family of rasch models..\nWang, WC. Yeh, YL.\n. Effects of anchor item methods on differential item functioning detection with the likelihood ratio test..\nWoods, CM.\n. Empirical selection of anchors for tests of differential item functioning..\nWoods, CM.\n. Dif testing for ordinal items with poly-sibtest, the mantel and gmh tests, and irt-lr-dif when the latent distribution is nonnormal for both groups..\nWoods, CM., Cai, L. Wang, M.\n. The Langer-improved wald test for DIF testing with multiple groups: Evaluation and comparison to two-group IRT..\n<https://doi.org/10.1177/0013164412464875>\nZhao, P. Yu, B.\n. On model selection consistency of lasso..\nZou, H.\n. The adaptive lasso and its oracle properties..\nFigures\n=======\n [fig:truepar]\n [fig:typeierror]\n [fig:power]\n [fig:fdr]\n [fig:biasax]\n [fig:biasdx]\n [fig:samplingdif]\n [fig:Varax]\n [fig:Vardx]\n [fig:SErecoveryax]\n [fig:SErecoverydx]\nTables\n======\n[ht!] [tab:algorithm1]\n[H] [alg:dscore]\n[1] Negative sample log-likelihood $\\ell(\\ppsi,\\eeta)$, penalty function *p**λ*(), and tuning parameters *λ* and *λ*′.\nEstimate $\\hat{\\xxi}$ using penalized ML as in Equation [eq:pmlikn] and partition $\\hat{\\xxi}$ into $\\hat{\\xxi}=(\\hat{\\ppsi}^\\top,\\hat{\\eeta}^\\top)^\\top$ Estimate $\\WW$ column by column\n \n$$\\begin{aligned}\n \\label{eq:w}\n \\hat{\\WW}\\_{\\*j}=\\operatorname\\*{\\arg\\min}\\_{{\\bf w}\\_j}\\frac{1}{2n}\\sum\\_i^n\\{\\nabla\\_{\\ppsi\\_j}\\ell\\_i(\\hat{\\xxi})-\\WW\\_{\\*j}^\\top\\nabla\\_{\\eeta}\\ell\\_i(\\hat{\\xxi})\\}^2+p\\_{\\lambda^\\prime}(\\WW\\_{\\*j})\n \\end{aligned}$$ \nCalculate the estimated descorrelated score function using\n \n$$\\begin{aligned}\n \\hat{\\ss}(\\ppsi,\\hat{\\eeta}) & = \\nabla\\_{\\ppsi}\\ell(\\ppsi,\\hat{\\eeta})-\\hat{\\WW}^\\top\\nabla\\_{\\eeta}\\ell(\\ppsi,\\hat{\\eeta})\n \\end{aligned}$$ \nReturn $\\hat{\\ss}(\\ppsi,\\hat{\\eeta})$\n[ht!] [tab:truepar]\nC1cm C1cm C1cm C1.5cmC1.5cmC1.5cmC1.5cmC1.5cmC1.5cm & & & & \n& & & Age & Gender & Product & Age & Gender & Product \n & & & *β*11& *β*12& *β*13& *β*01& *β*02&*β*03 \n1 & 0.00 & 2.00 & 0.20 & 0.50 & 0.20 & 0.20 & -0.50 & -0.20 \n2 & 1.20 & 1.20 & -0.20 & -0.50 & 0.00 & -0.20 & 0.25 & 0.00 \n3 & -0.20 & 2.00 & -0.25 & 0.25 & 0.10 & -0.15 & -0.25 & -0.15 \n4\\* & 1.50 & 1.50 & 0.20 & -0.50 & -0.20 & 0.20 & 0.50 & 0.20 \n5\\* & 1.20 & 1.20 & -0.20 & -0.50 & 0.00 & -0.20 & 0.25 & 0.00 \n6\\* & 1.10 & 1.90 & -0.25 & 0.25 & -0.10 & -0.15 & -0.25 & 0.15 \n7 & -1.80 & 2.40 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \n8 & 0.50 & 1.50 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \n9 & 0.60 & 1.40 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \n10 & -2.00 & 1.80 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \n11 & 0.60 & 2.30 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \n12 & 1.60 & 1.80 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \n$\\ggamma$& \n$\\ddelta$& \n \n---\n1. See for an exception.[↩](#fnref1)\n2. These assumptions are documented in the Appendix [ch:Appendixb] and are not yet verified in the latent variable models. The intuition of the assumption verification is discussed in the Appendix [ch:Appendixb][↩](#fnref2)\n3. We are grateful to Dr. Paul Barrett for granting us access to the data[↩](#fnref3)\n4. A pilot study found that BIC calculated based on the penalized EM parameter estimates always select a relatively smaller *λ* value when the DIF effect size is large, which results in too many FDR. This phenomena persists even when sample size is large *n* = 2, 500 or *n* = 5, 000. We suspect that the less optimal *λ* selection is due to parameter bias due to the regularization. This can be verified that when the *λ* value is selected by the BIC value calculated based on the maximum likelihood estimates (i.e., the model refitting method), the FDR rate is more controlled especially under the large sample size condition.[↩](#fnref4)\n5. Recovery of SEs measure for the model refit method based on non-zero DIF effects only has strange large values. Similar findings are reported from[↩](#fnref5)\n\nSTATISTICAL INFERENCE FOR THE PENALIZED EM ALGORITHM TO TEST DIFFERENTIAL ITEM FUNCTIONING\n==========================================================================================\nRecent advancements in testing differential item functioning (DIF) have greatly relaxed restrictions made by the conventional multiple group item response theory (IRT) model with respect to the number of grouping variables and the assumption of predefined DIF-free anchor items. The application of the *L*1 penalty in DIF detection has shown promising results in identifying a DIF item without a priori knowledge on anchor items while allowing the simultaneous investigation of multiple grouping variables. The least absolute shrinkage and selection operator (LASSO) is added directly to the loss function to encourage variable sparsity such that DIF parameters of anchor items are penalized to be zero. Therefore, no predefined anchor items are needed. However, DIF detection using LASSO requires a non-trivial model selection consistency assumption and is difficult to draw statistical inference. Given the importance of identifying DIF items in test development, this study aims to apply the decorrelated score test to test DIF once the penalized method is used. Unlike the existing regularized DIF method which is unable to test the statistical significance of a DIF item selected by LASSO, the decorrelated score test requires weaker assumptions and is able to provide asymptotically valid inference to test DIF. Additionally, the deccorrelated score function can be used to construct asymptotically unbiased normal and efficient DIF parameter estimates via a one-step correction. The performance of the proposed decorrelated score test and the one-step estimator are evaluated via a Monte Carlo simulation study.\nDifferential item functioning, Moderated nonlinear factor analysis, LASSO, *L*1 penalty, LASSO, Score test, Penalized EM algorithm\nIntroduction\n============\nA major goal of educational and psychological assessment is to create interpretable scores on a relevant construct being measured. A well-constructed test score representing a latent trait (e.g., proficiency level), should be reliable and valid. One major threat to the validity of the intended use of scores from an instrument is the lack of measurement invariance or existence of *differential item functioning* (DIF). Sometimes, characteristics of a test may introduce unintended systematic score differences between individual test takers or subgroups of examinees of the same ability level, which can result in invalid score interpretation. Therefore, DIF detection should be conducted for routine operations of psychological and educational assessment to ensure valid interpretations of test scores.\nItem response theory (IRT), a collection of mathematical and statistical models, wasand still remainsthe foundational modeling system in educational assessment. One advantage of an IRT model in conducting a DIF analysis is that it offers a formal statistical framework for assessing DIF. Despite the popularity and advantages of DIF detection using IRT models, the inherent assumptions therein make practical applications of DIF analyses challenging. For example, IRT DIF detection methods usually explicitly or implicitly assume the existence of *predefined* DIF-free items[1](#fn1), also known as anchor items. Typically, DIF-free items are needed to anchor the latent scale so that DIF can be distinguished from the between group difference in latent trait distributions. Correctly specifying anchor items is crucial in correctly identifying a DIF item. Violations to the DIF-free anchor assumption may lead to inflated false detection rates in finding DIF items. A practical obstacle is locating DIF-free anchor items without a priori information. Another challenge is detecting DIF items associated with multiple grouping variables. Conventional IRT DIF detection methods typically investigate DIF with respect to one categorical grouping variable (e.g., gender, ethnicity, SES) at a time. However, it is impossible to investigate the complex nature of DIF due to interconnected background characteristics and thus it is unlikely to truly uncover the source of DIF.\nAs a solution to these practical difficulties encountered by conventional IRT DIF methods, regularization has been applied to the latent regression modeling framework to detect DIF items. In particular, the use of the *L*1 penalty or the LASSO has shown promising results in DIF detection due to its efficiency in variable selection <e.g., >bauer2017more,belzak2020improving,magis2015detection,tutz2015penalty. For example, the *L*1 penalty has been successfully applied to moderated nonlinear factor analysis (MNLFA) model to identify DIF items. Such a modeling framework offers greater flexibility to detect DIF for multiple grouping variables simultaneously. Item parameters and population distribution parameters, upon applying suitable link functions, can be expressed as linear or nonlinear functions of multiple person characteristics. Consequently, DIF detection can be treated as a variable selection problem—non-zero coefficients are only assigned to covariates that cause DIF. The goal of the penalty procedure is to encourage sparseness as much as possible with respect to DIF parameters such that DIF parameters of anchor items are penalized to be zero while those of DIF items are non-zero. Hence, no predefined anchor items are needed as items with DIF parameters penalized to be zero are used as anchors.\nDespite the flexibility of the *L*1 penalty for DIF detection, several concerns need to be addressed carefully. First, the accuracy of DIF detection based on the *L*1 penalty is contingent on non-trivial assumptions. In order for LASSO to select the right covariate for each item, and thus, items exhibiting true DIF, a variable selection consistency condition is needed. In the current case, failing to meet this condition could result in mistakingly flagging non-DIF items or miss real DIF items even in large samples. Second, with a finite sample size especially those encountered in the social sciences, inferential statistics such as confidence interval estimates and p-values are extremely important to differentiate a true DIF item from a sampling error. For example, a common practice in DIF detection in educational and psychological assessments is effect size reporting after DIF is detected. In this case, an uncertainty measure may prove to be helpful in deciding whether a DIF effect size is truly different from zero. Nevertheless, it is difficult to draw statistical inference based on the LASSO-type estimator due to shrinkage resulting from penalization. Lastly, LASSO creates biased parameter estimates and non-normal limiting distribution of the parameter estimates, which makes subsequent analyses difficult. For example, DIF effect sizes, item parameter estimates, and standard error estimates cannot be directly used to quantify uncertainty in latent score estimates.\nPurpose of the Study\n====================\nThe current study aims to fill the gap in the literature to make inferential claims once LASSO is used. Specifically, a decorrelated score test is proposed to detect DIF for binary response data with multiple covariates after the *L*1 penalty is used. The decorrelated score test could potentially be extended to accommodate continuous and discrete response data and multidimensional latent variables. In the current study, the simplest and the most commonly used case was considered to better demonstrate the extension of the decorrelated score test to detect DIF. Unlike the existing regularized DIF method, the decorrelated score test does not require the variable selection consistency assumption and is able to provide valid inference on DIF effects. Specifically, a sparse score vector with respect to the focal parameter is estimated consistently so that the resulting score test statistic has an asymptotically normal reference distribution. Additionally, an asymptotic unbiased estimator can be constructed using a one-step bias correction using the decorrelated score function.\nA Monte Carlo simulation study is conducted to examinee the finite sample behavior of the decorrelated scodre test. Its performance in controlling the Type I error rate, establishing sufficient power, controlling the false detection rate in identifying a DIF item under different DIF-related conditions are compared with three methods: (1) regularization method based on LASSO selection only, (2) a naive model refitting method, and (3) the oracle solution assuming known anchors.\nDecorrelated Score Test to Test DIF\n===================================\nMathematically, DIF can be defined as follows. Considering *n* independent and identically distributed multivariate vectors, $\\mathfrak{Y}=(\\YY\\_1,\\cdots,\\YY\\_n)^\\top$ and $\\YY\\_i=(Y\\_{i1},\\dots, Y\\_{iJ})^\\top$, following a statistical model $\\mathcal{P}=\\{P\\_{\\xxi}:\\xxi \\in \\Xi\\}$, where $\\xxi$ is a *d* − dimensional vector of unknown parameters and $\\Xi \\subset \\real^d$ is the parameter space. The categorical random variable *Y**i**j* representing the item response from person *i*, *i* = 1, ⋯, *n* to item *j* ∈ J = {1, …, *J*}, define the conditional item response function, denoted as $f\\_j(y|\\theta,\\xx)=P(Y\\_{ij}=y|\\theta\\_i=\\theta,\\xx\\_i=\\xx)$, as the probability of endorsing a particular answer conditional on the person’s latent ability level $\\theta\\_i=\\theta\\in\\real$ and person covariate $\\xx\\_i\\in\\real^K$. More broadly, item *j* exhibits DIF if\n \n$$\\begin{aligned}\n \\label{eq: nonpar-dif}\n f\\_j(y|\\theta,\\xx) \\neq f\\_j(y|\\theta, \\tilde{\\xx}),\\end{aligned}$$ \nfor some $\\xx \\neq \\tilde{\\xx}\\in\\real^K$. Equation [eq: nonpar-dif] indicates that DIF exists when the conditional item response function *f**j* differs for any $\\xx \\neq \\tilde{\\xx}$ after controlling for the same level of latent ability. In other words, measurement invariance holds for item *j*, only when $ f\\_j(y|\\theta,\\xx) =\nf\\_j(y|\\theta, \\tilde{\\xx})$ for all $\\xx \\neq \\tilde{\\xx}$.\nModerated Nonlinear Factor Analysis Model\n-----------------------------------------\nThe moderated nonlinear factor analysis model (MNLFA) assumes that the item response for item *j* from person *i*, (i.e., *Y**i**j* ∈ {0, 1}) follows a Bernoulli distribution ($ Y\\_{ij}|\\theta\\_i\n\\sim \\hbox{Bern}(P\\_{ij}(\\theta\\_i))$), where *P**i**j* denotes the probability of endorsing item *j* for person *i*. Incorporating vector of person covariates $\\xx\\_i$ into the conditional item response function (cIRF) to affect both the item response and the latent distribution offers greater flexibility in testing DIF across different levels of both continuous and categorical variables. It also permits testing DIF for multiple grouping variables as well as their interaction effects (e.g., gender × age) at the same time. Using the same notation, the cIRF *f**j* is written as\n \n$$\\begin{aligned}\n\\label{eq:cIRF}\n f\\_j(Y\\_{ij}|\\theta,\\xx)&=P\\_{ij}^{Y\\_{ij}}(1-P\\_{ij})^{1-Y\\_{ij}},\\,\n\\hbox{where}\\nonumber\\\\\n P\\_{ij}=P(Y\\_{ij}&=1|\\theta\\_i=\\theta,\n\\xx\\_i=\\xx)=\\frac{1}{1+\\exp[-\\alpha\\_j(\\xx)-\\beta\\_j(\\xx)\\theta]},\n \\end{aligned}$$ \nwhere $\\alpha\\_j(\\xx)$ and $\\beta\\_j(\\xx)$ are the item intercept and item slope functions, respectively, which can be further expressed as functions of the vector of person covariate $\\xx\\in \\real^{K}$. Latent ability *θ**i* is assumed to be normally distributed: $\\theta\\_i\\sim \\mathcal{N}(\\mu(\\xx),\\sigma^2(\\xx))$, in which the latent population parameters: mean (*μ*) and variance (*σ*2) can also be expressed as functions of the person covariate vector $\\xx$ defined as follows:\n \n$$\\begin{aligned}\n \\label{eq:blatent}\n \\mu(\\xx)=\\ggamma^\\top\\xx \\quad \\sigma^2(\\xx)=\\exp(\\ddelta^\\top\\xx),\\end{aligned}$$ \nwhere *K*-dimensional vectors $\\ggamma$ and $\\ddelta$ represent effects of the person covariate vector $\\xx$ on the latent mean and variance on a logarithmic scale. These two vectors are further referred to as latent population parameters. Also, item intercept and item slope functions are expressed as\n \n$$\\begin{aligned}\n \\alpha\\_j(\\xx)&=d\\_j+\\bbeta\\_{0j}^\\top\\xx, \\label{eq:unif} \\\\\n \\beta\\_j(\\xx)&=a\\_j+\\bbeta\\_{1j}^\\top\\xx \\label{eq:nonunif},\n \\end{aligned}$$ \nwhere *d**j* and *a**j* are the item location and item slope parameters. The *K*-dimensional vectors $\\bbeta\\_{0j}$ and $\\bbeta\\_{1j}$ stand for person covariate effects on the item intercept and item slope parameter, respectively. To detect DIF for item *j*, a composite hypothesis test is typically conducted to test against the null hypothesis\n \n$$\\begin{aligned}\n \\label{eq:null}\n \\hbox{H}\\_0: \\, \\bbeta\\_{0j}=\\bbeta\\_{1j}=\\bf{0}.\n \\end{aligned}$$ \nTypically, a minimal constraint is needed to identify the MNLFA model. This requires that there exists at least one item for each column of $\\bbeta\\_{0j}$ and $\\bbeta\\_{1j}$ is zero/anchor. Without predefined anchor items or constrained parameters, the MNLFA model can be estimated using the Penalized Expectation Maximization (EM) Algorithm and DIF items are those items with remaining DIF parameters.\nRegularized Differential Item Functioning\n-----------------------------------------\nThe marginal likelihood function $f(\\yyi|\\xx\\_i)$ of an observed individual response vector $\\yyi=(y\\_{i1},\\dots,y\\_{1J})^\\top$ assuming conditional independence among item responses given the latent variable and person covariates $\\xx\\_i=(x\\_{i1},\\dots,x\\_{iK})^\\top$ is expressed as\n \n$$\\begin{aligned}\n \\label{eq:marginal}\n f(\\yyi|\\xx\\_i)=\\int\\prod\\_{j=1}^J f\\_j(y\\_{ij}|\\theta\\_i,\\xx\\_i)\\phi(\\theta\\_i|\\xx\\_i)d\\theta\\_i,\n \\end{aligned}$$ \nin which *ϕ*( ⋅ ) is the probably density function of a normal distribution governed by the population parameters $\\ggamma$ and $\\ddelta$. Assuming that the item response vectors Y is independent and identically distributed (i.i.d), the sample likelihood function is written as\n \n$$\\begin{aligned}\n \\label{eq:likelihood}\n f\\_n(\\YY|\\XX)=\\prod\\_{i=1}^nf(\\yyi|\\xx\\_i),\n \\end{aligned}$$ \nwhere $\\YY=(\\yy\\_1,\\cdots,\\yy\\_n)^\\top$ and $\\XX=(\\xx\\_1,\\cdots,\\xx\\_n)^\\top$ represent observed data matrices. Subsequently, the corresponding marginal log-likelihood is expressed as\n \n$$\\begin{aligned}\n \\label{eq:ln}\n \\ell\\_n(\\xxi; \\YY|\\XX)=\\frac{1}{n}\\log f\\_n(\\YY|\\XX)=\\frac{1}{n}\\sum\\_{i=1}^n\\log f(\\yyi|\\xx\\_i),\n \\end{aligned}$$ \nwhere $\\xxi = (a\\_1, d\\_1, \\bbeta\\_{01}^\\top, \\bbeta\\_{11}^\\top, \\dots, a\\_J, d\\_J, \\bbeta\\_{0J}^\\top, \\bbeta\\_{1J}^\\top,\\ggamma^\\top,\\ddelta^\\top)^\\top$ is a collection of model parameters. To estimate the model using the penalized EM algorithm, the *L*1 penalty function is added directly to the loss function\n \n$$\\begin{aligned}\n \\label{eq:pmlikn}\n p\\_n(\\xxi; \\YY, \\lambda|\\XX) =-\\ell\\_n(\\xxi;\\YY|\\XX)+\\lambda\\|\\bbeta\\_j\\|\\_1,\n \\end{aligned}$$ \nwhere $\\bbeta\\_j=(\\bbeta\\_{0j}^\\top,\\bbeta\\_{1j}^\\top)^\\top$ denotes all DIF parameters, $\\ell\\_n(\\xxi;\\YY|\\XX)$ is the sample log-likelihood expressed in Equation [eq:ln] and ∥∥1 indicates the *L*1 norm that sums absolute values of all elements weighted by the penalty weight *λ*. Then the local minimizer of $ p\\_n(\\xxi; \\YY, \\lambda|\\XX)$ with respect to $\\xxi$\n \n$$\\begin{aligned}\n \\hat{\\xxi} &= \\operatorname\\*{\\arg\\min}\\_{\\xxi} p\\_n(\\xxi;\\YY,\\lambda|\\XX)\\end{aligned}$$ \ncan be estimated by the penalized EM algorithm, which is documented in Appendix A.\nIn the upcoming section, the decorrelated score test will be introduced. The intuition about the decorrelated score test resembles the efficient score test when the model is identified. Thus, the efficient score test for an identified model will first be introduced followed by the decorrelated score test when the model is not identified with redundant parameters.\nDecorrelated Score Test\n-----------------------\nPartition model parameters $\\xxi$ into $\\xxi =(\\ppsi,\\eeta^\\top )^\\top$, where $\\ppsi$ is the *d*0 − dimensional vector of focal parameters and $\\eeta^\\top $ stands for *d*1 = (*d* − *d*0) − dimensional vector of nuisance parameters. Given the loss function $\\ell(\\xxi,\\YY)=-\\ell\\_n(\\xxi,\\YY)$, which is the average sample log-likelihood, define $\\II=\\operatorname{\\mathbb{E}}\\_{\\xxi}(\\nabla^2\\ell(\\xxi,\\YY))$. Let $\\xxi^\\*$ be the true value of $\\xxi$. Similarly, denote $\\II^\\*=\\operatorname{\\mathbb{E}}\\_{\\xxi^\\*}(\\nabla^2\\ell(\\xxi^\\*,\\YY))$. Note that for the rest of the study, *d* is fixed. Let’s first consider the case when the model is identified. For example, an MNLFA model with sufficiently large subset of known anchors. The asymptotic normality of the ML estimator [$\\hat{\\xxi}=\\hbox{argmin}\\_{\\xxi \\in \\Xi}\\ell(\\xxi,\\YY)$] follows from a first-order Taylor series expansion of the score function at $\\xxi^\\*$,\n \n$$\\begin{aligned}\n \\bold{0}&=\\nabla\\_{\\xxi}\\ell(\\hat{\\xxi},\\YY)=\\nabla\\_{\\xxi}\\ell({\\xxi^\\*},\\YY)+\\nabla^2\\_{\\xxi,\\xxi}\\ell({\\xxi^\\*},\\YY)(\\hat{\\xxi}-\\xxi^\\*)+R\\_n\\end{aligned}$$ \nwhere the remainder term is of the form $R\\_n=\\frac{1}{2}(\\hat{\\xxi}-\\bar{\\xxi})^T\\nabla\\_{\\xxi,\\xxi}^3\\ell(\\bar{\\xxi},\\YY)(\\hat{\\xxi}-\\bar{\\xxi})$ and $\\bar{\\xxi}$ is in between $\\hat{\\xxi}$ and $\\xxi^\\*$. By rearranging this equation,\n \n$$\\begin{aligned}\n \\sqrt{n}\\,(\\hat{\\xxi}-\\xxi^\\*)&=-\\sqrt{n}\\,\\II^{\\*-1}\\nabla\\_{\\xxi}\\ell({\\xxi^\\*},\\YY)+\\sqrt{n}R\\_n\\\\\n \\Leftrightarrow \\sqrt{n}\\,\\begin{pmatrix}\n \\hat{\\ppsi} -\\ppsi^\\*\\\\ \\hat{\\eeta} -\\eeta^\\*\n \\end{pmatrix}&=-\\sqrt{n}\\,\\begin{pmatrix}\n \\II^\\*\\_{\\ppsi,\\ppsi}& \\II^\\*\\_{\\ppsi,\\eeta}\\\\\n \\II^\\*\\_{\\eeta,\\ppsi}&\\II^\\*\\_{\\eeta,\\eeta}\n \\end{pmatrix}^{-1}\\begin{pmatrix}\n \\nabla\\_{\\ppsi}\\ell({\\xxi^\\*},\\YY)\\\\\n \\nabla\\_{\\eeta}\\ell({\\xxi^\\*},\\YY)\n \\end{pmatrix}-\\sqrt{n}R\\_n,\\end{aligned}$$ \nwhere subscripts of $\\II^\\*$ are corresponding partitions of the matrix. Drop $\\YY$ from the rest of the notations. As $\\II\\_{\\eeta,\\eeta}$ is invertible, using the block inverse formula, it follows that\n \n$$\\begin{aligned}\n \\sqrt{n}\\,(\\hat{\\ppsi}-\\ppsi^\\*)=-\\sqrt{n}\\,\\II^{\\*-1}\\_{\\ppsi|\\eeta}s({\\xxi^\\*})+\\sqrt{n}R\\_n,\\quad R\\_n= o\\_p(1/\\sqrt{n})\n \\end{aligned}$$ \nwhere $\\ss({\\xxi^\\*})=\\nabla\\_{\\ppsi}\\ell(\\xxi^\\*)-\\II^\\*\\_{\\ppsi\\eeta}\\II^{\\*-1}\\_{\\eeta\\eeta}\\nabla\\_{\\eeta}\\ell(\\xxi^\\*)$ and $\\II^\\*\\_{\\ppsi|\\eeta}=\\II^\\*\\_{\\ppsi\\ppsi}-\\II^\\*\\_{\\ppsi\\eeta}\\II^{\\*-1}\\_{\\eeta\\eeta}\\II^\\*\\_{\\eeta\\ppsi}$ are the efficient score and efficient information, respectively. Note that efficient score can be interpreted as the projection of the $\\nabla\\_{\\ppsi}\\ell(\\xxi)$ to the orthogonal complement of the score function with respect to the nuisance parameters. Under $H\\_{0}: \\ppsi^\\*=\\mathbf{0}$, it holds that $n\\nabla\\_{\\ppsi}(\\mathbf{0},\\hat{\\eeta})^{\\top}[\\hat{\\II}\\_{\\ppsi|\\eeta}]^{-1}\\nabla\\_{\\ppsi}(\\mathbf{0},\\hat{\\eeta})\\overset{\\mathcal{D}}{\\rightarrow}\\chi^2\\_{d\\_0}$, where $\\hat{\\eeta}=\\hbox{argmin}\\_{\\eeta}\\ell(\\mathbf{0},\\eeta)$ is constrained parameter estimates. It is straightforward to see that the estimated efficient score ($\\ss(\\mathbf{0},\\hat{\\eeta})$) equals $\\nabla\\_{\\ppsi\n}(\\mathbf{0},\\hat{\\eeta})\n$ due to the fact that $\\nabla\\_{\\eeta}\\ell(\\mathbf{0},\\hat{\\eeta})=\\mathbf{0}$.\nHowever, when the model is not identified, $\\II\\_{\\eeta,\\eeta}$, in general, is not invertible. Intuitively, there exists redundant parameters in the model. A natural extension of the above case when the model is not identified is estimating an “efficient score” so that the influence of entries of the redundant parameters in the nuisance score is minimized. Following the same logic as the efficient score, a sparse score can be estimated by projecting $\\nabla\\_{\\ppsi}\\ell(\\xxi)$ to the orthogonal complement of a low-dimensional subspace spanned by the non-redundant part of the nuisance score vector. Therefore, a sparse vector/matrix is needed to find the best sparse linear combination of $\\nabla\\_{\\eeta}\\ell(\\xxi)$ to approximate $\\nabla\\_{\\ppsi}\\ell(\\xxi)$. Mathematically, the decorrelated score, or the extension of the efficient score when the model is not identified, has no correlation with the score function with respect to the nuisance score (i.e., $\\operatorname{\\mathbb{E}}(\\ss(\\xxi)^\\top\\nabla\\_{\\eeta}\\ell(\\xxi))=\\mathbf{0}$). Geometrically, the decorrelated score has the same interpretation which is the projection of $\\nabla\\_{\\ppsi}\\ell(\\xxi)$ to the orthogonal complement of the linear space spanned by the nuisance score function $\\nabla\\_{\\eeta}\\ell(\\xxi)$. Following this intuition, define the decorrelated score function as\n \n$$\\begin{aligned}\n \\label{eq:dscore}\n\\ss(\\xxi) & = \\nabla\\_{\\ppsi}\\ell(\\xxi)-\\WW^\\top\\nabla\\_{\\eeta}\\ell(\\xxi), \\hbox{ where } \\WW^{\\top}= \\II\\_{\\ppsi\\eeta}\\II^{-1}\\_{\\eeta\\eeta} \\in \\real^{d\\_0\\times d\\_1}.\n \\end{aligned}$$ \nTo find the projection of $\\nabla\\_{\\ppsi}\\ell(\\xxi)$ to the orthogonal complement of the linear space span by the nuisance score function $\\nabla\\_{\\eeta}\\ell(\\xxi)$, the sparse matrix $\\WW$ can be estimated using Algorithm [alg:dscore] (see Table [tab:algorithm1]). As can be seen, the key is to estimate a sparse matrix $\\hat{\\WW}=(\\hat{\\WW}\\_{\\*1},\\cdots,\\hat{\\WW}\\_{\\*d\\_0})$ column by column to construct the decorrelated score ($\\ss(\\ppsi,\\eeta)$) so that the additional and redundant parameters do not influence the decorrelated score. Further the Dscore test statistic can be constructed\n \n$$\\begin{aligned}\n \\hat{T}\\_{Dscore}=n\\hat{\\ss}(\\mathbf{0},\\eeta)^{\\top}[\\hat{\\II}\\_{\\ppsi|\\eeta}]^{-1}\\hat{\\ss}(\\mathbf{0},\\eeta),\\end{aligned}$$ \nwhere $\\hat{\\II}\\_{\\ppsi|\\eeta}=\\nabla^2\\_{\\ppsi,\\ppsi}\\ell(\\hat{\\xxi})-\\hat{\\WW}^{\\top}\\nabla^2\\_{\\eeta,\\ppsi}\\ell(\\hat{\\xxi})$. Under some technical assumptions[2](#fn2), it can be proved that\n \n$$\\begin{aligned}\n\\label{eq:Fdscore}\n \\hat{T}\\_{Dscore}\\xrightarrow{\\cal{D}} \\chi^2\\_{d\\_0}.\\end{aligned}$$ \nMoreover, a one-step asymptotical unbiased estimator $\\tilde{\\ppsi}$ can be constructed using a single Newton step as\n \n$$\\begin{aligned}\n \\label{eq:onestepubiased}\n \\tilde{\\ppsi}=\\hat{\\ppsi}-\\hat{\\II}^{-1}\\_{\\ppsi|\\eeta}\\hat{\\ss}(\\hat{\\xxi})\\end{aligned}$$ \nand with assumptions [asp:1] to [asp:4] documented in Appendix [ch:Appendixb], showed that\n \n$$\\begin{aligned}\n \\label{eq:Funbiased}\n\\sqrt{n}(\\tilde{\\ppsi}-\\ppsi^\\*)\\xrightarrow{\\cal{D}}{{\\cal N}(0,\\II\\_{\\ppsi|\\eeta}^{\\*-1})}.\\end{aligned}$$ \nSubsequently, the (1 − *α*) × 100% confidence interval can be constructed for a linear combination of $\\ppsi^\\*$ (i.e., $\\mathbf{c}^{\\top}\\ppsi^\\*$ where **c**⊤ is a *d*0 − dimensional constant vector) as $[\\tilde{\\ppsi}-n^{-1/2}\\Phi^{-1}(1-\\alpha/2)(\\mathbf{c}^\\top\\hat{\\II}\\_{\\ppsi|\\eeta}\\mathbf{c})^{-1/2},\\tilde{\\ppsi}+n^{-1/2}\\Phi^{-1}(1-\\alpha/2)(\\mathbf{c}^\\top\\hat{\\II}\\_{\\ppsi|\\eeta}\\mathbf{c})^{-1/2}]$.\nMonte Carlo Simulation\n======================\nIn order to investigate the finite sample behavior of the proposed Dscore test in testing DIF under different conditions, a Monte Carlo study is conducted. Performance of the proposed Dscore test is evaluated in comparison to three methods: (1) the Wald test assuming known and correctly specified anchors, (2) the Reg-DIF method (i.e., results based on the LASSO selection only), and (3) and the naive model refitting approach. The simulation study design and evaluation criteria are discussed in Section [ss:studydesign] and Section [ss:evaluationcriterion], respectively. All computations are performed in R. The source code will be made publicly available through the Open Science Framework (OSF) upon acceptance of the manuscript.\nStudy Design\n------------\nBinary response data are generated from a unidimensional MNLFA model under two conditionswith or without DIF items. Two factors are manipulated including (1) the total sample size (*n* = 500, 1, 000, & 2, 500) and (2) the percentage of DIF items (0%, 25%, & 50%). As a result, there are 3 × 3 = 9 fully crossed conditions. These manipulating factors are selected due to their relevance to the DIF detection mechanism of the Dscore test. Moreover, actual values of the manipulated factors were chosen to conform to real-world data analytic scenarios and align with previous methodological studies <e.g.,>bauer2017more,belzak2020improving. Due to the limited DIF research on the topic, model parameters were generated based on a real data analysis using the UK normative sample data of the Revised Eysenck Personality Questionnaire <EPQ-R, >eysenck1985revised[3](#fn3). True data generating values of these DIF items are described in details in the following section.\nThe True Data Generation Model\n------------------------------\nThe binary response for examinee *i* for item *j*, denoted *Y**i**j* ∈ {0, 1}, *i* = 1, …, *n*, *j* = 1, …, *J*, was generated from a MNLFA model (see Equation [eq:cIRF]). The total number of items was fixed at *J* = 12. Model parameters of the data generating model are tabulated in Table [tab:truepar]. Specifically, item discrimination and intercept parameters are the ML estimates of item parameters using the EPQ-R data with five items identified as anchors. Three person covariates are considered in the study which mimic gender, age, and their product. Specially, the dichotomous grouping variable is generated from a Bernoulli distribution with success probability of.5 (i.e., $x \\sim \\hbox{Bern}(0.5)$). Then, for those in category 1, age is generated from N(0.2, 1). Otherwise, age is generated from N(0, 1), which creates a correlation between age and gender of 0.1. An interaction effect is then created by multiplying age and gender. By including an interaction effect, the correlations between the interaction effect and the two variables is large ( > .6). Furthermore, three types of DIF items were generated. Under the 25% DIF condition, items 1 to 3 are DIF items representing items with large, medium, and small DIF effect sizes. When the proportion of DIF items is increased to 50%, items 4 to 6 are added to the DIF set and their DIF effect sizes replicate those of items 1-3.\nTo visualize the generated items, Figure [fig:truepar] displays the true probability of each person endorsing items with the condition of 50% DIF items and *n* = 2, 500. Plots on the left correspond to DIF items whereas those on the right visualize non-DIF items. Gender influence on the item response can be seen from different colors and shapes.\nLastly, the latent variable follows a normal distribution ${\\cal N}(\\ggamma^\\top \\xx,\\ddelta^\\top \\xx)$ conditional on $\\xx$, and similarly true population parameter values are shown in Table [tab:truepar], which are also enlightened by real data. A total of 500 replications for each condition are implemented ensure that the 95% normal approximation confidence band at the nominal level 0.05 to be [0.031, 0.069].\nEstimation\n----------\nFor each replication under each condition, the following four methods are fitted to the binary response data to test DIF at the item level and the parameter level. MNLFA models are estimated using penalized maximum likelihood for the Reg-DIF method and the Dscore method or ML for the refit method and oracle solution. For both estimation methods, integration with respect to the latent variable is approximated by a 49-point Gauss–Hermite quadrature. The four methods are discussed in detail here.\nFirst, each item is tested for DIF based on LASSO selection only (i.e., the Reg-DIF method). The MNLFA model fitted with all three grouping covariates is estimated using the penalized EM algorithm. To select the optimal tuning parameter under each replication, typically BIC is used in latent variable models. However, the current study fixed *λ* at an estimated optimal value (i.e., $c\\sqrt{1/N}$ where the constant *c* is estimated by $\\hat{c} = \\sqrt{n}\\sum^{R}\\_{r=1}(\\hat{\\lambda}\\_r)/R$)[4](#fn4). The conditions with *n* = 500 were used to estimate the constant. Results show that *ĉ* = 0.8291, 0.6883, 0.5727 when the number of DIF items is 0, 3, and 6, respectively. Therefore, *λ* is fixed at the estimated optimal value for each condition. For the reg-DIF method, an item does not exhibit DIF if both $\\bbeta\\_{0j}$ and $\\bbeta\\_{1j}$ are zero vectors. Otherwise, the item is considered as a DIF item.\nNext, the naive model refitting method can be applied by refitting the same MNLFA model with the anchors selected using method 1. This model is estimated using the marginal ML estimation method with the EM algorithm. The marginal likelihood function can be approximated using the same configuration as in the penalized EM algorithm. The convergence tolerance for the log-likelihood change for the penalized EM algorithm and the M-step are set to be 10− 4 and 10− 6, respectively. The maximum number of iterations is set to be 500. Additionally, the Dscore test is conducted after the initial run (method 1) using the penalized EM algorithm. Details of the Dscore test was described in Section [ss:dscoretest] and thus are not repeated here. A critical step to conduct the Dscore test is to estimate $\\hat{\\WW}$ as displayed in Equation [eq:w]. As the theory only requires that *λ* and *λ*′ are of the same rate, *λ*′ will be set to the selected *λ* value in the initial parameter estimation step to speed up the computation and save time. and have found that the decorrelated score test is not sensitive to *λ*′ and both fixed $\\lambda^\\prime=0.5\\sqrt{\\log d/n}$ in their simulation studies. Furthermore, asymptotic unbiased parameter estimates can be estimated using the one-step bias correction (see Equation [eq:onestepubiased]). Note that although the focal parameters can be multidimensional or unidimensional (i.e., the debias step can be conducted at the item level or parameter level), the current study investigates the one-step bias correction by parameter type. For instance, for each item, the bias correction treats $(a\\_j, d\\_j, \\bbeta\\_{0j}, \\bbeta\\_{1j})^\\top$ as the focal parameters and everything else as nuisance parameters. The population parameter estimates are corrected at the same time by treating all item parameters as nuisance parameters.\nLastly, to compare the performance of the Dscore test with the oracle solution assuming anchors are known, Wald tests are performed to test DIF at the item-level. For each replication and each condition, item 11 and item 12 are treated as anchors. In addition, final model parameter estimates and their corresponding standard errors are estimated using anchors selected by the Wald test.\nEvaluation Criteria\n-------------------\nThe comparative inferential performance of the four methods is evaluated in terms of (1) hypothesis testing in testing a DIF item measured by rejection rates at the 0.05 level, (2) parameter recovery including bias and variance of model parameter estimates including a-DIF and d-DIF, and (3) recovery of standard errors. Standard error estimates are evaluated by comparing the square root of the mean of the variance of parameter estimates against the empirical standard errors (i.e., Monte Carlo standard deviation of the parameter estimates).\nResults\n=======\nResults: Hypothesis Testing\n---------------------------\nIn this section, the Type I error rate, false positive rate (FDR), and power of detecting a DIF item are reported. To visualize the comparative performance, the empirical rejection rate at the nominal level 0.05 was plotted under (1) the null condition when there were no DIF items (i.e., Type I error), (2) the alternative condition when there was a mix of DIF and DIF-free items for DIF items only (i.e., Power), and (3) the alternative condition when there was a mix of DIF and DIF-free items for DIF-free items only (i.e., FDR).\n### Type I Error Rate\nFigure [fig:typeierror] shows the Type I error rate of detecting a DIF item under the null condition when there were no DIF items. The horizontal dashed lines show the 0.05 nominal level and the dotted lines represent the 95% normal approximation confidence band at the nominal level 0.05 across 500 replications. If the rejection rate falls within the 95% confidence band (i.e., 95% confidence interval =[0.031, 0.069]), the specific method is considered to have well-controlled Type I error rate of incorrectly detecting a DIF item. Larger values than the upper bound of the confidence band at the nominal level 0.05 represent over-rejecting the null hypothesis whereas smaller values than the lower bound of the confidence band indicate under-rejecting the null hypothesis.\nOverall, the Reg-DIF, model refit method, and decorrelated score test showed controlled Type I error under all manipulated sample size conditions as the rejection rate of all these methods fall within the 95% confidence band.\n### Power\nFigure [fig:power] shows the power of detecting DIF items under different conditions. The power can be negatively impacted by the sample size while more robust to the number of true DIF items. Overall, the decorrelated score test was more powerful than the Reg-DIF and the model refit methods across all conditions. The difference was more obvious under more challenging conditions where the sample size is small. The performance of the Reg-DIF method and the model refit method was nearly identical when the total number of DIF items was small. The difference between the two methods was more obvious when the number of DIF items was large. Specifically, the model refit method was slightly less powerful than Reg-DIF by design. The reason is that once an item is identified as an anchor item, its DIF parameters are excluded in the refitted model. In other words, this item can no longer be identified as a DIF item.\n### False Detection Rate\nFinally, results for incorrectly detecting an anchor item as a DIF item can be visualized using the empirical rejection rate for the true anchor item under alternative conditions. As displayed in Figure [fig:fdr], the performance of the decorrelated score test is outstanding as the FDR of the decorrelated score test falls within the 95% normal-approximation confidence band, indicating controlled FDR. Conversely, the Reg-DIF and model refit method under-rejected the null hypothesis when the number of DIF items is small. This is understandable, as was previously shown that the power of identifying the true DIF item was in general smaller than the decorrelated score test and the oracle solution. However, when the number of DIF items was large, the Reg-DIF method over-rejected the null hypothesis. As was previously illustrated, the model refit method is more conservative in rejecting a null hypothesis as compared to the Reg-DIF method. Intuitively, it makes sense that the FDR is more controlled as compared to the Reg-DIF method. Nevertheless, it is still overly rejecting the null hypothesis when the sample size is not sufficiently large. As the sample size increases, the FDR using the model refit method falls within the 95% confidence band at the nominal 0.05 level.\nResults: Parameter Recovery\n---------------------------\nTo investigate parameter recovery, the bias and variance of the parameter estimates using each method were computed. Discussion is focussed on the DIF parameters, the parameters that inferences will be drawn on. Bias and variance of other model parameter estimates including item slope (*a**j*), item intercept (*d**j*), and population parameters ($\\ggamma,\\ddelta$) using different methods are included in the supplementary materials. For all methods other than the one-step debiased estimator using the decorrelated score function, DIF parameters penalized to be 0 or not included in the final model are treated as 0 to compute bias and variance of an estimator.\n### Bias\nWhen there was no DIF, all methods recover model parameters well with bias generally less than 0.05. This is understandable because when there was no DIF, the Type I error rate, as was previously shown, was well-controlled. The penalized EM performed similarly as the EM since in most cases because *λ* was large enough to penalize all DIF parameters to be 0. Even if this does not happen, the model refit method is always fitting the correct model under the null condition.\nWhen there was a mix of DIF and non-DIF items, the performance of different methods varied. The Reg-DIF method often resulted in biased DIF parameters for the true DIF items (i.e., items 1 to 3 in the 3 DIF item condition and item 1 to 6 in the 6 DIF item condition) due to the shrinkage. The bias for the DIF parameters for the anchor items were relatively small. Compared with the Reg-DIF method, the model refit method reduced bias for the true DIF items by 3% to 92% depending on the item. Note that bias remained relatively large for the a-DIF parameter even sample size was large (see last row of the Figure [fig:biasax]) due to failing to select the right DIF-effect. It was more clear when the sampling distribution of non-zero DIF parameters was investigated (see Figure [fig:samplingdif]). As can be seen, the sampling distribution of the non-zero DIF effect (blue line) using the model refit method was bi-modal with one mode at 0 indicating that the effect was not selected by the LASSO and another non-zero mode.\nAs a comparison, the one-step debiased parameter estimate using the decorrelated score function performed remarkably well. DIF parameters were recovered well as the bias ranged from -0.03 to 0.06 across conditions. Its advantages in reducing bias due to shrinkage were more obvious as compared with the model refit method under smaller sample size or more DIF items conditions. Under the most difficult condition (i.e., small sample size and large number of DIF items), it performed equally well as compared to the oracle solution. More importantly, as the sample size increased, the bias decreased.\n### Variance of DIF Parameter Estimates\nFigure [fig:Varax] and Figure [fig:Vardx] present the variance of the DIF parameter estimates. Large values indicate more uncertainty of the parameter estimates while smaller values indicate less uncertainty. As the sample size increases, variance should, in general, decrease. Among all methods, the Reg-DIF method produced the least variable parameter estimates. The model refit method produced similarly variable a-DIF parameter estimates but notably more variable d-DIF parameter estimates as compared to the Reg-DIF especially for the DIF items. Lastly, the one-step debiased parameter estimate using the decorrelated score function was the most variable estimator for the a-DIF parameter as compared to all other methods especially when the sample size was small. However, it was less variable for the d-DIF parameter for the DIF item as compared with the model refit method.\nResults: Standard Error Estimates\n---------------------------------\nThe recovery of SE was calculated as the ratio between the square root of the mean of the variance of parameter estimates over the empirical standard deviation of the Monte Carlo parameter estimates. If the ratio is closer to 1, the method has a more valid uncertainty measure. One caveat when calculating the SE using the model refit method is that there is no valid SE estimates for DIF parameters penalized to be 0. If the Reg-DIF method does not select a specific DIF effect for a specific replication, the corresponding DIF covariate is excluded from the model refit method. In such cases, SE estimates of the unselected DIF effect is treated as 0. These arbitrary 0 SEs would artificially decrease the average of the estimated SEs across replications and thus should be interpreted with caution. Alternatively, omitting DIF effects penalized to be 0 will often exclude too many DIF effects across replications resulting in instability of the empirical SEs and inaccurate recovery SE measure[5](#fn5).\nThe SE recovery for the a-DIF and d-DIF parameters can be visualized in Figure [fig:SErecoveryax] to [fig:SErecoverydx]. As there are no valid SE estimates from the penalized EM alogrithm, the Reg-DIF method currently presented in the following figures are based on the SE estimates using the model refit method. Note that as some DIF parameters of certain items were never selected by the Reg-DIF method leaving the recovery of SEs unavailable for the model refit method. This is actually one of the disadvantages of using the model refit method that inference can only be drawn on the selected DIF parameters. Again, the model refit method tended to underestimate SEs of DIF parameters. The underestimation of the SEs of DIF effects could contribute to incorrectly detecting an anchor item as a DIF item as was shown previously shown in Figure [fig:fdr]. As a comparison, the decorrelated score method not only can recover SEs of non-zero DIF effects but also zero DIF effects. For a-DIF and d-DIF, SEs ranged from 0.95 to 1.08 and 0.90 to 1.02, respectively. Note that it appears that SEs of a-DIF and d-DIF effects for item 7 were underestimated. However, given that variances of a-DIF and d-DIF parameters for item 7 were strangely large due to the skewness of the sampling distribution, it makes sense that SEs of DIF parameters of item 7 seem to be underestimated.\nDiscussion and Conclusion\n=========================\nSummary\n-------\nOne of the fundamental issues and remaining challenges inherent in DIF detection is finding the correct anchor items. Although recent development in the DIF detection literature (such as using regularization in the MNLFA modeling framework) has shown promising results in detecting DIF items without predefined anchor items, issues such as inflated false detection rate and inability to draw valid inference still remain.\nThe goal of the current study is to apply the decorrelated score test to test DIF items based on the *L*1 − penalized maximum likelihood solution under the MNLFA modeling framework. Unlike DIF detection based on regularization only, the decorrelated score test is valid for all DIF and DIF-free items across all covariates. Additionally, it has been shown that the decorrelated score function can be further used to construct an asymptotically unbiased estimator. Furthermore, the simulation study has shown the comparative performance of the decorrelated score test, DIF detection using regularization only, the model refit method, and the Wald test assuming the correct anchor items in hypothesis testing and parameter recovery.\nOverall, the decorrelated score is advantageous than the other two anchor-free DIF detection methods in three aspects. First, the decorrelated score test shows more statistical power in detecting a DIF item while maintaining controlled false detection rate even compared with the oracle solution with two anchor items available. As summarized and highlighted by previous studies, the Reg-DIF method and the model refit method often result in inflated false detection rate in identifying a true DIF item especially when the sample size is small. Similar to these studies, the current simulation study also shows inflated false detection rate especially under the most difficult yet realistic conditions (i.e., when the sample size is small and the number of DIF items is large). As was previously discussed, the current simulation study also shows that the model refit method has similar performance as the decorrelated score test in correctly identifying a true DIF item while controlling for the Type I error when the sample size is large.\nAnother important finding from the current simulation study is that the one-step debiased estimator using the decorrelated score function yields less biased item and DIF parameter estimates. Specifically, as compared with the model refit method, the one-step debiased estimator recovers the item slopes, item intercepts, a-DIF, and d-DIF parameters well and sometimes even comparable to the oracle solution. As expected, the model refit method reduces the bias for model parameters estimated from the penalized EM algorithm. However, bias still remains. This is expected as LASSO seems to be more sensitive to detecting d-DIF rather than the a-DIF, a point that we return to at the end of this section, it is likely that a-DIF parameters are likely to be mistakenly unselected. Accordingly, the model refit method fits an incorrect model and, therefore, bias remains. Given the less biased item and DIF parameter estimates, it is recommended to use the one-step debiased estimator to produce reliable point estimates without fitting an additional MNFLA model using EM. However, less bias often indicates more parameter variability. This is evidenced by the fact that there are more uncertainty in the item parameter and DIF parameter estimates using the one-step debiased estimator based on the decorrelated score function. Another interesting observation is that consistent with the model refit method, the one-step debiased estimator always recover item intercept parameters better as compared to item slope parameters (see the supplementary materials). This finding is also aligned other studies <see>[for examples]bauer2020simplifying, belzak2021using that d-DIF parameters are in general recovered better than a-DIF parameters, which could potentially explain the superiority of the item intercept parameter recovery. Consequently, a-DIF and item slope parameters have better initial parameter estimates to be plugged into the bias correction function.\nLastly, the decorrelated score test is able to draw reliable statistical inference on a-DIF and d-DIF parameters for zero and non-zero effects, respectively, where the model refit method may fail or is not able to generate one when the DIF effect is zero. As discussed earlier, SEs cannot be computed if DIF effects are penalized to be zero. To this end, the one-step debiased estimator is the only method considered in the present study that can produce reliable statistical inference for the zero DIF effects. As for the remaining DIF effects that are not penalized to be zero, the model refit method often underestimate the SEs of the DIF effects leading to incorrect inferences. This may be of concern, if substantive researchers are interested in testing a-DIF and d-DIF separately. If so, the decorrelated score test offers promising and valid statistical test to identify DIF at both the item and the parameter levels.\nFuture Studies\n==============\nAlthough the proposed decorrelated score test and the one-step debiased estimator have shown promising results in detecting DIF items without predefined anchor items and providing valid uncertainty measures for DIF effects, limitations and issues still remain to be addressed by future studies.\nFirst, the simulation study has suggested that the one-step debiased estimator does not seem to reduce the bias of the penalized ML estimator and that the associated SEs are unreliably estimated using the efficient information. Given that the ill recovery with the uncertainty measure also happens with unpenalized item parameters, it is conjectured that additional penalty weights might be needed for the population parameter or a more careful selection of penalty weights is needed. Some preliminary investigation suggests that if the population parameter is penalized in the initial *L*1 penalization stage, the bias correction using the decorrelated score function shows better performance. Future studies are needed to examine the behavior of the decorrelated score function when population parameters are penalized. Alternatively, finding the optimal *λ*′ for the population parameter for each iteration might also improve the unpredicted behavior of SEs. In the current simulation study, *λ*′ is set to the fixed *λ* based on the estimated rate as the theory suggests $\\lambda \\asymp \\sqrt{1/n}$ and $\\lambda^{\\prime}\\asymp\\sqrt{1/n}$ are approximately the same rate to ensure the *l*2 error bound for initial parameter estimates. It might worth to explore the influence of *λ*′ on the SE recovery of the unpenalized parameter. Although and have reported that the performance of the decorrelated score test is insensitive to the actual *λ*′ value in the high-dimensional regression model and proportional hazard model, it is not clear whether *λ*′ will impact the one-step debiased estimator especially when there are unpenalized parameters in the model.\nSecond, as *λ* is critically important for the initial parameter estimates and the accuracy of DIF detection using the Reg-DIF method and the model refit method, future research is encouraged to investigate different model selection critera in addition to BIC. As mentioned in Section [s:simulation], a pilot study found that BIC calculated based on the penalized EM parameter estimates always selects a relatively smaller *λ* value when the DIF effect size is relatively large and thus results in too many false positives. The current study used a fixed *λ* value estimated from the rate calculated from the *n* = 500 condition based on the ML estimator which potentially avoids the issue of poor performance of BIC. This may also explains the relatively smaller inflated false detection rate and power as compared with other Reg-DIF studies <e.g.,>bauer2020simplifying,belzak2020improving. Given the sparse literature on the model selection accuracy in the regularized DIF framework, future research is needed to find viable and efficient penalty selection approaches.\nThird, future research is needed to investigate the impact of different penalty functions on the performance of the decorrelated score test in DIF detection and the asymptotic behavior of the debiased parameter estimator. The general theory provided in can be directly applied to many penalty functions <e.g., adaptive LASSO,>[]zou2006adaptive. Our simulation study along with the other studies <e.g.,>belzak2020improving showed similar differential performance in recovering different parameters and SEs. In addition, applying different penalty weights for a-DIF and d-DIF may yield better initial parameter estimates and thus improve the performance of the one-step debias estimator.\nLastly, DIF effect size measures can be critically important when deciding to remove, modify, or keep a flagged DIF item. In practice it might be more desirable or more realistic to flag items with large DIF impact, which makes effect size reporting more essential in DIF detection. Latent regression models such as MNLFA models provide a natural DIF effect size measure (i.e., the person covariate effect on the item slope $\\bbeta\\_{1j}$ and item intercept $\\bbeta\\_{0j}$). However, more thorough and careful study on the meaningful cut-off values are needed to facilitate decision making. Alternatively, item level or scale level effects can also be helpful and can be extended to the MNLFA modeling framework. For example, average unsigned difference <see>woods2011dif which calculates weighted differences in the expected response functions between the focal and reference groups can be extended to the MNLFA model for selected values or levels of person covariate to evaluate the magnitude of the DIF effect at the item level. In addition to the item level DIF impact, the differential test function (DTF) index or the expected total test score difference due to DTF can be computed to inform the overall DIF effect on the scale level.\nIn sum, the proposed decorrelated score test and its one-step debiased estimator based on the regularized moderated nonlinear factor analysis model offers a promising solution to the practical obstacles encountered by conventional IRT methods. Valid inference at the DIF parameter level opens doors for more complicated substantive research questions such as investigating the complex nature of DIF due to interconnection of background characteristics.\nAppendix A\n==========\nPenalized Expectation-Maximization Algorithm\n--------------------------------------------\nThe Expectation-Maximization (EM) algorithm is used to find the local minimizer of $ p\\_n(\\xxi; \\YY, \\lambda|\\XX)$ with respect to $\\xxi$. The basic idea of the penalized EM algorithm is to repeatedly approximate the upper bound of the penalized negative sample log-likelihood $p\\_n(\\xxi;\\YY,\\lambda|\\XX) \\leq q\\_n(\\xxi;\\xxi^{(r)}), r=0,1,2,\\cdots$ defined in Equation [eq:q] at each iteration (E-step) and obtain the optimizer (M-step). As a result, the alternation between the E-step and the M-step produces a sequence of parameter updates $\\xxi^{(r)}$. The final parameter estimates $\\hat{\\xxi}$ is referred to as the penalized ML estimator. A critical property of the EM algorithm is that the parameter estimate updates the penalized negative sample log-likelihood in a non-increasing fashion (i.e., $p\\_n(\\xxi^{(r+1)};\\YY,\\lambda|\\XX)-p\\_n(\\xxi^{(r)};\\YY,\\lambda|\\XX)\\leq q\\_n(\\xxi^{(r+1)},\\xxi^{(r)}) - q\\_n(\\xxi^{(r)},\\xxi^{(r)})\\leq 0$ where $q\\_n(\\xxi,\\xxi^{r})$ can be viewed as the upper bound of $p\\_n(\\xxi;\\YY,\\lambda|\\XX)$). Specifically, at iteration *r*, $q\\_n(\\xxi;\\xxi^{(r)})$ is defined as\n \n$$\\begin{aligned}\n \\label{eq:q}\n q\\_n(\\xxi;\\xxi^{(r)}) &= \\mathbb{E}\\_{(\\theta\\_i|\\xxi^{(r)},\\YY,\\XX)}\\left[p\\_n(\\xxi; \\YY;\\theta\\_i|\\XX)\\right]\\\\\n &=-\\frac{1}{n}\\sum\\_{i=1}^n\\mathbb{E}\\_{(\\theta\\_i|\\xxi^{(r)},\\yy\\_i,\\xx\\_i)}\\left[\\log f(\\xxi; \\yy\\_i;\\theta\\_i|\\xx\\_i)\\right] +\\ssum p\\_\\lambda(\\bbeta\\_j)\\\\\n &=-\\frac{1}{n}\\sum\\_i^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl(\\log \\left( f(\\yy\\_i|\\theta\\_i,\\xxi,\\xx\\_i)f(\\theta\\_i|\\xxi,\\xx\\_i) \\right) \\biggl) d \\theta\\_i \\nonumber\\\\\n \\quad &+\\ssum p\\_\\lambda(\\bbeta\\_j)\\end{aligned}$$ \nDropping $\\ssum p\\_\\lambda(\\bbeta\\_j)$ for simplicity, the first term of the right hand side (RHS) equals\n \n$$\\begin{aligned}\n &\\hbox{First term of the RHS}\\\\\n &=-\\frac{1}{n}\\sum\\_{i=1}^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl(\\sum\\_{j=1}^J\\log f\\_j(\\yy\\_{ij}|\\theta\\_i,\\xxi\\_j,\\xx\\_i)+\\log f(\\theta\\_i|\\ggamma,\\ddelta,\\xx\\_i) \\biggl) d \\theta\\_i\\\\\n&=-\\frac{1}{n}\\sum\\_{i=1}^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl(\\sum\\_{j=1}^J\\log f\\_j(\\yy\\_{ij}|\\theta\\_i,\\xxi\\_j,\\xx\\_i) \\biggl) d \\theta\\_i \\\\\n&\\quad -\\frac{1}{n}\\sum\\_{i=1}^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl ( \\log f(\\theta\\_i|\\ggamma,\\ddelta,\\xx\\_i) \\biggl) d \\theta\\_i\\\\\n&=-\\frac{1}{n} \\sum\\_{j=1}^J \\sum\\_{i=1}^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl(\\log f\\_j(\\yy\\_{ij}|\\theta\\_i,\\xxi\\_j,\\xx\\_i) \\biggl) d \\theta\\_i\\\\\n&\\quad -\\frac{1}{n}\\sum\\_{i=1}^{n} \\int f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i,\\xx\\_i)\\biggl ( \\log f(\\theta\\_i|\\ggamma,\\ddelta,\\xx\\_i) \\biggl) d \\theta\\_i.\\end{aligned}$$ \nFurther, denote *θ**i**q*, *q* = 1, ⋯, *Q* and *w**i**q* as quadrature nodes and weights for person *i*, respectively. The intractable integral in the $q\\_n(\\xxi;\\xxi^{(r)})$ can be approximated by summations on this quadrature grid as\n \n$$\\begin{aligned}\n q\\_n(\\xxi;\\xxi^{(r)}) &\\approx \n -\\frac{1}{n} \\sum\\_{j=1}^J \\sum\\_{i=1}^{n} \\sum\\_{q=1}^Q e\\_{iq}^{(r)}\\biggl(\\log f\\_j(\\yy\\_{ij}|\\theta\\_{iq},\\xxi\\_j,\\xx\\_i) \\biggl) +\\sum\\_{j}^J p\\_\\lambda(\\bbeta\\_j) \\\\\n&\\quad -\\frac{1}{n} \\sum\\_{i=1}^{n} \\sum\\_{q=1}^Q e\\_{iq}^{(r)}\\biggl ( \\log f(\\theta\\_{iq}|\\ggamma,\\ddelta,\\xx\\_i) \\biggl),\\end{aligned}$$ \nwhere $ e\\_{iq}^{(r)}=f(\\theta\\_i|\\ggamma^{(r)},\\ddelta^{(r)},\\yy\\_i, \\xx\\_i)$ is the posterior probability of *θ**i* at iteration *r*, which is approximated by\n \n$$\\begin{aligned}\n \\label{eq:ewt}\n e\\_{iq}^{(r)} = \\frac{\\prod\\_{j=1}^J f\\_j(y\\_{ij}|\\theta\\_{iq},\\xxi\\_j^{(r)},\\xx\\_i)w\\_{iq}}{\\sum\\_{q'=1}^ Q\\prod\\_{j=1}^J f\\_j(y\\_{ij}|\\theta\\_{iq'},\\xxi\\_j^{(r)},\\xx\\_i)w\\_{iq'}}.\\end{aligned}$$ \nFinally, the Bock"},"data_id":{"kind":"string","value":"arxiv_0002844"}}},{"rowIdx":696,"cells":{"content":{"kind":"string","value":" rich representation of each input word: this is a dense, *H*-dimensional real vector. Obviously, a very large data set is required to properly train such a model, and in Word2Vec’s case that was GoogleNews (100 billion words).\nOther word embedding algorithms followed-up and improved upon the Word2Vec concept, such as GloVE. However, the next significant milestone was the development of context-sensitive word embedding networks, which were able to generate different vector representations for a single word depending on its context, i.e., its position in a sentence. ELMo was able to achieve this using an LSTM architecture, but it was Bidirectional Encoder Representations from Transformers (BERT) which originally exploited the power of Transformers in order to significantly advance the relevant SoA. Such DNNs that learn to generate contextualized word embeddings, by being trained to predict the next word of an input sentence, are essentially *language models*.\n### BERT\nBERT is a bidirectional, deep Transformer Encoder, without an attached Decoder. It receives an input sequence consisting of numerical representations of a text’s tokens/words (e.g., non-semantic indices to a supported vocabulary), to generate a corresponding sequence of refined, semantically encoded output vectors as this text’s representation. Initially, the special token [CLS] is externally appended to the beginning of the overall input sequence, so that its output semantic representation will aggregate global contextual information about the entire textual sequence. Also, the special token [SEP] is inserted after each input sentence to separate consecutive sentences. The ordered token representations are then transformed by a preliminary, trainable embedding layer. Subsequently, a learned dense ``segment embedding\" vector and a learned position embedding vector[1](#fn1) are added to the vector representation of each input token. Finally, the first actual Transformer layer receives these ordered vector sums as its input sequence. The segment embedding of a token belonging to the *i*-th sentence simply indicates whether *i* is odd or even[2](#fn2).\nThe output word embeddings have specialized semantic content in comparison to corresponding Word2Vec representations, i.e., adapted to the context of the specific input sentence. This is because the network processes a complete input text at each iteration of its training stage, taking into account all its words and their order at a single pass, through the self-attention mechanism. Thus, during inference, it processes each input word in the context of its bilaterally ordered phrasal contexts. Therefore, the same word in the context of different sentences, or even placed at a different position in an otherwise identical sentence, can be mapped to different embedding vectors by the pretrained network.\nMore difficult objectives are employed for pretraining BERT, compared to simply predicting the previous and next words of the current sentence, as in Word2Vec. These objectives are tailored to the Transformer’s special features. The most common goal is to predict a few randomly chosen masked words of a complete input sentence based on its remaining words. This is a “Masked Language Model” (MLM) pretraining objective, inspired by the Cloze task, where a subset of the input’s tokens are randomly masked, i.e., each one is replaced by a special [MASK] token, and the goal is to predict the original words based on the context. The MLM objective encourages the model to fuse the left and the right context and, thus, is particularly suited to deep bidirectional Transformers. A complementary pretraining objective is to classify two jointly fed input sentences as either consecutive or non-consecutive ones (Next Sentence Prediction, NSP). This teaches BERT to understand longer-term dependencies across sentences. In both cases, after training the final prediction head or classification layers are discarded and only the trained Encoder is retained, to generate the word embeddings of input sentences.\nA shallow, pretrained Word2Vec model encodes a fixed 1-1 mapping of words to representations/embedding vectors and, thus, can be used in a variety of NLP tasks only for feature extraction at a preprocessing stage. In contrast, a pretrained deep BERT model can optionally be adapted itself to the desired downstream NLP task, by appending appropriate additional final neural layers (usually fully connected or recurrent) and performing minimal fine-tuning on the overall network. During inference, BERT must be given as input a complete sentence, so as to generate the most contextually appropriate embeddings of its words. Due to the fixed maximum sequence size accepted by Transformers, BERT is restricted to receiving at most 512 tokens as its input text. Finally, a pretrained BERT comes with a dictionary of words it has been trained to embed. Thus, if during inference it is found out that the current word is unknown, it is broken into subwords which are considered by the network as different, consecutive elements of the input sequence (*subword tokenization*). In the case of compound words, such a subword is indeed potentially known, otherwise the token decomposition eventually reaches the level of the individual alphanumeric characters of the original word, which always belong to the known dictionary. The final representation of an input word can be derived as the vector average of the embeddings of its subwords. Thus, BERT can easily handle “unsupported” out-of-vocabulary words. Figure [fig::BERTInput] depicts how the input to BERT is constructed.\n[fig::BERTInput]\n### RoBERTa and ELECTRA\nVarious improvements over the original BERT have been proposed over the past few years. For instance, “Robustly Optimized BERT pre-training Approach” (RoBERTa) aggregated several minor improvements in the training process and slightly modified the MLM pretraining objective, by dynamically changing the masking pattern applied to the training data.\nOne of the most significant recent improvements over BERT is “Efficiently Learning an Encoder that Classifies Token Replacements Accurately” (ELECTRA), which was also motivated by the shortcomings of the MLM self-supervised pretraining objective. The latter “corrupts” the input sentence by replacing a small subset of its tokens with [MASK] and then training the model to reconstruct/predict the original ones. This requires a lot of training iterations/sample to be effective, since the task is essentially defined only over the masked tokens. Thus, ELECTRA proposed “Replaced Token Detection” (RTD) as an alternative self-supervised pretraining task. In RTD, the model learns to distinguish real input tokens from plausible but synthetically generated replacements: a randomly selected subset of the input tokens are replaced with synthetic alternatives (typically the output of a small language model), instead of being masked. The pretraining task is defined over all input tokens, since each of them has to be classified as original or synthetic. Learning from all input tokens leads to a learning process which is both much faster and more efficient; thus, ceteris paribus, ELECTRA achieves increased performance in downstream tasks due to better contextual word embeddings. Figure [fig::MLM-RTDTasks] graphically compares the MLM and the RTD tasks.\n[fig::MLM-RTDTasks]\n### T5\nT5 is a renowned, general-use Transformer architecture for NLP. The modus operandi of T5 is to transform each desired NLP task, either discrimnative or generative, into a sequence-to-sequence mapping problem (text-to-text). Thus, it follows the standard Encoder-Decoder architectural paradigm, a design choice that allows it to easily handle input and output sequences of different sizes. This is unlike BERT, which is a bidirectional Encoder-only Transformer. Similarly to other language models, T5 is first pretrained on a large-scale corpus with a self-supervised objective, before being finetuned on the desired, typically supervised downstream task. Pretraining enables the model to learn general language patterns and representations, while downstream finetuning specializes it to task-specific nuances. Both training modes fall under the text-to-text format, therefore a common maximum likelihood loss function is employed regardless of the task.\nThe text-to-text nature of the architecture allows T5 to be pretrained with a masked language modelling objective that uses targets composed of multiple consecutive words per each [MASK] token. Unlike BERT, this is done through random corruptions of text segments with varying masking ratios and segment sizes. A simplified example of the T5 self-supervised objective is depicted in Figure [fig::T5objective]. The use of a complete Transformer Encoder-Decoder architecture, coupled with this fitting self-supervised pretraining task and a very large-scale pretraining dataset, jointly permit T5 to achieve remarkable performance across a variety of different language tasks (e.g., document summarization, sentiment analysis, etc.), after proper finetuning.\n[fig::T5objective]\n### BART\nSimilarly to T5, BART is also a Transformer language model that follows a full Encoder-Decoder architecture and is pretrained with a self-supervised sequence-to-sequence mapping objective. As in BERT, the Encoder is bidirectional and pretraining consists in mapping arbitrarily corrupted textual sentences to their original form. However, BART generalizes the BERT objective since a single [MASK] token can cover several consecutive target tokens (as in T5), while alternative corruptions are also possible. After pretraining, downstream finetuning can take place without any corruptions to the input. With this setup, BART achieves very good results in a variety of downstream language analysis tasks.\nNotably, both T5 and BART utilize a full Transformer Encoder-Decoder architecture with an autoregressive Decoder and an inherent suitability to text-to-text mapping. An important consequence of this choice is that they can easily be finetuned for generative NLP tasks (text generation, e.g., for document summarization) without *any* architectural additions after pretraining, while this is not possible with the bidirectional Encoder-only BERT. In this sense, BERT and its derivatives can be considered deprecated as of 2023.\n### GPT and Large Language Models\nRecently, NLP algorithms employing pretrained GPT (Generative Pretrained Transformer) neural models have been gaining a lot of traction. Similarly to other SoA language models, GPT is a large-scale autoregressive Decoder-only Transformer pretrained in a self-supervised manner and on a gigantic corpus for predicting the next token in a sequence. Due to this traditional causal language modelling objective, pretrained GPT models can be directly employed for text generation without any finetuning: they are given a text *prompt* as input, and they generate a corresponding textual response as an output. Although pretrained GPT models can be finetuned in a typical supervised manner for a specific downstream task, their most impressive capability is that of zero/few-shot downstream language task execution during inference, through careful prompting and without any finetuning. In this case, the desired task is essentially described in natural language on-the-fly via the prompt. However, this setting falls outside the scope of this article.\nWhen it comes to long document analysis, the latest openly available pretrained GPT model is GPT 2.0, which remains regularly outperformed by smaller language models, such as BART, in the finetuning setting. However, the publicity GPT models have attracted due to their text generation capabilities makes them a serious competitor for BERT and its variants. GPT 3.0 achieves more impressive results, but is not openly available to the public. GPT 4.0 is even more powerful, but similarly closed-source, like a number of its competitors such as and the recently announced Gemini. Open-source alternatives are BLOOM and OPT. The scale factor, i.e., highly complex Decoder-only Transformer DNN architectures and a gigantic pretraining corpus/dataset, has proven instrumental for obtaining good performance in all the above cases of Large Language Models (LLMs). However, the observation that scaling up the DNN architecture is only beneficial when accompanied by a corresponding increase in the dataset size has recently led to more reasonably sized models of this nature, i.e., less complex ones, such as Chinchilla and LLaMA.\nTable [tab::ModelComplexity] showcases the escalation of language model complexity in recent years.\n |p3cm|p3cm|p3cm|p3cm| **Name** & **Year** & **#Parameters** & **Input Size (#tokens)** \nBERT & 2019 & 110M & 512 \nBART & 2019 & 140M & 1024 \nT5 & 2019 & Up to 11B & 512 \nGPT & 2018 & 120M & - \nGPT-2 & 2018 & 1.5B & 1024 \nGPT-3 & 2020 & 175B & 4096 \nBLOOM & 2022 & 176B & - \nOPT & 2022 & 125M to 175B & - \nPALM & 2022 & 540B & 3072 \nChinchilla & 2022 & 70M to 16B & - \nLLaMa & 2023 & 7B to 65B & 2048 \nGPT-4 & 2023 & Undisclosed & 128,000 as of writing \nGemini & 2023 & 1.8B to undisclosed & 30720 \nRumoured at 1.76 trillion. The original GPT-4 paper does not disclose any details about the size or architecture of the model.\n[tab::ModelComplexity]\nLong Document Analysis\n======================\nDefinition\n----------\nThis article focuses exclusively on document-level analysis, or long text/long document classification and summarization. There is, however, currently no commonly-accepted definition of what constitutes a ``long text/document“. Most distinctions in current literature focus on specific sub-genres of text (for example, algorithms considering only research papers or books as input). This has led to a fuzzy and unclear definition, where texts such as tweets, product and movie reviews are considered ``short documents”, while research papers, legal documents and books are considered ``long documents“. Texts such as news reports, individual chapters of books and research paper abstracts are arbitrarily classified as both. An example of these qualitative definitions is presented in, where the authors argue that short texts are often characterized by their brevity, their structured nature with explicit facts presented in a logical order and the absence of long-range causal relationships and dependencies, making the task much easier for an NLP model. This highlights the difficulty of defining long documents with just quantitative criteria, since even ``long” documents may actually be easily handled if their structure and contents are ``simple\" enough.\nTo reach a practical, strict definition of long texts/documents for the purposes of this article, various types of documents that are typically considered to be long have been taken into account. This includes research papers, which are often utilized as input for long document analysis tasks and that typically feature a word count in the range 3, 000 to 10, 000 words. This also includes books, which have an average of 80, 000 to 170, 000 words, and long document datasets such as arXiv and Pubmed (see Section [sec::Datasets]). Thus, the following definition is utilized in the sequel:\n**A ``long document\" dataset contains documents with an average word count of at least 2000 words per document, reaching potentially up to 170, 000 words or more. Additionally, each long document must not be semantically segmented, in the sense that context and entity relationships may persist across different paragraphs and sections**. Long document NLP algorithms must be able to cope with the relatively large average size of these documents, but also handle cases with sizes way above that average. They must also be able to hold and analyze dependencies and contexts across vast sections of text.\nThis definition covers the typical document types outlined above (tweets, reviews, research papers, books, etc.), while also clearly distinguishing between fuzzier datasets. For example, despite being of identical domain and very similar content, the CNN/Daily Mail,, 20NewsGroups and Hyperpartisan datasets do not fall under the same category. The first one does not meet the quantitative or qualitative criteria (average word threshold/long-span dependencies) of the proposed definition, as opposed to both 20NewsGroups and Hyperpartisan that are indeed frequently used for long document analysis research.\nShared Challenges\n-----------------\nLong document analysis algorithms, irrespectively of the specific task, inevitably face various challenges due to the nature of the inputs. A subset of them are common for all input texts, while others are specific to the case of long documents. Below, follows a non-exhaustive list of the most significant relevant challenges.\n* **The curse of dimensionality**. As the supported vocabulary size grows, the number of possible test-stage word combinations grows exponentially, unlike the fixed training dataset size. Thus, in semantic document analysis, the longer the test text, the more likely it is for the DNN to encounter sentences unseen during training. This is a challenge for DNNs in the common case of data distribution shifts that lower generalization ability and degrade inference accuracy.\n* **Polysemy**. A word may express multiple different meanings in different contexts. For example, the word ``bank\" may mean a financial institution or a building. DNN models must learn to appropriately differentiate the exact meaning of a word, depending both on the local context (sentence, paragraph) and the global context (document).\n* **Homonymy**. Homonyms are words that either share spelling (homographs), pronunciation (homophones) or both, but are not semantically related. For instance, the term ``bank\" may refer to a financial institution or to the shore of a river. The difference between homonymy and polysemy is that homonyms are not at all related to each other in meaning.\n* **Figurative language**. Sarcasm, irony and metaphors pose serious challenges to NLP, since the real meaning of a phrase is different from the immediately obvious one. Figurative language is common to both short and long texts, but in the latter case it can take the form of sustained allegories (e.g., in literary books).\n* **Unstructured text**. Long texts may be very unstructured for the most part, while communicating information in rather indirect and abstract ways. For example, consider a novel, partitioned in a few dozens of chapters, and contrast it with short texts, such as product reviews or tweets. In the first scenario, crucial information and layers of meaning may be distributed across numerous very long chapters. Furthermore, long documents tend to follow constantly evolving textual and contextual conventions, leading to a higher chance of shifts between learned and actual probability distributions. This is especially problematic for literary works, which are fluid and susceptible to changing cultural biases.\n* **Foreign language datasets**. There is limited availability of training content for languages other than the most popular ones (e.g., English, French, Spanish, etc.). Given that both the word embedding networks and the main, task-specific NLP DNNs are typically trained on one language at a time, the lack of extensive datasets in rarer languages leads to less optimal embeddings and, as a result, reduced performance in downstream tasks. The situation is aggravated by limited training sets for the downstream tasks themselves. Thus, less data-intensive machine learning algorithms may be preferred over DNNs for rare-language texts, such as decision tree classifiers, or the existing datasets can be artificially augmented.\nAdditionally, certain challenges are tied to the nature of the existing DNN architectures. Below, follows a list of such issues that are common across most document analysis tasks (e.g., classification, summarization, etc.), but they are prominent in the case of long document inputs, in contrast to short texts.\n* **Input size**. The size of the input document is the most significant challenge in long document analysis. For example, the computational overhead of CNNs tends to grow proportionally to their inputs; thus, they are unable to process documents with thousands of sentences, without specialized hardware. On the other hand, RNNs/LSTMs struggle to conserve information and identify long-term dependencies over huge spans of text. Transformer DNNs, such as BERT and most of its variants, are typically constrained by a rather low maximum token sequence length; usually 512 or 1024 tokens. This limits their training to relatively short texts, forcing brutal token selection during inference. As a result, their performance on long texts is severely limited. Additionally, vanilla Transformer architectures rely on the typical *global self-attention* mechanism, comparing each token to all other input tokens at each self-attention head, leading to a quadratic (O(*n*2)) increase in computational and memory requirements with respect to input size.\n* **Long-range dependencies**. SoA DNN architectures for long document analysis do not rely entirely on a global context, due to computational and qualitative constraints, thus they may attempt to reduce the input’s segments where they consider local context. However, this strategy may not work sufficiently well for documents with long-range context dependencies spanning over multiple, and often highly separated sections of text. This is mostly an issue with unstructured inputs (e.g., novels), but in fact it may arise even with highly structured documents if they are long enough.\nIgnoring these challenges is equivalent to turning one’s back to a plethora of very important use cases and real-world applications. As stated in, small-to-medium document summarization is not as important as long document summarization, given that smaller texts can be reasonably read by a human without the need for AI automation. In today’s era of pretrained LLMs, this is also a task commonly designated to them via prompting, but this case falls outside the scope of this article. Additionally, document classification is impossible in important domains such as books (e.g., genre identification, library management ), academic papers and legal documents without specialized models.\nCommon Strategies\n-----------------\nIn order to cope with the above challenges, DNNs specifically targeting long document analysis frequently use a combination of the following strategies:\n* **Sliding Window**, where embeddings are created using a sliding window over the input tokens. This is frequently used to capture local context from long spans of text.\n* **Truncation**, which involves minimizing the input text or input embedding size in order to conform to input size limits. This is usually achieved by raw text selection, but can be applied to embeddings as well.\n* **Sparse Attention**, where the underlying Transformer architecture is changed in order to allow for larger inputs.\n* **Iterative Embedding Creation**, where embeddings are iteratively combined and aggregated in order to finally produce a fixed-sized document embedding. This is most often met in hierarchical architectures.\nThese strategies are frequently combined with Transformer architectures, in order to allow them to cope with high document length. Further details can be found in the next sections.\nCategorization and Evolution of Long Document Deep Neural Architectures\n-----------------------------------------------------------------------\nThis subsection offers an original taxonomy of SoA deep neural methods for long document analysis. This taxonomy is composed of two main categories: neural architectures which abide by the Transformer token input limit, and ones that modify the underlying Transformer architecture to allow for very large token limits.\nThe first category contains methods employing one of two main strategies: i) either carefully selecting the most representative text and feeding it as-is to the model (**Feature Selection**) or, ii) transforming the input in order to encode as much information as possible given the fixed token limit (**Hierarchical Models**). Feature selection may be as simple as using user-defined rules from common heuristics, or as complex as using DNNs and Information Retrieval (IR) theory to select the passages with the largest information gain. Hierarchical Models, on the other hand, use stacked neural layers to iteratively aggregate and combine knowledge from distinct parts of the text, finally generating a ``document embedding\" that is fed to a typical Transformer (usually BERT).\nAnother valid strategy is bypassing the token limitation all-together. This is achieved by two main strategies: i) the attention mechanism may either be modified to allow for linear growth of model parameters with respect to the input tokens (frequently called **Sparse Attention**), or ii) the Transformer itself may be modified to contain hidden states much like an RNN, allowing the model to access previous knowledge from arbitrary points in the past. Using the terminology presented in, the latter approach is called a **Recurrent Transformers** in this article.\nA complete view of the presented taxonomy can be found in Figure [fig::taxonomy]. Note that certain DNNs may belong to more than one categories, borrowing ideas from both strategies. The taxonomy includes the approaches that are being detailed in further sections of this article. The evolution of these methods over time is summarized, with respect to this established taxonomy, in Figure [fig::timeline].\n[fig::taxonomy]\n[fig::timeline]\nDocument Classification\n=======================\nDocument classification is the automatic assignment of one or more class labels to a piece of text, based purely on its contents and assuming a known finite set of discrete labels. In the long document case, the text in question is typically an entire document. As an example, a book of fiction could automatically be assigned a genre label by a text classifier deployed in a library, supporting labels such as “Romance” or “Science Fiction”. However, a classification-specific challenge is that of **multiple labels**: the longer an input document is, the more probable is that it belongs to multiple semantic classes concurrently.\nOverall, recent methods for neural long text/document classification mostly rely on pretrained Transformer language models (e.g., BERT) with a classification head appended at the end. The language model may be finetuned for the desired downstream classification task, while training the head. This approach is suitable for most relevant tasks . However, Transformers may be suboptimal for specialized tasks or domains, allowing CNN and RNN architectures to occasionally outperform them on long documents. Relevant issues and solutions are discussed below.\nSolutions\n---------\nTraditional machine learning algorithms, such as Naive Bayes, SVMs (Support Vector Machines) and ID3 decision trees have been repeatedly applied for document classification , being competitive even in large literary texts. Ensemble algorithms have also proven capable in such tasks especially in the context of hybrid schemes. However, the field has generally moved decidedly towards DNNs and this subsection presents the relevant SoA. The innovations of the various existing approaches mostly fall under the following categories, which are more or less tied to the challenges faced by long document classification.\n### Multi-label Classification\nMultiple labels are essential for many real-world applications , thus many methods attempt to identify a document as belonging to multiple classes concurrently. This may be trivial to achieve in itself, using well-known DNN models that have been properly adapted. The varying number of labels per document and class imbalance issues, which may lead to learning extremely biased patterns from the training data, can be handled by applying weighted loss functions. However, multi-label classification poses unique challenges with regard to how accuracy is measured.\nOne representative SoA approach attempting to tackle this scenario is, where multiple different classifiers (including a pretrained BERT) are comparatively evaluated in multi-label multi-class document classification, using the book genre recognition task as a benchmark. The steps taken in to build a successful genre recognizer are telling of more general difficulties with long document classification. Regarding the issue of multiple labels, a modified accuracy metric is exploited to measure classification accuracy: the true positive and true negative classifier predictions are first computed separately per class label, with the proposed metric being the weighted average of correctness over all the label classes. Per-class weights are proportional to the class label frequency and normalized to sum to 1. Similarly, the loss function employed during training (Binary Cross Entropy) is modified to include weights for each label, so that each class is scaled inversely to the prominence of the class in the training set.\n### Feature Selection and Pruning\nThis is the traditional way to handle long textual inputs. It involves systematically discarding as much text as possible at the preprocessing stage, while minimizing information loss. In order to cope with the potentially huge input size in long document analysis, almost all relevant NLP methods are prefaced by a preprocessing stage, where less useful input elements are removed. Appropriately reducing the input size improves both computational efficiency and accuracy.\n#### Text Sampling\nThe simplest way to achieve this is by sampling the document, with a combination of user-defined rules and random chance. For instance, the book genre recognizer of uses an aggressive sampling method which first discards:\n* “stopwords”, i.e., known, extremely frequent words such as “a”, “and”, “he”, etc.,\n* words that appear less than 20 times,\n* words that appear in more than 75% of all books,\n* words that appear in more than 75% of all classes,\n* paragraphs where the frequency of the remaining words is less than a certain threshold,\n* the contents of each remaining paragraph after the 512th token, because of technical limitations imposed by BERT.\nThus, each remaining paragraph is represented by a subset of its words that fall within a restricted, book-level set of ``keywords\". Finally, the sampled paragraphs are only those with the highest concentration of keywords. The goal of this ruthless sampling is to keep the input’s length manageable for documents containing many thousands of pages. Methods similar or identical to this one are present in most NLP models specialized for long texts; thus, they are utilized extensively in most research cited in this paper.\nOn the other hand, the book genre recognizer of employs only minimal preprocessing, just enough to allow the DNN to run relatively efficiently. It relies on an index, consisting of the 5000 most common words from the supported vocabulary. No other preprocessing/sampling takes place, under the intuition that exposing the DNN to even statistically unimportant words facilitates an understanding of patterns and grammatical modifiers, such as tense and plurality. Words are represented by typical neural embeddings. Different strategies were evaluated for feeding the document to the classification DNN:\n* Feed only the first 5000 words of the document.\n* Feed only the last 5000 words of the document.\n* Feed only 5000 random words of the document.\n* Feed the entire document, split into chapters.\n* Utilize a traditional bag-of-words approach.\nThese strategies were then employed on a series of different architectures. Ultimately, the random word selection proved to consistently outperform other selection strategies for a multitude of classifiers, with the best performer being a CNN-Kim architecture.\n#### Using Information Retrieval (IR)\nA more advanced and recent approach to text/feature selection is presented in. It relies on local query-block pre-ranking, allowing the selection of blocks of text that hold the largest semantic significance; these can then be fed to a computationally demanding DNN such as BERT.\nSegmentation of the document into blocks is accomplished using the CogLTX Block Decomposition method. CogLTX is a dynamic programming segmentation approach which emphasizes text containing punctuation marks such as “.” and “!”. It uses an upper token limit on blocks, ensuring a manageable and uniform structure. The ranking of each individual block is based on their Retrieval Status Value (RSV), which is measured either by BM25 (a ranking function frequently used by search engines ) or by simple cosine similarity.\nKeyB(vBERT) is a BERT variant whose inputs are derived by the above feature selection process. KeyB(PARADEk) instead uses PARADE, a SoA set of IR methodologies (involving DNNs) that compute a query-document representation. The latter one is fed alongside the text to a pretrained BERT. Both approaches outperform vanilla ones on IR, although concerns over the memory requirements of KeyB(PARADEk) are raised in.\nSimilar IR-based methods for feature selection may significantly contribute to the use of full Transformer architectures, in parallel with Hierarchical and Sparse Attention Transformer variants that follow a different rationale for bypassing computational/memory limitations (see below).\nAs a side-note, it must be pointed out that such IR-based methods for feature selection are not identical to the use of pretrained extractive summarization DNNs. While in principle the preprocessing steps are indeed similar, their computational and memory requirements are significantly higher, while they are likely to be outperformed by modern, dedicated summarization DNNs (see Section [sec::Summarization]).\n### Sparse Attention Transformers\nGiven the quadratic computational complexity of Transformer DNNs and their prominence in recent years, attempts have been made to reduce their cost. Such approaches try to achieve linear complexity with respect to the input size, by keeping track of a small, fixed-size window of neighboring tokens around each input token, instead of considering all *n* tokens of the input sequence within each self-attention head. However, the unavoidable trade-off is a reduced ability of the DNN to capture long-term dependencies, which is as important for long texts as the complexity with respect to the input size. It must be noted that such efficient Transformer architectures are task-agnostic and generic ones for long input sequence analysis, not tied to a specific task, although they are presented here in the context of document classification. Below, the terms token and element are used interchangeably, although strictly speaking only the first Transformer layer receives actual tokens as input.\n#### Sparse Transformer\nThe Sparse Transformer was introduced in, to lower asymptotic complexity from O(*n*2) to $\\mathcal{O}(n\\sqrt{n})$. This is done by applying sparse factorizations to each *self-attention matrix* computed by the Transformer during inference, which contains the current attention weights of each input element against each of the other elements of the input sequence. Given that such a matrix is separately constructed at each self-attention head of each layer, it is important to effectively reduce the computational and memory cost of this operation. The sparse attention mechanism relies on replacing full attention with several small, optimized attention operations, which jointly approximate the original mechanism sufficiently. Essentially, each token attends to only a subset of the overall input sequence tokens. Additional optimizations, such as efficient sparse attention kernels and improved weight initialization, contribute to a less demanding architecture, which is thus able to handle longer sequential inputs. The Sparse Transformer is a general model suitable for various applications involving very long input sequences, ranging from image compression to document analysis.\n[fig::longformer]\n#### Longformer\nSurpassing the Sparse Transformer’s computational gains, the Longformer achieves a linear asymptotic cost and is specialized for long document NLP. Its self-attention mechanism can be directly incorporated within other Transformer models and operates by employing a *dilated sliding attention window*, which is a modification of the classical sliding window approach for *local attention*. In the typical case, each token’s window consists of *w* tokens surrounding it: $\\frac{w}{2}$ tokens to its left and right, respectively. The underlying intuition is that the semantic context of a token can be mostly derived from its neighboring tokens, while the computational complexity of this operation is O(*n* × *w*). In a sense, the end effect resembles the local receptive fields of neurons in convolutional layers. Similarly to CNNs, stacking *l* Longformer layers gradually increases this receptive field, so that representations of tokens faraway from the query can be attended within the later layers.\nThe *dilated sliding window* approach modifies the sliding window with a predetermined dilation factor *d* added at each step, which determines the spacing between adjacent window positions and allows data analysis at multiple scales simultaneously. Thanks to the introduced “gaps” in the attention pattern, a larger range of tokens within the self-attention matrix can be covered without increasing calculations and computation time. Thus the effective receptive field of the window is of length *l* × *d* × *w*, while memory and processing costs remain steady.\nSince windowed and dilated attentions may not be enough to extract a suitable sequence representation, certain additional, pre-selected locations of the full self-attention matrix are also allowed to be considered as global context. These global tokens attend to/are attended by the entire sequence as normal, but their number is kept constant and relatively small to limit complexity to O(*n*). The [CLS] token is a good candidate for being selected as a global one, since its generated representations should convey information about the entire sequence. Overall, as a result of Longformer’s innovations, it has been successfully evaluated with inputs of length up to 4096 tokens (compared to BERT’s maximum of 512 tokens). A comparison between full, sliding, dilated sliding and global-dilated sliding window self-attention patterns is depicted in Figure [fig::longformer].\n#### BigBird\nBigBird can be seen as a variation of the Longformer, attempting to support much longer input sequences by reducing the complexity of self-attention from quadratic to linear. It also employs a sliding window, since in typical NLP tasks the semantic context of a token can be mostly derived by its neighboring tokens. Instead of the Longformer’s dilated sliding windows, it employs random token selection to reach faraway tokens and provide context. This complements the global attention and the regular sliding window attention schemes, adopted and adapted from the Longformer. It has been demonstrated mathematically that this modified sparse attention mechanism satisfies many theoretical properties of the original full attention mechanism, such as universal approximation and Turing-completeness, but seems to require more layers to retain accuracy comparable to full attention architectures.\n#### Performer\nPerformers are Transformer architectures which can estimate regular full-rank-attention Transformers with provable accuracy, but using only linear space and time complexity. They achieve this through two key aspects, described below.\nTo efficiently compute approximate attention, introduces the FAVOR+ (Fast Attention Via positive Orthogonal Random features) algorithm. It enables linear-time and constant-memory complexity, making it more scalable for long sequences of data such as long documents. As established in Section [ssec::Transformers], the standard attention mechanism involves calculating an attention matrix **A**, which has a quadratic time and space complexity with respect to the sequence length *L*. In FAVOR+, the attention matrix **A** is approximated through a randomized mapping *ϕ* and a kernel function *K*. The kernelized attention is defined as **A**(*i*, *j*) = *K*(**q***i**T*, **k***j**T*) where **q***i**T* and **k***j**T* are the *i*-th query and *j*-th key row-vectors, respectively. The key innovation lies in the use of random feature maps *ϕ*, with the mapping function defined as $\\phi(x) = \\frac{h(x)}{\\sqrt{m}} (f\\_1(\\omega\\_1^T x), \\cdots, f\\_l(\\omega\\_1^T x)$ allowing the modeling of various kernels used in practice.\nThe resulting attention mechanism, denoted as FAVOR+ attention, is characterized by significantly improved time and space complexity compared to regular attention. FAVOR+ has a space complexity of O(*L**r* + *L**d* + *r**d*)O(*L**r* + *L**d* + *r**d*) and a time complexity of O(*L**r**d*), where *r* is the dimensionality of the random features. This represents a substantial reduction from the quadratic complexities associated with standard attention mechanisms. Moreover, FAVOR+ addresses stability concerns associated with negative dimension-values in random feature maps. By introducing a robust mechanism, it mitigates potential issues and generally provides a more reliable and effective solution for approximating attention scores.\nThe last step of FAVOR+ utilizes the standard Gram-Schmidt orthogonalization procedure in order to transform the *ω**i*, *i* = 1, ⋯, *m* vectors to orthogonal vectors, reducing the variance of the attention mechanism’s estimator. The authors prove that by utilizing orthogonal features, the variance reduction applies to any dimensionality *d*, instead of previous results necessitating large values of *d*.\nA recent improvement on Performer is the Permuteformer architecture that adds a position embedding vector to the Performer’s input, while keeping its linear scaling for long sequences. The permutations are achieved by applying a position-aware transformation on query features and key features to encode positional information. The cost of these permutations is negligible compared to the overall computational cost of the Performer, resulting in these two being almost equally efficient and requiring almost identical amount of training time and resources. Regarding results, Permuteformer improves upon Performer in all classification tasks and accelerates its convergence.\n### Hierarchical Transformers\nHierarchical architectures attempt to handle the large input size of long texts by appropriately building upon original Transformers, instead of modifying them. As before, they are task-agnostic and generic ones, but presented here in the context of document classification. There are two main approaches to building a hierarchical Transformer:\n* By transforming the Transformer’s input via a suitable DNN (e.g., a CNN or RNN/LSTM). The latter one generates a single document representation (document embedding) able to fit into a standard Transformer (e.g., BERT), thus bypassing the input size limitations.\n* By segmenting the input document. The input segments are independently truncated to the Transformer’s input size and the network (e.g., BERT) generates embeddings for each of these segments. A separate DNN then combines the segment embeddings and predicts the document label.\n#### Hierarchical Attention Network (HAN)\nThe first attempt towards a Hierarchical Transformer was built on top of standard BERT. During inference, the input document is split into paragraphs that are truncated to fit into BERT. The latter one generates independent paragraph embeddings, which are combined by the succeeding HAN into a single output. The end result is an increased ability to handle long texts, by exploiting the natural partitioning of documents into paragraphs.\nA similar approach was later used in, where the base models were BERT, RoBERTa, DeBERT, Legal-BERT and CaseLaw-Bert. In long text analysis tasks, the hierarchical versions proved competitive with Sparse Attention Transformers, such as Longformer and BigBird.\nThe method in, relying on BERT or on Universal Sentence Encoders (USEs) (models learning sentence representations for transfer learning), follows up on this methodology, but replaces the final HAN with a CNN or an LSTM architecture. The BERT+LSTM combination proved to be the best among the evaluated hierarchical approaches, which however were generally outperformed by Sparse Attention Transformers.\n#### Hi-Transformer\nThe main alternative to and its variants is to essentially reverse the process: that is, to employ a hierarchical DNN for obtaining a fixed-size document-level representation/embedding from the entire input, which is then fed to a regular classification DNN. *Hi-Transformer* achieves this via a Transformer acting as a “Sentence Encoder” (SE), i.e., aggregating and projecting the regular, individual word embeddings of the input document into sentence-level embeddings. Once every sentence has passed through SE, these embeddings are subsequently ordered by positional embedding and fed as input to a subsequent Transformer called “Document Encoder” (DE). The latter’s output is a context-aware document embedding, which is fed as input to the next SE along with the original sentences. Thus, the generated, revised sentence embeddings are aware of the global, document-level context. This process is repeated by stacking multiple such layers, until the final output is passed to a pooling layer that extracts the final document embedding. According to, two such layers seem to be sufficient for outperforming Sparse Attention Transformers in long text analysis. This can be attributed to the readily accessible global/document-level context. Since the final classifier has to analyze a fixed-size document representation and the SE’s complexity is linear to the number of sentences, computational demands are not higher than in the sparse attention case. More impressively, Hi-Transformer’s accuracy has been observed to actually increase with longer documents, since more relevant context can be extracted.\n#### Hierarchical Sparse Transformer\nHierarchical Transformers have been combined with the sparse attention mechanism, in order to ameliorate the latter’s inability to capture all the necessary global/document-level context, in the case of long texts. This inability arises from the low level of global context diffusion in the generated token representations, since the global tokens are limited in number. The *ERNIE-SPARSE* architecture merges the two schools of thought by using hierarchical attention, which is then fed to a Sparse Transformer in order to increase the information extracted from the global context.\nERNIE-SPARSE utilizes a modified attention mechanism, where each query is allowed to attend to a limited number of additional representative tokens within each fixed-size window of the self-attention matrix, along with regular global and local tokens from the input sequence. These inserted representative tokens carry global context from the entire document and are derived in two steps: first, the regular Sparse Transformer is applied to the unaltered input sequence and, then, regular full self-attention is applied to a small collection of selected outputs. ERNIE-SPARSE has been shown to outperform the competition on well-known document classification datasets. It can be considered both a hierarchical and a sparse attention approach, since it modifies the Transformer’s internal architecture. Figure [fig::ERNIE] depicts the modified attention mechanism of ERNIE-SPARSE, in comparison to that of the Sparse Transformer.\n[fig::ERNIE]\n#### Hierarchical BERT-Based dynamic fusion\nAnother approach taken by for processing very long documents combines hierarchical information through nodes capturing nested relationships and semantic structures in text data, contextual information through pretrained Transformer models such as BERT or GPT, and a novel dynamic fusion approach. The latter combines outputs from previous stages with an external model, enhancing overall performance. The architecture proceeds via the following stages:\n* **Linguistic Feature Extraction** where features such as Part-of-Speech (POS) Tagging, Dependency Parsing, Named Entity Recognition (NER) and Coreference Resolution are extracted.\n* **Hierarchical Node Construction with Domain-Specific Knowledge** The Hierarchical Graph is first defined as follows: a graph *G**i* is constructed for each document *t**i*, where *N**i* represents nodes (document and sentences) and *E**i* represents hierarchical relationships. Document nodes are formalized as *N**i* = {*n*1, *n*2, ⋯, *n**m**i*}, where *n*1 is the document node, and *n*2 to *n**m**i* are sentence nodes. Edges *E**i* represent relationships between the document and its sentences. For each document node, a document graph is created in a similar fashion: domain-specific knowledge graphs, such as WordNet, are integrated to enhance the capture of relationships and semantic meanings between words. Word nodes are formalized as *V**i* = {*v*1, *v*2, ⋯, *v**m**i*}, where *v**i* is the node of the i-th word. The edge *E**i* = < *v**k*, *v**l* > exists only if a domain-specific relationship exists between the *v**k* and *v**l* words. The two graphs are then fused into a single hierarchical graph.\n* **Dynamic Text Sequential Feature Interaction** which the Dynamic Time Warping (DTW) algorithm to align word embeddings, facilitating clearer distinction of temporal relationships between words. The algorithm identifies the optimal path through the sequence of embeddings, minimizing the distance between two sequences. In this context, the involved sequences are the word embeddings for a specific document and a reference sequence, typically represented by average embeddings for a designated category. The integration of the DTW algorithm markedly enhances the model’s capacity to capture temporal relationships between words, augmenting the model’s understanding of the sequential structure inherent in textual data. This holds particular significance in text classification scenarios, where discerning the temporal order of words is imperative for tasks such as sentiment analysis.\n* **Hierarchical Graph and Knowledge Graph Fusion**. The fusion process involves combining the hierarchical graph and knowledge graph to obtain an integrated graph. Several approaches can be employed for fusion, such as graph convolutional networks or attention mechanisms.\n* **Model Functionality**. The hierarchical/linguistic representation is fed into a pretrained BERT model, which generates a contextual vector that is processed by an MLP. The latter one outputs the class labels.\nThe architecture achieves SoA results in many different benchmark datasets.\n### Recurrent Transformers\nThe fixed input size of baseline Transformers leads, during both training and inference, to *context fragmentation* in long document analysis: essentially, the DNN analyzes the input as independent, consecutive segments. Thus, a complementary direction of the SoA is to allow the Transformer to learn and extract long-term dependencies beyond the preset fixed input sequence size, while maintaining linear computational complexity and retaining local and global context. As before, they are task-agnostic and generic architectures, but presented here in the context of document classification.\n#### Transformer-XL and XR-Transformer\nThe Transformer-XL architecture introduces recurrence into the Transformer, thus integrating the RNN concept of retained hidden states. These states are passed within the Transformer from one segment to the succeeding one, thus effectively transmitting the previously established context and allowing the identification of long-term dependencies across segments. Unlike RNN/LSTM recurrence, where each layer’s stored state is exploited at the next time step by the same layer, the corresponding Transformer-XL mechanism shifts the transmitted state one layer downwards. For example, layer *l* = *n* has access to 2 hidden states from layer *l* = *n* − 1, 4 hidden states from layer *l* = *n* − 2 and 2*n* from layer *l* = 1, which are the original input word embeddings.\nThis mechanism guarantees O(*n* × *l*) cost, where *l* are the number of layers, and significantly faster inference compared to baseline Transformer, since representations computed for previous segments are re-used instead of being computed from scratch for each new segment. It also allows different input sequence sizes between the training and the inference stage. Ultimately, Transformer-XL achieves competitive results on extremely long document analysis, while being significantly more efficient than the baseline Transformer.\nAnother similar approach aimed at extreme multi-label text classification is the XR-Transformer which employs a three-stage framework, those being partitioning, shortlisting, and ranking. During the partitioning stage, the set of labels is divided into clusters or groups based on certain criteria, in the shortlisting stage a subset of candidate labels are selected from the entire label set, allowing the XR-Transformer to focus on a smaller, more manageable set of potential relevant labels. Finally, the ranking stage involves assigning a priority or order to the selected labels based on their relevance or likelihood of being correct. The XR-Transformer also employs a structure similar to recurrence, but additionally introduces a Hierarchical Label Tree (HLT) that captures the relationships and dependencies between different labels in a structured manner. The HLT is constructed during the training phase through recursive label clustering, forming a tree-like, hierarchical organization of labels that allows training the Transformer on multi-resolution objectives.\n#### ERNIE-Doc\nERNIE-Doc builds upon Transformer-XL but makes two passes over the input sequence, similarly to how humans first “skim” a document before paying attention to important sections. To achieve this, the cached hidden state is computed as follows:\n \n$$\\hat{\\mathbf{H}} = [\\hat{\\mathbf{H}}^{1}\\_{1:T};...;\\hat{H}^{N}\\_{1:T}] \\text{ (skim phase)}$$ \n \n$$\\tilde{\\mathbf{h}}\\_{r+1}^{n-1} = [\\tilde{\\mathbf{H}};\\mathbf{h}\\_{r+1}^{n-1}] \\text{ (retrospective phase)},$$ \nwhere *T* is the number of consecutive document segments, *L* is the length of the document, *N* is the number of layers. $\\hat{\\mathbf{H}}^{i}\\_{1:T} = [\\hat{\\mathbf{h}}^{i}\\_1;...;{\\mathbf{h}}^{i}\\_T]$ is the concatenation of the *T* cached hidden states initially derived in the skimming phase for the *i*-th layer. Thus, $\\tilde{\\mathbf{H}} \\in \\mathbb{R}^{(LTN) \\times d}$ is the concatenation of all $\\hat{\\mathbf{H}}^{i}\\_{1:T}$ (one per layer), Thus, context from the entire document can be exploited in the retrospective phase in order to form the extended hidden state $\\tilde{\\mathbf{h}}^{n}\\_t$.\nThis skimming architecture is incompatible with the recurrence mechanism from and, due to its linear cost with regard to the number of layers, it may be prohibitive for long documents. To solve this, “flattens” the hidden state dependency from one-layer-downwards recurrence to same-layer recurrence, similarly to RNNs. Additionally, a novel, document-level task for self-supervised pretraining is introduced that is called “Segment-Reordering Objective” (SRO). It entails randomly splitting a long document into *m* segments, shuffling them, then letting the Transformer reorganize them in order to learn their interrelations. After pretraining on both the MLM task and on SRO, ERNIE-Doc proved highly competitive in several long document analysis datasets for a range of tasks such as classification, question answering and key-phrase extraction. Figure [fig::ERNIEDoc] depicts the modified recurrent mechanism of ERNIE-Doc, in comparison to that of basic Recurrent Transformers.\n[fig::ERNIEDoc]\nSparse, Hierarchical or Recurrent Transformers?\n-----------------------------------------------\nSparse attention, hierarchical and recurrent approaches dominate the current literature for long document analysis using Transformers. However, a number of challenges remain, such as:\n* Sparse Attention Transformers tend to suffer in accuracy, due to their reliance on a small number of global tokens that must encapsulate all document-level context.\n* Hierarchical Transformers are particularly susceptible to context fragmentation, since the individual segment embeddings are derived either without sufficient local context, or with no access to the global document context.\nIn general, there is no consensus on which approach performs better, nor any empirical rule on which tasks each architecture performs best in. Given that each approach features its own benefits and drawbacks, as well as potentially domain-specific strengths, the literature as of 2024 recommends manual architecture selection by the user, taking into account the specific problem. However, at a high level of abstraction, hybrid algorithms and recurrence seem to be promising avenues for advancing long text analysis.\nDocument Summarization\n======================\nAutomatic Text Summarization (ATS) is the task of generating a short summary from a source input document. In the context of long documents, such as medical and legal records, it is often called “document summarization”. The generated summary must be coherent, concise and avoid redundancy, while accurately maintaining the meaning of the most important information in the source material. There are two main approaches to ATS: *extractive* and *abstractive* summarization.\n**Extractive summarization** generates its summary by selecting verbatim sentences found in the source text. This practically reduces summarization into finding a binary selection vector **s** = [*s*1, *s*2, ..., *s**n*]*T*, *s**i* ∈ {0, 1}∀*i* ∈ [1, *n*], where *s**i* = 1/*s**i* = 0 if the *i*-th source input text sentence is/is not included in the summary, respectively. The simplicity and the tolerable computational cost of this approach have made it very common for on-line article summarization. However, its applicability is inherently limited by the fact that not all document types can be summarized well by just a few of their sentences taken verbatim and re-arranged. For example, there is no way to describe the content of most books by using a selection of their own sentences.\n**Abstractive summarization** is the alternative approach of generating new text. It is the method humans use when trying to manually summarize text, by reading the source, extracting its meaning and then writing a condensed document which contains as much of the source’s meaning as possible. As expected, modern abstractive algorithms are DNNs. Compared to extractive alternatives, they generate more meaningful and condensed summaries, while being more adaptable to the input document type. However, there is a much larger risk of the output summary misrepresenting facts and/or not being coherent at all. The most commonly used method for abstractive summarization involves employing a sequence-to-sequence mapping (Seq2Seq) model, via an Encoder-Decoder DNN architecture. RNNs and Transformers have both been employed in this context, where the Encoder captures suitable representations of the document’s tokens and feeds them to the Decoder that generates a summary.\nDocument summarization comes with its own set of task-specific challenges, essentially applying almost exclusively to the long input document case, since there is typically no reason to summarize a short text.\n* **Multiple topics**: The length of the input document gives rise to significant issues which go beyond simple computational and hardware requirements. Extractive DNNs particularly struggle with long documents ; the greater the length of the input, the more topics it typically covers, thus the harder it is for a DNN to generate a summary covering effectively all of them.\n* **Evaluation metrics**: The ROUGE evaluation metric, which is the most commonly used metric for summarization tasks, might not be sufficient for abstractive DNNs. It has been claimed to perform considerably worse than in the case of extractive summarization, since it lacks semantic understanding of the words it compares.\n* **Sentence extraction**: Framing extractive summarization as a binary decision task assumes that the meaning of individual sentences is independent of that of others, leading to high redundancy. Even accounting for this fact, such DNNs are still vulnerable to greedily picking general sentences, instead of finding a combination of specific sentences which together summarize the document much more effectively.\n* **Repetition**: One of the most common issues of abstractive DNNs dealing with long texts is sentence repetition; this is potentially caused by the over-reliance of a Decoder to its input, which gives rise to an endless loop of phrase repetition. Internal issues concerning RNNs with attention mechanisms have been pointed out as the root cause of both this issue and of false information appearing in the output summary.\n* **Hallucinations**: Abstractive DNNs additionally suffer from the opposite issue of hallucinations, i.e., generating summaries with content that is not actually present in the original document. According to recent research, hallucinations tend to appear during summary generation when the DNN is uncertain about how to continue, thus favouring common phrases it has memorized from its training set.\nSolutions\n---------\nThis section reviews the commonly employed ROUGE evaluation score, as well as extractive and abstractive text summarization methods. The evolution of relevant algorithms, as well as how each family attempted to handle the challenges it faces, are briefly discussed.\n### Evaluation Metrics\nROUGE is a very commonly used metric for evaluating ATS algorithms, featuring as many as 192 variations. The base ROUGE metric measures the number of overlapping n-grams/sequences between the generated and ground-truth summary. This is sufficient for many extractive summarization tasks, but faces severe challenges when used for abstractive summarization. ROUGE is unable to distinguish between different grammatical versions of the same word and synonyms, while being agnostic towards sentence ordering. Most importantly, the underlying assumption made by ROUGE, that the quality of a summary is correlated to its coverage compared to the human-generated summary, may not be statistically significant.\nThe reasons why ROUGE is still popular despite such severe drawbacks are multiple, including the unavailability of proven competitors and the lack of easy-to-use software implementations or standardized benchmarks for ATS evaluation. Besides, alternative metrics may suffer from similar weaknesses, since they typically do not account for factual consistency between the source document and its evaluated summary. For a detailed discussion of text summarization metrics, the reader is referred to. For the purposes of this article, a number of common and/or recent alternatives to ROUGE are briefly mentioned below:\n* METEOR (recommended by ), functions similarly to ROUGE, while being largely invariant to grammatical variants and synonyms.\n* Sem-nCG is a recent, semantically-aware metric focused on extractive summarization.\n* BERTScore uses the similarity of the contextual embeddings between the generated and the ground-truth summaries.\n* *S*3 combines other evaluation metrics through a model, in order to produce an aggregated evaluation of the summary.\n* BLANC is an unsupervised evaluation metric which does not depend on human-generated ground-truth summaries, by comparing BERT’s performance on the Cloze task (on the source document) when it has and when it has not additional access to the generated summary.\n* BARTScore offers multiple directions (Faithfulness, Recall, Precision, F-score) which make it a flexible metric for different evaluation scenarios.\n* Contextualized Topic Coherence Metrics are a group of metrics which evaluate the coherence and semantic similarity among the topics covered by a generated text.\nMany more evaluation metrics have been proposed, but none has taken a foothold yet. The question of a capable successor to ROUGE is still an open one.\nBesides summarization evaluation metrics themselves, factuality metrics have also recently proven instrumental in abstractive summarization due to their application in evaluating or fighting hallucinations. Three important and relevant recent metrics are the following ones:\n* FactCC (Factual Consistency Checking) is essentially a BERT model that has been finetuned on predicting, as a binary output, the factual consistency of each summary sentence with the original document.\n* SummaC (Summary Consistency) is somewhat similar to FactCC in conception, but is implemented as a Natural Language Inference (NLI) DNN that checks whether each summary-document sentence pair is consistent. Per-sentence scores can be combined into a single score per summary.\n* CTCScore (Compression, Transduction and Creation Scores) is a suite of various metrics for evaluating different aspects of natural language generation, including a consistency aspect, based on pretrained DNN language models that are utilized for evaluating information alignment between different texts. The particular property of CTCScore is that, by design, it can be exploited for evaluating performance in many different generative NLP tasks.\n### Extractive summarization\nThis Subsection presents the gradual evolution of extractive summarization DNNs towards the current SoA.\n#### Building Summaries Using Hierarchies\nThe first attempts to use DNNs for extractive summarization, while certainly obsolete today, developed core concepts that were later re-used and improved.\nThus, the method in developed a neural Encoder-Decoder architecture, which is trained in a supervised manner by preprocessing human-written summaries to construct extractive ground-truth summaries (binary selection vectors). Architecturally, the Encoder is composed of a shallow CNN, called “Convolutional Sentence Encoder”, and a subsequent LSTM, called “Recurrent Document Encoder”. The first one analyzes a matrix of Word2Vec word embeddings, separately for each sentence of the source text, aggregating them to dense sentence embeddings. The second one sequentially receives these embeddings as input and generates a document-level embedding in the form of a fixed-size vector.\nThe Decoder is one of two models following different, mutually exclusive strategies: a “Sentence Extractor”, which approaches the extraction task as a sentence selection problem, and a “Word Extractor” which approaches it as a word generation task. The latter one is practically a limited-efficiency abstractive summarizer, with its supported vocabulary constrained to that of the source document, so it will not be detailed here. The Sentence Extractor is an LSTM-MLP combination, sequentially labeling individual source sentences for inclusion in the summary based on two criteria: i) whether they are relevant, and ii) whether they are mutually redundant with other sentences. The selection decision for the *t*-th sentence is computed as follows:\n \n$$\\mathbf{\\hat{h}\\_t} = LSTM(p\\_{t-1}\\mathbf{s\\_{t-1}}, \\mathbf{\\hat{h}\\_{t-1}}),$$ \n \n$$p\\_t = \\sigma(MLP(\\mathbf{\\hat{h}\\_t}; \\mathbf{h\\_t})),$$ \nwhere $\\hat{\\mathbf{h}}\\_t$/**h***t* is the Extractor’s/Encoder LSTM’s hidden state for the *t*-th sentence/time step, respectively. *p**t* − 1 represents the assigned probability that the previous sentence **s***t* − 1 belongs to the summary, and ; is the concatenation operation.\nFollowing-up on, “SummaRuNNer” was proposed as an extractive summarization RNN with the useful ability to be directly trainable on abstractive ground-truth summaries. It contains multiple layers of bidirectional GRUs and a hierarchical Encoder. Similarly to, the first subnetwork generates sentence embeddings from individual word representations, which are sequentially fed as input to the second GRU. The latter one analyzes them and outputs a document-level embedding from the sentence representations; this is done by average pooling the second subnetwork’s hidden states. The Decoder is simply a logistic binary classifier, assigning to each source sentence a “summary”/“non-summary” label. To be directly trainable on datasets with abstractive ground-truth summaries, a separate RNN Decoder can be optionally attached to the Encoder *only* during training and tasked to emit the most probable word at each time step. Figure [fig::SummaRunner] depicts the SummaRuNNer architecture.\n[fig::SummaRunner]\nThe method “LSTM-Ext” improves on and by considering both global and local context information along with each sentence, in order to decide on its inclusion into the generated summary. Inspired by how humans read lengthy text, it was reasonably hypothesized that hierarchical information could benefit ATS. The “LSTM-minus” algorithm is applied to generate local context-aware embeddings at the section level. Similarly to and, an Encoder-Decoder architecture is employed. The Encoder is again composed of a Sentence Encoder (SE) and a Document Encoder (DE), but now also includes a “Topic Encoder” (TE) that provides access to local (section) context surrounding each sentence.\nSE is a GRU that maps input word embeddings to a fixed-size sentence-level vector, while DE is a bidirectional GRU that encodes the sentence embeddings into a document-level vector representation. The final document embedding is computed as the concatenation of the final hidden states of the backward and the forward GRU. TE separately embeds local context about each sentence’s section by employing the LSTM-minus method; the result is an additional vector per section. Subsequently, for each sentence, the embeddings of itself, of its section and of the entire document are concatenated, optionally with weights assigned by an attention mechanism, and fed to the Decoder. The latter one is simply a binary MLP classifier that predicts whether each sentence should be included in the summary or not.\nThe architecture’s efficiency seems to increase with increasing document length, making it a powerful architecture for long documents, while the most benefit comes from the use of local rather than global context. Thus, LSTM-Ext was among the first modern document summarization methods that explicitly handle the fact that longer texts tend to cover multiple topics. On the other hand, it is not designed to avoid unnecessary repetitions in the generated summary, resulting in outputs with potentially high redundancy.\n#### Hierarchical Summaries using BERT\nIn recent years, Transformer and BERT variants gradually replaced RNN and Word2Vec/GloVe variants in extractive summarization. However, older ideas have been retained and improved, with their efficiency increased, in an attempt to resolve long-standing challenges of document summarization.[sssec: HierBERT]\nThe method in uses BERT for both extractive and abstractive summarization (BERTSUM-EXT and BERTSUM-ABS, respectively). Instead of using sentence-level vector representations/embeddings, BERT inherently generates individual token representations. Thus, BERTSUM-EXT inserts special [CLS] tokens in-between sentences as separators, to notify the DNN of distinct sentences and capture sentence-level context within the respective generated token representations. This modified pretrained BERT is utilized as the Sentence Encoder component within an Encoder-Decoder architecture. The Document Encoder is a Transformer that receives the sentence-level embeddings of the [CLS] tokens that are derived by BERT and transforms them so as to contain richer document-level context. The final resulting sentence embeddings are fed to a shallow, fully connected binary classifier that predicts whether each sentence is to be included in the summary or not. All input documents are truncated to a fixed number of supported tokens. The overall architecture is finetuned in a supervised manner, using ground-truth binary selection vectors.\nDespite the relatively strong performance of BERTSUM-EXT, it faces the typical challenges encountered by Transformers in long document analysis, such as varying and diverse topics. To counter this, the approach in extends BERTSUM-EXT by taking advantage of the usually strong hierarchical structure of long texts (individual sentences compose titled sections, which jointly form the document). Thus, it computes hierarchical positional embeddings that capture the relative position of a token within each section and the position of that section in the document. These representations, along with additional representations of the section titles, are incorporated into the sentence-level embeddings by the Document Encoder, which feeds the final shallow, fully connected binary classifier. Of course, the overall “HiStruct+” architecture is finetuned for extractive summarization. The method seems to perform well in long scientific articles with a rigid section structure, while still achieving SoA performance in less formal documents of short-to-medium length. Finally, by evaluating the method using pretrained Transformer language models other than BERT, the Longformer seems to come at the top.\nAnother approach aimed at long document extractive summarization is GBT-EXTSUM (Global BERT-based Transformer for Extractive Summarization). It is an architecture composed of stacked propagation layers, each consisting of a Transformer layer, a BiGRU for inter-block information propagation and an MLP. GBT-EXTSUM achieves scalability to long document sequences without relying on attention mechanisms, by introducing a hierarchical structure with recurrent hierarchical modules and Bidirectional Gated Recurrent Units (BiGRU). This design allows for the propagation of information between text blocks at different levels, eliminating the need for attention mechanisms and providing an effective solution for processing lengthy documents. Furthermore, it eliminates the need for positional encoding on block representations, contributing to the efficiency of summarization tasks.\n#### HyperGraph Transformers\nHyperGraph Transformers are a recently proposed approach to long document extractive summarization, exploiting ideas from previous hierarchical methods and implementing them using Graph Neural Networks (GNNs).\nThe method in attempts to deal with the Transformer’s quadratic complexity and context retention problems by developing a hybrid architecture called “HEGEL” (HypErGraph transformer for Extractive Long document summarization), relying on the so-called *Hypergraph Transformer*. HEGEL models the document as a graph *G* = (*V*, *E*), where *V* is a set of nodes and *E* a set of *hyperedges*: an edge that can connect two *or more* nodes. We use the notations *u* ∈ *e* and $u \\notin e$ to represent whether a node is connected to a hyperedge in the graph *G* or not, respectively.\nIn HEGEL, the graph nodes are the document’s sentences and extractive summarization is modelled as a node classification task. Each sentence is initially semantically embedded to a dense vector using a pretrained BERT. There are three kinds of hyperedges in this document graph:\n* A *section hyperedge* connects all sentences that belong to the same section. For instance, in the context of a scientific article, a section could be the problem statement, the methodology or the conclusions. An incidence matrix *A**s**e**c* is defined as:\n \n$$\\mathbf{A}^{sec}\\_{i,j} =\n \\begin{cases}\n 1, \\text{ if } s\\_{i} \\in e^{sec}\\_{j} \\\\\n 0, \\text{ if } s\\_{i} \\notin e^{sec}\\_{j},\n \\end{cases}$$ \nwhere *s**i* ∈ *e**j**k**w* means the *i*-th sentence belongs in the *j*-th section.\n* A *topic hyperedge* connects all sentences sharing the same topic. Topic clustering is implemented by using Latent Dirichlet Allocation. The incidence matrix **A***t**o**p**i**c* is defined as:\n \n$$\\mathbf{A}^{topic}\\_{i,j} =\n \\begin{cases}\n 1, \\text{ if } s\\_{i} \\in e^{topic}\\_{j} \\\\\n 0, \\text{ if } s\\_{i} \\notin e^{topic}\\_{j},\n \\end{cases}$$ \nwhere *s**i* ∈ *e**j**k**w* means that the *i*-th sentence belongs to the *j*-th topic.\n* A *keyword hyperedge* connects all sentences containing the same keywords. These keywords are extracted using Key-BERT from the training sample. The incidence matrix **A***k**w* is defined as:\n \n$$\\mathbf{A}^{topic}\\_{i,j} =\n \\begin{cases}\n 1, \\text{ if } s\\_{i} \\in e^{kw}\\_{j} \\\\\n 0, \\text{ if } s\\_{i} \\notin e^{kw}\\_{j},\n \\end{cases}$$ \nwhere *s**i* ∈ *e**j**k**w* means that the *i*-th sentence contains the *j*-th keyword.\nThe overall incidence matrix *A* ∈ R*n* × *m* is generated by concatenating the three incidence matrices as:\n \n**A** = **A***s**e**c*; **A***t**o**p**i**c*; **A***k**w*. \nHEGEL receives as its input both the sequence of initial sentence embeddings and the overall incidence matrix. It adopts the hierarchical positional embeddings from and defines an alternative “hypergraph attention” mechanism, along with the corresponding multihead hypergraph attention module, instead of regular self-attention heads. As in the vanilla Transformer, these components are contained within a deep neural architecture with standard fully connected layers, skip connections and layer normalization mechanisms. This Hypergraph Transformer extracts semantically rich sentence-level representations that are well aware of local context, global context and semantic interrelations between different parts of the document. The overall architecture is trained end-to-end for binary sentence classification using a ground-truth binary selection vector per document.\nHEGEL manages to outperform all extractive and abstractive baselines on several datasets of scientific articles, demonstrating the potential of GNNs for long document summarization. It remains to see if such architectures can retain this level of performance on less rigid and hierarchical document types. Figure [fig::HEGEL] depicts schematically HEGEL’s architecture.\n[fig::HEGEL]\n### Abstractive Summarization\nAbstractive summarization is typically approached as a sequence-to-sequence language modeling task, where a neural Decoder predicts the next word of the summary at each time step. The vast majority of modern relevant methods are trained in a supervised manner, using ground-truth summaries. This is arguably a more advanced task than its extractive alternative.\n#### Pointer-Generator Networks\nOne of the first truly successful abstractive summarizers is essentially a hybrid one: a DNN that dynamically chooses whether to select a source sentence verbatim, as an extractive method, or synthesize the next word in the summary. Inspired by the human ability to refer to unknown objects by pointing at them, such Pointer-Generator Networks (PGNs) are an improvement over the “Word Extractor” proposed by.\nPGN can copy out-of-vocabulary (OOV) words from the source document, making it possible to generate text with rare words and keep a smaller vocabulary, thus reducing computational and memory costs. Essentially, it is a modified, attention-augmented sequence-to-sequence Encoder-Decoder architecture implemented similarly to, where both the Encoder and the Decoder are one-layer, bidirectional LSTMs. The Decoder ends at a softmax layer and at each time step generates a new word, originating either in the source document or in a global supported vocabulary. Thus, at each time step of the Encoder’s/Decoder’s unfolding, the next token is read/outputted from the input/to the generated summary, respectively. In order to decide on the course of action for the next word, the PGN defines its output distribution over the union of the global and the document’s vocabulary (called the “extended vocabulary”), thus deciding at each time step whether it should copy or generate the next word. If a word is OOV, it will be copied from the source document by sampling from the attention distribution, while if it does not exist in the document a new word will be generated.\nThe attention mechanism is defined as in. During inference, a *coverage vector* is updated at each Decoder time step, containing the sum of attention distributions over all previous time steps. It is provided as an additional input to the attention mechanism, so that the latter avoids giving attention to already extracted words and, thus, repeating them in the summary. To enhance this behaviour, PGN is also guided during training by an extra *coverage loss*: a regularizer that penalizes repeated attention to the same input tokens. Thus, the overall loss function computed at the *t*-th Decoder’s time step is:\n \n*L**o**t* = *L**M**L**t* + *λ*∑*i**m**i**n*(*a**i**t*, *c**i**t*), \nwhere *λ* is a scalar weighting hyperparameter, **a***t* is the current attention distribution, index *i* traverses the Encoder’s time steps, **c***t* is the current coverage vector, while *L**M**L**t* is the primary loss term at the *t*-th Decoder’s time step. Typically, *L**M**L**t* is the negative log likelihood of the *t*-th ground-truth summary word according to the final Decoder output, i.e., a generated probability distribution defined over all supported words of the extended vocabulary. The overall loss is summed over all the Decoder’s time steps. PGN does not use any pretrained word embeddings, learning instead token representations from scratch. At the time of its appearance it showed remarkable success in handling OOV words, while also featuring a smaller vocabulary size and faster training times. It thus successfully handled most of summarization challenges. However, although historically significant, PGN cannot handle long documents well. As a result, more efficient and specialized architectures were later developed.\n#### BERT & Sparse Attention Transformers\nAs described in Subsection [sssec:extractive], BERT has been adapted for extractive summarization in the form of BERTSUM-EXT. That neural architecture has also been employed for abstractive summarization in the form of BERTSUM-ABS.\nThe Encoder of BERTSUM-ABS is a pretrained BERTSUM-EXT model, a choice which provides efficiency gains, due to transfer learning, and task-specific gains, due to emerging synergies between extractive and abstractive summarization. The Decoder is a regular, randomly initialized 6-layer Transformer. Different finetuning and training schedules are utilized for the Encoder and the Decoder, respectively, to avoid stability issues during optimization. Overall, the approach attained only marginal performance gains over competitors.\nSection [ssec::ClassificationSolutions] reviewed the issues vanilla Transformers face with very long documents, with the biggest challenge being the quadratic cost of the self-attention mechanism. The various sparse attention solutions that attempt to bypass this restriction can also be employed for document summarization, by replacing a regular Encoder with a Sparse Transformer variant. One such architecture is the “Longformer-Encoder-Decoder” (LED), i.e., where Longformers are utilized both for the Encoder and the Decoder. Critically, the Decoder pays full attention to the entire Encoder and to already decoded positions of the input. As expected, LED scales linearly with the input size, and has been shown to achieve highly competitive performance.\nSimilarly, BigBird has also been employed for abstractive document summarization. The sparse attention mechanism is used only in the Encoder, which handles the long input, while the Decoder exploits full attention, as the length of the summary (in tokens) is small by definition. BigBird achieved significant performance gains on long document summarization and competitive performance on smaller ones.\nBigBird makes use of Gap-Sentences Generation (GSG) for abstractive summarization. It is a self-supervised pretraining objective especially constructed to aid downstream finetuning for text summarization. Essentially being a variant of T5’s pretraining objective, it consists in masking an entire sentence with a single [MASK] token. Then, during pretraining, each input is a set of consecutive sentences, with a subset of them selected and hidden according to GSG. The DNN is tasked to generate the masked sentences jointly, in the form of a pseudo-summary, based on the seen ones. The selection of which input sentences to mask is not random, but based on the ROUGE score of each sentence with the remaining input text. Thus, the ones finally hidden with [MASK] tokens are those that have been automatically judged as the most important ones. It must be noted that GSG was originally introduced as the main novelty of the PEGASUS neural language model, following a Transformer Encoder-Decoder architecture, like BART or T5.\nGSG and a variant of the sparse attention mechanism is also employed for abstractive document summarization in LongT5, which is a modification of the basic T5 neural architecture. Thus, the LongT5 Encoder contains self-attention sublayers where each input token attends to its local neighbourhood plus to all global tokens. The latter ones are being dynamically constructed before each attention operation, by summing and normalizing different, non overlapping subsets of the input token embeddings. Despite the approximate nature of this sparse attention mechanism, the resulting ability to process longer inputs leads to overall improved accuracy in long document summarization.\nAnother variant of the Sparse Attention Transformer mechanism is the EMMA Memory-Augmented Transformer Summarizer, which was developed for low resource systems and uses text segmentation in order to better process the text in chunks. It then feeds the data into a modified version of BART which utilizes a long-term memory matrix for each layer, combining the previous and current memory matrices in order to avoid losing long-term details. This modified BART model also features cross-memory attention in order to consider information from previous chunks when processing the current chunk. This results in the EMMA model generating more contextually coherent summaries, while retaining its low space complexity of O(*L**c*2) (where *L**c* is the predetermined max chunk size).\n#### Local Attention and Content Selection\nThe majority of modern abstractive summarizers learn to implicitly select pertinent content from the input in order to generate the summary. Recently, attempts have been made to combine the advantages of Sparse Attention Transformers, which allow the handling of longer documents, with explicit content selection before the abstractive summary generation.\nThe main abstractive DNN in is a modified version of BART, called LoBART. Vanilla BART supports input sequences with a length up to 1024 tokens. LoBART modifies its layers so as to employ the sparse attention scheme of Longformer, with the difference that only local and no global attention is utilized. This allows LoBART to drop its memory requirements significantly, at the cost of missing context over large spans of text. The reduced costs result in a sliding window of *W* = 700 tokens, which is calculated as 2 × *D* where *D* is the empirical average attention distance of *D* ∈ [250, 350] tokens for an attention mechanism with uniform weights, attending to a distance of 1024. Even with this compromise, important context may still be missed, especially in long documents.\nAs a result, the method in feeds LoBART with suitably preprocessed input sequences, so as to effectively increase its context span. This preprocessing takes place separately during the training and the test stage. At training, the input data are first passed through a content selection algorithm. This can either be an ``oracle“ (ORC), which selects sentences based on the ground-truth (the ROUGE score of the selected input vs its abstract), or a hierarchical, extractive DNN pretrained for multitask content selection (MCS). The straightforward ”ORC“ preselection keeps an average of 21.3% of sentences, drastically reducing the input’s size and leads to 56% of the input documents being shorter than the typical BART input length of 1024. Missing sentences are padded with random sentences sampled from the input document. On the other hand, MCS is a GRU-based, attention-augmented Encoder-Decoder architecture followed by a classification layer. It employs a hierarchical design similar to that of (see Section [ssec::ClassificationSolutions]), but in this case it is only utilized to preprocess the input document. In order to exploit synergies between extractive and abstractive summarization, similarly to, MCS is jointly pretrained (in a multitask fashion) both for predicting the ground-truth summary token-by-token and for binary sentence classification as ”summary“ or ”non-summary\".\nDuring inference, each sentence is first scored for retention by suitably combining the two different predictions (one per task) made by MCS. The top-scoring ones are fed to LoBART as the input sequence. LoBART has been trained for abstractive summarization in the usual manner, but the two-stage content preselection stage (explicit during preprocessing, implicit during the main summarization process) allows the method to efficiently handle longer documents.\n#### Combining Top-Down and Bottom-Up Inference\nAn innovative abstractive methodology relying on a composite Transformer architecture is proposed in, aiming to facilitate the sharing of global and local context between different token representations. This is done by allowing information to flow both from the individual token representations to the document embedding (bottom-up) and vice versa (top-down).\nThe Encoder in features a hierarchical approach similar to certain ones utilized for extractive summarization (e.g., ): individual token representations computed by an initial (bottom-level) deep Transformer are pooled in segments to form top-level embeddings of coarse granularity at a paragraph/section-level. This is the so-called “bottom-up” route. The preliminary top-level embeddings (one per section) are then transformed via a separate, shallow Transformer. However, the main novelty of is that these final top-level representations are then injected to the individual token embeddings via a separate, subsequent deep“top-down” Transformer. The latter’s layers are equipped with regular self-attention, as well as with cross-attention between the top-level representations and the individual token embeddings. The final output token representations are attended by the Decoder to generate the desired abstractive summary in the usual manner.\nBy properly adapting Sparse Attention Transformers, the bottom-level and the top-down layers are assigned efficient local attention mechanisms only. Top-level layers can afford to be assigned full attention due to the much smaller number of sections, compared to the number of tokens. Thus, the top-down information flow enriches the lower-level embeddings with global context and the ability to capture long-range dependencies: individual token representations effectively attend to aggregated global information from the entire document, despite the local attention-only mechanisms of bottom-level layers.\nThe methodology is rather generic and model-agnostic, allowing it to be used with efficient Sparse Attention Transformers. This fact and the hierarchical design facilitate exceptional summarization performance in long documents, even at book-length texts. Exploring its applicability to other tasks, such as document classification, is a promising research direction.\n#### Instruction-tuned summarization\nAn orthogonal direction of recent research is that of supervised finetuning of pretrained language models on ``instruction\" and corresponding ground-truth summary pairs. The instruction is a command to summarize in natural language, followed by the long text to be summarized. The goal is to then use, during inference, the finetuned DNN like one would use a pretrained LLM under a zero/few-shot prompting mode. This can be done both with Encoder-Decoder Transformer architectures pretrained on masked token objectives and with LLMs, which typically are Decoder-only Transformer architectures pretrained on traditional next token prediction objectives. Both T5-based architectures and LLM-based architectures have recently followed this approach, typically under a multitask instruction tuning setting where text summarization is one among multiple tasks. According to common automated evaluation metrics, this approach typically leads to subpar summarization results in common public benchmarks, compared to SoA dedicated summarization DNNs. However, such summaries generated from instruction-tuned LLMs can actually be better than ones obtained from dedicated summarizers and even surpass the ground-truth ones in quality, according to manual, subjective evaluation. Although instruction-tuned summarizers tend to perform a lot worse with long text inputs, there have been recent breakthroughs such as the Longform dataset, constructed via diverse instruction generation methods; finetuning LLMs on Longform has recently led to good results on various downstream tasks. Overall, this is a still an open area of cutting-edge research, where progress in automated evaluation metrics is a crucial prerequisite for further advances (as discussed in Subsection [sss::SummarizationEvaluation]).\n#### Dealing with hallucinations\nA few dedicated methods have appeared in the abstractive summarization literature, concerned with how to reduce the appearance of hallucinations in the generated summary. Thus, these methods are not summarizers themselves, but can be employed in conjunction with an abstractive summarization DNN in order to improve its output.\nThere are two main families of hallucination containment methods: *post-processing* approaches and *faithfulness-aware training*. In the first case, a separate DNN is appended after the summarizer and evaluates the latter one’s outputs. An important such method is, where a sequence-to-sequence summarizer is followed by a scoring RoBERTa model. The summarizer outputs multiple candidate summaries and the scoring model is trained to rank the candidates according to their ROUGE evaluation score with regard to the ground-truth summary, using contrastive learning. Thus, the two models are being trained separately in a supervised manner, while during the test phase the scoring model predicts quality scores without access to ground-truth. The candidate summary with the highest score is finally selected. An alternative post-processing approach models the issue as a trade-off between faithfulness to the original document and abstractiveness (in contrast to extraction), and then attempts to generate a summary that transcends said trade-off by being both faithful and abstractive. This is achieved by training a follow-up selector DNN to explicitly choose among a set of candidate summaries, at different levels of extractiveness, the one that is both the most abstractive and faithful.\nIn contrast to such approaches, faithfulness-aware training trains the summarizer itself with additional objectives for hallucination containment. Several methods have appeared in this direction, for example focusing on filtering the training data . A more recent method modifies the GSG pretraining objective of the PEGASUS architecture, where the employed ROUGE score is combined with the factuality metric FactCC. Downstream finetuning for summarization is also modified in three complementary ways: i) the [MASK] token from the GSG pretraining objective is inserted into the downstream training document as well, so that factuality knowledge acquired during pretraining is better retrained, ii) hallucinations are explicitly removed from the ground-truth summaries of the downstream dataset, and iii) an additional finetuning objective, borrowed from the contrastive self-supervised learning literature, encourages the summarizer to internally generate summary representations that are close to the embedding of the original document and away from those of artificially constructed non-factual summaries.\nAn alternative method for faithfulness-aware training explicitly quantifies faithfulness as the ratio of the summary’s named entities derived from the original document. This is measured for all training documents, with respect to the corresponding ground-truth summaries, and converted to a suitable code representation which is given to the summarizer as a conditioning mechanism during training. During inference, the code denoting high faithfulness is given as input along with the test document. The method in also focuses on the consistency of named entities, attempting to eliminate hallucinated entities that do not appear in the original document, while retaining ones that do appear in the ground-truth summary. This is achieved by incorporating the proposed ``SpanCopy\" mechanism into the summarizer’s Decoder, assuming an Encoder-Decoder architecture. SpanCopy directly copies in the output entities from the full-length source document and its activation is controlled by a final classifier that decides whether the next span of text will be copied or generated in the usual manner; thus, the method bears similarities to the PGN architecture, but operates the named entity level. The entire DNN is trained end-to-end in a supervised manner.\nA different aspect has been explored in, where alternative strategies by which the Decoder decides on the exact next output token during summary generation are compared with regard to their effect on hallucinations. There are various such decoding strategies, beyond the default greedy selection of the most probable output token, which are commonly utilized in text generation tasks. The most widespread alternatives to greedy selection are *nucleus sampling* and *beam search*, with the latter one identified in as the one that generates the most faithful summaries. The authors modify it to more explicitly consider hallucinations containment in two ways: i) by internal re-ranking of the candidates, based on summarization evaluation metrics that reward faithfulness, and ii) by looking ahead into the future contingent summary that can be generated based on the current partial summary and scoring it for faithfulness; this added heuristic is incorporated into the search process.\nOverall, dealing with hallucinations in abstractive summaries is a currently very active research topic that overlaps with factuality/constistency checking and NLI.\nExtractive or Abstractive Summarization?\n----------------------------------------\nSoA summarization models for long texts are divided into 2 categories, abstractive and extractive summarizers, each with their own strengths and weaknesses.\n* Extractive summarization DNNs tend to be more simple, factually consistent and have a much smaller risk of introducing errors, since the extracted sentences come directly from the source text. However, they lack creativity and may suffer from redundant, repeated information.\n* Abstractive summarization DNNs solve the problems that extractive summarizers present, generating more creative and diverse summaries without significant repetitions, and are also able to condense information more efficiently. However, they suffer from hallucinations and lack of factual consistency, introducing a much bigger risk of containing inaccurate information. As a result, it is more complex and difficult to properly utilise them in practical situations.\nGenerally, both approaches have their strengths and SoA architectures can employ both to achieve good results. For instance, certain recent approaches for fighting hallucinations in the abstractive case essentially attempt to force the DNN to become more extractive. Extractive summarization DNNs seem to outperform abstractive ones in relevancy and informativeness, however longer text input size seems to narrow that gap down. Hybrid models seem to be the most promising avenue for long text summarization in most tasks where the human-like element that abstractive summarizers provide is important. A thorough review and comparison of the two types of architectures specifically for long texts can be found in.\nSentiment Analysis\n==================\nOpinion mining (OM) or sentiment analysis (SA) is the extraction of *polarity sentiment* (positive/negative) from a piece of text using NLP algorithms. It is typically approached as a particular case of text classification, although sentiment score regression is also common. Thus, possible opinions are modelled either as a set of discrete classes (e.g., “positive”, “neutral”, “negative”), or as a continuous range of sentiment scores (e.g., in the [ − 1, 1] range). Additionally, emotional dimensions besides polarity are occasionally considered (e.g., classes such as “fear”, “joy”, “sadness”, etc.), potentially by relying on psychological theory (e.g., ). Common use-cases mainly concern SA of short texts, such as social media posts, product reviews and online discussions, with obvious applications in marketing and brand management or political analysis. For an in-depth review of SA applications, the reader is referred to.\nSA for long texts is not the most common task, since context and tone can change throughout a lengthy passage, making it difficult to maintain a constant sentiment. Moreover, the presence of complex language structures and nuanced expressions in long texts can pose challenges for SA models. Additionally, in certain domains SA is meaningless due to the typical lack of emotionally charged words (e.g., in academic articles, medical or legal documents, etc.). However, it is important for automatic analysis of long journalistic articles, to extract their political viewpoint, financial texts, to predict how markets will react to the presented news, or literary books, in order to process narrative hints. For example, it can be used to determine characters’ emotional states, or for identifying key narrative points, sections and anticipated reader feelings. These predictions are potentially useful in themselves, but they can also be exploited as auxiliary input for particular book classifications tasks (e.g., automatic genre recognition in library software).\nSolutions\n---------\nSA has peculiarities that distinguish it from vanilla document classification. Traditional SA algorithms rely on ``sentiment lexicons“, i.e., predefined vocabularies of words that are heavily weighed towards positive or negative emotions. For example, words such as ``good”, ``bad“, ``great”, ``terrible“, etc. are likely candidates for inclusion in such as lexicon. Algorithms relying on lexicons and/or traditional classifiers (e.g., Naive Bayes, SVMs, ensembles, etc.) were common before the advent of DNNs . However, texts with complicated vocabulary often lack ``simple” keywords that are included in most lexicons. In such cases, most lexicon-based algorithms struggle to classify accurately.\nAs a result, long document SA has moved decisively towards DNNs in the past few years. The main body of relevant research approaches the field as simply an instance of document classification, so the content of Section [sec::Classification] will not be repeated here. The relevant challenges are common or analogous to those found in most long document analysis tasks, such as prohibitive computational costs, difficulty to process long input sequences, long-range dependencies that are hard to identify and sentiment variation across different sections or paragraphs.\nFor example, utilizes a BiLSTM with an embedding layer and a group-wise enhancement mechanism, which is an alternative to a ``traditional\" attention mechanism. The group-wise enhancement mechanism adjusts the significance of sub-features within specific spatial and semantic groups in the text. It employs attention factors for each spatial position in semantic groups, allowing independent enhancement of learned expressions and suppression of noise. Inspired by, it utilizes feature weighting schemes on LSTM and GRU to extract informative features. Its goal is to leverage general information from critical regions, approximating the semantic vector for each group. It does so by dividing feature groups, obtaining global statistical features through spatial averaging, and scaling context vectors based on generated importance coefficients. This approach aims to refine representation and discriminative capabilities in DNNs, particularly concerning uninformative features.\nSA can greatly benefit from sarcasm/figurative language (S/FL) detection, to a degree much greater than vanilla document classification. Neural S/FL detection has mostly been conducted using Bi-LSTMs, GRU or CNNs for short-to-medium length texts, therefore integrating such approaches into document/book-level analysis is an open challenge. An interesting avenue would be to exploit related short text semantic detectors of high specificity (e.g., S/FL detection) at the sentence-level, given that such tools are mature and are indeed being used to aid short text SA, aggregating their results in a hierarchical fashion to facilitate document-level SA. However, this research direction has not been significantly exploited.\nExisting deep neural methods for long text SA that offer a particular angle, compared to simple document classification, are reviewed below. The majority of these and other similar methods have not been evaluated on actually long documents, due to the lack of relevant public annotated datasets, but they can in principle support them. Overall, however, long document SA is more of a near-future vision, rather than a currently blooming field.\n### Hybrid architectures\nInitial efforts tackled documents that were a few paragraphs long, instead of book-length. For example, a composite, hierarchical deep neural architecture is proposed in for classifying long user reviews (e.g., of movies, restaurants, etc.). In order to handle polysemy, the method exploits the “principle of compostitionality”, which states that the meaning of a body of text (a sentence, a paragraph or a document) depends on the meanings of its constituents, by using an LSTM to generate sentence-level embeddings for the entire document. These are sequentially fed to a subsequent GRU that encodes their relations and semantics, outputting a document representation that is classified by a final softmax layer. The GRU was explicitly selected due to its virtual lack of an output gate, since it was deemed necessary for all available sentence-level information to be used in the construction of the document representation. The overall architecture is trained end-to-end for classification and receives dense Word2Vec embeddings at its input layer.\nA similar method combines a CNN and a bidirectional LSTM to extract the sentiment of newspaper articles. It uses Doc2Vec to pre-generate paragraph-level embeddings that form the input matrix for the CNN. An alternative hierarchical method is presented in, where an initial bidirectional LSTM sequentially receives word embeddings per sentence and transforms them to sentence embeddings. A second bidirectional LSTM equipped with an attention mechanism receives the latter representations to generate a document-level embedding in the form of a matrix, which is then fed to a final 2D CNN followed by the softmax classifier layer. As a general statement, it can be said that such straightforward hierarchical approaches (e.g., ) have become a de facto baseline in long text SA, with the recurrence of LSTMs typically exploited to handle the variable and large number of tokens, sentences and paragraphs in long documents.\n### Sentiment Analysis with Transformers\nSoA models make use of Transformers, with BERT and its variations being very commonly used for SA. The employed pretrained language model is typically finetuned on the desired SA dataset. For instance, this is how RoBERTa is utilized in for SA of financial texts. However, the document-level sentiment classification task is implemented in an ensemble learning fashion, using weighted joint prediction from multiple independent classifiers, and then combined with a separate pipeline for *key entity detection*, so that specific entities (e.g., companies, countries, etc.) are identified in the text. Each of these entities is individually fed to an instance of an appropriately finetuned RoBERTA model (different than the SA ensemble participants) along with the document, in order for the DNN to predict whether each recognized entity is critical for its meaning.\nPublic Long Document Datasets\n=============================\nLong text analysis is not a well-explored field, particularly for document-level analysis of book-length texts (e.g., literary fiction). As a result, compared to small text analysis, the amount of relevant public, annotated datasets remains fairly limited. Furthermore, there is a clear imbalance with regards to the different NLP tasks; document summarization datasets dominate the field, while long document datasets for SA are nearly non-existent.\nTraditionally, concerning document summarization, the most commonly used dataset is **CNN/DailyMail**, featuring over 330k articles and their respective highlights, as written by journalists at CNN and the Daily Mail. On average, each article has 781 input tokens and 56 summary (target) tokens. While the dataset is certainly large and ships with convenient APIs, its individual articles are not be long enough for modern long document summarization. This has not prevented its continuing use as a benchmark, for measuring the efficiency of contemporary summarizers in short-to-medium length documents.\nTwo of the most prominent long document datasets at the time of writing are the **arXiv** and **PubMed** datasets. Both are composed of scientific research articles, and are used mainly for summarization. They feature a huge number of documents, remarkable document length and user-curated summaries in the form of article abstracts. Their success is evident from their inclusion in many popular machine learning libraries and from their popularity as benchmarks in current document summarization literature. Besides these ``main\" datasets, many variations have been made available. For example SumPubMed is a pre-processed, curated variant of the Pubmed dataset and which aims at making summarization more challenging.\nCertain datasets are notable for their focus on literary texts, rather than scientific articles which are the subject of most commonly used long-document datasets. An example would be the Project Dialogism Novel Corpus (PDNC dataset), which comprises 22 English novels. A modern large-scale dataset for literary fiction texts is also **LitBank**, a dataset of 100 English literature works that fully includes word annotations and co-reference relations between words that belong in different groups.\nRecently, two more challenging datasets have been released: **BookSum** and **SummScreen**. BookSum covers fiction books from multiple domains in literature, including short stories, novels and plays. SummScreen includes pairs of movie transcripts and respective human-written summaries. They both require the summarizer to combine narrative plot events from implicit subplots, potentially progressing in parallel, forcing it to rely on extremely long context spans and dependencies.\nOther notable datasets include a subset of long texts, such as the **OPUS** dataset for machine translation, while domain-specific datasets are crucial for domain-specific NLP algorithms, either for training or for domain finetuning. Below is a brief list of such datasets, featuring primarily long documents for classification or summarization tasks:\n* **LexGLUE**, a classification dataset containing a collection of smaller datasets of legal texts.\n* **ContractNLI**, a classification dataset for document-level Natural Language Inference (NLI). Each data point is a contract text and a set of hypotheses, with the three classes indicating whether each hypothesis agrees with, contradicts or is not mentioned by the contract.\n* **20 Newsgroup**, a dataset containing approximately 20,000 Usenet newsgroup documents partitioned evenly across 20 different newsgroups. This dataset provides a great amount and variety of data, but its articles tend to be significantly shorter than dedicated long document datasets.\n* **Hyperpartisan News Detection**, which contains documents for political standpoint classification. This dataset contains both long and short articles, therefore a split must be performed based on the text’s length for long document analysis.\nThe above-mentioned datasets are summarized in Table [tab::Dataset]. The posted dataset statistics have been either made available on their original citation, computed for the purposes of this article, or mentioned in other, properly cited papers. Given that these datasets are of gigantic size, utilize different formats and present drifts due to occasional expansions, certain sources may conflict on the exact size of these datasets.\n |p2cm|p2cm|p4cm|p2cm|p2.5cm|p1.5cm| **Name** & **Domain** & **Tasks** & **Documents** & **Average Document Size (words)** & **Language** \nCNN/Daily Mail (benchmark) & News Articles & Document Summarization & 300K & 781 & English \nLitbank & Literature & Named Entity Recognition & 100 & 2,000 & English \nPDNC & Literature (Old) & Quote Attribution, Speaker identification & 35,978 & 79,745 & English \narXiv & Academic & Document Summarization, Text Retrieval & 1.7M & 8,029.19 & English \nPubmed & Academic & Language Modeling, Document Classification, Document Summarization & 19,717 & 4,000 & English \nSubPubPubmed & Academic & Document Summarization & 33,772 & 4,227/1,578/1,891 depending on version & English \nBookSum & Literature & Text (Narrative) Summarization & 405 & 112,885.15 (BookSum Full) & English \nSummScreen & Screenplays & Document Summarization & 22,503 & 7,605.4 (FD subset) / 6,420.7 (TMS subset) & English \nLexGLUE (only classification subsets) & Legal & Multiclass Document Classification & 184,214 & 4,761.9 & English \nContractNLI & Legal & Multiclass Document Classification, Natural Language Inference & 607 & 2,254.3 & English \n20 Newsgroup & News & Multiclass Document Classification & 50,139 & 2,036.3 & English \nHyperpartisan & News & Document Classification & 750,645 & 4,848.17 & English \n[tab::Dataset]\nConclusions\n===========\n |p3cm|p3cm|p9cm| **Method** & **Addressed Issues** & **Details** \n \nWeighted Multi-label Classification & Varying Count of Labels Per Document, Class Imbalance & By using multi-label metrics and loss functions using weighting, effective multi-label multi-class classifiers can be trained and accurately evaluated. \nFeature Pruning & Dataset Size Computational Cost & Ruthless pruning and careful selection / sampling can significantly cut down on the computational/memory costs of long documents. On the other hand, even words traditionally considered semantically useless can be significant for a model to understand subtle grammatical and syntactical rules. \n \nSparse Transformers & O(*n*2) Transformer Cost & By utilizing a sparse attention mechanism, and with various mathematical optimizations, a standard Transformer can become viable at parsing entire long document inputs. \nDilated Sliding Window (Longformer) & O(*n*2) Transformer Cost Context Fragmentation & Using local self-attention with a dilated sliding window, while also allowing a limited number of global tokens, the Longformer becomes capable of successfully parsing ever-increasing-length documents, while keeping track of local context. \nSliding Window + Random Selection + Global Attention (BigBird) & O(*n*2) Transformer Cost Context Fragmentation & Combining Longformer’s sliding window, block attention scheme and random token selection can encapsulate local and global context more efficiently while keeping computational costs low. \n \nHierarchical Transformers combined with traditional networks & Transformer Input Limit & Instead of using Transformers for the entire document, they can be used to produce segment embeddings, which are then combined by traditional networks (HAN, LSTM, CNN). \nHierarchical Transformers with BERT (Hi-Transformer) & Global Context Context Fragmentation & BERT can be used to produce sentence embeddings, then document embeddings which can be fed to new, context-aware sentence embeddings iteratively. \nHierarchical Sparse Transformers (ERNIE-Sparse) & O(*n*2) Transformer Cost Local/Global Context issues & By using a hierarchical architecture embedded in a Sparse Transformer, ERNIE-SPARSE imparts token representations with additional global context. \n \nRecurrent Transformers & O(*n*2) Transformer Cost Context Fragmentation Global Context & By keeping the context of previously inputted segments in hidden states and by utilizing relative positional embeddings, the Recurrent Transformer effectively supports much longer input sequences during inference, irrespective of training-stage sequence length. \nSkimming Recurrent Transformers & O(*n*2) Transformer Cost Context Fragmentation Global Context Forward context & Forward and global context of greater quality can be obtained by passing through the input text twice, once by peeking small text segments (skimming phase), and then by performing a full pass (retrospective phase), fully utilizing the recurrence mechanism. \n[tab::textclasstable]\n |p3cm|p3cm|p9cm| **Method** & **Addressed Issues** & **Details** \n \nHierarchical Summarization & DNN Use for Summarization Document Structure Under-utilization & Traditional Encoder-Decoder DNN architectures attempt to explicitly incorporate context and hierarchical elements to the features used by CNN and RNN/LSTM extractive models. \nBERTSUM-Ext & Poor Traditional DNN PerformanceWeaknesses in Sentence Embeddings & By using BERT as Encoder and attaching stacked Transformer layers, BERT’s expressive power leads to accurate extractive summaries. \nHiStruct+ & Segment/Document Context LossUnder-utilization of Document Structure & By encoding the hierarchical and topical structure of a document, the Transformer can leverage more information about rigidly structured documents (e.g., research articles). \nHEGEL & Under-utilization of Document StructureO(*n*2) Transformer CostLong-span Dependencies & Encoding each document as a hypergraph under multiple different views leads to the model being exceptionally adept at summarizing very long and rigidly hierarchical documents. \n \nPointer-Generator Networks & Word RepetitionVocabulary LimitationsLong Document Computational Cost & By allowing the network to choose whether to copy the next summary word from the source text or predict its own, the model can benefit from a significantly expanded vocabulary, reduce computation costs by keeping a smaller vocabulary and bypass the word repetition problem found in many extractive summarization models. \nSparse Attention Transformers & O(*n*2) Transformer Cost Vocabulary Limitations & By using a modified Sparse Transformer as the Encoder, the model can leverage the expressive power of Transformers on long documents. \nLocal Attention and Content Selection & O(*n*2) Transformer Cost Long Document Computational Cost & Pretrained models can be used for content selection in order for a Sparse Attention model to be fed only useful data, both in training and inference. \n[tab::textsumtable]\nAnalyzing long documents currently presents a daunting challenge, even with the recent NLP breakthroughs. While traditional machine learning and statistical methods provide limited functionality on these tasks, they ultimately prove less adaptable and less accurate than modern DNNs. However, the latter ones need to face long-standing issues, such as the curse of dimensionality, limited datasets, prohibitively large inputs and difficult writing formats (e.g., in literary fiction).\nMost major issues are common to all long document analysis tasks and currently tend to require similar solutions, such as limiting the model’s vocabulary, modifying existing Transformer architectures to handle longer input sequences, efficiently exploiting the hierarchical structure of documents and artificially augmenting the training datasets. Task-specific solutions have also been proposed, such as using content preselection and specialized GNNs to capture long-spanning context in summarization, or passing over the input twice and using document-level representations to enrich sentence-level representations in classification. The identified issues and existing solutions are summarized in Tables [tab::textclasstable] and [tab::textsumtable], concerning document classification and summarization, respectively. Overall, further investigation of hybrid and recurrent Transformer architectures, as well as of the top-down Transformer, seem to be promising directions for near-future improvements. Incorporation of such advances into generic LLMs is also an avenue worth exploring, along with the development of novel, improved evaluation metrics for automatic text summarization.\nWhile this survey focuses only on document-level analysis for long texts, it becomes apparent how attempts at solving specific tasks can lead to solutions or insights with impact to all NLP fields. However, it is important to note that existing SoA solutions rely on useful trade-offs and, therefore, cannot be used universally. Finally, there is significant research potential for research into literary sentiment analysis, in collaboration with the digital/computational Humanities community. The necessary theoretical background already exists, while the mentioned breakthroughs in long document NLP have not been yet applied.\nAutomated analysis of long documents is essentially a field still in its infancy, with breakthroughs leading to large jumps in computational efficiency and accuracy still frequent. Since 2015 neural network models have evolved from being constrained to (occasionally artificially) small-sized documents, to being able to effectively parse large articles and books. It is not unreasonable to expect that continuing research will help bypass most of the current limitations within the next few years.\nAcknowledgement\n===============\nThe research leading to these results has received funding from the European Union’s Internal Security Fund under grant agreement No 101103298 (KLEPTOTRACE). This publication reflects only the authors’ views. The European Commission is not responsible for any use that may be made of the information it contains.\nAppendix\n========\n**Relevant Previous Review/Survey Articles** \n[H]\n |p4cm|p0.6cm|p0.6cm|p0.6cm|p0.6cm|p0.6cm|p0.6cm|p0.6cm|p0.6cm|p0.6cm|p2cm| **Review**&&&&&&&&&&This article \nDocument Classification&&&&&&&&&& \nDocument Summarization&&&&&&&&&& \nSentiment Analysis&&&&&&&&&& \nRelevant Neural Networks&&&&&&&&&& \nIssue Discussion&&&&&&&&&& \nIssue Aggregation&&&&&&&&&& \nLong Document Datasets&&&&&&&&&& \nLLMs&&&&&&&&&& \n[tab::reviewtable]\nDespite the existence of many overview/survey articles pertaining to NLP or relevant neural architectures (e.g., efficient Transformers), there is a distinct scarcity of recent such publications that specifically concern long document analysis and overlap with this article (i.e., focus on classification or summarization). This section briefly presents a selection of existing overlapping surveys and highlights the points of differentiation from this article.\nIn, the authors present an experimental comparative evaluation of traditional text classification methods and deep neural approaches, in the context of long document classification. However, they do not cover recent neural architectures such as efficient Transformers, public long document datasets, document summarization or sentiment analysis. Thus, they do not synthesize their findings across different NLP tasks and their article is removed from today’s SoA. In, a thorough survey of SoA models, datasets and metrics for document summarization is offered, which does cover a number of early efficient Transformer architectures and systematically extracts insights regarding open issues and potential solutions. However, it does not correlate its findings with long document classification and does not elaborate on how the presented neural architectures operate, while its coverage is slightly outdated as of 2024. The survey in is more up-to-date regarding long document summarization while also presenting relevant metrics and datasets in brief, but otherwise it also does not extend to document classification and does not detail the employed neural architectures.\nSentiment analysis, also known as *opinion mining*, typically concerns short documents, such as reviews, comments and social media posts. However, a review of modern approaches to a specific subtask of sentiment analysis is conducted in, where the goal is to analyze entire literary works with respect to their story and emotions. Due to its particular scope, at the intersection of NLP and literary analysis with a strong emphasis on emotion recognition, this survey does not cover generic document classification or document summarization, does not elaborate on the operation of the discussed neural architectures and does not present public long document datasets. In contrast, surveys text classification in general across multiple datasets and classification subtasks (such as sentiment analysis), including in long document datasets, focusing on recent Transformer architectures. However, it does not specialize to the peculiarities of long texts, it does not cover document summarization and does not present the discussed neural architectures in-depth, as its focus is on identifying the currently optimal method per dataset. Finally, focuses on Large Language Models (LLMs) in contrast to this article that explicitly does not, emphasizing instead task-specific efficient Transformer architectures.\nA number of recent surveys/reviews focus on efficient Transformer neural architectures, regardless of the application domain, and thus are overlapping with this article from a different perspective, given that such DNNs can better analyze lengthy input sequences. For instance, the survey in covers efficient Transformers extensively and in-depth, but does not specialize in any long document analysis task and, therefore, does not present relevant datasets or domain-specific issues. The situation is similar for and, although the latter’s coverage of efficient Transformers is much less extensive. Moreover, does present benchmark datasets for lengthy input sequences, but only minimally touches upon document classification. Finally, significant overlap exists with the neural architectures covered in, but that survey’s emphasis is on efficient Transformers for NLP with limited computational resources (not on long document analysis).\nWhile all of the above surveys/reviews overlap with this article, there is no previous effort for aggregating and comparing the issues and challenges specifically posed by long documents across both classification and summarization, while focusing exclusively on modern DNNs. This article attempts to expose common issues and encourage the sharing of tried-and-tested solutions across tasks. Additionally, due to the in-depth coverage of the presented neural architectures, it possesses a tutorial value which is lacking from the majority of related published surveys. In summary, a simplified comparison between this article and notable recent review publications pertaining to long document classification or summarization can be found in Table [tab::reviewtable]. The row ``Issue Discussion“ indicates that the corresponding paper specifically discusses open issues with long document analysis, while the row ``Issue Aggregation” indicates that issues are aggregated and compared across different NLP tasks. Only officially published articles have been included in this comparison, i.e., no arXiv-uploaded preprints.\n---\n1. This differs from the original Transformer’s statically defined positional encoding vector.[↩](#fnref1)\n2. The baseline BERT supports only 2 sentences, but each of these is a segment of consecutive textual content and not an actual individual sentence in the linguistic sense.[↩](#fnref2)\n\nNeural Natural Language Processing for Long Texts: A Survey on Classification and SummarizationAnswers to Reviewers’ comments\n=============================================================================================================================\nWe kindly thank the reviewers for their careful review and insightful comments. We have revised the manuscript accordingly and the revised sentences are highlighted in red. Additionally, we would like to clarify that the article contains an introductory/tutorial section (Section 2 in the revised manuscript, which was Section 3 in the original submission) that briefly presents neural building blocks for document analysis in general (not long-document-specific). The title of this section was erroneous in the original submission, which understandably caused confusion to most of the reviewers. We believe that we have fixed this in the revised version.\nAnswers to individual comments follow below.\nReviewer #1\n===========\n**Comment #1** \nThe technical components of the paper are sound. While there were several references to related material, I would suggest that the author(s) incorporate the following reference that would add to their literature review and take into consideration regarding machine learning: \n1. Hierarchical Graph-based Text Classification Framework with Contextual Node Embedding and BERT-based Dynamic Fusion, Journal of King Saud University- Computer and Information Sciences, https://doi.org/10.1016/j.jksuci.2023.101610 \n2. GTR-GA: Harnessing the Power of Graph-Based Neural Networks and Genetic Algorithms for Text Augmentation. Expert Systems with Applications, 120908. \n3. SRL-ACO: A Text Augmentation Framework Based on Semantic Role Labeling and Ant Colony Optimization, Journal of King Saud University- Computer and Information Sciences, https://doi.org/10.1016/j.jksuci.2023.101611 \n4. Bidirectional convolutional recurrent neural network architecture with group-wise enhancement mechanism for text sentiment classification. Journal of King Saud University- Computer and Information Sciences, https://doi.org/10.1016/j.jksuci.2022.02.025 \n5. Two-stage topic extraction model for bibliometric data analysis based on word embeddings and clustering. IEEE Access, 7, 145614-145633. \n6. Ensemble of keyword extraction methods and classifiers in text classification. Expert Systems with Applications, 57, 232-247. \n7. Consensus clustering-based undersampling approach to imbalanced learning. Scientific Programming, 2019. \n8. A feature selection model based on genetic rank aggregation for text sentiment classification. Journal of Information Science, 43(1), 25-38. \n9. A hybrid ensemble pruning approach based on consensus clustering and multi-objective evolutionary algorithm for sentiment classification. Information Processing and Management, 53(4), 814-833. \n10. Sentiment analysis on product reviews based on weighted word embeddings and deep neural networks. Concurrency and Computation: Practice and Experience, 33(23), e5909. \n11. Mining opinions from instructor evaluation reviews: a deep learning approach. Computer Applications in Engineering Education, 28(1), 117-138. \n12. Sentiment analysis on massive open online course evaluations: a text mining and deep learning approach. Computer Applications in Engineering Education, 29(3), 572-589. \n13. An ensemble scheme based on language function analysis and feature engineering for text genre classification. Journal of Information Science, 44(1), 28-47. \n14. A term weighted neural language model and stacked bidirectional LSTM based framework for sarcasm identification. IEEE Access, 9, 7701-7722. \n15. Topic-enriched word embeddings for sarcasm identification. In Computer Science On-line Conference (pp. 293-304). Springer, Cham \n**Answer** \nBased on this and other similar reviewer comments, several new references have been included in the revised version, including many of the ones mentioned in this comment. We opted not to include all 15 of them, however, since a subset of them was not directly relevant to the topics covered in the article and/or did not pertain specifically to *long* document analysis. \nReviewer #2\n===========\n**Comment #1** \nI think it would be interesting to present the problem in its generality prior to dive into specific fields as document classification. What are the limitations of current deep architectures regarding document length? Are there many text collections containing long documents or is it a marginal problem? You can rely on studies and arguments provided in the references you provide for that. Additionally, I recommend you read and discuss the following reference: \n[1] The power of selecting key blocks with local pre-ranking for long document information retrieval. M Li, DN Popa, J Chagnon, YG Cinar, E Gaussier. ACM Transactions on Information Systems 41 (3), 1-35. \n**Answer** \nBased on this comment, a new subsection titled ``Shared Challenges\" has been added in the revised manuscript (new Section 3.2). Moreover, the reference mentioned by the reviewer has been incorporated into the article.\nRegarding text collections containing long documents, we believe that the revised long documents datasets Table near the end of the revised article clarifies the situation. \n**Comment #2** \nIt would also be nice to present the main families of approaches that have been followed by different authors on different fields. If I don’t believe that there is an approach outperforming all other approaches on all fields, a few main ideas (truncation, sliding window, sparse attention,...) underlie most approaches. \n**Answer** \nBased on this comment, a new subsection titled ``Common Strategies\" has been added in the revised manuscript (new Section 3.3). \n**Comment #3** \nI do believe that you should add IR to your analysis. As far as I know, there haven’t been many studies on long documents for IR and there is no need to have an analysis as long as the ones you have for document classification and summarisation. I furthermore believe most information can be found in [1]. If you don’t do taht, please consider changing your title by specifying the subfields of NLP you are focusing on. \n**Answer** \nBased on this comment, we have slightly changed the title of the article to ``Neural Natural Language Processing for Long Texts: A Survey on Classification and Summarization“. Additionally, we have added a relevant brief subsection titled ``Using Information Retrieval” (new Section 4.1.2.2), describing the use of IR for feature selection and pruning (typically in document pre-processing). \n**Comment #4** \nThere is an important reference missing for text summarisation. Please add it and discuss it. \n[2] Globalizing BERT-based transformer architectures for long document summarization. Q Grail, J Perez, E Gaussier. Proceedings of the 16th conference of the European chapter of the Association for Computational Linguistics. \n**Answer** \nDiscussion of the proposed paper has been added under the ``Extractive summarization\" subsection. \n**Comment #5** \nLSTM are a special case of recurrent networks. 3.1.4 could thus be a sub-section of 3.1.3. \n**Answer** \nThe reviewer is, of course, correct in this observation. However, based on another reviewer’s comment (Reviewer 3, Comment 3), we opted to more drastically compress the entirety of Section 3.1 (that has become Section 2.1 in the revised manuscript) and focus it only on attention mechanisms and Transformer architectures. This better reflects the state-of-the-art for long document analysis. \n**Comment #6** \nThe way you describe the curse of dimensionality in Section 4.1 is not standard and I am not sure what you mean exactly. Can you specify the challenges proper to long documents in this section? \n**Answer** \nThe employed definition for the curse of dimensionality in the case of NLP has been lifted from the seminal paper ``A Neural Probabilistic Language Model“ (Y. Bengio, R. Ducharme, P. Vincent, C. Jauvin, ``Journal of Machine Learning Research”, vol. 3, pp. 1137-1155, 2003) where it is stated:\n``This is intrinsically difficult because of the curse of dimensionality: a word sequence on which the model will be tested is likely to be different from all the word sequences seen during training. For example, if one wants to model the joint distribution of 10 consecutive words in a natural language with a vocabulary V of size 100,000, there are potentially 10000010 - 1 = 1050 - 1 free parameters.\"\nAlthough this definition comes from statistical language modeling, we believe it applies well to semantic long document analysis where the longer the test text, the more possible it is for the algorithm to stumble upon sentences unseen during training. This is challenging for learning models if there are data distribution shifts that impede good generalization. In response to this comment, the mentioned subsection has been revised and a reference to the 2003 paper has been added. If an issue is particular to (or particularly important in) the case of long texts, it is explicitly mentioned as such within its corresponding paragraph. Finally, this subsection has now been moved into the new subsection ``Shared Challenges\" (Section 3.2 of the revised manuscript), as all of its contents are common to both classification and summarization tasks. \nReviewer #3\n===========\n**Comment #1** \nOn page 3, the assertion that document classification, sentiment analysis, and document summarization constitute the “main NLP tasks for long documents” is a rather assertive statement and may not be universally agreed upon. It would be prudent to support this claim with citations or provide a more compelling rationale. For example, why sentiment analysis should be more relevant than information extraction for Long NLP? \n**Answer** \nIn response to this comment, we have limited the declared scope of the revised paper to document classification and summarization (sentiment analysis is treated as a particular form of classification). The title has been changed accordingly and we have removed the overly assertive statement mentioned by the reviewer. A paragraph concretizing the rationale behind choosing to focus on these three tasks has been added to Section 1. \n**Comment #2** \nThe high-level comparison with prior review-level related studies presented in Table 1 lacks detailed information and appears to be incomplete. It is not very informative. The inclusion of reviews pertaining to non-long NLP makes little sense, given the extensive body of literature available. Meanwhile, it’s worth noting that important works like [1] have not been referenced in the paper. \n \n1 \n Tay, Y., Dehghani, M., Bahri, D., and Metzler, D. (2020). Efficient Transformers: A Survey. ACM Computing Surveys, 55, 1 - 28. \n**Answer** \nIn response to this comment, we removed from the section in question two surveys that did not explicitly pertain to long document analysis and we have incorporated a few recent reviews concerning efficient Transformer architectures (given that the state-of-the-art in long text NLP uses efficient Transformers almost exclusively). These new additions include the reference mentioned by the reviewer. Additionally, we have incorporated a few other recent surveys regarding long text analysis. However, we did not find many recent published surveys specifically on long document NLP (at least in reputable venues) that overlap with our article (i.e., specifically concern long document classification or summarization). Finally, the discussion of the existing overlapping surveys has been enhanced and more than doubled in size.\nIn response to a different comment by another reviewer (Reviewer 4, Comment 2), we have moved this section to an appendix in the revised manuscript. \n**Comment #3** \nThe authors state that their primary focus is on ``DNN-enabled long document NLP“ and reviewing state-of-the-art works that specifically address the intricacies of long documents. However, a significant portion of the paper, approximately 14 out of 45 content pages, is dedicated to more general NLP architectures, including outdated solutions such as MLPs, CNNs, and RNNs/LSTMs. The authors acknowledge that these architectures have limitations, with CNNs having computational overhead issues and RNNs/LSTMs struggling with long-term dependencies in extensive text. These architectures are considered unconventional for contemporary Long NLP and appear to be distant from the current state-of-the art. The inclusion of these architectures among ``the most effective neural architectures for NLP” requires a robust defense and contextualization. Moreover, the architecture discussion presented in the paper does not provide specific insights into long document processing. It reiterates well-known introductory concepts that can be found in many other surveys. While this introductory content may be convenient for standalone publication and introductory purposes, it cannot be counted as contribution, does not contribute to the paper’s focus on Long NLP and instead veers toward a compressed task-independent NLP landscape overview. In its current form, without specialization for long documents, this section could be omitted, with readers referred to existing reviews. By dedicating space to non-essential architecture explanations, the paper overlooks many long-specific aspects that are expected in a survey on Long NLP. These aspects include memory-augmented transformers [2], performers [3], relative position embeddings [4], segmentation, retrieval, and fusion-in-decoder [5,6]. \n \n2 \n Moro G, Ragazzi L, Valgimigli L, Frisoni G, Sartori C, Marfia G. Efficient Memory-Enhanced Transformer for Long-Document Summarization in Low-Resource Regimes. Sensors. 2023; 23(7):3542. https://doi.org/10.3390/s23073542 \n \n3 \n Choromanski, K., Likhosherstov, V., Dohan, D., Song, X., Gane, A., Sarlós, T., Hawkins, P., Davis, J., Mohiuddin, A., Kaiser, L., Belanger, D., Colwell, L.J., and Weller, A. (2020). Rethinking Attention with Performers. ArXiv, abs/2009.14794. \n \n4 \n Chen, P. (2021). Permuteformer: Efficient relative position encoding for long sequences. EMNLP. \n \n5 \n Moro, G., and Ragazzi, L. (2022). Semantic Self-Segmentation for Abstractive Summarization of Long Documents in Low-Resource Regimes. Proceedings of the AAAI Conference on Artificial Intelligence, 36(10), 11085-11093. https://doi.org/10.1609/aaai.v36i10.21357 \n \n6 \n De Jong, M., Zemlyanskiy, Y., Ainslie, J., FitzGerald, N., Sanghai, S.K., Sha, F., and Cohen, W. (2022). FiDO: Fusion-in-Decoder optimized for stronger performance and faster inference. ArXiv, abs/2212.08153. \n**Answer** \nIn response to this comment, we drastically compressed the entirety of Section 3.1 (that has become Section 2.1 in the revised manuscript) and focused it on attention mechanisms and Transformer architectures. This better reflects the state-of-the-art for long document analysis. However, long-text-specific architectures or modifications are presented in the next sections (3, 4, 5, 6). Section 2 of the revised manuscript (which was Section 3 in the original submission) is still an introductory section having an additional tutorial value, which is a point of differentiation from existing surveys. To better highlight this, we have now changed its title to ``Deep Neural Networks Used in Document Analysis\".\nFinally, we have incorporated most of the references mentioned by the reviewer into the performed survey. \n**Comment #4** \nThe paper lacks clear and consistent inclusion criteria for the presented models, such as ELECTRA and T5. For instance, the reason for the absence of specific models like RoBERTa, which is briefly mentioned on page 18, is not adequately explained. The presentation choices in the paper appear to be somewhat subjective, chaotic, and not grounded on a clear scientific methodology. \n**Answer** \nModels such as ELECTRA, RoBERTa or T5 are basic neural NLP building blocks covered in the introductory/tutorial Section 2 of the revised manuscript (which was Section 3 in the original submission)l; they are not long-document-specific. So, given the need to keep the article at a reasonable length, we opted to not make separate subsections for each of these. For instance, RoBERTa and ELECTRA are covered in the same subsection of Section 2, as different modifications/improvements of the original BERT that have revised its training objective in different ways. \n**Comment #5** \nThe performance comparisons made between generative GPT-2 models and representational BERT models on page 18 lack fairness due to the absence of task-specific details and context. Furthermore, these comparisons are presented without proper citations. In a review paper, it is essential that each bold assertion is supported and substantiated. \n**Answer** \nIt seems highly possible that the reviewer refers to a statement in page 17 of the originally submitted manuscript, where it was written that ``GPT 2.0 remains regularly outperformed by BERT-derived language models, such as RoBERTa, in the finetuning setting.... GPT 3.0 achieves similar results to most BERT-based language models, but is not openly available to the public.\". Thus, in response to this comment, we have added the clarification that the performance comparison is in the task of domain-specific generation of very long text. Additionally we spotted a serious error in the original manuscript, as the actual comparison was not between GPT-2 and RoBERTa (as erroneously stated in our original submission), but between GPT-2 and BART. Of course, vanilla BART can easily be finetuned for text generation, unlike vanilla BERT-derived models. We have now fixed this in the revised version and thank the reviewer for their extremely useful observation.\nThis statement is supported in the article by a citation to the paper ``Progressive Generation of Long Text with Pretrained Language Models\" from a well-recognized venue (Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 4313–4324). Indeed, the arXiv preprint version was cited in the original version, but we haved fixed this in the revision. \n**Comment #6** \nThe sections on document classification, sentiment analysis, and document summarization suffer from poor organization, offering only shallow discussions. They frequently include task-unspecific references and lack clear criteria for the inclusion of models. Furthermore, they do not provide valuable practical insights relevant to the reader for the specific task being discussed. As observable in Section 2, a survey’s distinctive feature is its coverage of downstream tasks, but the quality of this part appears to be insufficient. \n(a) Document classification \n(i) The presentation of task-independent challenges and architecture weaknesses related to long-form NLP within Section 4.1, which is dedicated to document classification, raises questions about the organizational choices made in the paper. It is not clear why issues such as vocabulary size, polysemy, homonymy, figurative language, and low-resource languages are discussed under this section. \n(ii) Multi-label classification stands out as the sole classification type treated, with only one cited work, e.g., hierarchical cases are neglected. This is somewhat perplexing as it is grouped with topics like feature pruning and sparse attention transformers, which raises questions about its placement in the context. Furthermore, sparse attention transformers seem out of place when discussing document classification and should be considered a task-independent approach for handling longer input sequences. In a broader context, efficient and recurrent transformers are widely recognized as foundational models, particularly for generative tasks. Placing these methods within the document classification context leads the reader to expect insights and available results regarding their specific application in this task. They should be moved earlier in Section 3 as suitable architectures for Long NLP and referenced in Section 4 for task-specific adaptions. \n(b) Document summarization \n(i) Missing citations, e.g., “concerns have been raised that...”, “ensembles of multiple extractive models”, page 28. \n(ii) No reference to decoding strategies for containing hallucinations and repetitions [7] \n(iii) Poor discussion about NLG metrics [8] specifically designed for document summarization and long documents, e.g., coherence quality dimensions [9]. For their importance in the document summarization field, the authors should include also BARTScore [10] and factuality metrics such as SummaC [11] and CTCScore [12]. \n(c) Sentiment Analysis \n(i) The discussion, particularly in comparison to classification and summarization, is subpar (2 pages only), which underscores that this task is not a standard one. \n \n7 \n Wan, D., Liu, M., McKeown, K., Dreyer, M., and Bansal, M. (2023). Faithfulness-Aware Decoding Strategies for Abstractive Summarization. Conference of the European Chapter of the Association for Computational Linguistics. \n \n8 \n Frisoni, G., Carbonaro, A., Moro, G., Zammarchi, A., and Avagnano, M. (2022). NLG-Metricverse: An End-to-End Library for Evaluating Natural Language Generation. International Conference on Computational Linguistics. \n \n9 \n Rahimi, H., Hoover, J.L., Mimno, D.M., Naacke, H., Constantin, C., and Amann, B. (2023). Contextualized Topic Coherence Metrics. ArXiv, abs/2305.14587. \n \n10 \n Yuan, W., Neubig, G., and Liu, P. (2021). BARTScore: Evaluating Generated Text as Text Generation. ArXiv, abs/2106.11520. \n \n11 \n Laban, P., Schnabel, T., Bennett, P.N., and Hearst, M.A. (2021). SummaC: Re-Visiting NLI-based Models for Inconsistency Detection in Summarization. Transactions of the Association for Computational Linguistics, 10, 163-177. \n \n12 \n Deng, M., Tan, B., Liu, Z., Xing, E.P., and Hu, Z. (2021). Compression, Transduction, and Creation: A Unified Framework for Evaluating Natural Language Generation. Conference on Empirical Methods in Natural Language Processing. \n**Answer** \nIn response to this comment, we revised the manuscript in the following manner: \n(a) Document classification \n(i) The subsection in question has now been moved into the new, task-agnostic subsection ``Shared Challenges“ (Section 3.2 of the revised manuscript), as all of its contents are common to both classification and summarization tasks. The only exception was the issue of multiple ground-truth labels, which has been incorporated into the main text of the section on classification. Similarly, the challenges listed in the sections on summarization and sentiment analysis have been revised to avoid repeting common, task-independent issues. \n(ii) The reviewer is corrent in observing that sparse attention and efficient Transformers are generic architectures for processing long sequences, therefore they are not specific to document classification. However, we intentionally avoided placing them into Section 3 (which is Section 2 in the revised manuscript) in order to keep that as an introductory/tutorial section which does not specialize on long documents (in the revision, we have correspondingly changed its title and shortened it, as a response to the reviewer’s suggestions). Thus, the ``meat” of the paper (including the various efficient Transformer architectures) is presented specifically in the context of classification and summarization on which the article is solely focused. Task-specific adaptations of such architectures are very minimal in the case of classification, so removing them from the section on classification would highly weaken the latter. However we have added a statement in the revised manuscript, explicitly declaring them as task-agnostic and generic architectures. \nRegarding the rationale behind grouping multi-label classifiers, feature pruning and efficient Transformers in the context of a single subsection, it is the following one: these are all solutions to the various challenges faced by DNNs for long text classification (with the majority of such challenges being common across many document-level/long text analysis tasks, besides classification). \n(b) Document summarization \n(i) The statements in question were backed-up by an older reference (from 2016), so we opted to removed them entirely. We thank the reviewer for their extremely useful observation. \n(ii) A new subsection has been added within the section on summarization, specifically for methods targeting hallucination containment (``Dealing with hallucinations\"). This includes the reference mentioned by the reviewer. \n(iii) The summarization/coherence metrics mentioned by the reviwer have been added, along with a separate list for factuality metrics. \n(c) Sentiment analysis \n(i) In response to this comment, the section on sentiment analysis has been extended and enriched in the revised version. Additionally, it has been higlighted more saliently in the article that this is a particular case of document classification and not an entirely different task. The reduced length is a consequence of the fact that this section only contains sentiment-specific issues/adaptations; otherwise, in general, what is mentioned within the section on classification also holds true for sentiment analysis. \n**Comment #7** \nThe survey does not offer a comprehensive literature organization in the form of a (original) taxonomy. Instead, it places more emphasis on concise model discussions, lacking specific selection criteria and quantitative statistics related to the various strategies, their relationships, adoption, and prevalence within top-tier venues. \n**Answer** \nIn response to this comment, a relevant taxonomy has been added under the new subsection titled ``Categorization and Evolution of Long Document Deep Neural Architectures\" (new Section 3.4, Figure 5). \n**Comment #8** \nThere is no timeline provided to illustrate the progression of approaches over time. \n**Answer** \nIn response to this comment, a relevant timeline has been added under the new subsection titled ``Categorization and Evolution of Long Document Deep Neural Architectures\" (new Section 3.4, Figure 6). \n**Comment #9** \nThe effectiveness comparison is solely conducted for document summarization, and even in this case, it is notably inadequate. \n**Answer** \nIt was very difficult to find just comparisons across many different state-of-the-art methods in the same datasets, using the same metrics. We tried to construct a similar table for document classification, containing all of the mentioned datasets, but it proved to be very sparse due to these inconsistencies in the literature. Thus, in the end, we removed it as rather meaningless. Regarding summarization, it was almost impossible to enrich the existing table because different papers may utilize different metrics (including different variants of ROUGE), even when being evaluated at the same datasets. To the best of our knowledge, other relevant surveys (except a few that conduct their own experimental comparative evaluation on a specific domain and task) also do not contain such comparative tables, most probably due to the same issue that we faced.\nAs a result, and also taking into account Comment 7 from Reviewer 4, in the revised manuscript we have entirely removed the performance comparisons Table on summarization. \n**Comment #10** \nThere is no comparison regarding efficiency factors such as training time, inference time, CO2 emissions, FLOPS with transparent hardware specifications and evaluation setup. \n**Answer** \nUnfortunately, we did not find such comparisons in publicly available sources and we did not have the resources necessary to conduct such comparisons ourselves. \n**Comment #11** \nThe section on long document datasets lacks an organized discussion and fails to provide in-depth statistical distributions or valuable insights within the dataset analysis. Computing the average document size when such information is not publicly available in Table 3 appears to be a minimal effort. Additionally, CNN/DM notoriously belongs to short document summarization. \n**Answer** \nIn response to this comment, the mentioned table with the datasets has been significantly enriched. CNN/DM is the only included dataset that does not contain long documents, but it has been included for reference purposes due to its popularity. \n**Comment #12** \n“It must be noted that, alternatively, recent instruction-tuned LLMs (see 5.2.3) can be leveraged successfully for these tasks, but they are not explicitly designed to support long document analysis. Thus, they typically face issues with lengthy textual inputs.” This statement is false and should be updated to reflect the recent developments and techniques proposed after the manuscript’s submission. \n**Answer** \nThe reviewer is correct and the statement in question has been removed. Additionally, the paragraph on instruction-tuned summarization has been revised. \nReviewer #4\n===========\n**Comment #1** \nIn section 1, authors have spent several words discussing contents that are either inconsequential or have been widely accepted by those interested in this topic. I suggest this section goes straight to the challenges of long texts in dealing with neural network problems. Readers may also wonder why they should care about this issue and its significance in the application they are developing. The authors should clarify their definition of long and short texts, mentioning whether they are measured in words, characters, tokens, or other techniques. \n**Answer** \nIn response to this comment, we have significantly shortened the introductory material in Section 1 of the revised manuscript and we have deleted a couple of paragraphs with definitions of widely known NLP terms. Additionally, a definition of long texts has been added in new Section 3 of the revised document, as suggested by the reviewer. \n**Comment #2** \nSection 2, a comparative content illustrating the difference between this survey and earlier ones, may not be necessary. It could be worth considering moving this content to an appendix. \n**Answer** \nDone; we have moved this section into an appendix for the revised manuscript. \n**Comment #3** \nSection 3 is a detailed survey of DNN models, from classic models like MLP, CNN, RNN to Transformers. However, the heading “Deep Neural Networks for Long Document Analysis” seems inappropriate, given that these models were not designed to handle long documents. Moreover, the section contains several paraphrased contents from original papers, which may not be required. I suggest paraphrasing only the parts directly supporting the authors’ arguments.\n**Answer** \nBased on this comment and on Comment 3 from Reviewer 3, we have significantly shortened Section 3.1 in the revised manuscript and now only cover the attention mechanism and Transformers, which power the state-of-the-art in document analysis. This is a brief, tutorial-level introduction to the employed neural building blocks for modern NLP (not long document analysis specifically), so we have correspondingly changed the section’s title. The actually modified/efficient Transformers (which are indeed utilized for long document analysis, with many of them designed specifically for this purpose) are covered in the following sections (mainly Section 4 and Section 5).\n**Comment #4** \nFurther, Section 3 lacks essential works on long text processing in specific domains. The authors could explore more focused papers for a comprehensive and updated survey. Some examples are: \n- PLM-ICD: Automatic ICD Coding with Pretrained Language Models \n- XR-Transformer: Fast Multi-Resolution Transformer Fine-tuning for Extreme Multi-label Text Classification \n- Paraformer: Attentive Deep Neural Networks for Legal Document Retrieval \n**Answer** \nThe reviewer is correct and the paper has been enriched with additional methods (e.g., XR-Transformer). However, in the revised version we have clearly focused only on summarization and classification tasks (based on Comment 3 from Reviewer 2), thus architectures such as Paraformer (which is for retrieval) are out of scope. \n**Comment #5** \nSections 4, 5, and 6 present three tasks - Document Classification, Document Summarization, and Sentiment Analysis - making the manuscript difficult to follow since the outline lacks a clear structure. Moreover, these tasks do not represent all tasks related to long text problems, such as translation, retrieval, and legal judgment prediction. Sentiment Analysis can also be considered a subtask of Document Classification. \n**Answer** \nIn response to this comment, we have more clearly articulated the article’s structure at the end of Section 1. Based on Comment 3 from Reviewer 2, the revised manuscript is now clearly limited only to summarization and classification (sentiment analysis is indeed discussed as a particular case of document classification). Section 2 (which was Section 3 in the original submission) is an introductory section having an additional tutorial value, which is a point of differentiation from existing surveys. It briefly presents basic neural NLP building blocks that are not long-document-specific. The newly added Section 3 introduces the issues, challenges and particularities of long document analysis in general. Then, Sections 4, 5 and 6 in the revised document present the state-of-the-art in long document classification, summarization and sentiment analysis, respectively. However, given that sentiment analysis is a sub-case of document classification, only sentiment-specific issues and particularities are presented in Section 6. \n**Comment #6** \nSection 7 provides several datasets that the authors believe could be useful for researching this issue. However, the analysis of these datasets is insufficient, and the unit of measure for the Average Document Size column in Table 3 is unclear. The authors should clarify whether 781 words are considered long text or not. This question could have been avoided if the authors had provided a clear definition of long text from the beginning. \n**Answer** \nIn response to this comment, the table with the datasets has been improved/enriched and a definition of long texts has been articulated within the newly added Section 2 of the revised manuscript. Additionally, it has been clarified in the datasets table that the CNN/DM dataset (the one with 781 words per document) is not a long text dataset, but is included for reference purposes due to its popularity. The case of CNN/DM not being a long document dataset is also explicitly stated in the newly added Section 3 of the revised manuscript. \n**Comment #7** \nSection 8, the summary, further complicates the manuscript. Is it necessary to have a summary section at this point? Could it be rewritten as part of section 9 or omitted? \n**Answer** \nIn response to this comment, the Summary section and the Conclusions section have been merged in the revised manuscript. \n"},"data_id":{"kind":"string","value":"arxiv_0002846"}}},{"rowIdx":697,"cells":{"content":{"kind":"string","value":"+, 0, *F*2(*x*+)) stable in the competition plane in region **(k)**. We selected three parameter values from this region and the first one is *x*⋆ = .1 and *a*2 = 3.5 (marked by a top blue \\* in region **(k)** of Figure [bifurcationdiagram]). The situation is depicted in Figure [Simubild11](a): The coexistence cycles that emerged from the transcritical bifucation between regions **(g)** and **(h)** still exists, but the cycles in the competition plane have disappeared. The projections of the coexistence limit sets on the various coordinate planes are still depicted.\nThe next value of interest is *x*⋆ = .18, *a*2 = 2.8. The parameter value is indicated with a blue \\* -mark in region **(k)** in Figure [bifurcationdiagram]). We have depicted the dynamical situation in Figure [Simubild11](b). We observe that the coexistence equilibrium has just lost its stability in the Hopf bifurcation predicted by Theorem [complexity-stability-relation], that all equilibria are unstable and that the system possesses coexistence cycles.\nWe are of course interested in whether more complex oscillations could exist and illustrate our findings with the parameter value *x*⋆ = .15 and *a*2 = 3.0 in Figure [Simubild11](c) (marked by a red \\*-mark in region **(k)** of Figure [bifurcationdiagram]). We note that the oscillations observed are a mixture of the competition cycles that emerged from the transcritical bifurcation in the competition plane and the Hopf-bifurcation occurring for the coexistence equilibrium after that the Hopf-bifurcation has occurred for the autotroph-herbivore equilibrium according to Theorem [complexity-stability-relation]. The stable and unstable manifold of two equilibria are essential for these oscillations and we start the description of the oscillation from the competition equilibrium (*x*+, 0, *F*2(*x*+)). It is stable in the competition plane but unstable with respect to herbivore invasion. The oscillation approaches this equilibrium close to the stable manifold (the competition plane) of this point and it leaves it along the its unstable manifold. Next the mixotroph carrying capacity has the *y**z*-plane as its stable manifold and an unstable manifold that intersects the two-dimensional stable manifold of the competition equilibrium.\nThe oscillations might come into the vicinity of the coexistence equilibrium before they are captured by the stable manifold of the mixotroph equilibrium and such an interaction is possible to observe in Figure [Simubild11](c). This interaction is, however, not essential. We now remove the coexistence equilibrium by selecting the parameter values *x*⋆ = .15 and *a*2 = 2.3 that is indicated by a red \\* -mark in region **(q)** in Figure [bifurcationdiagram]. We see that a very similar type of oscillations persist despite that the coexistence equilibrium is removed (Figure [Simubild11](d)). The two-dimensional stable manifolds of (*x*+, 0, *F*2(*x*+)) and (0, 0, *c*) interact with their one-dimensional unstable manifolds. We have so far no reliable indications of chaotic behavior for this system. All computations of Lyapunov exponents have so far resulted in either negative or values close to zero (from cyclic/quasiperiodic oscillations). Moreover, the possible route to chaos remains so far unclear in our simulations. The oscillations that occur without the presence of an interior fixed point in region **(q)** and **(w)** collapse through a transcritical of cycles bifurcation into autotroph-herbivore cycles on the boundary between the regions **(q)** and **(w)** and the cyan regions **(r)** and **(x)** that correspond to autotroph-herbivore cycles.\nA similar transcritical of cycles bifurcations might occur at the boundaries between the regions **(v)** and **(w)**, **(t)** and **(u)**, and **(k)** and **(p)**, respectively. However, at these boundaries we have just vague numerical evidence for such transitions without theoretical support. The observed transition might be caused by numerical instabilities or can be of more complex nature than just a transcritical of cycles bifurcation. On the other hand, more theoretical evidence for transcritical bifurcations along the boundaries between **(q)** and **(r)** and **(w)** and **(x)** exists, the transcritical boundary meets e. g. the transcritical bifurcation of the coexistence equilibrium between the areas **(n)** and **(s)**. We select still a number of parameter values for separate study in Figure [Simubild3]. We start with the parameter value *x*⋆ = .02, *a*2 = .6 in region **(u)** in Figure [Simubild3](a). Here an unstable coexistence equilibrium coexists with a coexistence cycle. Figure [Simubild3](b) demonstrated the parameter value *x*⋆ = .05, *a*2 = .5 in Region **(w)**. The coexistence cycle exists, but no coexistence equilibrium. Figure [Simubild3](c) demonstrates with the parameter value *x*⋆ = .2, *a*2 = .5 in region **(x)** how the coexistence cycle has collapsed into the herbivore-autotroph cycles in the predator-prey plane. No competition equilibrium exists in this region. Figure [Simubild3](d) demonstrates a similar transcritical collapse in region **(r)**, where a competition equilibrium exists.\nEnding up at the expectations of the title of this paper, first we have a typical qualitatively destabilizing region at region **(j)** in Figure [bifurcationdiagram] and here fixed point dynamics is replaced by limit cycle dynamics as mixotrophs invade. Second, we have a typical qualitatively stabilizing region at region **(m)** in Figure [bifurcationdiagram]. Here limit cycle dynamics is replaced by fixed point dynamics as mixotrophs invade. Third, we have typical region where mixotrophic invasion causes multiple attractors and initial value dependent behavior, see Figure [multipleattractors](d).\n(350,350)(0,0) (-56,-30) (365,0)*x*⋆ (-7,285)*a*2 (-13,245)**(c)** (4,248)(1,0)5 (50,285)**(a)** (250,285)**(b)** (60,235)**(e)** (250,235)**(f)** (70,197)**(i)** (250,197)**(j)** (85,170)**(l)** (250,130)**(o)** (86,133)**(m)** (110,110)**(n)** (50,80)**(q)** (97,75)**(r)** (150,60)**(s)** (60,22)**(x)** (250,25)**(y)** (18,220)**(d)** (11,194)**(g)** (40,201)**(h)** (18,160)**(k)** (35,32)**(w)** (25,59)**(u)** (25,57)(-1,-1)10 (-15.5,70)**(p)** (4,73)(1,0)5 (3,22)**(t)** (16,-7)**(v)** (22,3)(-1,2)5\n[bifurcationdiagram]\n[bifurcationtable]\n(350,350)(0,0) (0,0) (80,150)(a) (280,150)(b) (80,0)(c) (280,0)(d) (35,240)*z* (35,90)*z* (210,230)*z* (210,90)*z* (80,170)*y* (280,165)*y* (110,25)*y* (285,22)*y* (115,165)*x* (320,170)*x* (145,30)*x* (320,30)*x*\n[Simubild22]\n(350,350)(0,0) (0,0) (80,150)(a) (280,150)(b) (80,0)(c) (280,0)(d) (38,230)*z* (38,88)*z* (210,230)*z* (210,88)*z* (112,164)*y* (285,165)*y* (110,25)*y* (288,25)*y* (145,168)*x* (321,168)*x* (145,30)*x* (315,30)*x* (105,246)(-1,0)5\n[Simubild11]\n(350,350)(0,0) (0,0) (80,150)(a) (280,150)(b) (80,0)(c) (280,0)(d) (35,245)*z* (35,93)*z* (210,230)*z* (210,93)*z* (80,170)*y* (280,165)*y* (110,25)*y* (280,25)*y* (115,165)*x* (315,170)*x* (148,30)*x* (318,30)*x*\n[Simubild3]\nDiscussion\n==========\nIn the present study, we have examined the role of invading mixotrophs on the stability of an autotroph-herbivore system. Specifically, we have chosen a spectrum of mixotrophic organisms by varying their competitive abilities. Mixotrophs are assumed as weak competitors compared to the pure autotrophs and the pure herbivores (Litchman et al. (2007); Zubkov and Tarran (2008)). We are able to formulate such a condition precisely for the mixotrophs in relation to the autotrophs, but in relation to the herbivores we end up with a too strong global condition and a too weak local condition. However, we did not encounter any obvious problem with our analysis as we used the local condition. We have analyzed the system with a limiting case of the chemostat instead of doing logistic approximations since the limiting case still preserves the complete information of the system exposed to mixotrophic invasion (Lindström and Cheng (2015)) similar to the logistic approximations (Kuang and Freedman (1988)).\nTwo main subsystems of our system, a competition system (autotroph-mixotroph) and a predator-prey system (autotroph-herbivore), describe the dynamical properties in different invariant coordinate planes. The predator-prey system has a *ω*-limit set that consists of either a unique globally stable equilibrium or a unique limit cycle. As the cycles grow larger, the system becomes increasingly sensitive to perturbation, cf. Rosenzweig (1971). The global analysis is not that complete for the competition system. It has at most five equilibria and both situations where all existing equilibria are unstable and two of them remain locally stable exist. As a result, this subsystem possesses both initial value dependent behavior and limit cycles.\nOur analysis also reveals the existence of a possible additional unstable equilibrium in the complete three-species model. This leads to a situation without any stable equilibrium and opens the possibility for complicated oscillations. However, we were unable to find any evidence for chaos at this stage but found that the oscillations observed typically spend a lot of time close to some of the saddle-type equilibria. Checking the existence of chaos needs a lot of effort in computing Lyapunov exponents in a reliable way and is thus out of the scope of this paper (Guckenheimer and Holmes (1983), and Wolf et al. (1985)).\nWe have analyzed the system by using two major bifucation parameters, one that controls the competitive ability of the mixotroph compared to the autotroph for limiting resources (*a*2-the mixotrophic link parameter), and the other one that controls the autotroph-herbivore interaction (*x*⋆-the equilibrium of the autotroph when interacting with herbivores). For small *x*⋆, the coexistence between autotrophs and herbivores is described by unique limit cycles, whereas as *x*⋆ becomes large, a globally stable fixed point describes the situation. On the other hand, when *a*2 remains small, autotroph appears as a better competitor for nutrient than the mixotroph and outcompetes the mixotroph. As *a*2 increases, first the mixotroph and the autotroph coexist in a stable equilibrium, and then a Hopf-bifurcation occurs resulting in autotroph-mixotroph cycles. Further increase in *a*2 results in the existence of multiple competition equilibria together with a stable equilibrium at the mixotrophic carrying capacity. For very large values of *a*2, mixotroph outcompetes the autotroph and the mixotrophic carrying capacity equilibrium (0, 0, c) becomes stable.\nThe most interesting result of this study is that invading mixotrophs may both stabilize and destabilize existing autotroph-herbivore dynamics. We have provided substantial analytical and numerical evidence for our claim. Globally stable two-species interaction may turn into competition cycles after the grazer has been outcompeted and the cyclic dynamics might be converted into either fixed points or oscillatory dynamics that may involve all, just two, or only one species. The appearance of multiple attractors is not only something that can occur for the autotroph-mixotroph relation, but a behavior that persists even in the complete three-species system. Note that, in a similar type of system but with relatively simpler predation term, Jost et al. (2004) found only stabilizing role of mixotrophy. Other modelling studies support the claim of stabilizing nature of mixotrophy (Hammer and Pitchford, 2005; Mitra et al. 2014). There is also evidence that the invasion of mixotrophs destabilizes the system by importing oscillations (Crane and Grover 2010). However, both stabilizing and destabilizing roles of invading mixotrophs in a single model has been found for the first time in the present study.\nThe model presented here was used to show only the fundamental mechanisms and must not be confused with quantitative simulations. The main purpose was to investigate the role of mixotrophs in a simple predator-prey system in order to understand the conditions under which invasion of mixotrophs is possible and how it affects the stability of the system and leads to more complicated dynamics. The occurrence of multi-stability and both stabilization and destabilization effects associated with the invasive mixotrophs indicate that nutrient limited lower latitude areas and higher latitude summer conditions which are ideal for mixotrophs to dominate, the community composition might be very sensitive to any kind of perturbations, i.e. a small change in the environmental condition can result into a huge change in the community composition and finally ecosystem functions. However, to quantify the real effects we need to run our system in a real physical setup using more realistic parameter values.\n#### Acknowledgements\nSome of the questions treated in this paper emanated from a discussion between Prof. em. Edna Granéli and Torsten Lindström that was connected to a joint research project proposal 2004, but national Swedish funding was never admitted. Consequently, it took some time to arrange possibilities for completing this study. Torsten Lindström thanks the Department of Mathematics at University of Helsinki for a stimulating discussion after a seminar lecture November 8, 2017, based on an earlier version of this paper. Subhendu Chakraborty was supported by the H. C. Ørsted COFUND postdoc fellowship.\n\n**Destabilization, stabilization, and multiple attractors in saturated mixotrophic environments**\n=================================================================================================\nThe ability of mixotrophs to combine phototrophy and phagotrophy is now well recognized and found to have important implications for ecosystem dynamics. In this paper we examine the dynamical consequences of the invasion of mixotrophs in a model that is a limiting case of the chemostat. The model is a hybrid of a competition model describing the competition between populations of autotroph and mixotroph for limiting resources, and a predator-prey type model describing the interaction between populations of autotroph and herbivore. Our results show that mixotrophs are able to invade in both autotrophic environments and environments described by interactions between autotrophs and herbivores. The interaction between autotrophs and herbivores might be in equilibrium or cycle. We find that invading mixotrophs have the ability to both stabilize and destabilize autotroph-herbivore dynamics depending on the competitive ability of mixotrophs. Moreover the invasion of mixotrophs can also result in multiple attractors. Therefore, our results reveal important consequences of mixotrophic invasions in ecosystems depending on environmental conditions.\nIntroduction\n============\nA large number of plankton taxa, known as mixotrophs, are able to simultaneously exploit both phototrophic and phagotrophic pathway of nutrition. These mixotrophs play important roles in exporting organic matters to higher trophic levels (Ward and Follows (2016)), carbon export to deep water (Mitra et al. (2014)), nutrient cycling (Stoecker et al. (2017)) and primary production (Chakraborty et al. (2019)). Mixotrophs are found almost everywhere in the illuminated water column, both in freshwater and marine environments (Hartmann et al. (2012); Stoecker et al. (2009)), oligotrophic and eutrophic systems (Burkholder et al., (2008)), and from polar to equatorial regions (Zubkov and Tarran (2008); Stoecker et al. (2009); Sanders and Gast (2012)). They are reported in all planktonic functional groups (except diatoms) and differ largely in their mixotrophic types depending on the availability of light, nutrient and/or prey or other particles upon which they feed (Flynn et a. (2013); Chakraborty et al. (2017); Berge et al. (2017)). How different types of mixotrophs with different competitive abilities affect the ecosystem dynamics is therefore an interesting question in ecology.\nIn most of the mathematical models of plankton community, plankton are mainly divided into phototrophs and heterotrophs. Among limited studies that include mixotrophs, only few of them investigated the impact of mixotrophs on the system dynamics. Jost et al. (2004) considered a simple food web model with four variables (nutrient, autotrophs, herbivores and mixotrophs) where mixotrophs can consume nutrients as well as graze on autotrophs, and investigated the importance of inclusion of the mixotrophic link in microbial food webs. By incorporating different types of mixotrophs in a phytoplankton-zooplankton system, Hammer and Pitchford (2005) examined the consequences of incorporation of mixotrophy on the system’s equilibrium structure, stability, short-term dynamics and productivity. Similarly, various physiological types of mixotrophs are incorporated separately in a nutrient-phytoplankton-zooplankton-detritus model by Stickney et al. (2000) to observe the effects of different types of mixotrophy on the trophic dynamics of ecosystems. Incorporating mixotrophs in a nutrient-phytoplankton-zooplankton-bacteria system with cell quotas and two essential nutrients, carbon and phosphorous, Crane and Grover (2010) discussed the role of the degree of mixotrophy on the system dynamics under different environmental conditions.\nThe present study goes further than its predecessors in several aspects. Similar to Jost et al. (2004), here we consider a chemostat model consisting of nutrient-autotroph-herbivore-mixotroph with more realistic resource and prey uptake term for mixotroph representing the shift in uptake preference depending on the resource and prey availability. Moreover, instead of doing logistic approximations that have already received much criticism (Kooi et al. (1998)), we analyze a limiting case of the chemostat that will still provide complete information of the system (Lindström and Cheng (2015)) similar to the logistic approximations (Kuang and Freedman (1988)). We choose a spectrum of mixotroph types by varying the competitive ability of mixotrophs compared to pure phototrophs and pure herbivores. Similar to the previous studies, here also we assume that mixotrophs are less competitive than pure phototrophs for inorganic resources (Litchman et al. (2007)) and less competitive than pure herbivores for prey (Zubkov and Tarran (2008)). This assumption resembles real observations that mixotrophs obtain benefit in competition over specialists mainly under limitation by multiple resources that favors generalists (Rothhaupt, (1996); Katechakis and Stibor, (2006)). The main aim of the present study is to investigate the dynamical consequences of mixotrophic invasion, especially how the invasion of mixotrophs affects the stability of the system. We perform a rigorous analytical study of the system and support our conclusions with numerical simulations.\nOur paper is organized as follows. The model is introduced in Section [modelsection] together with its basic properties and parameter restrictions. In Section [saturatedcase] we justify our selection of bifurcation parameters. In Section [selectionofproperties] we derive some properties of the bifurcation diagram that will help validating the numerical part of this paper. In Section [numbereq] the number of equilibria in the model is analyzed and in Section [stabboundeq]-[stabinteq] the stability of the equilibria is analyzed. The analytical study is corroborated and accomplished with a numerical study in Section [numres]. We end up with a discussion of the consequences of our work in Section [dicussion].\nThe model\n=========\nIn this paper we study an important special case of the following chemostat based model for mixotrophs\n \n$$\\begin{aligned}\n\\frac{dS}{dT}&=&CD-DS-\\frac{A\\_1SX}{1+A\\_1B\\_1S}-\\frac{A\\_2SZ}{1+A\\_2B\\_2S+A\\_4B\\_4X},\\nonumber\\\\\n\\frac{dX}{dT}&=&\\frac{M\\_1A\\_1SX}{1+A\\_1B\\_1S}-DX-\\frac{A\\_3XY}{1+A\\_3B\\_3X}-\\frac{A\\_4XZ}{1+A\\_2B\\_2S+A\\_4B\\_4X},\\nonumber\\\\\n\\frac{dY}{dT}&=&\\frac{M\\_3A\\_3XY}{1+A\\_3B\\_3X}-DY,\\label{chemostat\\_mixo\\_orig\\_alt2}\\\\\n\\frac{dZ}{dT}&=&\\frac{(M\\_2A\\_2S+M\\_4A\\_4X)Z}{1+A\\_2B\\_2S+A\\_4B\\_4X}-DZ.\\nonumber\\end{aligned}$$ \nHere, *S* stands for nutrient concentration, *X* for concentration of autotrophs, *Y* for concentration of herbivores, and *Z* for concentration of mixotrophs. The parameter *C* represents the nutrient input concentration, *D* is the dilution rate, *A**i*’s are the search rates, *B**i*’s are the handling times, and *M**i*’s are the conversion factors. A division of the time-budget for the mixotrophs’ searching for nutrient and autotrophs is assumed, but no preference between different recourses is considered as in the case of optimal foraging theory models, see e. g. Krebs and Davies (1993), Lindström (1994), or Boukal and Krivan (1999). When the Holling (1959) arguments are repeated for this situation, the amount of nutrient ${\\cal{S}}$ and autotrophic organism ${\\cal{X}}$ consumed during the food-gathering period ${\\cal{T}}$ is given by\n \n$$\\begin{aligned}\n{\\cal{S}}&=&A\\_2S({\\cal{T}}-B\\_2{\\cal{S}}-B\\_4{\\cal{X}}),\\\\\n{\\cal{X}}&=&A\\_4X({\\cal{T}}-B\\_2{\\cal{S}}-B\\_4{\\cal{X}}).\\end{aligned}$$ \nUsing Cramer’s rule we can directly find a solution of this linear system in the form\n \n$$\\begin{aligned}\n\\frac{{\\cal{S}}}{{\\cal{T}}}&=&\\frac{A\\_2S}{1+A\\_2B\\_2S+A\\_4B\\_4X},\\\\\n\\frac{{\\cal{X}}}{{\\cal{T}}}&=&\\frac{A\\_4X}{1+A\\_2B\\_2S+A\\_4B\\_4X}.\\end{aligned}$$ \ncorresponding to the functional responses used in ([chemostatmixoorigalt2]).\nIn this paper we are particularly interested in the special case *M*1 = *M*2 = *M*3 = *M*4 = 1 since it allows for a more complete analysis than other cases. Indeed, consider the functional *H*(*S*, *X*, *Y*, *Z*) = *S* + *X* + *Y* + *Z* − *C*. We get\n \n$$\\frac{dH}{dT}=CD-DS-DX-DY-DZ=-DH.$$ \nThat is, *H* → 0 along the solution curves of ([chemostatmixoorigalt2]). Therefore, we can hope that removal of one of the variables can provide some asymptotic information regarding the system. By substituting *S* = *C* − *X* − *Y* − *Z* in ([chemostatmixoorigalt2]), we get\n \n$$\\begin{aligned}\n\\frac{dX}{dT}&=&\\frac{A\\_1(C-X-Y-Z)X}{1+A\\_1B\\_1(C-X-Y-Z)}-DX-\\frac{A\\_3XY}{1+A\\_3B\\_3X}\\nonumber\\\\\n&&-\\frac{A\\_4XZ}{1+A\\_2B\\_2(C-X-Y-Z)+A\\_4B\\_4X},\\nonumber\\\\\n\\frac{dY}{dT}&=&\\frac{A\\_3XY}{1+A\\_3B\\_3X}-DY,\\label{chemostat\\_mixo\\_asymp\\_altII}\\\\\n\\frac{dZ}{dT}&=&\\frac{A\\_2(C-X-Y-Z)Z+A\\_4XZ}{1+A\\_2B\\_2(C-X-Y-Z)+A\\_4B\\_4X}-DZ.\\nonumber\\end{aligned}$$ \nThe last argument is not always valid, see e. g. Thieme (1992). The asymptotic information provided by the reduced system like ([chemostatmixoasympaltII]) will not always reflect all details of the limit sets of the original system ([chemostatmixoorigalt2]). Hence, there are questions at several instances related to the selection of our special case with no natural death rates and all conversion factors equal to one. Our hope is anyway that the special cases selected for detailed analytical study here contain substantial information of the properties of nearby more realistic models.\nIn order to be a valid model, ([chemostatmixoasympaltII]) needs to satisfy some conditions. We express these conditions as follows:\n* The autotrophs should be better competitors for nutrient than the mixotrophs. A criterion for this can be formulated by considering the dynamical properties of the system ([chemostatmixoasympaltII]) when *Y* = 0 and *A*4 = 0 or\n \n$$\\begin{aligned}\n\\frac{dX}{dT}&=&\\frac{A\\_1(C-X-Z)X}{1+A\\_1B\\_1(C-X-Z)}-DX,\\nonumber\\\\\n\\frac{dZ}{dT}&=&\\frac{A\\_2(C-X-Z)Z}{1+A\\_2B\\_2(C-X-Z)}-DZ.\\label{competitive\\_system\\_auto\\_mixo}\\end{aligned}$$ \nWe end up (see Lemma [lemma1] below) with a condition that involves the fixed points of the systems\n \n$$\\frac{dX}{dT}=\\frac{A\\_1(C-X)X}{1+A\\_1B\\_1(C-X)}-DX$$ \nand\n \n$$\\frac{dZ}{dT}=\\frac{A\\_2(C-Z)Z}{1+A\\_2B\\_2(C-Z)}-DZ,$$ \nrespectively. This inequality then reads\n \n$$\\frac{A\\_1C(1-DB\\_1)-D}{A\\_1(1-DB\\_1)}>\\frac{A\\_2C(1-DB\\_2)-D}{A\\_2(1-DB\\_2)}.\n\\label{fixed\\_pt\\_larger\\_cond\\_orig}$$ \nStandard algebraic manipulation reduces this inequality to\n \n*A*1 − *A*2 > *D*(*A*1*B*1 − *A*2*B*2). \nThe consequences of this inequality for the competitive dynamics between the autotroph and the mixotroph are given in Lemma [lemma1] below.\n* The herbivore is a better grazer on autotrophs than the mixotroph. Criteria for this can be formulated by considering the dynamical properties of the system ([chemostatmixoasympaltII]) when *A*2 = 0 or\n \n$$\\begin{aligned}\n\\frac{dX}{dT}&=&\\frac{A\\_1(C-X-Y-Z)X}{1+A\\_1B\\_1(C-X-Y-Z)}-DX-\\frac{A\\_3XY}{1+A\\_3B\\_3X}\\nonumber\\\\\n&&-\\frac{A\\_4XZ}{1+A\\_4B\\_4X},\\nonumber\\\\\n\\frac{dY}{dT}&=&\\frac{A\\_3XY}{1+A\\_3B\\_3X}-DY,\\label{competitive\\_system\\_herbi\\_mixo}\\\\\n\\frac{dZ}{dT}&=&\\frac{A\\_4XZ}{1+A\\_4B\\_4X}-DZ.\\nonumber\\end{aligned}$$ \nIn order to make sure that the ratio *Y*/*Z* always increases, we have\n \n$$\\frac{A\\_3X}{1+A\\_3B\\_3X}>\\frac{A\\_4X}{1+A\\_4B\\_4X}, \\:\\: 0< X\\leq C$$ \nwhich is equivalent to *A*3 − *A*4 > 0 and\n \n*A*3 − *A*4 > *A*3*A*4*C*(*B*3 − *B*4). \nThe above condition holds if *A*3 > *A*4 (the herbivore searches for autotrophs more efficiently than the mixotroph) and *B*3 < *B*4 (the mixotroph needs more handling time for the autotroph than the herbivore). This global condition can certainly be improved, see e. g. Kustarov (1986) and Lindström (1994, 2000). We shall derive a local condition for this, as well. The above condition ensures that the herbivore is a better grazer on autotroph in equilibrium in the presence of autotroph (without looking whether this equilibrium is stable or not). We note that the last two equations for the subsystem of ([competitivesystemherbimixo]) take the form\n \n$$\\begin{aligned}\n\\frac{dY}{dT}&=&\\frac{A\\_3XY}{1+A\\_3B\\_3X}-DY=\\Psi\\_1(X)Y,\\nonumber\\\\\n\\frac{dZ}{dT}&=&\\frac{A\\_4XZ}{1+A\\_4B\\_4X}-DZ=\\Psi\\_2(X)Z.\\label{chemostat\\_mixo\\_asymp\\_row2\\_3}\\end{aligned}$$ \nThis means that the last row vector of the Jacobian matrix evaluated at the unique interior fixed point at the *X*, *Y*-plane takes the form (0, 0, Ψ2(*X*⋆)) assuming that *X*⋆ is the positive solution of Ψ1(*X*) = 0. In other words, the eigenvalue corresponding to invasion of the mixotroph near this equilibrium is given by Ψ2(*X*⋆). Requiring that the mixotroph cannot invade near this equilibrium is now equivalent to\n \n*A*3 − *A*4 > *D*(*A*3*B*3 − *A*4*B*4) \nwhich is our local condition. Obviously, we have $(\\ref{global\\_condition\\_B})\\Rightarrow (\\ref{local\\_condition\\_B})$, since ([localconditionB]) means that\n \nΨ2(*X*⋆) < 0 = Ψ1(*X*⋆) \nwhereas ([globalconditionB]) means that\n \nΨ2(*X*) < Ψ1(*X*), ∀*X* ∈ ]0, *C*]. \nWe experience later that ([localconditionB]) can be formulated as a restriction of one of the bifurcation parameters of the system, cf. Remark [remarkupperbound].\nConditions derived in (A) and (B) imply that there is a trade-off between the flexibility of being a mixotroph and specializing as an autotroph or as grazer. Such assumptions have substantial support for many organisms, see Stearns (1992), Litchman et al. (2007), and Zubkov and Tarran (2008). We describe the dynamical consequences of ([fixedptlargercond]) in the following lemma.\nIf *([fixedptlargercond]), *then the fixed point** \n \n$$\\left(\\frac{A\\_1C(1-DB\\_1)-D}{A\\_1(1-DB\\_1)},0\\right)\n\\label{fixedpt\\_comp\\_syst\\_stable}$$ \nattracts solutions of all initial conditions in the triangle *X* > 0, *Z* ≥ 0, *X* + *Z* ≤ *C* for the system *([competitivesystemautomixo]). *[lemma1]** \nWe first prove that the triangle mentioned in the conditions remains invariant. We have solutions at *X* = 0, *Z* = 0 and by uniqueness of solutions they cannot be intersected. Next consider the functional *H*(*X*, *Z*) = *X* + *Z*. The total time derivative of this functional with respect to ([competitivesystemautomixo]) is given by\n \n$$\\dot{H}=\\frac{A\\_1(C-X-Z)X}{1+A\\_1B\\_1(C-X-Z)}-DX+\\frac{A\\_2(C-X-Z)Z}{1+A\\_2B\\_2(C-X-Z)}-DZ.$$ \nThis quantity is negative at the line *X* + *Z* = *C*, for *X* > 0 and *Z* > 0. The Jacobian of this system evaluated at a generic point of the system ([competitivesystemautomixo]) is given by\n \n$$J(X,Z)=\\left(\\begin{array}{cc}\nJ\\_{11}(X,Z)&\\frac{-A\\_1X}{(1+A\\_1B\\_1(C-X-Z))^2}\\\\\n\\frac{-A\\_2Z}{(1+A\\_2B\\_2(C-X-Z))^2}& J\\_{22}(X,Z)\n\\end{array}\\right).$$ \nThe off-diagonal elements are clearly negative and thus, ([competitivesystemautomixo]) is competitive. If a two dimensional system is competitive, then all its solutions converge towards a fixed point, see Smith (1995). The system has one equilibrium at (0,0) and it is unstable exactly when the equilibria at ([fixedptcompsyststable]) and\n \n$$\\left(0,\\frac{A\\_2C(1-DB\\_2)-D}{A\\_2(1-DB\\_2)}\\right)$$ \nhave one positive co-ordinate, each. By ([fixedptlargercond]), there are no other equilibria. A sketch of the phase-portrait (see Figure [kurva]) or a check of the eigenvalues shows that ([fixedptcompsyststable]) is locally stable. Hence, it attracts all initial conditions in the positive quadrant.\n(232,232)(0,0) (60,10)(1,0)250 (60,10)(0,1)180 (210,10)(-1,1)150 (170,10)(-1,1)110 (150,10)(-1,1)90 (170,10) (60,100) (60,10) (60,40)(0,1)10 (60,120)(0,-1)10 (60,150)(0,-1)10 (80,30)(1,1)10 (90,10)(1,0)10 (150,10)(1,0)10 (200,10)(-1,0)10 (140,20)(1,0)7 (110,50)(1,0)7 (80,80)(1,0)7 (150,30)(0,-1)7 (120,60)(0,-1)7 (90,90)(0,-1)7 (130,40)(1,-1)10 (95,75)(1,-1)10 (185,35)(-1,-1)10 (135,85)(-1,-1)10 (85,135)(-1,-1)10 (205,0)*C* (50,155)*C* (290,-2)*X* (45,170)*Z*\n[kurva]\nThe interpretation of ([fixedptlargercond]) is thus, that in the absence of grazers, the autotroph is a better competitor for nutrient than the mixotroph. Another interpretation of this result is that we have verified Fisher’s (1930) maximum principle for this model.\nWe introduce dimensionless parameters and variables through the changes\n \n$$x=\\frac{A\\_1(1-DB\\_1)}{A\\_1C(1-DB\\_1)-D}X,\\:\\:\ny=\\frac{A\\_1(1-DB\\_1)}{A\\_1C(1-DB\\_1)-D}Y,$$ \n \n$$z=\\frac{A\\_1(1-DB\\_1)}{A\\_1C(1-DB\\_1)-D}Z,\\:\\:\n\\tau=(A\\_1C(1-DB\\_1)-D)T,$$ \n \n$$0\\leq\\kappa\\_1=\\frac{B\\_1(A\\_1C(1-DB\\_1)-D)}{1-DB\\_1}\\leq A\\_1B\\_1C=\\gamma\\_1,$$ \n \n$$0\\leq\\kappa\\_2=\\frac{A\\_2B\\_2(A\\_1C(1-DB\\_1)-D)}{A\\_1(1-DB\\_1)}\\leq A\\_2B\\_2C=\\gamma\\_2,$$ \n \n$$0<m=\\frac{A\\_3(1-DB\\_3)}{A\\_1(1-DB\\_1)}\\leq\\frac{A\\_3}{A\\_1(1-DB\\_1)}=a\\_1,$$ \n \n$$0<a\\_2=\\frac{A\\_4(1-DB\\_4)}{A\\_1(1-DB\\_1)},$$ \n \n$$0\\leq b\\_1=\\frac{A\\_3B\\_3(A\\_1C(1-DB\\_1)-D)}{A\\_1(1-DB\\_1)},$$ \n \n$$0\\leq b\\_2=\\frac{A\\_4B\\_4(A\\_1C(1-DB\\_1)-D)}{A\\_1(1-DB\\_1)},$$ \n \n$$0<\\frac{A\\_1D(1-DB\\_1)}{A\\_3(1-DB\\_3)(A\\_1C(1-DB\\_1)-D)}=x\\_\\star<1,$$ \n \n$$0<k=\\frac{A\\_2(1-DB\\_2)}{A\\_1(1-DB\\_1)}<1,\\:{\\rm{and}}$$ \n \n$$0<c=\\frac{A\\_1(1-DB\\_1)}{A\\_2(1-DB\\_2)}\\cdot\\frac{A\\_2C(1-DB\\_2)-D}{A\\_1C(1-DB\\_1)-D}<1.$$ \nAs a result, system ([chemostatmixoasympaltII]) becomes\n \n$$\\begin{aligned}\nx^\\prime&=&\\frac{x(1-x-y-z)}{1+\\gamma\\_1-\\kappa\\_1(x+y+z)}-\\frac{a\\_1xy}{1+b\\_1x}-\\frac{a\\_2 xz}{1+\\gamma\\_2-\\kappa\\_2(x+y+z)+b\\_2 x},\\nonumber\\\\\ny^\\prime&=&\\frac{a\\_1}{1+b\\_1x\\_\\star}\\cdot\\frac{x-x\\_\\star}{1+b\\_1x}y,\\label{chemostat\\_mixo\\_orig\\_scaled\\_altII}\\\\\nz^\\prime&=&\\frac{k z\\left(c-x-y-z\\right)+a\\_2 xz}{1+\\gamma\\_2-\\kappa\\_2(x+y+z)+b\\_2 x}.\\nonumber\\end{aligned}$$ \nThere exists an explicit relationship between *a*1 and *m* and it is used in ([chemostatmixoorigscaledaltII]) above for removing the parameter *m*. We also have\n \n$$\\frac{\\gamma\\_2}{\\kappa\\_2}=\\frac{\\gamma\\_1}{\\kappa\\_1}$$ \nbut prefer to use the expression at the right-hand side in the equations ([chemostatmixoorigscaledaltII]) in order to reduce the number of parameters in the involved functions.\nThe conditions for a couple of the inequalities above are less clear than the others and are stated in a lemma. Its proof follows standard algebraic procedures.\nIf *([fixedptlargercond]), *then we have 0 < *k* < 1, and 0 < *c* < 1. [lemmakc]** \nWe use ([fixedptlargercond]) to prove the inequality for *k*. The inequality *c* < 1 follow more easily and directly from ([fixedptlargercondorig]) than from ([fixedptlargercond]).\nWe end this section by proving the following theorem.\nSolutions of *([chemostatmixoorigscaledaltII]) *remain positive and bounded.** \nThe solutions remain positive since we have solutions in the planes *x* = 0, *y* = 0, and *z* = 0. Uniqueness of solutions grants positive solutions. Next consider the Lyapunov functions *H*(*x*, *y*, *z*) = *x* + *y* + *z* for *x* + *y* + *z* ≥ 1. We have from ([chemostatmixoorigscaledaltII])\n \n$$\\begin{aligned}\nH^\\prime&=&\\frac{x(1-x-y-z)}{1+\\gamma\\_1-\\kappa\\_1(x+y+z)}-\\frac{a\\_1xy}{1+b\\_1x}-\\frac{a\\_2 xz}{1+\\gamma\\_2-\\kappa\\_2(x+y+z)+b\\_2x}\\\\\n &&+\\frac{a\\_1}{1+b\\_1x\\_\\star}\\cdot\\frac{x-x\\_\\star}{1+b\\_1x}y+\\frac{k z\\left(c -x-y-z\\right)+a\\_2 xz}{1+\\gamma\\_2-\\kappa\\_2(x+y+z)+b\\_2x}\\\\\n &\\leq&-\\frac{a\\_1xy}{1+b\\_1x}+\\frac{a\\_1}{1+b\\_1x\\_\\star}\\cdot\\frac{x-x\\_\\star}{1+b\\_1x}y\\leq 0.\\end{aligned}$$ \nThus, all solutions of interest will eventually be located in the simplex *x* ≥ 0, *y* ≥ 0, *z* ≥ 0, and *x* + *y* + *z* ≤ 1.\nThe saturated almost logistic case\n==================================\nThe main topic of this paper is the saturated almost logistic case *B*1 = *B*2 = 0, i.e. *γ*1 = *κ*1 = *γ*2 = *κ*2 = 0. As a result, ([chemostatmixoorigscaledaltII]) takes the form\n \n$$\\begin{aligned}\nx^\\prime &=&x(1-x-y-z)-\\frac{a\\_1xy}{1+b\\_1x}-\\frac{a\\_2 xz}{1+b\\_2x},\\nonumber\\\\\ny^\\prime &=&\\frac{a\\_1}{1+b\\_1x\\_\\star}\\cdot\\frac{x-x\\_\\star}{1+b\\_1x}y,\\label{chemostat\\_mixo\\_sat\\_scaled\\_altII}\\\\\nz^\\prime &=&\\frac{k z}{1+b\\_2x}\\left(c-x-y-z+\\frac{a\\_2}{k} x\\right).\\nonumber\\end{aligned}$$ \nNote that ([chemostatmixoorigscaledaltII]) differs from logistic approximations in the terms − *x**y* and − *x**z* in the equation for *x*′ and that these terms makes the model a limiting case of the chemostat possessing explicit resource dynamics. It corresponds to the limiting cases *κ**i* = 0, *γ**i* = 0, *i* = 1, 2 studied in Lindström and Cheng (2015) where the complete global information regarding the dynamics of the subsystem describing the situation when *z* = 0 was ensured. We start by doing the nonlinear change\n \n$$\\frac{d\\tau}{dt}=\\frac{1+b\\_2x}{k}\n\\label{time\\_trans}$$ \nof the independent variable and rewriting the above systems in the following isocline form\n \n$$\\begin{aligned}\n\\dot{x}&=&f\\_i(x)F\\_i(x)-yf\\_1(x)-zf\\_2(x),\\nonumber\\\\\n\\dot{y}&=&y\\psi(x),\\label{chemostat\\_mixo\\_sat\\_isocline\\_altII}\\\\\n\\dot{z}&=&z(G(x)-y-z).\\nonumber\\end{aligned}$$ \nwhere\n \n$$f\\_i(x)=\\frac{x+\\frac{a\\_ix}{1+b\\_ix}}{\\frac{k}{1+b\\_2x}},\\:\\:F\\_i(x)=\\frac{(1-x)(1+b\\_ix)}{1+a\\_i+b\\_ix},\\:\\: i=1,2$$ \nand\n \n$$\\psi(x)=\\frac{\\frac{a\\_1}{1+b\\_1x\\_\\star}\\cdot\\frac{x-x\\_\\star}{1+b\\_1x}}{\\frac{k}{1+b\\_2 x}},\\:\\: G(x)=c-x+\\frac{a\\_2}{k}x.$$ \nWe begin with a lemma stating the most important properties of the functions involved here.\nWe have that *f**i*′(*x*) > 0, *x* ≥ 0, *i* = 1, 2, and *ψ*′(*x*⋆) > 0.\nSince *ψ* is differentiable and (*x* − *x*⋆)*ψ*(*x*) is positive definite, *ψ*′(*x*⋆) > 0. For *f*2 we get\n \n$$f\\_2(x)=\\frac{x}{k}(1+a\\_2+b\\_2x),\\: {\\rm{and}}\\: f\\_2^\\prime(x)=\\frac{1}{k}(1+a\\_2+2b\\_2x).$$ \nIt is obvious that *f*2 increases for *x* ≥ 0. For *f*1 we have\n \n$$f\\_1(x)=\\frac{x(1+a\\_1+b\\_1x)(1+b\\_2x)}{k(1+b\\_1x)}=\\frac{x+a\\_1x+b\\_1x^2+b\\_2x^2+a\\_1b\\_2x^2+b\\_1b\\_2x^3}{k(1+b\\_1x)}.$$ \nWe differentiate and get\n \n$$\\begin{aligned}\nk(1+b\\_1x)^2f\\_1^\\prime(x)&=&(1+a\\_1+2b\\_1x+2b\\_2x+2a\\_1b\\_2x+3b\\_1b\\_2x^2)(1+b\\_1x)\\\\\n&&-b\\_1(x+a\\_1x+b\\_1x^2+b\\_2x^2+a\\_1b\\_2x^2+b\\_1b\\_2x^3) \\\\\n &=&1+a\\_1+2b\\_1x+2b\\_2x+2a\\_1b\\_2x+3b\\_1b\\_2x^2\\\\\n &&+b\\_1x+a\\_1b\\_1x+2b\\_1^2x^2+2b\\_1b\\_2x^2+2a\\_1b\\_1b\\_2x^2+3b\\_1^2b\\_2x^3\\\\\n &&-b\\_1x-a\\_1b\\_1x-b\\_1^2x^2-b\\_1b\\_2x^2-a\\_1b\\_1b\\_2x^2-b\\_1^2b\\_2x^3\\\\\n &=&1+a\\_1+2b\\_1x+2b\\_2x+2a\\_1b\\_2x\\\\\n &&+b\\_1^2x^2+4b\\_1b\\_2x^2+a\\_1b\\_1b\\_2x^2+2b\\_1^2b\\_2x^3>0,\\\\\\end{aligned}$$ \nfor *x* ≥ 0.\nWe continue with an analysis of the various subsystems. If *y* = 0 and *z* = 0 we get the logistic equation $\\dot{x}=x(1-x)$. For *x* = 0, *z* = 0, we get\n \n$$\\dot{y}=-\\frac{\\frac{a\\_1}{k}x\\_\\star y}{1+b\\_1x\\_\\star}$$ \nand for *x* = 0, *y* = 0 we get $\\dot{z}=z(c-z)$, respectively. We return to our logistic friend also in the *y**z*-plane. Now we assume *z* = 0 and consider\n \n$$\\begin{aligned}\n \\dot{x} &=& f\\_1(x)\\left(F\\_1(x)-y\\right),\\nonumber\\\\\n \\dot{y} &=& \\psi(x)y.\\label{sat-pp}\\end{aligned}$$ \nThis system turns out to be a Gause-type predator-prey system and its qualitative dynamics is completely known, see Lindström and Cheng (2015). We may, if necessary, consider the system without the transformation of the independent variable ([timetrans]). Indeed, the sign of the derivative of *F* determines a lot of its dynamical properties and the derivative is given by\n \n$$F\\_1^{\\prime}(x)=\\frac{-1-a\\_1+a\\_1b\\_1-2b\\_1(a\\_1+1)x-b\\_1^2x^2}{(1+a\\_1+b\\_1x)^2}.$$ \nIts denominator is always positive, but its nominator has zeros at\n \n$$x=\\frac{-a\\_1-1\\pm\\sqrt{a\\_1^2+a\\_1+a\\_1b\\_1}}{b\\_1}.$$ \nOne of these zeros is always negative and there exits a positive zero if $b\\_1>1+\\frac{1}{a\\_1}$. The following lemmas follow from Lindström and Cheng (2015).\nConsider *([sat-pp]). *If either $b\\_1\\leq 1+\\frac{1}{a\\_1}$ or** \n \n$$0<\\frac{-a\\_1-1+\\sqrt{a\\_1^2+a\\_1+a\\_1b\\_1}}{b\\_1}\\leq x\\_\\star$$ \nthen the equilibrium (min(*x*⋆, 1), max(0, *F*1(*x*⋆))) is globally asymptotically stable in the cone *x* > 0, *y* > 0. [globstab]\nIf\n \n$$0<x\\_\\star<\\frac{-a\\_1-1+\\sqrt{a\\_1^2+a\\_1+a\\_1b\\_1}}{b\\_1}$$ \nthen *([sat-pp]) *possesses a unique limit cycle that is stable. [uniquelimitcycle]** \nWe are now ready with the subsystem at the *z* = 0 plane and turn over to the subsystem at the *y* = 0 plane. We obtain the competition system\n \n$$\\begin{aligned}\n\\dot{x} &=& f\\_2(x)\\left(F\\_2(x)-z\\right),\\nonumber\\\\\n\\dot{z}&=&z\\left(G(x)-z\\right).\\label{sat\\_comp\\_altII}\\end{aligned}$$ \nOf course, the properties of *F**i*, *i* = 1, 2 are identical. The solutions *G*(*x*) − *F*2(*x*) = 0 determine the number of interior fixed points in the *x**z*-plane. Since cubic terms do not enter in this equation, there exist at most two interior fixed points in the positive quadrant of this plane and these fixed points can be computed explicitly. We have now identified the following equilibria of ([chemostatmixosatscaledaltII]): (0, 0, 0) (washout/extinction), (1, 0, 0) (carrying capacity/survival), (0, 0, *c*) (mixotroph carrying capacity/mixotroph survival), (*x*⋆, *F*1(*x*⋆), 0) (predator-prey) that always exist. In addition, we have possibly two competition equilibria (*x*±, 0, *F*2(*x*±)). The values *x*± are the solutions of the quadratic equation\n \n$$k c\\left(a\\_2-\\frac{1-c}{c}\\right)+\\left(a\\_2-(k-1)\\right)a\\_2 x-b\\_2 k(1-c)x+a\\_2 b\\_2 x^2=q(x)=0,\n\\label{multiple\\_eq\\_alt\\_II}$$ \nin the unit interval. Finally we have possibly one coexistence equilibrium that we denote by (*x*⋆, *y*⋆, *z*⋆). The last two coordinates of the equilibria are given by the solutions of the linear system\n \n$$\\begin{aligned}\nx\\_\\star(1-x\\_\\star)&=&f\\_1(x\\_\\star)y+f\\_2(x\\_\\star)z, \\\\\nG(x\\_\\star)&=&y+z.\\end{aligned}$$ \nHere, Cramer’s rule provides the solutions\n \n$$\\begin{aligned}\ny\\_\\star&=&\\frac{f\\_2(x\\_\\star)(F\\_2(x\\_\\star)-G(x\\_\\star))}{f\\_1(x\\_\\star)-f\\_2(x\\_\\star)}=\\frac{k(1-x\\_\\star)-\\left(1+\\frac{a\\_2}{1+b\\_2x\\_\\star}\\right)\\left(k(c-x\\_\\star)+a\\_2x\\_\\star\\right)}{\\frac{k}{x\\_\\star}\\left(\\frac{a\\_1x\\_\\star}{1+b\\_1x\\_\\star}-\\frac{a\\_2x\\_\\star}{1+b\\_2x\\_\\star}\\right)},\n\\\\\nz\\_\\star&=&\\frac{f\\_1(x\\_\\star)(G(x\\_\\star)-F\\_1(x\\_\\star))}{f\\_1(x\\_\\star)-f\\_2(x\\_\\star)}=\\frac{\\left(1+\\frac{a\\_1}{1+b\\_1x\\_\\star}\\right)\\left(k(c-x\\_\\star)+a\\_2x\\_\\star\\right)-k(1-x\\_\\star)}{\\frac{k}{x\\_\\star}\\left(\\frac{a\\_1x\\_\\star}{1+b\\_1x\\_\\star}-\\frac{a\\_2x\\_\\star}{1+b\\_2x\\_\\star}\\right)}.\\end{aligned}$$ \nWe conclude first that the denominators in the expressions for *y*⋆ and *z*⋆ are positive.\nEach of the following are equivalent:\n1. [*(i) *] The local condition *([localconditionB])***\n2. [*(ii)*] *f*1(*x*⋆) > *f*2(*x*⋆).**\n3. [*(iii)*] *a*1 − *a*2 > (*a*2*b*1 − *a*1*b*2)*x*⋆**\n[lemmaloccond]\nWe start from *f*1(*x*⋆) > *f*2(*x*⋆), or (ii). After cancelling of some positive factors and terms, this condition is equivalent to\n \n$$\\frac{a\\_1}{1+b\\_1x\\_\\star}>\\frac{a\\_2}{1+b\\_2x\\_\\star}\n\\label{parametric\\_local\\_cond\\_B}$$ \nphrased in our current parameters. Since the denominators are positive, this condition is equivalent to (iii) by standard algebraic procedures. Direct substitution of the original parameters in ([parametriclocalcondB]) gives now\n \n$$\\frac{\\frac{A\\_3}{A\\_1(1-DB\\_1)}}{1+\\frac{B\\_3D}{(1-DB\\_3)}}>\\frac{\\frac{A\\_4}{A\\_1(1-DB\\_1)}}{1+\\frac{A\\_4B\\_4D}{A\\_3(1-DB\\_3)}}.$$ \nStandard algebraic procedures reduces this expression to ([localconditionB]) and these steps can be carried out in the opposite direction, too. Hence (i), (ii), and (iii) are equivalent.\n*We note that ([parametriclocalcondB]) puts an upper bound for *a*2 given by* \n \n$$a\\_2<a\\_1\\frac{1+b\\_2x\\_\\star}{1+b\\_1x\\_\\star}.\n\\label{upper\\_bound\\_a2}$$ \nSince *a*2 is going to be a bifurcation parameter in our subsequent numerical study, such bounds are quite useful. [remarkupperbound] \nWe conclude that the interior equilibrium exists (phrased in terms of the mixotrophic parameters) if and only if *ă*2 < *a*2 < *ã*2 with\n \n$$\\begin{aligned}\n\\breve{a}\\_2&=&\\frac{k(1-x\\_\\star)-k(c-x\\_\\star)\\left(1+\\frac{a\\_1}{1+b\\_1x\\_\\star}\\right)}{x\\_\\star+\\frac{a\\_1x\\_\\star}{1+b\\_1x\\_\\star}},\\\\\n\\tilde{a}\\_2&=&\\frac{-(1-\\frac{k}{1+b\\_2x\\_\\star})x\\_\\star-\\frac{kc}{1+b\\_2x\\_\\star}+\\sqrt{\\left(\\left(1-\\frac{k}{1+b\\_2x\\_\\star}\\right)x\\_\\star+\\frac{kc}{1+b\\_2x\\_\\star}\\right)^2+\\frac{4x\\_\\star k(1-c)}{1+b\\_2x\\_\\star}}}{\\frac{2x\\_\\star}{1+b\\_2x\\_\\star}}\\end{aligned}$$ \nand that a similar criterion in terms of the involved functions can be stated as\n \n*F*1(*x*⋆) < *G*(*x*⋆) < *F*2(*x*⋆), \nsee Figure [F2GF1].\nA selection of properties of the bifurcation diagram\n====================================================\nThe last conclusions in the previous section states that *a*2 and *x*⋆ are important bifurcation parameters not only in the competition plane and the predator-prey plane, respectively, but also for the entire system ([chemostatmixosatscaledaltII]). For instance, there are possibilities that an interval with respect to the mixotrophic link exists, granting equilibrium coexistence. A number of results that specify whether different regions in the bifurcation diagram exist or not can now be formulated. They will be used later on for selecting parameters for a sufficiently general numerically computed bifurcation diagram and for validating our numerical results.\nWe begin with formulating a condition for *ă*2 < *ã*2 indicating that restrictions must be put on *k* if *c* is small in order to ensure an interior equilibrium point.\nDefine\n \n$$\\hat{k}=\\left\\{\\begin{array}{cc}\n \\infty &,{\\rm{if}}\\:\\: c\\geq x\\_\\star+\\frac{1-x\\_\\star}{1+\\frac{a\\_1}{1+b\\_1x\\_\\star}}, \\\\\n \\frac{x\\_\\star a\\_1}{\\frac{1-x\\_\\star}{1+a\\_1}+x\\_\\star-c} &,{\\rm{if}} \\:\\: c<x\\_\\star+\\frac{1-x\\_\\star}{1+\\frac{a\\_1}{1+b\\_1x\\_\\star}}.\n \\end{array}\\right.$$ \nWe have *ă*2 < *ã*2 ⇔ 0 < *k* < min(1, *k̂*). *[exist-paramsat]* \nWe commence by introducing the auxiliary variable\n \n$$s=\\frac{1-x\\_\\star-\\left(1+\\frac{a\\_1}{1+b\\_1x\\_\\star}\\right)(c-x\\_\\star)}{(1+b\\_2x\\_\\star)\\left(1+\\frac{a\\_1}{1+b\\_1x\\_\\star}\\right)}\n=\\frac{1-x\\_\\star}{(1+b\\_2x\\_\\star)\\left(1+\\frac{a\\_1}{1+b\\_1x\\_\\star}\\right)}-\\frac{c-x\\_\\star}{1+b\\_2x\\_\\star}.\n\\label{s\\_def\\_sat}$$ \nThe inequality *ă*2 < *ã*2 is then equivalent to\n \n$$\\begin{aligned}\n&&2ks+\\left(1-\\frac{k}{1+b\\_2x\\_\\star}\\right)x\\_\\star+\\frac{kc}{1+b\\_2x\\_\\star}\\nonumber\\\\&<&\\sqrt{\\left(\\left(1-\\frac{k}{1+b\\_2x\\_\\star}\\right)x\\_\\star+\\frac{kc}{1+b\\_2x\\_\\star}\\right)^2\n+\\frac{4x\\_\\star k(1-c)}{1+b\\_2x\\_\\star}}.\\label{s\\_ineq\\_sat}\\end{aligned}$$ \nThe radical expression in ([sineqsat]) is positive by Lemma [lemmakc]. Inequality ([sineqsat]) is therefore equivalent to ([sineq2sat]) after that the squares have been cancelled and the common positive factor 4*k* has been cancelled. Further reorganization of the inequality and substitution of the expression ([sdefsat]) gives\n \n$$\\begin{aligned}\nks^2+s\\left(\\left(1-\\frac{k}{1+b\\_2x\\_\\star}\\right)x\\_\\star+\\frac{kc}{1+b\\_2x\\_\\star}\\right)&<&\\frac{x\\_\\star (1-c)}{1+b\\_2x\\_\\star}\\label{s\\_ineq2\\_sat}\\\\\n&\\Updownarrow&\\nonumber\\\\\nks\\left(s+\\frac{c-x\\_\\star}{1+b\\_2x\\_\\star}\\right)+sx\\_\\star&<&\\frac{x\\_\\star (1-c)}{1+b\\_2x\\_\\star}\\nonumber\\\\\n&\\Updownarrow&\\nonumber\\\\\nks\\frac{1-x\\_\\star}{(1+b\\_2x\\_\\star)\\left(1+\\frac{a\\_1}{1+b\\_1x\\_\\star}\\right)}+&&\\nonumber\\\\\n+\\left(\\frac{1-x\\_\\star}{(1+b\\_2x\\_\\star)\\left(1+\\frac{a\\_1}{1+b\\_1x\\_\\star}\\right)}-\\frac{c-x\\_\\star}{1+b\\_2x\\_\\star}\\right)x\\_\\star&<&\\frac{x\\_\\star(1-c)}{1+b\\_2x\\_\\star}\\nonumber\\\\\n&\\Updownarrow&\\nonumber\\\\\nks\\frac{1-x\\_\\star}{1+\\frac{a\\_1}{1+b\\_1x\\_\\star}}+\n\\left(\\frac{1-x\\_\\star}{1+\\frac{a\\_1}{1+b\\_1x\\_\\star}}-c+x\\_\\star\\right)x\\_\\star&<&x\\_\\star(1-c)\\nonumber\\\\\n&\\Updownarrow&\\nonumber\\\\\nks\\frac{1-x\\_\\star}{1+\\frac{a\\_1}{1+b\\_1x\\_\\star}}+\\frac{1-x\\_\\star}{1+\\frac{a\\_1}{1+b\\_1x\\_\\star}}x\\_\\star&<&x\\_\\star(1-x\\_\\star)\\nonumber\\\\\n&\\Updownarrow&\\nonumber\\\\\n\\frac{ks}{1+\\frac{a\\_1}{1+b\\_1x\\_\\star}}+\\frac{x\\_\\star}{1+\\frac{a\\_1}{1+b\\_1x\\_\\star}}&<&x\\_\\star\\nonumber\\\\\n&\\Updownarrow&\\nonumber\\\\\n\\frac{ks}{1+\\frac{a\\_1}{1+b\\_1x\\_\\star}}&<&x\\_\\star-\\frac{x\\_\\star}{1+\\frac{a\\_1}{1+b\\_1x\\_\\star}}=\\frac{\\frac{x\\_\\star a\\_1}{1+b\\_1x\\_\\star}}{1+\\frac{a\\_1}{1+b\\_1x\\_\\star}}\\nonumber\\\\\n&\\Updownarrow&\\nonumber\\\\\nks&<&\\frac{x\\_\\star a\\_1}{1+b\\_1x\\_\\star}\\label{last\\_s\\_ineq\\_sat}\\end{aligned}$$ \nThe required restriction follows now from ([lastsineqsat]) if *c* is sufficiently small. For sufficiently large *c* no restriction follows since we have 0 < *k* < 1 by Lemma [lemmakc].\n(350,350)(0,0) (-7.5,-8) (34,45)(0, *c*) (33,53)(-1,1)10\n(23,140)(*x*−, *G*(*x*−)) (36,135)(0,-1)50\n(222,93)(*x*⋆, *F*1(*x*⋆)) (220,101)(-3,2)10 (153,250)*x* = *x*⋆ (188,253)(1,0)10 (363,-5)*x* (-12,280)*y* + *z* (45,170)*y* + *z* = *F*2(*x*) (85,163)(2,-3)12 (220,165)*y* + *z* = *G*(*x*) (253,173)(-1,2)10 (290,125)*y* + *z* = *F*1(*x*) (323,132)(-1,4)3\n(303,215)(*x*+, *G*(*x*+)) (321,223)(-1,4)3\n[F2GF1]\nIt holds that lim*x*⋆ → 1*ă*2 = lim*x*⋆ → 1*ã*2 = *k*(1 − *c*) = *ǎ*2. [limitlemma]\nThe computation of the limit for *ă*2 is straightforward after cancelling of the common factor 1 + *a*1/(1 + *b*1). The computation of the limit for *ã*2 leads to the square\n \n$$\\left(1+\\frac{k}{1+b\\_2}-\\frac{kc}{1+b\\_2}\\right)^2$$ \nunder the radical and consequently, since 0 < *c* < 1, to a straightforward computation.\nIf *x*⋆ > *c*, then it holds that $\\tilde{a}\\_2\\leq\\frac{1-c}{c}=\\hat{a}\\_2$. [lemma4]\n*We always have that *ǎ*2 = *k*(1 − *c*) < (1 − *c*)/*c* = *â*2.* \nIt follows from the estimate\n \n$$-w+\\sqrt{w^2+u}\\leq\\frac{u}{2w},\\:\\:u\\geq -w^2,\\: w>0\n\\label{sqrt-estimate}$$ \nwith\n \n$$w=\\frac{(1-\\frac{k}{1+b\\_2x\\_\\star})x\\_\\star+\\frac{kc}{1+b\\_2x\\_\\star}}{\\frac{2x\\_\\star}{1+b\\_2x\\_\\star}},\\:\\: u=\\frac{k(1-c)(1+b\\_2x\\_\\star)}{x\\_\\star}$$ \nthat\n \n$$\\tilde{a}\\_2\\leq\\frac{k(1-c)}{(1-\\frac{k}{1+b\\_2x\\_\\star})x\\_\\star+\\frac{kc}{1+b\\_2x\\_\\star}}\\leq\\frac{k(1-c)}{c}\\leq\\frac{1-c}{c}=\\hat{a}\\_2.$$ \nWe have a chain of three inequalities in the above expression and justify each of them separately. The first inequality holds by ([sqrt-estimate]), the second since the expression in the denominator is a mean between *x*⋆ and *c* and by the assumption *x*⋆ > *c*. The last inequality holds since *k* ≤ 1.\nIf *x*⋆ > ( < )(*a*1 − *â*2)/(*b*1*â*2), then *ǎ*2 < ( > )*ă*2. [lemma5]\nThe inequality *ǎ*2 < ( > )*ă*2 is equivalent to the inequality\n \n$$\\left(1-c-\\frac{a\\_1c}{1+b\\_1x\\_\\star}\\right)(1-x\\_\\star)>(<)0$$ \nThe second factor is always positive. The sign of the first factor depends on *x*⋆. For large *x*⋆ it is positive and the limit for *x*⋆ is the one given in the assumption. If *a*1 < *â*2, then all values 0 < *x*⋆ < 1 gives *ǎ*2 < *ă*2.\nCombining Lemma [lemma4] and [lemma5], we get\nIf *x*⋆ > max(*c*, (*a*1 − *â*2)/(*b*1*â*2)) and 0 < *k* < min(1, *k̂*) then ∅ ≠ (*ă*2, *ã*2) ⊂ (*ǎ*2, *â*2)\nOn the number of competition equilibria\n=======================================\nWe return for deriving the conditions for the existence of a certain number of competition equilibria. Define\n \n$$\\check{x}=\\frac{-a\\_2^2-a\\_2(1-k)+b\\_2k(1-c)}{2a\\_2b\\_2}\n\\label{min\\_pt\\_q}$$ \nand consider the quadratic equation\n \n$$\\begin{aligned}\n -a\\_2^2-a\\_2(1-k)+b\\_2k(1-c)&=&-(a\\_2-a\\_2^+)(a\\_2-a\\_2^-)=0,\\end{aligned}$$ \nalong with the quartic equation\n \n$$q\\_\\ast(a\\_2)=4a\\_2b\\_2q(\\check{x})=4a\\_2b\\_2kc\\left(a\\_2-\\frac{1-c}{c}\\right)-((a\\_2-a\\_2^+)(a\\_2-a\\_2^-))^2=0.\n\\label{4e\\_gradare}$$ \nWe note that\n \n$$a\\_2^\\pm=\\frac{-(1-k)\\pm\\sqrt{(1-k)^2+4b\\_2k(1-c)}}{2},$$ \n*a*2+ > 0, *a*2− < 0 and begin our analysis with the following lemma.\nAssume that *a*2+ > (1 − *c*)/*c*. In each of the intervals ] − ∞, *a*2−], [*a*2−, 0], [(1 − *c*)/*c*, *a*2+], and [*a*2+, ∞], the equation *([4egradare]) *has exactly one real root. [intervalrootlemma]** \nWe have\n \n$$\\begin{aligned}\n\\lim\\_{a\\_2\\rightarrow -\\infty}q\\_\\ast(a\\_2)&=&-\\infty,\\\\\n q\\_\\ast(a\\_2^-)&>&0, \\\\\n q\\_\\ast(0)&<&0, \\\\\n q\\_\\ast\\left(\\frac{1-c}{c}\\right) &<&0, \\\\\n q\\_\\ast(a\\_2^+)&>&0, \\:\\:{\\rm{and}}\\\\\n \\lim\\_{a\\_2\\rightarrow\\infty}q\\_\\ast(a\\_2)&=&-\\infty,\\end{aligned}$$ \nimplying that the specified location of the roots follows by the intermediate value theorem.\nWe denote the root of ([4egradare]) in the interval [(1 − *c*)/*c*, *a*2+] by *a*2\\* and put\n \n$$\\underline{a}\\_2=\\frac{-(1-k+b\\_2)+\\sqrt{(1-k+b\\_2)^2+4b\\_2k(1-c)}}{2}.$$ \nWe can now state the following lemma. It simplifies the formulation of the next theorem and the range of possible dynamical scenarios for our competition system ([competitivesystemautomixo]). Indeed, for increasing *a*2 an additional competition equilibrium can bifurcate from the mixotroph carrying capacity (0, 0, *c*) through a transcritical bifurcation only.\nWe have that\n \n$$\\underline{a}\\_2<\\frac{1-c}{c}=\\hat{a}\\_2.$$ \n[lemma12]\nWe shall use ([sqrt-estimate]) for proving the inequality. We have with *w* = (1 − *k* + *b*2)/2 and *u* = *b*2*k*(1 − *c*) that it is enough to show\n \n$$\\underline{a}\\_2<\\frac{b\\_2k(1-c)}{1-k+b\\_2}<\\frac{1-c}{c}=\\hat{a}\\_2$$ \nThe first inequality holds by ([sqrt-estimate]). It remains to prove the second inequality. It is equivalent to\n \n − (1 − *k*) < *b*2(1 − *k**c*) \nwhich is identically true since its left-hand side is negative and its right-hand side is positive.\nWe now have the following bi-stability theorem.\nIf *k*(1 − *c*) < *a*2 < (1 − *c*)/*c* then *([satcompaltII]) *has one interior equilibrium. If** \n \n$$\\frac{1-c}{c}<a\\_2<a\\_2^\\ast\n\\label{2\\_fixed\\_int\\_comp}$$ \nthen *([satcompaltII]) *has two interior equilibria. Otherwise, *([satcompaltII]) *has no interior equilibria.**** \nConsider the quadratic equation *q*(*x*) = 0 given by ([multipleeqaltII]). The condition is *q*(0)*q*(1) < 0 becomes\n \n$$kc(a\\_2+b\\_2+1)\\left(a\\_2-\\frac{1-c}{c}\\right)(a\\_2-k(1-c))<0.$$ \nThus, we have precisely one competition equilibrium when\n \n$$\\check{a}\\_2=k(1-c)<a\\_2<\\frac{1-c}{c}=\\hat{a}\\_2.$$ \nWe notice that the condition for the presence of precisely one competition equilibrium is independent from saturation. The first statement of our bi-stability theorem follows.\nFor the second statement, the first necessary condition is that the function *q* must have either a local minimum or a local maximum in the unit interval. Since the leading term of *q* is positive, only local minima are possible. A minimum requires\n \n$$\\begin{aligned}\n q(0) &=& kc\\left(a\\_2-\\frac{1-c}{c}\\right)>0 \\\\\n q(1) &=& (a\\_2+b\\_2+1)(a\\_2-k(1-c))>0\\end{aligned}$$ \nor\n \n$$a\\_2>\\frac{1-c}{c}>k(1-c).$$ \nThe minimum point *x̌* of *q* was defined by ([minptq]) and we require 0 < *x̌* < 1. That is\n \n$$\\begin{aligned}\n -a\\_2^2-a\\_2(1-k)+b\\_2k(1-c)&>&0, \\\\\n -a\\_2^2-a\\_2(1-k+b\\_2)+b\\_2k(1-c)&<&0,\\end{aligned}$$ \nwhich in terms of the mixotrophic parameter *a*2 can be stated as $\\underline{a}\\_2<a\\_2<a\\_2^+$.\nFinally, we need *q*(*x̌*) < 0. It follows from ([4egradare]) and Lemma [intervalrootlemma] that the additional requirement (1 − *c*)/*c* < *a*2 < *a*2\\* must hold. By Lemma [lemma12], ([satcompaltII]) has two equilibria when ([2fixedintcomp]) holds.\nThe above theorem does not specify what saturation levels actually are needed in order to grant an interval of type ([2fixedintcomp]). The following necessary condition turned out to be helpful.\nIf an interval of type *([2fixedintcomp]) *exist, then** \n \n$$cb\\_2>\\frac{1}{k}-1.\n\\label{ness\\_crit}$$ \nIn order to have an interval of type ([2fixedintcomp]) we need at least *a*2+ > (1 − *c*)/*c*. From ([sqrt-estimate]) we have\n \n$$a\\_2^+=\\frac{k-1+\\sqrt{(k-1)^2+4b\\_2k(1-c)}}{2}<\\frac{b\\_2k(1-c)}{1-k}$$ \nand a necessary criterion for two competition equilibria is therefore ([nesscrit]).\nMultiple competition equilibria are therefore, expected for *k* close to one and *c**b*2 large. Remember that 0 < *k* < 1, 0 < *c* < 1, and *b*2 > 0.\n*In Figure [facescase1] we make a special study of the case *c* = .2, *k* = .95, *x*⋆ = .05, *a*1 = 8.5, *b*1 = 50, *a*2 = 4.5, and *b*2 = 55. This case allows for two competition equilibria (a), and unstable equilibrium in the predator-prey plane (b), and a coexistence equilibrium, cf Figure [F2GF1]. Consequently, the function *q* has two zeros in the unit interval and a minimum point. The function *q*\\* has been plotted in (d). It has zeros in the prescribed intervals and in this case 4 = (1 − *c*)/*c* < *a*2 < *a*2\\* ≈ 5.11 < *a*2+ ≈ 6.44.* \n(350,350)(0,0) (-42.5,-16) (10,297)*z* (160,178)*x* (82,230)*G* (86,240)(0,1)10 (62,267)*F*2 (65,265)(0,-1)10 (115,250)(*x*+, *G*(*x*+)) (147,262)(0,1)22 (30,290)(*x*−, *G*(*x*−)) (49,285)(-1,-3)20 (38,200)(0, *c*) (35,207)(-2,1)10 (50,230)(1,0)10 (110,267)(1,0)10 (90,267)(0,-1)7 (167,300)(-1,0)10 (171,294)(0,1)5\n(215,297)*y* (365,178)*x* (320,222)(*x*⋆, *F*1(*x*⋆)) (318,227)(-2,1)10 (304.5,210)(1,0)10 (370,253)(0,1)10 (304.5,260)(-1,0)10 (304.5,290)(-1,0)10 (264.2,215)(0,-1)10\n(160,12)*x* (21,120)*x*− (26,118)(0,-1)40 (142,120)*x*+ (147,118)(0,-1)40 (10,137)*q*\n(365,12)*a*2 (212,137)*q*\\* (260.5,158)*a*2− (265.5,155)(0,-1)10\n(312,158)$\\underline{a}\\_2$ (317,155)(0,-1)10 (332.5,158)$\\frac{1-c}{c}$ (340.5,152)(0,-1)7\n(353,158)*a*2+ (358,155)(0,-1)10\n[facescase1]\nThe stability of the boundary equilibria\n========================================\nThe generic Jacobian of ([chemostatmixosatisoclinealtII]) takes the form\n \n$$J(x,y,z)=\\left(\\begin{array}{ccc}\n\\begin{array}{c} f\\_1^\\prime(x)(F\\_1(x)-y)\\\\+f\\_1(x)F\\_1^\\prime(x)-zf\\_2^\\prime(x)\\end{array} & -f\\_1(x) & -f\\_2(x)\\\\\ny\\psi^\\prime(x) & \\psi(x) & 0\\\\\nzG^\\prime(x) & -z & G(x)-y-2z\n\\end{array}\\right).$$ \nIn the following we calculate the Jacobian for the different equilibrium points:\n1. The washout Jacobian *J*(0, 0, 0) takes a diagonal form with the diagonal elements 1/*k* > 0, − *a*1*x*⋆/(*k*(1 + *b*1*x*⋆)) < 0 and *c* > 0. Therefore this equilibrium is always a saddle.\n2. Similarly, the carrying capacity equilibrium takes an upper triangular form with diagonal − 1/*k* < 0, *ψ*(1) > 0, *G*(1). It has a transcritical bifurcation at\n \n*a*2 = *k*(1 − *c*) \ncorresponding to the sign-change of *G*(1). Hence this equilibrium is always a saddle.\n3. The mixotroph carrying capacity equilibrium (0, 0, *c*) takes lower triangular form with diagonal elements\n \n(1 − (1 + *a*2)*c*)/*k*, − *a*1*x*⋆/(*k*(1 + *b*1*x*⋆)) < 0, − *c* < 0 \nand is a saddle if *a*2 < (1 − *c*)/*c* and a stable node if *a*2 > (1 − *c*)/*c*.\n4. The Jacobian at the predator-prey equilibrium (*x*⋆, *F*1(*x*⋆), 0) takes the block-diagonal form\n \n$$J(x\\_\\star,F\\_1(x\\_\\star),0)=\\left(\\begin{array}{ccc}\n f\\_1(x\\_\\star)F\\_1^\\prime(x\\_\\star) & -f\\_1(x\\_\\star) & -f\\_2(x\\_\\star) \\\\\n F\\_1(x\\_\\star)\\psi^\\prime(x\\_\\star) & 0 & 0 \\\\\n 0 & 0 & G(x\\_\\star)-F\\_1(x\\_\\star)\n \\end{array}\\right)$$ \nwith a 2 × 2 block matrix reflecting the stability properties in the predator-prey plane that are already known and a single eigenvalue reflecting the behavior in the mixotroph direction of magnitude *G*(*x*) − *F*1(*x*). Criterion ([isointeriorcrit]) is visible here.\n5. We are now ready to proceed to the possible competition equilibria at (*x*±, 0, *G*(*x*±)). Now we consider the Jacobian\n \n$$J(x\\_\\pm,0,G(x\\_\\pm))=\\left(\\begin{array}{ccc}\nf\\_2(x\\_\\pm)F\\_2^\\prime(x\\_\\pm) & -f\\_1(x\\_\\pm) & -f\\_2(x\\_\\pm)\\\\\n0 & \\psi(x\\_\\pm) & 0\\\\\nG(x\\_\\pm)G^\\prime(x\\_\\pm) & -G(x\\_\\pm) & -G(x\\_\\pm)\n\\end{array}\\right).\n\\label{Jaco\\_comp\\_eq}$$ \nThe corner elements form a 2 × 2-block matrix reflecting the stability properties in the competition plane in terms of the intersections of *F*2 and *G*. Indeed, the determinant of the corner elements is *f*2(*x*±)*G*(*x*±)(*G*′(*x*±) − *F*2′(*x*±)). Since *G*′(*x*−) − *F*2′(*x*−) < 0, it follows that (*x*−, 0, *G*(*x*−)) is a saddle in the *x**z*-plane if it exists. Similarly, (*x*+, 0, *G*(*x*+)) is never a saddle in the *x**z*-plane. Consequently (*x*+, 0, *G*(*x*+)) is stable in the *x**z*-plane if *f*2(*x*+)*F*2′(*x*+) − *F*2(*x*+) < 0 (Note that *F*2(*x*+) = *G*(*x*+)). The last eigenvalue of the competition equilibria is *ψ*(*x*±). The second inequality of ([isointeriorcrit]) implies max(0, *x*−) < *x*⋆ < *x*+ and therefore we always have invasion of herbivores in the vicinity of (*x*+, 0, *F*2(*x*+)) if (*x*⋆, *y*⋆, *z*⋆) exists.\nOur program is now to eliminate some regions that are of limited interest for a subsequent numerical study. We begin by the following theorem.\nConsider the competition system *([satcompaltII]). *If *G* decreases, then all solutions converge towards an equilibrium. [nocompcyclesG]** \nWe note that the Jacobian of ([satcompaltII]) has the off-diagonal elements\n \n$$-f\\_2(x)<0\\:{\\rm{and}}\\: zG^\\prime(x)<0$$ \nmeaning that the system is competitive, ie its solutions are backwards monotone. All solutions of a backwards monotone two-dimensional system converge towards an equilibrium, see e. g. Smith (1995).\nThe next theorem excludes cycles in the competition plane for low saturation levels. In particular, the unsaturated competition system ([satcompaltII]) with *b*2 = 0 cannot possess competition cycles. This excludes a large parameter region that becomes of limited interest for the subsequent numerical part in the end of this paper.\nConsider the competition system *([satcompaltII]). *If *F*2 decreases, then all solutions converge towards an equilibrium. [nocompcycles]** \nConsider ([satcompaltII]) and the Dulac function $B(x,z)=\\frac{1}{zf\\_2(x)}$. The divergence of the system\n \n$$\\begin{aligned}\n \\dot{x} &=& \\frac{F\\_2(x)}{z}-1 \\\\\n \\dot{z} &=& \\frac{G(x)-z}{f\\_2(x)}\\end{aligned}$$ \ncan now be estimated as\n \n$$\\frac{F\\_2^\\prime(x)}{z}-\\frac{1}{f\\_2(x)}<-\\frac{1}{f\\_2(x)}<0.$$ \nBy Dulac’s theorem (Brauer and Castillo-Chávez (2001) and Ye (1986)), ([satcompaltII]) cannot possess periodic orbits. By the Poincaré-Bendixson’s theorem (Hirsch et al. (2013) and Wiggins (2003)), possible limit sets are equilibria or orbits connecting equilibria. The origin is not a saddle and cannot belong to a limit set consisting of equilibria and orbits connecting them. If a saddle exist in the interior of the positive quadrant, then *G* must decrease between *x*− and *x*+. In this case, *G* decreases on the unit interval and the solutions converge towards an equilibrium by Theorem [nocompcyclesG]. Assume therefore, that the saddles of ([satcompaltII]) are located either at the *x*-axis or at the *z*-axis. In both cases, either its stable or unstable manifolds are located along the axes. Possible saddle-connections violate therefore, uniqueness of solutions. Consequently, all solutions converge towards an equilibrium.\n(350,350)(0,0) (0,0) (89,155)(a) (5,285)*z* (162,165)*x* (50,219)(0,1)10 (100,207)(0,1)10 (150,191)(0,1)10 (58,246)(-1,0)10 (108,265)(-1,0)10 (156,284)(-1,0)10 (140,240)(-1,1)10 (70,280)(-1,-1)10 (50,195)(1,1)10 (295,155)(b) (212,285)*z* (369,165)*x* (290,200)(1,0)10 (247,220)(1,0)10 (240,281)(0,-1)10 (280,257)(0,-1)10 (320,228)(0,-1)10 (360,198)(0,-1)10 (250,195)(1,1)10 (250,245)(1,-1)10 (270,295)(-1,-1)10 (330,275)(-1,-1)10 (365,250)(-1,-1)10 (89,-10)(c) (5,116)*z* (162,-4)*x* (160,60)(-1,0)10 (130,48)(0,1)10 (35,110)(0,-1)5 (44,102)(1,0)5 (18,102)(0,1)5 (50,50)(1,1)10 (150,110)(-1,-1)10 (100,120)(-1,-1)10 (295,-10)(d) (212,100)*z* (270,5)*y* (305,7)*x*\n[multipleattractors]\nWe eliminate the last large region of limited interest by the following theorem.\nConsider the competition system *([satcompaltII]). *If no competition equilibria exist, then all solutions converge towards an equilibrium.** \nSince, *F*2 and *G* have no intersections, we have either that *G* > *F*2, or *F*2 > *G*. In each of the cases, and all positive solutions converge towards (0, *c*) or towards (1, 0), respectively, by phase-plane arguments, see Figure [multipleattractors](a)-(b).\nWe conclude this section by proving that Theorem [nocompcyclesG] implies the presence of parameter values giving rise to multiple attractors. Indeed, the parameter values *c* = .8, *k* = .75, *a*2 = .4, and *b*2 = 20 gives rise to multiple attractors for the competition system ([satcompaltII]), see Figure [multipleattractors](c). The equilibria (0, *c*) and (*x*+, *F*(*x*+)) are both locally attracting and no cycles exist by Theorem [nocompcyclesG]. This situation persists for ([chemostatmixosatscaledaltII]) if *x*⋆ > *x*+ is selected. The situation is depicted in Figure [multipleattractors](d) for *c* = .8, *k* = .75, *x*⋆ = .4, *a*1 = 8.5, *b*1 = 50, *a*2 = .4, and *b*2 = 20. The equilibria (0, 0, *c*) and (*x*+, 0, *F*2(*x*+)) are still both locally stable.\nThe stability of the coexistence equilibrium\n============================================\nWe proceed to the interior equilibrium (*x*⋆, *y*⋆, *z*⋆). Its Jacobian matrix is given by\n \n$$J(x\\_\\star,y\\_\\star,z\\_\\star)=\\left(\\begin{array}{ccc}\n\\begin{array}{c} f\\_1(x\\_\\star)F\\_1^\\prime(x\\_\\star)\\\\-z\\_\\star\\left(f\\_2^\\prime(x\\_\\star)-f\\_1^\\prime(x\\_\\star)\\frac{f\\_2(x\\_\\star)}{f\\_1(x\\_\\star)}\\right)\\end{array} & -f\\_1(x\\_\\star) & -f\\_2(x\\_\\star)\\\\\ny\\_\\star\\psi^\\prime(x\\_\\star) & 0 & 0\\\\\nz\\_\\star G^\\prime(x\\_\\star) & -z\\_\\star & -z\\_\\star\n\\end{array}\\right).$$ \nwith\n \n$$y\\_\\star=\\frac{f\\_2(x\\_\\star)(F\\_2(x\\_\\star)-G(x\\_\\star))}{f\\_1(x\\_\\star)-f\\_2(x\\_\\star)},\\:\\:z\\_\\star=\\frac{f\\_1(x\\_\\star)(G(x\\_\\star)-F\\_1(x\\_\\star))}{f\\_1(x\\_\\star)-f\\_2(x\\_\\star)}.$$ \nIn order to continue formulating stability conditions for this equilibrium, we first give the Routh (1877)-Hurwitz (1895) conditions (see also May (1974) and Wiggins (2003)) in matrix form. The matrixes *A**i**i* are the submatrixes obtained when the *i*th row and column are deleted from *A*. Sometimes they are referred to as principal submatrices, see e. g. Strang (2006).\nThe eigenvalues of the 3 × 3 matrix *A* have negative real parts if and only if *(i) *${\\rm{Tr}{A}}<0$, *(ii) *${\\rm{det}}A<0$, and**** \n \n$${\\rm{(iii)}}\\:\\:{\\rm{Tr}{A}}({\\rm{det}}A\\_{11}+{\\rm{det}}A\\_{22}+{\\rm{det}}A\\_{33})-{\\rm{det}}A<0.$$ \nIf some of the eigenvalues have zero real part, then equality must hold in either *(ii) *or *(iii). *In particular, if some of the eigenvalues are zero, then equality holds in *(ii).*[RouthHurwitzMatrixForm]****** \nIdentification of the coefficients of the characteristic polynomial gives the matrix form of the criterion. Since det*A* is the product of the eigenvalues, a zero eigenvalue gives equality in (ii). The next possibility for zero real parts is a real eigenvalue *λ*1 = *λ* < 0 and complex conjugate pair *λ*2, 3 = ± *ω**i*. The relation between roots and coefficients implies that\n \n$$\\begin{aligned}\n {\\rm{Tr}}A &=& \\lambda\\_1+\\lambda\\_2+\\lambda\\_3=\\lambda<0\\\\\n {\\rm{det}}A\\_{11}+{\\rm{det}}A\\_{22}+{\\rm{det}}A\\_{33} &=& \\lambda\\_1\\lambda\\_2+\\lambda\\_2\\lambda\\_3+\\lambda\\_1\\lambda\\_3 \\\\\n &=&\\lambda\\omega+\\omega^2-\\lambda\\omega=\\omega^2>0\\\\\n {\\rm{det}}A &=& \\lambda\\_1\\lambda\\_2\\lambda\\_3=\\lambda\\omega^2<0\\end{aligned}$$ \nCriterion (iii) now becomes\n \n$${\\rm{Tr}{A}}({\\rm{det}}A\\_{11}+{\\rm{det}}A\\_{22}+{\\rm{det}}A\\_{33})-{\\rm{det}}A=\\lambda\\omega^2-\\lambda\\omega^2=0.$$ \nA real 3 × 3-matrix *A* cannot lose its stability by passing through an equality in *(i) *in Lemma *[RouthHurwitzMatrixForm] *above. [stability-loss]**** \nIf det*A* < 0 and ${\\rm{Tr}}A=0$, then (iii) becomes\n \n$${\\rm{Tr}{A}}({\\rm{det}}A\\_{11}+{\\rm{det}}A\\_{22}+{\\rm{det}}A\\_{33})-{\\rm{det}}A=-\\det A>0$$ \ncontradicting the possibility for stability losses at ${\\rm{Tr}}A=0$.\n*Even if stability losses at ${\\rm{Tr}}A=0$ are excluded by the above corollary, criterion (i) is still a necessary criterion for stability in Lemma [RouthHurwitzMatrixForm]. Compare e. g. with the case *λ*1 = − 1, $\\lambda\\_2=\\frac{1}{2}$, $\\lambda\\_3=\\frac{3}{2}$. We get* \n \n$$\\begin{aligned}\n {\\rm{Tr}}A &=& \\lambda\\_1+\\lambda\\_2+\\lambda\\_3=-1+\\frac{1}{2}+\\frac{3}{2}=1>0 \\\\\n \\det A &=& \\lambda\\_1\\lambda\\_2\\lambda\\_3=-1\\cdot\\frac{1}{2}\\cdot\\frac{3}{2}=-\\frac{3}{4}<0 \\\\\n \\det A\\_{11}+{\\rm{det}}A\\_{22}+{\\rm{det}}A\\_{33} &=& \\lambda\\_1\\lambda\\_2+\\lambda\\_1\\lambda\\_3+\\lambda\\_2\\lambda\\_3= -\\frac{1}{2}-\\frac{3}{2}+\\frac{3}{4}=-\\frac{5}{4}\\end{aligned}$$ \nmeaning that\n \n$${\\rm{Tr}{A}}({\\rm{det}}A\\_{11}+{\\rm{det}}A\\_{22}+{\\rm{det}}A\\_{33})-{\\rm{det}}A=\n1\\cdot\\left(-\\frac{5}{4}\\right)-\\left(-\\frac{3}{4}\\right)=-\\frac{1}{2}<0.$$ \nWe see that the associated Jacobian matrix is unstable and that criteria (ii) and (iii) in Theorem [RouthHurwitzMatrixForm] are satisfied. \nWe now formulate the stability criteria of the interior fixed point in terms of criteria needed in addition to the global stability condition of the the autotroph-herbivore equilibrium in the *x**y*-plane.\nThe coexistence equilibrium (*x*⋆, *y*⋆, *z*⋆) is locally stable if\n1. [*(i)*] *F*1′(*x*⋆) < 0**\n2. [*(ii)*] *G*′(*x*⋆) > 0**\n3. [*(iii)*] *f*1(*x*⋆)*f*2′(*x*⋆) − *f*1′(*x*⋆)*f*2(*x*⋆) > 0**\nThe coexistence equilibrium loses its stability through a Hopf-bifurcation, if it exists. [complexity-stability-relation]\n*The requirement that the herbivore-autotroph equilibrium is stable in the plane *z* = 0 is a visible in condition (i). The requirement comes from Lemma [globstab]. Criterion (ii) is easy to check, *a*2 > *k* is sufficient.* \n*Criterion (iii) is strongly related to the local condition ([localconditionB]) and the upper bound for *a*2 in ([upperbounda2]). However, they are not equivalent and the ratio between *b*1 and *b*2 determines the direction of the implication between Criterion (iii) and ([localconditionB]). Criterion (iii) can be modified into the equivalent parametric condition* \n \n*a*1*b*1 − *a*2*b*2 + *a*1*a*2(*b*1 − *b*2) + 2*b*1*b*2(*a*1 − *a*2)*x*⋆ − *b*1*b*2(*a*2*b*1 − *a*1*b*2)*x*⋆2 > 0. \nLemma [lemmaloccond] (iii) can now be used to obtain the estimate\n \n$$\\begin{aligned}\na\\_1b\\_1-a\\_2b\\_2+a\\_1a\\_2(b\\_1-b\\_2)+2b\\_1b\\_2(a\\_1-a\\_2)x\\_\\star-b\\_1b\\_2(a\\_2b\\_1-a\\_1b\\_2)x\\_\\star^2&>&\\label{par\\_ineq}\\\\\na\\_1b\\_1-a\\_2b\\_2+a\\_1a\\_2(b\\_1-b\\_2)+b\\_1b\\_2(a\\_1-a\\_2)x\\_\\star&>&0.\\nonumber\\end{aligned}$$ \nWe see that all terms in the last expression are positive for *b*1 > *b*2. Indeed, ([upperbounda2]) gives *a*2 < *a*1 in this case. Therefore, ([localconditionB]) ensures Criterion (iii) in Theorem [complexity-stability-relation] for *b*1 > *b*2. If *b*1 < *b*2, we have *a*2 < *a*1*b*2/*b*1. We now use this relation for deriving a stricter but very similar condition as ([upperbounda2]). The second inequality in ([parineq]) gives\n \n$$a\\_2<\\frac{a\\_1b\\_1+b\\_1b\\_2a\\_1x\\_\\star}{b\\_2+a\\_1(b\\_2-b\\_1)+b\\_1b\\_2x\\_\\star}=a\\_1\\frac{b\\_1}{b\\_2}\\frac{1+b\\_2x\\_\\star}{1+\\frac{a\\_1}{b\\_2}(b\\_2-b\\_1)+b\\_1x\\_\\star}<a\\_1\\frac{1+b\\_2x\\_\\star}{1+b\\_1x\\_\\star}.\n\\label{a2\\_estimate}$$ \nThe last inequality compares this estimate with ([upperbounda2]). Since our estimate increases with *x*⋆, worst cases can be computed with *x*⋆ = 0. In many cases such estimates makes criterion (iii) redundant in comparison to the essential criterion (i). The situation for *b*1 = 50, *b*2 = 55, *a*1 = 8.5 is illustrated in Figure [a2limits] \n(350,350)(0,0) (0,0) (365,0)*x*⋆ (123,150)(-1,0)10 (128,147)$x\\_\\star=\\frac{-a\\_1-1+\\sqrt{a\\_1^2+a\\_1+a\\_1b\\_1}}{b\\_1}$ (149,235)(-1,2)5 (150,230)$a\\_2=\\frac{a\\_1b\\_1+b\\_1b\\_2a\\_1x\\_\\star}{b\\_2+a\\_1(b\\_2-b\\_1)+b\\_1b\\_2x\\_\\star}$ (49,262)(0,1)17 (30,255)$a\\_2=a\\_1\\frac{1+b\\_2x\\_\\star}{1+b\\_1x\\_\\star}$ (0,290)*a*2\n[a2limits]\nWe proceed with the proof of Theorem [complexity-stability-relation].\nWe use Lemma [RouthHurwitzMatrixForm] to derive stability conditions for the interior equilibrium. Indeed, we have\n \n$${\\rm{Tr}}J(x\\_\\star,y\\_\\star,z\\_\\star)=f\\_1(x\\_\\star)F\\_1^\\prime(x\\_\\star)-z\\_\\star\\left(f\\_2^\\prime(x\\_\\star)-f\\_1^\\prime(x\\_\\star)\\frac{f\\_2(x\\_\\star)}{f\\_1(x\\_\\star)}\\right)-z\\_\\star<0.$$ \nNext\n \n$$\\begin{aligned}\n{\\rm{det}}J(x\\_\\star,y\\_\\star,z\\_\\star)&=&-y\\_\\star\\psi^\\prime(x\\_\\star)\\left|\\begin{array}{cc}\n-f\\_1(x\\_\\star) & -f\\_2(x\\_\\star)\\\\\n-z\\_\\star & -z\\_\\star\\end{array}\\right|\\\\\n&=&-y\\_\\star z\\_\\star\\psi^\\prime(x\\_\\star)\\left(f\\_1(x\\_\\star)-f\\_2(x\\_\\star)\\right)<0.\\end{aligned}$$ \nThe last inequality holds because our local condition ([localconditionB]). It follows that the coexistence equilibrium cannot have zero eigenvalues and can thus, lose its stability through Hopf-bifurcations only. In order to formulate condition (iii) of Lemma [RouthHurwitzMatrixForm], we first compute the principal subdeterminants\n \n$$\\begin{aligned}\n{\\rm{det}}J\\_{11}(x\\_\\star,y\\_\\star,z\\_\\star)&=&0,\\\\\n{\\rm{det}}J\\_{22}(x\\_\\star,y\\_\\star,z\\_\\star)&=&z\\_\\star\\biggl(f\\_2(x\\_\\star)G^\\prime(x\\_\\star)-f\\_1(x\\_\\star)F\\_1^\\prime(x\\_\\star)\\\\&&+z\\_\\star \\left(f\\_2^\\prime(x\\_\\star)-f\\_1^\\prime(x\\_\\star)\\frac{f\\_2(x\\_\\star)}{f\\_1(x\\_\\star)}\\right)\\biggr),\\\\\n{\\rm{det}}J\\_{33}(x\\_\\star,y\\_\\star,z\\_\\star)&=&y\\_\\star f\\_1(x\\_\\star)\\psi^\\prime(x\\_\\star),\\end{aligned}$$ \nand note that (iii) of Lemma [RouthHurwitzMatrixForm] is equivalent to\n \n$$\\begin{aligned}\n {\\rm{Tr}}J(x\\_\\star,y\\_\\star,z\\_\\star)\\det J\\_{22}(x\\_\\star,y\\_\\star,z\\_\\star)+&&\\\\+{\\rm{Tr}}J(x\\_\\star,y\\_\\star,z\\_\\star)\\det J\\_{33}(x\\_\\star,y\\_\\star,z\\_\\star)-\\det J(x\\_\\star,y\\_\\star,z\\_\\star)=&& \\\\\n {\\rm{Tr}}J(x\\_\\star,y\\_\\star,z\\_\\star)\\det J\\_{22}(x\\_\\star,y\\_\\star,z\\_\\star)+&&\\\\\n +f\\_1(x\\_\\star)F\\_1^\\prime(x\\_\\star)y\\_\\star f\\_1(x\\_\\star)\\psi^\\prime(x\\_\\star)-z\\_\\star y\\_\\star f\\_1(x\\_\\star)\\psi^\\prime(x\\_\\star)\\left(f\\_2^\\prime(x\\_\\star)-f\\_1^\\prime(x\\_\\star)\\frac{f\\_2(x\\_\\star)}{f\\_1(x\\_\\star)}\\right)&&\\\\-z\\_\\star y\\_\\star f\\_1(x\\_\\star)\\psi^\\prime(x\\_\\star)+y\\_\\star z\\_\\star\\psi^\\prime(x\\_\\star)\\left(f\\_1(x\\_\\star)-f\\_2(x\\_\\star)\\right)=&&\\\\\n \\underbrace{{\\rm{Tr}}J(x\\_\\star,y\\_\\star,z\\_\\star)}\\_{<0}\\det J\\_{22}(x\\_\\star,y\\_\\star,z\\_\\star)\\underbrace{-y\\_\\star z\\_\\star f\\_2(x\\_\\star)\\psi^\\prime(x\\_\\star)}\\_{<0}+&&\\\\\n \\underbrace{y\\_\\star F\\_1^\\prime(x\\_\\star)(f\\_1(x\\_\\star))^2\\psi^\\prime(x\\_\\star)}\\_{<0}\\underbrace{-y\\_\\star z\\_\\star \\psi^\\prime(x\\_\\star)(f\\_1(x\\_\\star)f\\_2^\\prime(x\\_\\star)-f\\_1^\\prime(x\\_\\star)f\\_2(x\\_\\star))}\\_{<0}.&&\\\\\\end{aligned}$$ \nThe whole quantity is therefore, negative when det*J*22(*x*⋆, *y*⋆, *z*⋆) > 0. Indeed, we have since *F*1′(*x*⋆) < 0\n \n$$\\begin{aligned}\n\\frac{f\\_1(x\\_\\star)}{z\\_\\star}\\det J\\_{22}(x\\_\\star,y\\_\\star,z\\_\\star)&>&f\\_1(x\\_\\star)f\\_2(x\\_\\star)G^\\prime(x\\_\\star)-(f\\_1(x\\_\\star))^2F\\_1^\\prime(x\\_\\star)\\\\&&+z\\_\\star \\left(f\\_1(x\\_\\star)f\\_2^\\prime(x\\_\\star)-f\\_1^\\prime(x\\_\\star)f\\_2(x\\_\\star)\\right)>0.\\end{aligned}$$ \nNumerical Results\n=================\nAccording to Theorem [complexity-stability-relation] existence of cycles in the predator-prey plane is an important criterion for an unstable coexistence equilibrium whenever such an equilibrium exist. In this section we fix the parameters as follows *c* = .2, *b*1 = 50, *b*2 = 55, *k* = .95, and *a*1 = 8.5. The parameter values selected here allows an interval of existence for the coexistence equilibrium for all parameter values of *x*⋆, 0 < *x*⋆ < 1 according to Lemma [exist-paramsat]. The upper bound for the efficiency of the mixotroph specified by ([upperbounda2]) (*a*2 ≈ *a*1 = 8.5) does not enter to the window of parameter values selected for this diagram, cf Figure [a2limits].\nThe major bifurcations of ([chemostatmixosatscaledaltII]) are now indicated in Figure [bifurcationdiagram]. Here we vary (i) the equilibrium value of *x*⋆ of the predator-prey equilibrium representing the interaction between the autotroph and the herbivore, and (ii) *a*2 that determines the competitive ability between the mixotroph and the autotroph for the limiting resources. Lemmas [globstab]-[uniquelimitcycle] gives the vertical line at *x*⋆ ≈ .2598. On the left-hand side of this line, the herbivore-autotroph system has a unique limit cycle (this corresponds to the areas **(a)**, **(c)**, **(d)**, **(e)**, **(g)**, **(h)**, **(i)**, **(k)**, **(l)**, **(m)**, **(p)**, **(q)**, **(r)**, **(t)**, **(u)**, **(v)**, **(w)**, and **(x)**) on the right hand side of this line (*x*⋆, *F*1(*x*⋆), 0) is globally stable (this corresponds to the regions **(b)**, **(f)**, **(j)**, **(n)**, **(o)**, **(s)**, and **(y)**).\nIn a similar manner, the horizontal lines correspond to dynamics in the competition plane. For low *a*2 (in our case *a*2 < *k*(1 − *c*) = .76), no competition equilibria exist and the carrying capacity equilibrium (1, 0, 0) is stable in the competition plane (regions **(t)**, **(u)**, **(v)**, **(w)**, **(x)**, and **(y)**). For moderate values of *a*2 (in our case.76 < *a*2 < 3.582) a unique competition equilibrium (*x*+, 0, *F*2(*x*+)) exists and is stable in the competition plane (regions **(k)**, **(l)**, **(m)**, **(n)**, **(o)**, **(p)**, **(q)**, **(r)**, and **(s)**). For still larger values of *a*2 (in our case 3.582 < *a*2 < 4) a unique competition equilibrium exists, but no equilibria are stable in the competition plane (regions **(g)**, **(h)**, **(i)**, **(j)**). For further increase in *a*2 (4 < *a*2\\* ≈ 5.1118) two competition equilibria exist and (0, 0, *c*) is stable in the competition plane (regions **(c)**, **(d)**, **(e)**, and **(f)**). For extreme values of *a*2 (*a*2 > *a*2\\*), no competition equilibria exist and the mixotroph carrying capacity (0, 0, *c*) is stable in the competition plane (regions **(a)** and **(b)**).\nThe blue and red curves corresponds to the existence boundaries of the coexistence equilibrium. This means that the coexistence equilibrium exist in regions **(d)**, **(g)**, **(h)**, **(k)**, **(m)**, **(n)**, **(p)**, **(t)**, and **(u)**. On the far right of the bifurcation diagram we confirm the conclusion of Lemma [limitlemma]. Below the blue curve, the coexistence equilibrium does not exist and at the blue curve it collides in a transcritical bifurcation with the predator-prey equilibrium. We see that the intersection of the blue curve with the horizontal line *a*2 = *k*(1 − *c*) = .76 in the far left of the diagram is in agreement with Lemma [lemma5]. Above the red curve it does not exist, either. The red curve has a maximum at *a*2 = *a*2\\*. At the left hand side of this maximum, the coexistence equilibrium collides in a transcritical bifurcation with (*x*−, 0, *F*2(*x*−)) and at the right-hand side of this maximum a similar collision with (*x*+, 0, *F*2(*x*+)) occurs. The yellow area in regions **(g)**, **(k)**, **(u)**, the cyan areas in regions **(p)** and **(t)**, the green areas in regions **(d)**, **(h)**, and possibly black areas on the boundaries of regions **(k)**, **(g)**, **(p)**, and **(t)** corresponds to the region where the coexistence equilibrium is unstable. In the yellow area in regions **(k)**, **(g)**, and **(u)** criterion (i) in Lemma [RouthHurwitzMatrixForm] is sufficient for deducing instability whereas criterion (iii) is needed in the black areas. We see that this occurs on the boundaries only precisely as predicted by Corollary [stability-loss].\nIt is of importance to elucidate the dynamical behavior of our system in regions that possess no stable equilibria. In region **(s)** and **(y)**, the predator-prey equilibrium (*x*⋆, *F*1(*x*⋆), 0) is stable. In regions **(m)** and **(n)**, the coexistence equilibrium (*x*⋆, *y*⋆, *z*⋆) is stable. In regions **(l)** and **(o)**, the competition equilibrium (*x*+, 0, *F*2(*x*+)) is stable. In regions **(a)**, **(b)**, **(c)**, **(d)**, **(e)**, and **(f)**, the mixotroph carrying capacity (0, 0, *c*) is stable. For *x*⋆ > .2598 region **(j)** is of interest. There, the predator-prey plane has a fixed point that is globally stable in the plane but unstable with respect to mixotrophic invasion. However, at least one stable limit cycle exist in the competition plane and it is not clear whether or not it (or these limit cycles) might be unstable with respect to hervibore invasion. If cycles are small, they spend a lot of time near the competition equilibrium that has a negative third eigenvalue and when they grow larger they spend a substantial amount of time in the vicinity of the saddle (0, 0, *c*) that is still more stable with respect to herbivore invasion. We conclude that the invasion of mixotrophs has a destabilizing impact on the dynamics in region **(j)** since the autotroph-herbivore dynamics results in global stability and invasion of mixotrophs leads to out-competition of the herbivores and resulting cyclic dynamics.\nWe selected the parameter values *x*⋆ = .26 and *a*2 = 3.9 for illustration of the destabilizing behavior in region **(j)**. The parameter value is marked with a blue circle in region **(j)** in our bifurcation diagram, Figure [bifurcationdiagram]. The result is visible in Figure [Simubild22](a). The simplex containing all solutions and the *x* = *x*⋆-plane are marked with dotted green lines. The isocline in the *x**y*-plane, *F*1 is marked with a dotted blue curve and the equilibrium (*x*⋆, *F*1(*x*⋆), 0) is marked with a blue \\* -mark. It is globally stable in the *x**y*-plane but unstable in the *z*-direction. The remaining equilibria (0, 0, 0) and (1, 0, 0) in the predator-prey plane are marked with red \\* -marks. In the *x**z*-plane, the isoclines *F*2 and *G* are marked with dotted blue curves and lines, respectively, and their intersection at (*x*+, 0, *F*2(*x*+)) is marked with a red \\* -mark. The equilibrium at (0, 0, *c*) is marked with a red ∘ -mark. The limit cycle in the *x**z*-plane is marked with a red curve. It approaches the point (0, 0, *c*) from its stable manifold along the *z*-axis and leaves it along its unstable manifold in the *x**z*-plane and spends a considerable amount of time in the vicinity of this saddle. The projection of the limit cycle onto the *x**y*-plane is marked with a cyan curve and the projection onto the *y**z*-plane is indicated in green. There is no need to visualize the projection onto the *x**z*-plane, but if this projection would not agree with the limit cycle itself, it would appear in magenta. The idea of selecting the value *x*⋆ = .26 very close to the bifurcation value *x*⋆ = .2598 was to increase the possibility for observing more complex oscillations involving all three species in this region. We did not find any oscillatory behavior of this type at this stage of our study.\nWe now turn over to the region *x*⋆ < .2598. In this case, the herbivore-autotroph coexistence is always described by a unique limit cycle. Therefore, we do not expect qualitatively destabilizing behavior after a mixotrophic invasion in the sense that we previously found in region **(j)**. The area corresponding to an unstable coexistence equilibrium is marked with yellow or black dots in regions **(g)**, **(k)**, and **(u)**, green dots in regions **(d)** and **(h)**, and cyan dots in regions **(p)** and **(t)**. If *a*2 > 4, (0, 0, *c*) is always stable (areas **(a)**, **(c)**, **(d)**, and **(e)**). No coexistence equilibrium nor competition equilibria exist in region **(a)**. In regions **(c)** and **(e)**, two competition equilibria exist but no coexistence equilibrium and in region **(d)**, two competition equilibria coexist together with an unstable coexistence equilibrium. We selected the parameter values *x*\\* = .05 and *a*2 = 4.5 to illustrate the situation in region **(d)** (marked by a black \\* in Figure [bifurcationdiagram]). The result is depicted in Figure [Simubild22](b). All positive solutions seem to converge towards the mixotroph carrying capacity (0, 0, *c*) that is denoted by a red ∘ -mark. The washout equilibrium (0, 0, 0), the carrying capacity equilibrium (1, 0, 0) and the two competition equilibria are denoted by red \\* -marks. The autotroph-herbivore equilibrium (*x*⋆, *F*1(*x*⋆), 0) is denoted by a blue \\* -mark. The coexistence equilibrium (*x*⋆, *y*⋆, *z*⋆) is denoted by a black \\* -mark together with its projections in the autotroph-herbivore plane (cyan), competition-plane (magenta), and in the *y**z*-plane (green). Finally, the autotroph-herbivore cycle in the predator-prey plane is plotted in blue and is unstable with respect to mixotropic invasion (cf. Table [bifurcationtable]).\nIn region **(h)**, we have again no stable equilibria so the resulting dynamics must be oscillatory. We also notice that neither (*x*+, 0, *F*2(*x*+)) nor (*x*⋆, *F*1(*x*⋆), 0) are stable in the competition plane and the autotroph-herbivore plane, respectively. We selected the parameter value *x*⋆ = .08 and *a*2 = 3.8. The parameter value is indicated with a ∘ -mark in region **(h)** in our bifurcation diagram Figure [bifurcationdiagram]. The resulting dynamics is depicted in Figure [Simubild22](c). We have an autotroph-herbivore cycle that is indicated in blue and is unstable with respect to mixotroph invasion. After invasion, the mixotroph outcompetes the herbivore and the resulting dynamics are the mixotroph-autotroph cycles indicated in red.\nAt the boundary between regions **(g)** and **(h)** there is a transcritical of cycles bifurcation so that the herbivore is not out-competed anymore after mixotrophic invasion in region **(g)**. We illustrate this situation with the parameter value *x*⋆ = .09 and *a*2 = 3.7. The selected parameter value is indicated with a blue \\* -mark in Figure [bifurcationdiagram]. The resulting dynamics is depicted in Figure [Simubild22](d). All the equilibria are still indicated and the unstable coexistence equilibrium is still indicated along with its projections on the various coordinate planes. The unstable competition cycle is indicated in red together with the autotroph-herbivore cycle in blue. The newly emerged coexistence cycle is indicated in black along with its projections in magenta, cyan, and green on the various coordinate planes.\nSimilarly, at the boundary between the regions **(g)** and **(k)**, a Hopf bifurcation occurs that makes the competition equilibrium (*x*+, 0, *F*2(*x*+)) stable in the competition plane in region **(k)**. We selected three parameter values from this region and the first one is *x*⋆ = .1 and *a*2 = 3.5 (marked by a top blue \\* in region **(k)** of Figure [bifurcationdiagram]). The situation is depicted in Figure [Simubild11](a): The coexistence cycles that emerged from the transcritical bifucation between regions **(g)** and **(h)** still exists, but the cycles in the competition plane have disappeared. The projections of the coexistence limit sets on the various coordinate planes are still depicted.\nThe next value of interest is *x*⋆ = .18, *a*2 = 2.8. The parameter value is indicated with a blue \\* -mark in region **(k)** in Figure [bifurcationdiagram]). We have depicted the dynamical situation in Figure [Simubild11](b). We observe that the coexistence equilibrium has just lost its stability in the Hopf bifurcation predicted by Theorem [complexity-stability-relation], that all equilibria are unstable and that the system possesses coexistence cycles.\nWe are of course interested in whether more complex oscillations could exist and illustrate our findings with the parameter value *x*⋆ = .15 and *a*2 = 3.0 in Figure [Simubild11](c) (marked by a red \\*-mark in region **(k)** of Figure [bifurcationdiagram]). We note that the oscillations observed are a mixture of the competition cycles that emerged from the transcritical bifurcation in the competition plane and the Hopf-bifurcation occurring for the coexistence equilibrium after that the Hopf-bifurcation has occurred for the autotroph-herbivore equilibrium according to Theorem [complexity-stability-relation]. The stable and unstable manifold of two equilibria are essential for these oscillations and we start the description of the oscillation from the competition equilibrium (*x*+, 0, *F*2(*x*+)). It is stable in the competition plane but unstable with respect to herbivore invasion. The oscillation approaches this equilibrium close to the stable manifold (the competition plane) of this point and it leaves it along the its unstable manifold. Next the mixotroph carrying capacity has the *y**z*-plane as its stable manifold and an unstable manifold that intersects the two-dimensional stable manifold of the competition equilibrium.\nThe oscillations might come into the vicinity of the coexistence equilibrium before they are captured by the stable manifold of the mixotroph equilibrium and such an interaction is possible to observe in Figure [Simubild11](c). This interaction is, however, not essential. We now remove the coexistence equilibrium by selecting the parameter values *x*⋆ = .15 and *a*2 = 2.3 that is indicated by a red \\* -mark in region **(q)** in Figure [bifurcationdiagram]. We see that a very similar type of oscillations persist despite that the coexistence equilibrium is removed (Figure [Simubild11](d)). The two-dimensional stable manifolds of (*x*+, 0, *F*2(*x*+)) and (0, 0, *c*) interact with their one-dimensional unstable manifolds. We have so far no reliable indications of chaotic behavior for this system. All computations of Lyapunov exponents have so far resulted in either negative or values close to zero (from cyclic/quasiperiodic oscillations). Moreover, the possible route to chaos remains so far unclear in our simulations. The oscillations that occur without the presence of an interior fixed point in region **(q)** and **(w)** collapse through a transcritical of cycles bifurcation into autotroph-herbivore cycles on the boundary between the regions **(q)** and **(w)** and the cyan regions **(r)** and **(x)** that correspond to autotroph-herbivore cycles.\nA similar transcritical of cycles bifurcations might occur at the boundaries between the regions **(v)** and **(w)**, **(t)** and **(u)**, and **(k)** and **(p)**, respectively. However, at these boundaries we have just vague numerical evidence for such transitions without theoretical support. The observed transition might be caused by numerical instabilities or can be of more complex nature than just a transcritical of cycles bifurcation. On the other hand, more theoretical evidence for transcritical bifurcations along the boundaries between **(q)** and **(r)** and **(w)** and **(x)** exists, the transcritical boundary meets e. g. the transcritical bifurcation of the coexistence equilibrium between the areas **(n)** and **(s)**. We select still a number of parameter values for separate study in Figure [Simubild3]. We start with the parameter value *x*⋆ = .02, *a*2 = .6 in region **(u)** in Figure [Simubild3](a). Here an unstable coexistence equilibrium coexists with a coexistence cycle. Figure [Simubild3](b) demonstrated the parameter value *x*⋆ = .05, *a*2 = .5 in Region **(w)**. The coexistence cycle exists, but no coexistence equilibrium. Figure [Simubild3](c) demonstrates with the parameter value *x*⋆ = .2, *a*2 = .5 in region **(x)** how the coexistence cycle has collapsed into the herbivore-autotroph cycles in the predator-prey plane. No competition equilibrium exists in this region. Figure [Simubild3](d) demonstrates a similar transcritical collapse in region **(r)**, where a competition equilibrium exists.\nEnding up at the expectations of the title of this paper, first we have a typical qualitatively destabilizing region at region **(j)** in Figure [bifurcationdiagram] and here fixed point dynamics is replaced by limit cycle dynamics as mixotrophs invade. Second, we have a typical qualitatively stabilizing region at region **(m)** in Figure [bifurcationdiagram]. Here limit cycle dynamics is replaced by fixed point dynamics as mixotrophs invade. Third, we have typical region where mixotrophic invasion causes multiple attractors and initial value dependent behavior, see Figure [multipleattractors](d).\n(350,350)(0,0) (-56,-30) (365,0)*x*⋆ (-7,285)*a*2 (-13,245)**(c)** (4,248)(1,0)5 (50,285)**(a)** (250,285)**(b)** (60,235)**(e)** (250,235)**(f)** (70,197)**(i)** (250,197)**(j)** (85,170)**(l)** (250,130)**(o)** (86,133)**(m)** (110,110)**(n)** (50,80)**(q)** (97,75)**(r)** (150,60)**(s)** (60,22)**(x)** (250,25)**(y)** (18,220)**(d)** (11,194)**(g)** (40,201)**(h)** (18,160)**(k)** (35,32)**(w)** (25,59)**(u)** (25,57)(-1,-1)10 (-15.5,70)**(p)** (4,73)(1,0)5 (3,22)**(t)** (16,-7)**(v)** (22,3)(-1,2)5\n[bifurcationdiagram]\n[bifurcationtable]\n(350,350)(0,0) (0,0) (80,150)(a) (280,150)(b) (80,0)(c) (280,0)(d) (35,240)*z* (35,90)*z* (210,230)*z* (210,90)*z* (80,170)*y* (280,165)*y* (110,25)*y* (285,22)*y* (115,165)*x* (320,170)*x* (145,30)*x* (320,30)*x*\n[Simubild22]\n(350,350)(0,0) (0,0) (80,150)(a) (280,150)(b) (80,0)(c) (280,0)(d) (38,230)*z* (38,88)*z* (210,230)*z* (210,88)*z* (112,164)*y* (285,165)*y* (110,25)*y* (288,25)*y* (145,168)*x* (321,168)*x* (145,30)*x* (315,30)*x* (105,246)(-1,0)5\n[Simubild11]\n(350,350)(0,0) (0,0) (80,150)(a) (280,150)(b) (80,0)(c) (280,0)(d) (35,245)*z* (35,93)*z* (210,230)*z* (210,93)*z* (80,170)*y* (280,165)*y* (110,25)*y* (280,25)*y* (115,165)*x* (315,170)*x* (148,30)*x* (318,30)*x*\n[Simubild3]\nDiscussion\n==========\nIn the present study, we have examined the role of invading mixotrophs on the stability of an autotroph-herbivore system. Specifically, we have chosen a spectrum of mixotrophic organisms by varying their competitive abilities. Mixotrophs are assumed as weak competitors compared to the pure autotrophs and the pure herbivores (Litchman et al. (2007); Zubkov and Tarran (2008)). We are able to formulate such a condition precisely for the mixotrophs in relation to the autotrophs, but in relation to the herbivores we end up with a too strong global condition and a too weak local condition. However, we did not encounter any obvious problem with our analysis as we used the local condition. We have analyzed the system with a limiting case of the chemostat instead of doing logistic approximations since the limiting case still preserves the complete information of the system exposed to mixotrophic invasion (Lindström and Cheng (2015)) similar to the logistic approximations (Kuang and Freedman (1988)).\nTwo main subsystems of our system, a competition system (autotroph-mixotroph) and a predator-prey system (autotroph-herbivore), describe the dynamical properties in different invariant coordinate planes. The predator-prey system has a *ω*-limit set that consists of either a unique globally stable equilibrium or a unique limit cycle. As the cycles grow larger, the system becomes increasingly sensitive to perturbation, cf. Rosenzweig (1971). The global analysis is not that complete for the competition system. It has at most five equilibria and both situations where all existing equilibria are unstable and two of them remain locally stable exist. As a result, this subsystem possesses both initial value dependent behavior and limit cycles.\nOur analysis also reveals the existence of a possible additional unstable equilibrium in the complete three-species model. This leads to a situation without any stable equilibrium and opens the possibility for complicated oscillations. However, we were unable to find any evidence for chaos at this stage but found that the oscillations observed typically spend a lot of time close to some of the saddle-type equilibria. Checking the existence of chaos needs a lot of effort in computing Lyapunov exponents in a reliable way and is thus out of the scope of this paper (Guckenheimer and Holmes (1983), and Wolf et al. (1985)).\nWe have analyzed the system by using two major bifucation parameters, one that controls the competitive ability of the mixotroph compared to the autotroph for limiting resources (*a*2-the mixotrophic link parameter), and the other one that controls the autotroph-herbivore interaction (*x*⋆-the equilibrium of the autotroph when interacting with herbivores). For small *x*⋆, the coexistence between autotrophs and herbivores is described by unique limit cycles, whereas as *x*⋆ becomes large, a globally stable fixed point describes the situation. On the other hand, when *a*2 remains small, autotroph appears as a better competitor for nutrient than the mixotroph and outcompetes the mixotroph. As *a*2 increases, first the mixotroph and the autotroph coexist in a stable equilibrium, and then a Hopf-bifurcation occurs resulting in autotroph-mixotroph cycles. Further increase in *a*2 results in the existence of multiple competition equilibria together with a stable equilibrium at the mixotrophic carrying capacity. For very large values of *a*2, mixotroph outcompetes the autotroph and the mixotrophic carrying capacity equilibrium (0, 0, c) becomes stable.\nThe most interesting result of this study is that invading mixotrophs may both stabilize and destabilize existing autotroph-herbivore dynamics. We have provided substantial analytical and numerical evidence for our claim. Globally stable two-species interaction may turn into competition cycles after the grazer has been outcompeted and the cyclic dynamics might be converted into either fixed points or oscillatory dynamics that may involve all, just two, or only one species. The appearance of multiple attractors is not only something that can occur for the autotroph-mixotroph relation, but a behavior that persists even in the complete three-species system. Note that, in a similar type of system but with relatively simpler predation term, Jost et al. (2004) found only stabilizing role of mixotrophy. Other modelling studies support the claim of stabilizing nature of mixotrophy (Hammer and Pitchford, 2005; Mitra et al. 2014). There is also evidence that the invasion of mixotrophs destabilizes the system by importing oscillations (Crane and Grover 2010). However, both stabilizing and destabilizing roles of invading mixotrophs in a single model has been found for the first time in the present study.\nThe model presented here was used to show only the fundamental mechanisms and must not be confused with quantitative simulations. The main purpose was to investigate the role of mixotrophs in a simple predator-prey system in order to understand the conditions under which invasion of mixotrophs is possible and how it affects the stability of the system and leads to more complicated dynamics. The occurrence of multi-stability and both stabilization and destabilization effects associated with the invasive mixotrophs indicate that nutrient limited lower latitude areas and higher latitude summer conditions which are ideal for mixotrophs to dominate, the community composition might be very sensitive to any kind of perturbations, i.e. a small change in the environmental condition can result into a huge change in the community composition and finally ecosystem functions. However, to quantify the real effects we need to run our system in a real physical setup using more realistic parameter values.\n#### Acknowledgements\nSome of the questions treated in this paper emanated from a discussion between Prof. em. Edna Granéli and Torsten Lindström that was connected to a joint research project proposal 2004, but national Swedish funding was never admitted. Consequently, it took some time to arrange possibilities for completing this study. Torsten Lindström thanks the Department of Mathematics at University of Helsinki for a stimulating discussion after a seminar lecture November 8, 2017, based on an earlier version of this paper. Subhendu Chakraborty was supported by the H. C. Ørsted COFUND postdoc fellowship.\n"},"data_id":{"kind":"string","value":"arxiv_0002849"}}},{"rowIdx":698,"cells":{"content":{"kind":"string","value":"Layer-by-layer assembly of multilayer optical lattices: Application to displaced dice lattice\n=============================================================================================\nWe propose methods for synthesizing multilayer optical lattices of cold atoms in a layer-by-layer manner, to unlock the potential of optical lattices in simulating the fascinating physics of multilayer systems. Central to the approach is to compress the beam profile of a red-detuned Gaussian laser beam from disklike to a thin line by a telescope with two cylindrical lenses. A highly tunable multilayer optical lattice is obtained by passing the compressed Gaussian beam through an optical device consisting of beam splitters, mirrors, and glass plates. We illustrate the proposal with the displaced dice lattice, which is a trilayer lattice that maps to the dice lattice when projected to the same layer. Both the dice model and its interesting variants may be realized. For a model of fermionic cold atoms, featuring an isolated flat band between two dispersive bands, we find valley-contrasting interband transitions involving the flat band.\nIntroduction\n============\nOptical lattices of cold atoms have become a fertile ground to simulate novel quantum systems and to explore fundamental open problems in physics. In the past 30 years, a great variety of optical lattices, ranging from zero dimensional quantum dots to three dimensional (3D) lattices, have been studied both theoretically and experimentally. Motivated partly by the keen interest from the condensed matter community, the two-dimensional (2D) optical lattices have attracted special attention. Besides purely 2D materials containing a single atomic layer, such as graphene and its analogue, the majority of 2D systems studied in condensed matter physics consist of several atomic layers and are actually multilayer (or, few-layer) quasi-2D (q-2D) materials. They host properties determined not only by the constituent layers but also by their stacking arrangement, thereby breeding a field rich in novel phases and striking phenomena. A notable example is the bilayer graphene, which opens a tunable band gap under a vertical electric field, and turns into a Mott insulator or unconventional superconductor after twisting the two layers by a magic angle. With the superior controllability of the optical lattices, it seems promising to fabricate multilayer optical lattices with cold atoms to explore physics far beyond the reach of present-day solid state physics. In reality, however, multilayer optical lattices are seldom studied.\nMost existing studies of multilayer optical lattices rely on restricting the number of layers of a 3D optical lattice in one direction (e.g., along the *z* axis). The geometry of individual layers and the stacking of consecutive layers are both fixed by the 3D optical dipole potential and not free to tune. This approach, extracting a q-2D multilayer lattice from a 3D lattice, is analogous to the mechanical exfoliation in condensed matter physics. Another powerful technique in condensed matter physics is the molecular-beam epitaxy (MBE), in which a multilayer lattice is synthesized layer by layer. The MBE does not require the existence of a bulk material consisting of weakly coupled 2D or q-2D units and is suitable for producing multilayers of both single crystals and complicated heterostructures.\nThe above comparison inspires us to explore an MBE-type layer-by-layer scheme of fabricating multilayer optical lattices. We illustrate in what follows that the layer-by-layer scheme for a multilayer optical lattice can be implemented through a combination of three sets of optical elements: The *beam shapers* to compress the beam profile (i.e., cross section) of the input Gaussian beam from disklike to a thin line, the *beam splitters* to split a single laser beam into several beams, and the *path locators* to guide different laser beams to the spot of the optical lattice along appropriate paths. The beam splitters and path locators also produce relative phase shifts and play the role of *phase shifters*. Fig. 1(a) is a schematic plot of the proposal.\nThe remaining part of the paper is organized as follows. In Sec.II, we explain the essential factors to materialize this proposal. Then, in Sec.III, we expound the proposal in more details in terms of a trilayer optical lattice, the displaced dice (or, T3) lattice. In Sec. IV we continue to show that the dice model and its close variants may be simulated by fermionic cold atoms in the displaced dice lattice. In a symmetrically biased dice model, which has an isolated flat band in between two dispersive bands, we find valley-contrasting interband transitions between the flat band and one dispersive band. The main results of the study are summarized in Sec. V. We also discuss in this section possible generalizations of the scheme to other novel multilayer lattices, such as the twisted multilayers. In order to focus on elucidating the physical pictures in the main text, we have put some relevant mathematical details to the Appendices (A through F).\ngeneral idea of the layer-by-layer scheme\n=========================================\n[fig1]\n \n \n \n \n \n \nThe key prerequisite of realizing the layer-by-layer assembling of multilayer optical lattices, in analogy to the MBE for condensed matter systems, is to generate individual purely 2D optical lattices. Usually, a 2D optical lattice is formed either by enhancing the depth of a 3D optical potential (or, by adding an additional trapping potential) along the third direction to pinch off the interlayer couplings and focus on a single layer of the 3D lattice, or by strongly constraining the atoms only in the in-plane directions and obtain a 2D lattice of tube-like quasi-1D units which extend much longer than the in-plane lattice constant. These approaches to a 2D optical lattice are clearly inapplicable to the layer-by-layer scheme for the multilayer optical lattices.\nTherefore, the beam shaper in Fig. 1(b), which compresses the input Gaussian laser beam to the output beam with a straight filamentary beam profile, plays a central role in realizing *truly 2D* optical lattices of point-like lattice sites and in implementing the layer-by-layer assembling of multilayer optical lattices. The beam shaper in Fig.1(b) is a cylindrical Keplerian telescope consisting of a pair of cylindrical thin lenses. A cylindrical Galilean telescope applies equally well. We assume an incident laser beam propagating along the telescope’s optical axis, which is parallel to the *x* axis of the coordinates defined in Fig. 1(b). This beam shaper does not change the *y*-axis distribution of the input laser beam, but compresses its *z*-axis distribution. Defining the focal lengths of the left (L1) and right (L2) cylindrical lenses as *f*1 and *f*2 (*f*1 > *f*2), the waist size for the *z*-axis amplitude distribution is reduced from the original value *ρ*0 to *ρ*0*z* = *ρ*0*f*2/*f*1 (see Appendix A for relevant formulas). *f*1/*f*2 is the telescope’s magnification along *z* direction. The larger the magnification is, the thinner the slab-like outgoing laser beam.\nThe waist radius *ρ*0 of the input Gaussian beam is in the order of 1 mm. To get the 2D limit, while it appears necessary to reduce the thickness of the slab-like output beam to approximately 1 *μ*m, comparable to the in-plane lattice parameters, a thickness of about 10 *μ*m is actually small enough for sufficiently deep optical potentials (see Appendix B). This amounts to reducing *ρ*0*z* of the output laser beam to about 5 *μ*m. When line beams of this thickness are used to make a realistically deep 2D optical lattice, the energy level spacing between the ground state and the first excited state of a single potential well can be made larger than the recoil energy *E**R* = *h*2/(2*m**λ*2) (see Appendix B). In the low-temperature limit and without strong driving, the resulting optical lattice can safely be considered as a truly 2D lattice. Compressing an input Gaussian beam with *ρ*0 ≈ 1 mm to *ρ*0*z* ≈ 5 *μ*m requires *f*1/*f*2 ≈ 200. Besides using a single cylindrical Keplerian telescope with *f*1/*f*2 ≈ 200, we may also cascade two cylindrical Keplerian (or, Galilean) telescopes with magnifications around 10 to 20. In the cascaded configuration, the whole device is much more compact.\nAfter accomplishing the above step, the multilayer optical lattices may in principle be realized by passing the line beam through a proper combination of standard optical elements, such as beam splitters, glass plates, and mirrors. In Fig. 1(a), these are listed separately as Beam Splitter and Path Locator. They will usually be intertwined with each other in actual devices. According to the geometries of various layers and the number of layers, we split the line beam into several beams with equal or unequal intensities, by using beam splitters with suitable splitting ratios. For example, a triangular optical lattice may be formed by three traveling wave laser beams. To synthesize a q-2D optical lattice containing *N* triangular layers, we split the single line beam into 3*N* beams. Guiding the various laser beams to a designated interference region, the multilayer optical lattice is created therein. The distances between consecutive layers may be controlled by the path locators listed in Fig. 1(c).\nTo ensure the stability and accuracy of the whole optical device, it is advantageous to integrate the optical components therein within a single platform that allows precise control over both the positions and the orientations of all components. In addition, it is preferable to have all the optical surfaces in the device superpolished. Although technically demanding, there is no fundamental restriction preventing the realization of the proposal.\nLoading precooled cold atoms to the multilayer optical lattice may follow either of two procedures. Firstly, we may produce the different layers of the optical lattice in sequence, and also load precooled cold atoms to these layers in a sequential manner, fully analogous to the MBE. Secondly, we may produce all the layers of the optical lattice at the same time but distant enough from each other. Then we load cold atoms to the detached layers and move them towards each other along the vertical direction. In both approaches, the distances between neighboring layers and their relative orientations should be controlled precisely in concord. The atoms loaded to different layers could be the same or different, fermionic or bosonic. When the same species of cold atoms are loaded to all layers, we may directly bring all the layers in place and then load the cold atoms to them simultaneously.\nIt is now imperative to demonstrate through a concrete example the feasibility and fine details of the proposal and the promised tunability of the model parameters for the target system. For this purpose, we consider the displaced dice (or, T3) lattice. The dice lattice and its variants (e.g., the *α*-T3 lattice) have been extensively studied theoretically in both the cold atoms community and condensed matter physics. While the dice lattice has been fabricated in superconducting networks and normal metal networks, the dice model may not be realized in them. Hopefully, the present proposal will facilitate the experimental study of the pure and the generalized dice models. The following analysis for the displaced dice lattice may easily be adjusted to apply to other multilayer optical lattices.\nDisplaced dice lattice: The optical lattice\n===========================================\nAs shown in Fig.2, the (displaced) dice lattice consists of three sublattices (A, B, and C) which separately form a triangular lattice. In the limit that the dice model or its close variants (e.g., the *α*-T3 model) are applicable, only the bonds connecting the atoms on the A and C sublattices (the rim sublattices) and the nearest-neighboring (NN) atoms of the B sublattice (the hub sublattice) are retained. Here, we propose to stagger the three sublattices along the direction perpendicular to the layer planes, so that the A and C layers are separately above and below the B layer, at the same or different distances from the B layer.\n[fig2] \nEach triangular layer may be formed by superimposing three traveling wave laser beams. Therefore, we need nine laser beams to construct the displaced dice lattice. We denote the wave vectors of the three laser beams for the *i*-th layer as **k***i*(*j*) (*i*, *j* = 1, 2, 3). Since the three triangular layers of the displaced dice lattice have the same primitive lattice vectors **a**1 and **a**2, we choose $\\mathbf{k}^{(1)}\\_{i}=k(\\sqrt{3}/2,-1/2,0)=\\mathbf{k}\\_{1}$, **k***i*(2) = *k*(0, 1, 0) = **k**2, and $\\mathbf{k}^{(3)}\\_{i}=k(-\\sqrt{3}/2,-1/2,0)=\\mathbf{k}\\_{3}$ for *i* = 1, 2, 3. For a laser of wavelength *λ*, the wave number *k* = 2*π*/*λ*. The electric fields of the nine laser beams are written as\n \n$$\\mathbf{E}\\_{i}^{(j)}(\\mathbf{r},t)=E\\_{i}\\hat{\\boldsymbol{\\epsilon}}\\_{j}\\cos(\\mathbf{k}\\_{j}\\cdot\\mathbf{r}-\\omega t+\\phi\\_{i}^{(j)})g\\_{i}^{(j)}(\\mathbf{r}).$$ \n*g**i*(*j*)(**r**) describes the beam profile of the compressed Gaussian beams. We assume that the cold atoms are loaded to the middle of the whole optical lattice, in a region characterized by *ρ*0*z* (i.e., 2*π**ρ*0*z*2/*λ* ≃ 10*π**ρ*0*z* for *ρ*0*z* ≃ 5*λ*) and much narrower than the width of the line beam (i.e., ∼ 2*ρ*0) (see Appendix A). In this case, the variation of *g**i*(*j*)(**r**) along the width of the line beam is negligible. It is then a good approximation to take\n \n*g**i*(*j*)(**r**) → *g**i*(*z*) ≈ *e*− (*z* − *z**i*)2/*ρ*0*z*2, \nsuppose the region is also centering around the waist of the *z*-axis amplitude distribution. *z**i* is the center of the *i*-th layer along the *z* axis. To be specific, we label the A, B, and C layers in sequence by *i* = 1, 2, and 3.\nSetting the polarizations $\\hat{\\boldsymbol{\\epsilon}}\\_{j}=\\hat{\\mathbf{z}}$ (*j* = 1, 2, 3), the optical dipole potential for the *i*th layer is \n \n$$U\\_{i}(\\mathbf{r})=-\\frac{\\epsilon\\_{0}}{4}\\alpha\\_{0}^{\\prime}E\\_{i}^{2}(\\mathbf{r}),$$ \nwhere\n \n$$\\begin{aligned}\n&&E\\_{i}^{2}(\\mathbf{r})=E\\_{i}^{2}g\\_{i}^{2}(z)\\{3+2[\\cos(\\mathbf{k}\\_{12}\\cdot\\mathbf{r}+\\phi\\_{i}^{(12)}) \\notag \\\\\n&&\\hspace{0.5cm} +\\cos(\\mathbf{k}\\_{23}\\cdot\\mathbf{r}+\\phi\\_{i}^{(23)})+\\cos(\\mathbf{k}\\_{13}\\cdot\\mathbf{r}+\\phi\\_{i}^{(13)})]\\}.\\end{aligned}$$ \n**k***i**j* = **k***i* − **k***j*, *ϕ**i*(*m**n*) = *ϕ**i*(*m*) − *ϕ**i*(*n*). *α*0′ is the real part of the polarizability. For red-detuned laser, *α*0′ > 0, the minima of the optical dipole potential reside at the maxima of *E**i*2(**r**). For 40K, red-detuned lasers with wavelength at 1064 nm and 1030 nm were used in experiments.\nWhen the three layers are isolated (i.e., *z*1 − *z*2 ≫ *ρ*0*z* and *z*2 − *z*3 ≫ *ρ*0*z*), the full optical potential *V*(**r**) is the sum of Eq.(3) over the three layers. As we move the three layers closer so that laser beams for NN layers start to overlap, interference between the laser beams of NN layers adds new interlayer interference terms to *V*(**r**). While these interlayer interference terms are interesting, we will neglect them in the following analysis and defer more serious consideration of their effects to later studies. In experiments, these interlayer interference terms may be eliminated by making the laser frequencies for different layers slightly different, through nonlinear optical effects such as the stimulated Raman scattering or optical parametric oscillation. In this way, while the interlayer interference terms average out with time, the slight difference in frequency does not lead to appreciable differences in the lattice parameters of different layers. The full optical potential is therefore adequately described as the sum of the three single-layer components defined by Eqs.(3) and (4).\nBy finely adjusting the phase parameters *ϕ**i*(*j*), we may align the projections of the three triangular layers to the *x**y* plane according to Fig.2. One way is to set *ϕ**i*(1) = *ϕ*1 + 2(*i* − 1)*θ*, *ϕ**i*(2) = *ϕ*2 + (*i* − 1)*θ*, and *ϕ**i*(3) = *ϕ*3. *ϕ**i* (*i* = 1, 2, 3) are arbitrary constants. *θ* determines the successive shift of the three triangular layers along the *x* direction. When *θ* = 2*π*/3, we get the displaced dice lattice shown in Fig. 2. Of course, each phase may be different from the above value by an integral multiple of 2*π*. The two reciprocal lattice vectors are **b**1 = **k**12 and **b**2 = **k**23. The lattice constant is related to the wavelength by *k* = 2*π*/*λ* = 4*π*/(3*a*), which gives *a* = 2*λ*/3.\n[fig3]\n \n \n \n \nTo realize the displaced dice lattice defined above, we pass the line beam output from the cylindrical Keplerian telescope through the optical device in Fig. 3 (see also Appendix D for another optical device). After splitting the line beam into three beams by the beam splitters BS1 and BS2 in Fig. 3(a), we inject each of the three beams into a compound optical element that we call the modified Michelson interferometer (MMI). The MMI shown in Fig.3(b) splits the incident laser beam into three parallel beams that are separated along the *z* direction but coincide with each other when projected along the *z* direction onto the *x**y* plane. The vertical distances between consecutive layers are controlled by the special Fresnel bimirrors with 90∘ intersection angle between the two mirrors, see also Fig.1(c). These bimirrors may also be replaced by right-angle prisms made of glass with a high refractive index (see Appendix C). Three glass plates (P1 to P3) are inserted to the light paths, to act as phase shifters. Besides using plates of varying thickness, we may also tilt the plates to control the phase shifts. If frequency modulation is required, we insert a suitable nonlinear optical element to the corresponding light path.\nTight-binding models for fermionic cold atoms in the displaced dice lattice\n===========================================================================\nInstead of exploring the full parameter ranges of the displaced dice lattice, we fix **k***i*(*j*) and *ϕ**i*(*j*) (*i*, *j* = 1, 2, 3) to the ideal values defined in the previous section. We also consider loading the same fermionic cold atoms, such as 40K or 6Li, to all three layers. The polarizations of the laser beams will be fixed to $\\hat{\\boldsymbol{\\epsilon}}\\_{j}=\\hat{\\mathbf{z}}$ (*j* = 1, 2, 3). In addition, the laser beam output from the cylindrical telescope has a fixed waist size *ρ*0*z* along the *z* axis. Under these restrictions, the remaining tunable parameters of the optical lattice include *E**i* and *z**i* (*i* = 1, 2, 3). Taking *E*2 and *z*2 = 0 for the B layer as references, and the remaining four parameters as free parameters, we are able to simulate a rich family of interesting models.\nAssuming all three layers have attained the 2D limit, and the ultracold atoms only occupy the lowest energy levels of the various potential wells. This justifies a tight-binding description, since these orbitals are deeply localized in the potential wells. For our displaced dice lattice, the single-orbital tight-binding model up to NN hopping has nine parameters, including three on-site energies, three intralayer NN hopping amplitudes, and three interlayer NN hopping amplitudes. A qualitative estimation of the parameters follows by making the harmonic approximation to the potential wells and taking the lowest eigenstates as the local Wannier orbitals (see Appendix E). The dice model and its variants require the AB and BC interlayer hopping amplitudes to dominate over the three intralayer and the AC interlayer hopping amplitudes. This is achieved in the present layer-by-layer scheme through the exponential decay of the NN hopping amplitudes with the relevant inter-site distances. Namely, according to Fig.2, the AB and BC interlayer hopping amplitudes may dominate over the intralayer hopping amplitudes because the *x**y* projection of the NN distance is $a/\\sqrt{3}$ for the former versus *a* for the latter. The AB and BC NN interlayer hopping amplitudes may dominate over the AC NN interlayer hopping amplitude because the projection of the NN bonds to the *z* direction for AC, *z*1 − *z*3, is about twice as large as *z*1 − *z*2 and *z*2 − *z*3 for AB and BC (i.e., in most cases of interest we have *z*1 − *z*2 ≃ *z*2 − *z*3). Therefore, by finely tuning the *z* coordinates of the three layers, we may make the AB and BC NN interlayer hopping amplitudes dominate over the others (see Appendix E). Together with the control over *E*1/*E*2 and *E*3/*E*2, we may arrive at the desired dice model and its variants.\nIn conclusion, by tuning four free parameters (*E*1/*E*2, *E*3/*E*2, *z*1, *z*3), we may realize the following family of four-parameter tight-binding models\n \n$$\\begin{aligned}\n\\hat{H}&=&\\sum\\limits\\_{\\langle i,j\\rangle,\\sigma}(t\\_{ba}b^{\\dagger}\\_{i\\sigma}a\\_{j\\sigma}\n+t\\_{bc}b^{\\dagger}\\_{i\\sigma}c\\_{j\\sigma}+\\text{H.c.}) \\notag \\\\\n&&+\\sum\\limits\\_{i,\\sigma}(\\varepsilon\\_{ab}a^{\\dagger}\\_{i\\sigma}a\\_{i\\sigma}\n+\\varepsilon\\_{cb}c^{\\dagger}\\_{i\\sigma}c\\_{i\\sigma}).\\end{aligned}$$ \nThe summation ⟨*i*, *j*⟩ runs over NN intersublattice sites. The index *σ* denotes the two spin states of the fermionic cold atoms. H.c. means the Hermitian conjugate of the terms explicitly written out. We have taken the on-site energy for the B layer as reference, so *ɛ**a**b* = *ɛ**a* − *ɛ**b* and *ɛ**c**b* = *ɛ**c* − *ɛ**b*. For *ɛ**a**b* = *ɛ**c**b* = 0 and *t**b**c* = *t**b**a*, we get the dice (T3) model. For *ɛ**a**b* = *ɛ**c**b* = 0 and *t**b**c* = *α**t**b**a* (0 < *α* < 1) we get the *α*-T3 model. These two families of models both have a flat band connecting linearly to one upper and one lower dispersive bands. For *ɛ**a**b* = *ɛ**c**b* ≠ 0 and *t**b**c* = *α**t**b**a* (0 < *α* ≤ 1), the flat band connects quadratically only to one dispersive band. We define the displaced dice lattice as *biased* if *ɛ**a**b**ɛ**c**b* < 0, in analogy to a semiconductor slab under an electric field perpendicular to the slab, i.e. an electric bias. The inversion symmetry is broken in the biased dice lattice. In previous proposals for the dice lattice, the symmetry of the optical potential allows only the parameters for the pure dice model or *ɛ**a**b* = *ɛ**c**b* ≠ 0 and *t**b**c* = *t**b**a*. This comparison highlights the flexibility of the multilayer optical lattices obtained by the layer-by-layer approach.\nTo further motivate the above highly tunable model, we study the interband transitions in a biased dice model for *t**b**a* = *t**b**c* = *t*0 and *ɛ**a**b* = − *ɛ**c**b* = Δ. Fermionic cold atoms in this model features a band structure with a flat band at *E*0(**k**) = 0 isolated from two dispersive bands at *E**ν*(**k**) = *ν**E*(**k**) (*ν* = ± ), with $E(\\mathbf{k})=\\sqrt{\\Delta^{2}+2|\\xi(\\mathbf{k})|^{2}}$ and *ξ*(**k**) = *t*0(1 + *e**i***k** ⋅ **a**1 + *e**i***k** ⋅ **a**2). In the hexagonal first Brillouin zone, the flat band approaches the two dispersive bands at $\\mathbf{K}=(\\frac{\\sqrt{3}}{2},-\\frac{1}{2})\\frac{4\\pi}{3a}$ and $\\mathbf{K}^{\\prime}=(\\frac{\\sqrt{3}}{2},\\frac{1}{2})\\frac{4\\pi}{3a}$. Referring to the valley-contrasting optical excitations in inversion-broken graphene and transition metal dichalcogenide monolayers, it is natural to ask if the present model also shows valley-contrasting interband excitations.\nTo trigger the interband transitions in the neutral fermionic cold atoms confined in the optical lattice, we exert a weak harmonic stimulus by sinusoidally shaking the lattice, which amounts to periodically modulating the phases of the laser beams. We consider a resonant circular driving to explore possible valley-contrasting excitations. The strength of the vertical interband transition at **k** is proportional to (see Appendix F)\n \n$$\\langle\\psi\\_{f}(\\mathbf{k})|[\\frac{\\partial h(\\mathbf{k})}{\\partial k\\_{x}}+i\\eta\\frac{\\partial h(\\mathbf{k})}{\\partial k\\_{y}}]\n|\\psi\\_{i}(\\mathbf{k})\\rangle,$$ \nwhere ∣*ψ**i*(**k**)⟩ and ∣*ψ**f*(**k**)⟩ are the initial and final states of the interband transition at **k**, *η* = ± marks the chirality of the circularly polarized stimulus. We consider 1/3-filled and 2/3-filled bands, for which the band edges situate at the two **K** points of the Brillouin zone. For circular modulations resonant with the transitions between the highest occupied states and the lowest empty states, that is for ℏ*ω*0 ≃ ∣Δ∣ with *ω*0 being the angular frequency of the shaking, only states close to the **K** points contribute. We thus concentrate on these states and introduce the relative momenta **q** = **k** − **K***τ*, where *τ* = ± , **K**+ = **K**′, and **K**− = **K**.\nFor 1/3-filled bands, *ψ**i* = *ψ*− and *ψ**f* = *ψ*0. For 2/3-filled bands, *ψ**i* = *ψ*0 and *ψ**f* = *ψ*+. For both cases and to the leading order of **q**, the matrix element defined by Eq.(6) turns out to be (see Appendix F)\n \n$$\\frac{\\sqrt{3}a}{2}|t\\_{0}|[\\text{sgn}(\\Delta)-\\eta\\tau]\\frac{q\\_{x}+i\\eta q\\_{y}}{q},$$ \nwhere the sign function sgn(*x*) = *x*/∣*x*∣ for *x* ≠ 0. For fixed Δ, the transition is nonvanishing only for the circular driving satisfying *η**τ* = − sgn(Δ). The chirality *η* changes sign as the valley index *τ* changes sign. Therefore, the interband transitions involving the flat band are valley contrasting. It is interesting to explore the valley-contrasting physics in this displaced dice model with an isolated flat band, in analogy to that in monolayer transition metal dichalcogenides and gapped graphene.\nSummary and discussion\n======================\nWe have illustrated a general scheme of assembling multilayer optical lattices of cold atoms in a layer-by-layer manner. This allows us to simulate the properties of novel multilayer systems both from and beyond the condensed matter systems. As an example, we propose the optical devices to synthesize the displaced dice lattice that may realize the pure and generalized dice models, which have not been realized in experiment despite intensive theoretical studies. For a symmetrically biased dice model, we find valley-contrasting interband transitions associated with a flat band. Hopefully, because all the involved optical elements are standard, experimental realization of the proposal will come true soon.\nThe optical device in Figure 3 for the displaced dice lattice can easily be adjusted to apply to other novel multilayer optical lattices. Firstly, by controlling the orientations of the MMIs and so the propagation directions of the output laser beams, we may realize multilayer optical lattices with other lattice geometries, such as the square lattice, the Kagome lattice, and so on. In addition, by combining more than one sets of MMIs, we may realize multilayer optical lattices in which the constituent layers have hybrid lattice geometries (e.g., a multilayer of alternate square lattices and triangular lattices). By removing the triad of BS5, P2 and BM1 from Fig.3(b), the ensuing reduced MMIs may be used to create bilayer optical lattices. By inserting more triads of beamsplitters, plates, and bimirrors to the MMIs of Fig.3(b), on the other hand, we can make optical lattices with more than 3 layers. Secondly, by controlling the phase parameters of the laser beams, we may achieve relative slipping between different layers along the layer plane. For the displaced dice lattice, this can be seen from the discussions in Sec.III and Eq.(F20) of Appendix F. Thirdly, which is of great current interest, we may twist the layers of the multilayer optical lattice by a continuously tunable angle. Referring to Fig.3 for the displaced dice lattice, by rotating the intersection edge of the bimirror in the *x**y* plane off the direction perpendicular to the propagation direction of the incoming laser beam, we may twist the reflected laser beam by an in-plane angle (see Appendix C). In this manner, we may twist a layer by rotating the propagation directions of the three laser beams for this layer to the same amount in the *x**y* plane, and consequently control the relative orientations between consecutive layers. Here, the twisting between consecutive layers may be implemented independently and up to a large twisting angle, which seem to be difficult in conventional solid state multilayer systems such as the bilayer graphene and thin films of the transition metal dichalcogenides.\nExploration of the above extensions to other multilayer optical lattices, and incorporation of many-body correlations, constitute highly intriguing future studies.\nTransformation of the Gaussian beam by the cylindrical Keplerian telescope\n==========================================================================\nThe beam shaper in our scheme is a cylindrical Keplerian telescope. In this section, we provide explicit mathematical definitions for the parameters characterizing the Gaussian beam input to and output from the cylindrical Keplerian telescope. This is done by summarizing well-known formulae for the transformations of the Gaussian beams by conventional (paraxial) optical elements.\nWe refer to Figure 1(b) in the main text for the setup. The apertures of L1 and L2 are commonly much larger than the diameter of the input Gaussian laser beam, which is determined by the output aperture of the laser device and typically in the order of 1 mm. Therefore, the diffraction of the laser beam by the lenses L1 and L2 should be minor effects and is ignored in the analysis of this work.\nWe consider the propagation of the laser beam parallel to the *x* axis, according to Fig. 1(b) of the main text. Before entering the beam shaper, the complex amplitude of the electric field of standard TEM00 mode Gaussian laser beam can be written as \n \n$$E(x,y,z)=\\frac{A\\_{0}}{\\rho(x)}\\text{exp}\\{-ik[x+\\frac{y^2+z^2}{2q\\_{1}(x)}]+i\\phi(x)\\},$$ \nwhere *k* = 2*π*/*λ* is the wave number, *A*0/*ρ*(*x*) is the amplitude of the electric field along the *x* axis (*y* = *z* = 0), $\\rho(x)=\\rho\\_{0}\\sqrt{1+(\\frac{\\lambda x}{\\pi\\rho\\_{0}^{2}})^2}$ is the *x*-coordinate dependent radius of the disklike laser beam profile. *ρ*0 is the waist radius at the waist plane (*x*0 = 0 here) of the input Gaussian beam. Usually, *ρ*0 is much larger than the wavelength of the laser (i.e., *ρ*0 ≫ *λ*), so that *ρ*(*x*) increases slowly as *x* departs from the waist of the beam. A parameter characterizing the increase of *ρ*(*x*) is the Rayleigh range *z**R*, which is the distance from the waist at which $\\rho(x\\_{0}+z\\_{R})=\\sqrt{2}\\rho(x\\_{0})$. From the above definition of *ρ*(*x*), we see that\n \n*z**R* = *π**ρ*02/*λ*. \nA section of the beam centering at the waist and with a length 2*z**R* is usually taken as the range over which the expansion of the Gaussian beam is small and negligible. In terms of the Rayleigh range, $\\rho(x)=\\rho\\_{0}\\sqrt{1+(\\frac{x}{z\\_{R}})^2}$. The phase factor $\\phi(x)=\\text{arctan}(\\frac{\\lambda x}{\\pi\\rho\\_{0}^{2}})=\\text{arctan}(\\frac{x}{z\\_{R}})$. Twice of the Rayleigh range (i.e., 2*z**R*), or sometimes the Rayleigh range *z**R* itself, is also called the confocal parameter of the Gaussian beam.\nBeing neither plane wave nor spherical wave, the Gaussian beam is characterized by the *q*1(*x*) parameter defined as \n \n$$\\frac{1}{q\\_{1}(x)}=\\frac{1}{R(x)}-i\\frac{\\lambda}{\\pi\\rho(x)^{2}}.$$ \n$R(x)=x[1+(\\frac{z\\_{R}}{x})^{2}]$ is the *x*-dependent curvature radius of the equiphase surface. The waist of the Gaussian beam at the *x* = *x*0 = 0 plane is special and has an infinite curvature, namely *R*(0) = ∞ and *q*1(0) = *i**z**R*. The beam profile determined by Eq.(A1) on the *x* plane is related to the beam profile on the *x* = 0 waist plane, through a free propagation of *x* distance in the free space. Namely, the field amplitude on the *x* = 0 plane,\n \n$$E(0,y,z)=\\frac{A\\_{0}}{\\rho\\_{0}}\\exp\\{-ik\\frac{y^2+z^2}{2q\\_{1}(0)}\\}=\\frac{A\\_{0}}{\\rho\\_{0}}\\exp\\{-\\frac{y^2+z^2}{\\rho\\_{0}^{2}}\\},$$ \nis transformed to\n \n$$E(x,y,z)e^{ikx}=\\frac{A\\_{0}}{\\rho\\_{0}}\\frac{1}{A+[B/q\\_{1}(0)]}\\exp\\{-ik\\frac{y^2+z^2}{2q\\_{1}(x)}\\},$$ \nwhere\n \n$$q\\_{1}(x)=\\frac{Aq\\_{1}(0)+B}{Cq\\_{1}(0)+D}.$$ \nThe four coefficients are the components of the ABCD matrix for a free-space propagation of distance *x*, which is known to be \n \n$$\\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix}=\\begin{pmatrix} 1 & x \\\\ 0 & 1 \\end{pmatrix}.$$ \nSubstituting the above coefficients into Eqs.(A5) and (A6), and defining the phase factor $\\phi(x)=\\arctan(\\frac{x}{z\\_{R}})$, we reproduce Eq.(A1).\nThe free space as an optical medium is isotropic and transforms the Gaussian beam identically in both the *y* direction and the *z* direction. In an anisotropic medium (i.e., an astigmatic medium), however, the transformation matrix may be different along different directions. In this case, we treat the two directions independently and get two independent factors of the electric field amplitude, which are then multiplied together to give the overall electric field amplitude of the transformed laser beam.\nThe cylindrical Keplerian telescope defined by Fig.1(b) of the main text, the beam shaper in our proposal, happens to be an astigmatic optical device. It reshapes only the *z* axis amplitude distribution of the input Gaussian beam. The transformation of the *y* axis amplitude distribution of the input Gaussian beam is just like the transformation in the free space. This anisotropic reshaping is conveniently represented by introducing two new *q* parameters, *q*2 and *q*3, so that the electric field amplitude of the output laser beam becomes\n \n$$\\frac{A^{\\prime}\\_{0}}{\\sqrt{\\rho\\_{2}(x)\\rho\\_{3}(x)}}\\text{exp}\\{-ik[x+\\frac{y^2}{2q\\_{2}(x)}+\\frac{z^2}{2q\\_{3}(x)}]+i\\phi\\_{23}(x)\\},$$ \nIf the loss of the laser in passing through the telescope is negligible, we may set *A*0′ = *A*0. *ρ*2(*x*) and *ρ*3(*x*) are associated to *q*2 and *q*3, just as *ρ*(*x*) is related to *q*1 according to Eq.(A3). Before passing the beam shaper, we have *q*2(*x*) = *q*3(*x*) = *q*1(*x*). After passing the beam shaper, *q*2(*x*) = *q*1(*x*) still holds. However, *q*3(*x*) is different from *q*1(*x*) and transforms from *q*1(0) according to the overall ABCD matrix of the telescope system in the *z* direction. The phase factor *ϕ*23(*x*) is the average of the phase factors related to the transformations in the *y* and the *z* directions,\n \n$$\\phi\\_{23}(x)=\\frac{\\phi\\_{2}(x)+\\phi\\_{3}(x)}{2}.$$ \n$\\phi\\_{2}(x)=\\arctan(\\frac{\\lambda x}{\\pi\\rho\\_{0}^{2}})$. *ϕ*3(*x*) is related to the components of the overall ABCD matrix of the cylindrical Keplerian telescope in the *z* direction through \n \n$$e^{-i\\phi\\_{3}(x)}=\\frac{A+[B/q\\_{3}(0)]}{|A+[B/q\\_{3}(0)]|},$$ \nwhere *q*3(0) = *q*2(0) = *q*1(0) = *i**z**R*.\nThe overall ABCD matrix of a composite optical system is the matrix multiplication of the ABCD matrices of the optical elements therein in sequence. We consider an input Gaussian laser beam with the waist at the *x* = 0 plane at a distance *l*1 to the left of L1, and consider the output laser beam at a distance *l*2 to the right of L2. The telescope system is equivalent to a composition of five optical elements: a free space of length *l*1, L1 of focal length *f*1, a free space of length *f*1 + *f*2 ≡ *L*, L2 of focal length *f*2, and a free space of length *l*2. The ABCD matrix for a thin lens whose focal length is *f* is known as \n \n$$\\begin{pmatrix} 1 & 0 \\\\ -\\frac{1}{f} & 1 \\end{pmatrix}.$$ \nTogether with the ABCD matrix for a free-space propagation for a length of *l*, defined by setting *x* = *l* in Eq.(A7), we get the overall ABCD matrix for the *z*-axis transformation of the cylindrical Keplerian telescope by multiplying the five ABCD matrices together in sequence as\n \n$$\\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix}=\\begin{pmatrix} -\\frac{1}{M} & L-\\frac{l\\_{1}}{M}-Ml\\_{2} \\\\ 0 & -M \\end{pmatrix},$$ \nwhere *M* = *f*1/*f*2 is the magnification of the telescope in the *z* direction. The *q* parameter at the waist of the incoming laser beam is *q*1(0) = *i**π**ρ*02/*λ* = *i**z**R*. After passing through the cylindrical Keplerian telescope, the *q* parameter in the *z* direction becomes\n \n$$q\\_{3}(l\\_{2})=\\frac{Aq\\_{1}(0)+B}{Cq\\_{1}(0)+D}=\\frac{q\\_{1}(0)+l\\_{1}+M^{2}l\\_{2}-ML}{M^{2}}.$$ \nBy writing\n \n$$\\frac{1}{q\\_{3}(l\\_{2})}=\\frac{1}{R\\_{3}(l\\_{2})}-i\\frac{\\lambda}{\\pi\\rho\\_{3}^{2}(l\\_{2})},$$ \nand comparing the two sides of Eq.(A14), we get\n \n$$R\\_{3}(l\\_{2})=\\frac{A^{2}z\\_{R}^{2}+B^{2}}{BD},$$ \n \n$$\\frac{\\pi\\rho\\_{3}^{2}(l\\_{2})}{\\lambda}=\\frac{A^{2}z\\_{R}^{2}+B^{2}}{ADz\\_{R}}.$$ \nFor fixed *l*1, the waist for the *z*-axis amplitude distribution of the output laser beam is determined by *R*3(*l̃*2) = ∞, which gives\n \n$$\\tilde{l}\\_{2}=\\frac{L-\\frac{l\\_{1}}{M}}{M}.$$ \nSubstituting into the expression for *ρ*3(*l*2), we get the waist size *ρ*0*z* for the amplitude distribution along the *z* axis for the output laser beam as\n \n$$\\rho\\_{0z}=\\rho\\_{3}(\\tilde{l}\\_{2})=\\frac{\\rho(0)}{M}=\\frac{\\rho\\_{0}}{M}.$$ \nTaking the two waist size parameters *ρ*0 and *ρ*0*z* as references, the laser beam is compressed by *M* times in the *z* direction. Therefore, the cylindrical Keplerian telescope transforms the disklike beam profile of the input Gaussian beam to the straight filamentary beam profile of the output beam. Equivalently, the rod-like input Gaussian laser beam becomes the slab-like output laser beam.\nNote that, the waist of the amplitude distribution along the *y* direction is not changed by the telescope system and is still located at the plane *l*1 to the left of L1. We may want to change the waist position for the amplitude distribution of the output laser beam along *y* to the same position as that along *z*. Or alternatively, we may want to make the output laser beam have a uniform distribution along the *y* direction. This objective can be realized by replacing the simple cylindrical telescope used in Fig.1(b) of the main text by more sophisticated optical systems. However, because of the excellent unidirectionality of laser, one may take as a very good approximation to ignore the departure from the unidirectional propagation along *y*. This should be allowed in particular in the laboratory where the optical distances are of limited range. Therefore, we adhere in this work to the simple cylindrical telescope shown in Fig.1(b) of the main text. The waist size parameters *ρ*0 and *ρ*0*z* are taken to measure the width and thickness of the beam profile of the output laser beam.\nBesides technical limitations, and to facilitate the accurate control over the interlayer distances of the multilayer optical lattices, the reduced Rayleigh range for the amplitude distribution along the *z* axis also requires us to relax the restriction in the thickness of the slab-like output laser beam. Because as *ρ*0*z* decreases, the far-field divergence angle of the beam along *z* increases as 1/*ρ*0*z*, the unidirectionality of the laser beam is weakened. As a result, the spatial extent, over which the thickness in the *z*-axis amplitude distribution of the laser beam could be regarded approximately constant, decreases monotonically with *ρ*0*z*. Quantitatively, the range (along *x*, centering at the waist) over which the thickness of the laser along the *z* direction may be regarded as constant is approximately twice the Rayleigh range, 2*z**R* = 2*π**ρ*0*z*2/*λ*. If *ρ*0*z* ≃ *λ* is achieved, the spatial region over which a uniform 2D optical lattice may be defined hosts only about 100 unit cells of the optical lattice. For *ρ*0*z* ≃ 5*λ*, on the other hand, the approximately uniform region of the resulting multilayer optical lattice may have more than 6 × 104 unit cells. This should be enough for most quantum simulations.\nFinally, we comment on the position of the *z*-axis waist of the compressed Gaussian beam. According to Eq.(A17), we have *l̃*2 ≃ *f*2 for *l*1 ≃ 0 and *M* ≫ 1. In the layer-by-layer scheme for the multilayer optical lattices, we wish to have the *z*-axis waist positions of the various beams approximately coincide with each other at the center of the designated location for the optical lattice. To be able to insert the optical elements in Figs. 3(a) and 3(b) in between L2 of the cylindrical telescope and the location for the optical lattice, *l̃*2 and thus *f*2 should be large enough. For practical reasons, *f*1 cannot be too large. Therefore, the magnification *M* cannot be large. This means that, to achieve the desired compression ratio along the *z* axis (i.e., about 200), we have to cascade two or more telescopes. In the last telescope, we can choose its *f*2 larger than the preceding telescopes and a smaller magnification factor. For example, we can cascade two cylindrical telescopes of magnification 10 and a third cylindrical telescope with a magnification of 2 to achieve a total magnification of 200. In the last telescope, we may choose *f*2 ≃ 10 cm, which gives *l̃*2 ≃ 15 cm for *l*1 ≃ 0. By minimizing the sizes of the optical elements in the device, we should be able to bring the *z*-axis waist positions of the various thin-line beams to the center of the designated spatial region [e.g., within the circle of Fig.3(a)]. Besides increasing *f*2, another method is to add a proper cylindrical lens to each MMI and refocus the outgoing beams, so that their *z*-axis waists are at or close to the center of the optical lattice. Still another, more radical, strategy is to use diffractionless laser beams (in particular, the Bessel beams ) with a thin-line beam profile. This strategy not only resolves the issue of small *l̃*2 but may also enlarge the area of the ensuing multilayer optical lattice. It however requires more analysis as regards how to achieve this novel non-diffracting thin-line beam with sufficient intensity, which we leave to future studies.\nThe 2D limit of an individual layer\n===================================\nIn terms of the line beam output from the beam shaper introduced in Fig.1(b) of the main text, we may form a quasi-2D optical lattice with reduced thickness characterized by *ρ*0*z*. In applying to our layer-by-layer assembling of multilayer optical lattices, it is crucial to know the upper limit of *ρ*0*z* required for bringing the thickness of the resulting optical lattice down to the 2D limit.\nPhysically, if the input Gaussian laser beam is not compressed in the *z* axis, each 2D site of the ensuing 2D optical lattice is in fact a quasi one-dimensional (q-1D) tube, for which the spectrum is a quasi-continuous 1D spectrum. As we compress the *z*-axis breadth (i.e., *ρ*0*z*) of the laser beam, the q-1D spectrum becomes more and more discrete. Let us label the bound states in each site (i.e., potential well) of the 2D optical lattice by *ε*0, *ε*1, *ε*2, …, in an order of increasing with the subscript. Then the site should be considered as a real zero-dimensional point rather than a q-1D tube when *ε*1 − *ε*0 > *E**R* = *h*2/(2*m**λ*2). *E**R* is the recoil energy of the optical lattice. If this condition is satisfied, only the lowest bound state of each quantum well is occupied. A tight-binding model for the cold atoms loaded to the optical lattice may be constructed by retaining only the lowest bound state for each site of the 2D lattice.\nTo be concrete, let us consider the optical dipole potential defined in Eqs.(1)-(4) of the main text for a single layer of the displaced dice lattice, which is a triangular lattice. By redefining the origin of coordinates, we may set the phase factors to zero. The optical dipole potential is written as\n \n$$U(x,y,z)=-V\\_{0}\\{1+4\\cos\\frac{bx}{2}[\\cos\\frac{bx}{2}+\\cos\\frac{\\sqrt{3}by}{2}]\\}g^{2}(z),$$ \nwhere $b=4\\pi/(\\sqrt{3}a)$ is the magnitude of the reciprocal lattice vector, *g*(*z*) = exp( − *z*2/*ρ*0*z*2). We consider a red-detuned laser, so that \n \n$$V\\_{0}=\\frac{\\epsilon\\_{0}}{4}\\alpha\\_{0}'E\\_{0}^{2}>0.$$ \n(*x*, *y*, *z*) = (0, 0, 0) is clearly the bottom of one potential well. For simplicity, we focus on this potential well. Since we are interested in the 2D limit of the optical potential, we assume sufficiently deep potential wells and that the low-lying bound states are confined close to the bottom of the potential well. According to this assumption, we expand *U*(*x*, *y*, *z*) to the leading order polynomials of *x*, *y*, and *z* close to (*x*, *y*, *z*) = (0, 0, 0). We get\n \n$$\\begin{aligned}\nU(x,y,z)&\\simeq&-9V\\_{0}+\\frac{8\\pi^{2}V\\_{0}}{a^{2}}(x^{2}+y^{2})+\\frac{18V\\_{0}}{\\rho\\_{0z}^{2}}z^{2} \\notag \\\\\n&=&-9V\\_{0}+U(x)+U(y)+U(z).\\end{aligned}$$ \nThe above potential energy term plus the kinetic energy term of the cold atom define an anisotropic 3D harmonic oscillator. We choose {*Ĥ**x*, *Ĥ**y*, *Ĥ**z*} as the complete set of commuting observables, where\n \n$$\\hat{H}\\_{\\alpha}=\\frac{p\\_{\\alpha}^{2}}{2m}+U(\\alpha),$$ \n*α* = *x*, *y*, *z*, *m* is the mass of the cold atoms in the optical lattice. The eigen-spectrum of the above anisotropic harmonic oscillator is\n \n$$E\\_{n\\_{x}n\\_{y}n\\_{z}}=-9V\\_{0}+(n\\_{x}+n\\_{y}+1)\\hbar\\omega\\_{xy}+(n\\_{z}+\\frac{1}{2})\\hbar\\omega\\_{z},$$ \nwhere *n**α* (*α* = *x*, *y*, *z*) are nonnegative integers quantifying the quantization of the energy spectrum. $\\omega\\_{xy}=\\frac{4\\pi}{a}\\sqrt{\\frac{V\\_{0}}{m}}$ and $\\omega\\_{z}=\\frac{6}{\\rho\\_{0z}}\\sqrt{\\frac{V\\_{0}}{m}}$ are separately the angular frequencies for the center-of-mass motion of the cold atom in the *x**y* plane and along the *z* direction.\nIn order for the single layer to be in the 2D limit, the separation between the lowest energy bound state and the second lowest energy bound state should be larger than the recoil energy of the optical lattice. We therefore require ℏ*ω**x**y* > *E**R* and ℏ*ω**z* > *E**R*. For our purpose, considering the continuous compression of *ρ*0*z* from *ρ*0, we originally have *ρ*0*z* ≫ *a* and correspondingly $\\omega\\_{z}/\\omega\\_{xy}=\\frac{3}{2\\pi}\\frac{a}{\\rho\\_{0z}}\\ll1$. We assume the ℏ*ω**x**y* > *E**R* condition is always satisfied for all *ρ*0*z*. As we compress *ρ*0*z* continuously, the 2D limit is attained when ℏ*ω**z* > *E**R*. Suppose ℏ*ω**x**y* = *γ**E**R* (*γ* ≫ 1), the condition ℏ*ω**z* > *E**R* amounts to\n \n$$\\rho\\_{0z}<\\frac{3\\gamma}{2\\pi}a.$$ \nFor our triangular lattice, $a=\\frac{2}{3}\\lambda$. For *γ* > 5*π* ≃ 15.71, *ρ*0*z* = 5*λ* satisfies the above constraint and the 2D limit is attained. For 40K cold atoms, red-detuned laser with a wavelength 1064 nm or 1030 nm were used in experiments. Correspondingly, *ρ*0*z* = 5*λ* amounts to *ρ*0*z* ≈ 5 *μ*m.\nFrom the definition of the angular frequency, ℏ*ω**x**y* = *γ**E**R* > 15.71*E**R* is equivalent to $V\\_{0}=\\frac{\\gamma^{2}}{18}E\\_{R}>13.71E\\_{R}$. This is an intermediate value for the lattice depth. However, it is 9*V*0 from the minima to the maxima of the optical dipole potential defined by Eq.(B1). Therefore, 9*V*0 is the overall lattice depth, which should satisfy 9*V*0 > 123.39*E**R*. As a result, we need to work with a very deep optical lattice to attain the 2D limit for an individual triangular optical lattice. Luckily, this depth is within the reach of experiments. Clearly, for even deeper optical dipole potential, *ρ*0*z* larger than 5*λ* may also lead to the 2D limit. Conversely, if we compress the Gaussian beam thinner, with *ρ*0*z* < 5*λ*, a shallower optical potential may also attain the 2D limit. For example, if we consider *ρ*0*z* = 3*λ*, 9*V*0 > 44.5*E**R* is enough to get the 2D limit for the triangular optical lattice.\nFinally, we should emphasize that the analysis in this section is based on the optical dipole potential of a single layer. In a multilayer optical lattice containing several contiguous layers, such as the displaced dice lattice consisting of three consecutive layers, the superposition of the optical potentials for the various layers may possibly change the actual depth of the resulting multilayer optical dipole potential. From Figure 2 of the main text for the lattice geometry and the expressions of the optical dipole potentials of the various layers, this superposition tends to reduce the actual depth of the optical potential for the displaced dice lattice. From the analysis carried out later in Appendix E, the superposition does not change *ω**z* in the leading order approximation. Therefore, the above criterion for the 2D limit is still applicable. However, the overall trilayer optical dipole potential confining the cold atoms in the displaced dice lattice is actually shallower than the potential of a single layer. The shorter the distances between the three layers, the shallower the trilayer optical dipole potential becomes. For other multilayer optical lattices synthesized through the layer-by-layer approach, the depth of the overall multilayer optical dipole potential may also be larger than the depth of a single layer. At the same time, the critical depth of the lattice for attaining the 2D limit may also change quantitatively.\nReflection of a ray by the Fresnel bimirror with about 90∘ intersection angle\n=============================================================================\nIn what follows, we call the special Fresnel bimirror with a 90∘ intersection angle as the right-angle bimirror. In Fig. 3 of the main text, the reflection plane of the ray of light is assumed to be perpendicular to the intersection edge of the right-angle bimirror. In this case, the retroreflected light propagates along the direction which is exactly opposite to the incident ray of light. This is always the case once the intersection edge of the right-angle bimirror is perpendicular to the incident ray of light, because by default the intersection edge is always perpendicular to the normal of the two mirrors. The lateral shift (i.e., the distance) between the incident and the retroreflected rays of light depends on the incident angle and the distance from the point of incidence to the intersection edge. As the orientation of the intersection edge has an additional 2*π* rotational freedom in the plane perpendicular to the incident ray of light, the lateral shift of the retroreflected ray can also occur at any direction away from the incident ray of light.\nBy twisting the intersection edge of the right-angle bimirror away from the plane perpendicular to the incident ray of light by a small angle *θ*, the reflection plane of the incident ray of light is also slightly deviated from perpendicular to the intersection edge. In this case, we may twist the propagation direction of the retroreflected ray away from exactly the reverse direction of the incident ray. Here, the intersection edge at an angle *θ* to the plane perpendicular to the incident ray of light also has a 2*π* rotational freedom, which may be characterized by an angle *φ*. As regards the retroreflected ray of light, we expect there to be a corresponding freedom in the deviation of its propagation direction with respect to the direction exactly opposite to the incident ray.\nInstead of plotting the full optical path, which is clumsy and not easy to show clearly the various interesting situations, we will express the reflection off the bimirror in terms of vector algebra. Because of the simplicity of the reflection by plane mirrors, exact analytical expressions for the outgoing reflected ray may be obtained.\nFor generality, we consider a Fresnel bimirror whose intersection angle *α* is close to but may be slightly different from 90∘. We notice that the reflected ray is uniquely determined by the position and orientation of the *α*-angle bimirror, and the propagation direction and point of incidence of the incident ray. In terms of vector algebra, these conditions are contained in the two equations for the two mirror planes and the equation for the incident ray.\n[fig4] \nTo proceed, we firstly define the coordinate system in reference to the bimirror shown in Figure 4 of this section. We consider a 3D Cartesian coordinate. We assume the intersection edge of the bimirror is along the *y* axis, so that the planes perpendicular to the intersection edge are parallel to the *x**z* plane. An arbitrary perpendicular plane of the intersection edge crosses with the two mirrors at two lines, which join each other at a point of the intersection edge and subtend an angle of *α*. We assume that the bimirror is divided equally by the *x**y* plane containing the intersection edge. We define the *x* axis to run opposite to the open-mouth direction of the *α*-angle bimirror, so that the incident ray propagates along a direction very close to the positive *x* direction (i.e., close to $\\hat{\\mathbf{x}}$, the unit vector along the + *x* direction). We set the origin *O* of the coordinate, (0, 0, 0), at the middle of the intersection edge. The two mirrors are assumed to be large enough so that the interesting incident rays are all reflected twice by the *α*-angle bimirror before leaving it. Finally, we assume that the crossing point between the incident ray and the first mirror (e.g., M1) is close to the intersection edge.\nUnder the above conditions, the equations for the two mirror planes are\n \n$$\\mathbf{r}\\cdot\\hat{\\mathbf{n}}\\_{i}=0,$$ \nwhere *i* = 1, 2 labels the two mirrors, $\\hat{\\mathbf{n}}\\_{1}=(\\sin\\frac{\\alpha}{2},0,\\cos\\frac{\\alpha}{2})$ and $\\hat{\\mathbf{n}}\\_{2}=(\\sin\\frac{\\alpha}{2},0,-\\cos\\frac{\\alpha}{2})$ are unit normal vectors for M1 and M2.\nIn the application of the main text, we fixed *α* = *π*/2 and the incident ray propagates along the positive *x* direction (i.e., $\\hat{\\mathbf{x}}$). In this case, the incident ray is exactly retroreflected, up to a lateral shift in the *z* direction. Here, we assume the propagation direction of the incident ray may be slightly away from the $\\hat{\\mathbf{x}}$ direction by a small angle *θ*. To characterize this direction, we introduce a spherical polar coordinate taking $\\hat{\\mathbf{x}}$ as the polar axis, *θ* as the polar angle, and define the azimuthal angle *φ* as the angle with respect to the $\\hat{\\mathbf{y}}$ direction in the *y**z* plane. In this polar coordinate, the propagation of the incident ray is along the unit vector\n \n$$\\hat{\\mathbf{u}}=(\\cos\\theta,\\sin\\theta\\cos\\varphi,\\sin\\theta\\sin\\varphi).$$ \nWithout losing generality, we assume that the incident ray shines at a point P on M1 of a distance *d* > 0 away from the intersection edge. The coordinate of P is written as\n \n$$\\mathbf{r}\\_{1}=(-d\\cos\\frac{\\alpha}{2},y\\_{1},d\\sin\\frac{\\alpha}{2}).$$ \nFor simplicity, we set *y*1 = 0. The equation of the incident ray is thus\n \n$$(\\mathbf{r}-\\mathbf{r}\\_{1})\\times\\hat{\\mathbf{u}}=\\mathbf{0}.$$ \nThe reflected ray leaves M1 also at P. Assuming the unit vector along its propagation direction to be $\\hat{\\mathbf{u}}\\_{1}$, the law of reflection dictates that\n \n$$\\begin{cases}\n\\hat{\\mathbf{u}}\\cdot\\hat{\\mathbf{n}}\\_{1}=-\\hat{\\mathbf{u}}\\_{1}\\cdot\\hat{\\mathbf{n}}\\_{1}, \\\\\n\\hat{\\mathbf{u}}\\_{1}\\cdot(\\hat{\\mathbf{u}}\\times\\hat{\\mathbf{n}}\\_{1})=0, \\\\\n\\end{cases}$$ \nwhere the first equation ensures the reflection angle to be equal to the incident angle, and the second equation says that the reflected ray lies in the same plane containing the incident ray and the unit normal vector of M1 passing P. Solution to Eq.(C5) gives\n \n$$\\hat{\\mathbf{u}}\\_{1}=\\hat{\\mathbf{u}}-2(\\hat{\\mathbf{u}}\\cdot\\hat{\\mathbf{n}}\\_{1})\\hat{\\mathbf{n}}\\_{1}.$$ \nFrom $\\hat{\\mathbf{u}}\\cdot\\hat{\\mathbf{u}}=\\hat{\\mathbf{n}}\\_{1}\\cdot\\hat{\\mathbf{n}}\\_{1}=1$, we have $\\hat{\\mathbf{u}}\\_{1}\\cdot\\hat{\\mathbf{u}}\\_{1}=1$. So $\\hat{\\mathbf{u}}\\_{1}$ is the unit vector along the propagation direction of the first reflected ray. The equation for the first reflected ray leaving M1 at P is thus\n \n$$(\\mathbf{r}-\\mathbf{r}\\_{1})\\times\\hat{\\mathbf{u}}\\_{1}=\\mathbf{0}.$$ \nThe coordinate **r**2 of the crossing point between the first reflected ray and M2 is determined by solving the following simultaneous equations\n \n$$\\begin{cases}\n(\\mathbf{r}-\\mathbf{r}\\_{1})\\times\\hat{\\mathbf{u}}\\_{1}=\\mathbf{0}, \\\\\n\\mathbf{r}\\cdot\\hat{\\mathbf{n}}\\_{2}=0. \\\\\n\\end{cases}$$ \nThe solution **r** = **r**2 = (*x*2, *y*2, *z*2) to the above equations is also the point at which the second reflected ray leaves M2 and the whole bimirror. Similar to the first reflection, the unit vector $\\hat{\\mathbf{u}}\\_{2}$ along the propagation direction of the second reflected ray is determined by\n \n$$\\begin{cases}\n\\hat{\\mathbf{u}}\\_{1}\\cdot\\hat{\\mathbf{n}}\\_{2}=-\\hat{\\mathbf{u}}\\_{2}\\cdot\\hat{\\mathbf{n}}\\_{2}, \\\\\n\\hat{\\mathbf{u}}\\_{2}\\cdot(\\hat{\\mathbf{u}}\\_{1}\\times\\hat{\\mathbf{n}}\\_{2})=0, \\\\\n\\end{cases}$$ \nwhich gives\n \n$$\\hat{\\mathbf{u}}\\_{2}=\\hat{\\mathbf{u}}\\_{1}-2(\\hat{\\mathbf{u}}\\_{1}\\cdot\\hat{\\mathbf{n}}\\_{2})\\hat{\\mathbf{n}}\\_{2}.$$ \nThe equation for the second (i.e., final) reflected ray outgoing from the bimirror is\n \n$$(\\mathbf{r}-\\mathbf{r}\\_{2})\\times\\hat{\\mathbf{u}}\\_{2}=\\mathbf{0}.$$ \nThe explicit expressions for **r**2 = (*x*2, *y*2, *z*2) and $\\hat{\\mathbf{u}}\\_{2}=(u\\_{2x},u\\_{2y},u\\_{2z})$ are\n \n$$\\begin{cases}\nx\\_{2}=-\\frac{\\cos\\theta\\sin\\frac{\\alpha}{2}+\\sin\\theta\\sin\\varphi\\cos\\frac{\\alpha}{2}} {\\cos\\theta\\sin\\frac{3\\alpha}{2}+\\sin\\theta\\sin\\varphi\\cos\\frac{3\\alpha}{2}}d\\cos\\frac{\\alpha}{2}, \\\\\ny\\_{2}=\\frac{\\sin\\theta\\cos\\varphi\\sin\\alpha}{\\cos\\theta\\sin\\frac{3\\alpha}{2}+\\sin\\theta\\sin\\varphi\\cos\\frac{3\\alpha}{2}}d, \\\\\nz\\_{2}=-\\frac{\\cos\\theta\\sin\\frac{\\alpha}{2}+\\sin\\theta\\sin\\varphi\\cos\\frac{\\alpha}{2}} {\\cos\\theta\\sin\\frac{3\\alpha}{2}+\\sin\\theta\\sin\\varphi\\cos\\frac{3\\alpha}{2}}d\\sin\\frac{\\alpha}{2}, \\\\\n\\end{cases}$$ \nand\n \n$$\\begin{cases}\nu\\_{2x}=\\cos\\theta\\cos(2\\alpha)-\\sin\\theta\\sin\\varphi\\sin(2\\alpha), \\\\\nu\\_{2y}=\\sin\\theta\\cos\\varphi, \\\\\nu\\_{2z}=\\cos\\theta\\sin(2\\alpha)+\\sin\\theta\\sin\\varphi\\cos(2\\alpha). \\\\\n\\end{cases}$$ \nThe lateral shift related to **r**2 − **r**1 is linearly proportional to *d* and could be tuned continuously. To characterize the change in the propagation direction, we study the direction cosines of $\\hat{\\mathbf{u}}\\_{2}$ with regards to the direction of $\\hat{\\mathbf{u}}$ and $\\hat{\\mathbf{u}}\\times\\hat{\\mathbf{y}}$, which turn out to be\n \n$$\\hat{\\mathbf{u}}\\_{2}\\cdot\\hat{\\mathbf{u}}=-1+2[1-\\sin^{2}\\alpha(1-\\sin^{2}\\theta\\cos^{2}\\varphi)],$$ \nand\n \n$$\\hat{\\mathbf{u}}\\_{2}\\cdot(\\hat{\\mathbf{u}}\\times\\hat{\\mathbf{y}})=\\sin(2\\alpha)(\\sin^{2}\\theta\\sin^{2}\\varphi+\\cos^{2}\\theta).$$ \nFrom these formulae we uncover the conditions for the following two especially interesting cases.\nIn the first case, which is the case we focused on in the main text, the reflected ray propagates in exactly the opposite direction to the incident ray. This is satisfied if\n \n$$\\hat{\\mathbf{u}}\\_{2}\\cdot\\hat{\\mathbf{u}}=-1,$$ \nwhich leads to *α* = *π*/2 and *φ* = ± *π*/2. That is, the ray incident on an *α* = *π*/2 right-angle bimirror propagates along a direction parallel to the *x**z* plane and close to $\\hat{\\mathbf{x}}$ (i.e., *θ* is small so that the incident ray undergoes two reflections off the bimirror).\nIn the second case, the reflected ray propagates almost opposite to the direction of the incident ray but deflected a little bit away from that opposite direction in the plane perpendicular to $\\hat{\\mathbf{u}}\\times\\hat{\\mathbf{y}}$. This case is determined by\n \n$$\\begin{cases}\n\\hat{\\mathbf{u}}\\_{2}\\cdot\\hat{\\mathbf{u}}+1=\\epsilon, \\\\\n\\hat{\\mathbf{u}}\\_{2}\\cdot(\\hat{\\mathbf{u}}\\times\\hat{\\mathbf{y}})=0, \\\\\n\\end{cases}$$ \nwhere 0 < *ε* ≪ 1. For *θ* ≃ 0 that we focus on, the only solution to the second equation of Eq.(C17) for 0 < *α* < *π* is *α* = *π*/2. Substituting *α* = *π*/2 into Eq.(C14), we have\n \n$$\\hat{\\mathbf{u}}\\_{2}\\cdot\\hat{\\mathbf{u}}+1=2\\sin^{2}\\theta\\cos^{2}\\varphi.$$ \nThen the first equation of Eq.(C17) is satisfied once *θ* is small but nonzero and *φ* ≠ ± *π*/2. For our purpose of using the bimirror, it is enough to fix *φ* = 0 or *φ* = *π*, so that $\\hat{\\mathbf{u}}$ is parallel to the *x**y* plane. In this case, suppose the rotation angle from $\\hat{\\mathbf{u}}$ to $\\hat{\\mathbf{u}}\\_{2}$ is *π* + *θ̃*, we have\n \n$$\\hat{\\mathbf{u}}\\_{2}\\cdot\\hat{\\mathbf{u}}=\\cos(\\pi+\\tilde{\\theta})=-\\cos\\tilde{\\theta}=2\\sin^{2}\\theta-1=-\\cos(2\\theta).$$ \nTherefore, rotating the incident ray in the *x**y* plane by an angle of *θ* away from $\\hat{\\mathbf{x}}$ leads to a rotation of the reflected ray by an angle of 2*θ* in the *x**y* plane, compared to the direction of exact retroreflection. This rotation is also easy to see by substituting *α* = *π*/2 and sin*φ* = 0 into the explicit expression for $\\hat{\\mathbf{u}}\\_{2}$ in Eq.(C13) and comparing it with the definition of $\\hat{\\mathbf{u}}$. The angle *θ̃* = 2*θ* may be called as the *twist angle*. The twist angle *θ̃* enables a continuous control over the twisting angle between consecutive layers of the multilayer optical lattice constructed in the layer-by-layer manner.\nFor both of the two cases considered above, the right-angle bimirror may be replaced by an isosceles right-angle triangular prism (in brief, a right-angle prism) made of low-loss glass with a high refractive index. To see this, notice that the above analyses may be considered as corresponding to the two total internal reflections inside the interior of the right-angle prism. M1 and M2 of the right-angle bimirror are interpreted here as the two mutually perpendicular rectangular faces of the prism, and the incident ray enters the right-angle prism through the rectangular bevel face in between the two perpendicular faces. The actual incident ray is connected to the ray incident on the M1 face at P by a refraction at the bevel face of the right-angle prism. The actual outgoing ray is connected to the ray leaving the M2 face at **r**2 by another refraction at the bevel face of the right-angle prism. For the first case studied above, it is easy to see that the retroreflected ray still reverses the direction of the incident ray. In simple words, two parallel rays are still parallel rays upon refraction by a planar dielectric interface. For the second case, the actual twisting angle is changed from 2*θ* to 2*θ*ʹ. The actual twisting angle 2*θ*ʹ is related to the twisting angle 2*θ* within the right-angle prism by Snell’s law\n \nsin*θ*ʹ = *n*sin*θ*, \nwhere *n* is the refractive index of the glass making up the right-angle prism. The refractive index for the air has been taken as 1.\nThe above analyses deal with a single ray of light. In considering the reflection of a laser beam, we regard it as an assembly of nearly parallel rays. The reflection of a line beam, whose long axis of the beam profile is within the *x**y* plane, by a right-angle bimirror or a right-angle prism follows the same rules explained above.\nFinally, we notice that, rotating the incident ray (or, the incident line beam) in the *x**y* plane with respect to the $\\hat{\\mathbf{x}}$ direction is equivalent to fixing the incident ray to propagate along $\\hat{\\mathbf{x}}$ and rotating the intersection edge of the right-angle bimirror in the *x**y* plane by the same angle in the opposite sense (i.e., clockwise versus counterclockwise, and vice versa). In experimental implementations, it seems more convenient to control the bimirrors. Therefore, it is preferable to perform the rotations by acting upon the bimirrors, rather than upon the incident rays.\nAnother optical device for the displaced dice lattice\n=====================================================\nIn application to the displaced dice lattice, the optical device in Fig. 3 of the main text firstly splits the single compressed Gaussian beam (i.e., the line beam) into three beams of equal intensities, and secondly splits each of the three beams again into three beams of equal or unequal intensities (but with the relative intensities the same for all three triplet sets) that are displaced from each other in the *z* direction.\nThe optical device in Fig. 3 of the main text permits independent manipulation over all the nine laser beams. However, our interest in this work has been restricted to the displaced dice lattices where the lattice vectors of all the three layers are parallel to each other. In this ‘simple’ configuration, we may split the single line beam into nine beams following the reverse sequence. Namely, we firstly split the line beam output from the beam shaper into three beams that have equal or unequal intensities and are displaced from each other along the *z* direction but aligned in their projection in the *x**y* plane, and then split this compound beam into three compound beams of equal intensities. By guiding the three compound beams to the designated spot for the optical lattice, we again obtain a trilayer optical lattice. An MMI defined in Fig. 3(b) of the main text is certainly capable of implementing the first step. Figure 5 of this section has taken an alternative combination that shifts the laser beams in terms of tilted glass plates (i.e., according to the left of Fig. 1(c) of the main text).\n[fig5] \nOne drawback of this alternative setup lies in the difficulty in adjusting the relative phase shifts of consecutive layers, which is crucial to lining up the three triangular lattices in the correct relative positions in the *x**y* plane. This is because, according to the analysis in the main text, the phase shifts should be implemented differently for the three beams inside a single compound beam. To implement such phase shifts, as shown on the right part of Fig. 5 of this section, we suggest to use two glass plates with a gradient in the refractive index. This kind of optical elements, although unconventional in comparison to the other optical elements employed (i.e., cylindrical thin lenses, beam splitters, mirrors, glass plates), are also standard optical elements in the subject of gradient-index (GRIN) optics. According to the gradient-index optics, GRIN1 and GRIN2 in Fig.2 of this section belong to the GRIN medium with an axial gradient. With finely tuned gradient in the refractive index, hopefully these GRIN optical elements are able to introduce the desired phase shifts. In even simpler multilayer optical lattices where no relative in-plane shifts between the consecutive layers are required, we do not need these GRIN optical elements and the scheme similar to that of Fig. 5 of this section may find more applications.\nEstimation of tight-binding parameters for the displaced dice lattice\n=====================================================================\nWe show in this section that, by tuning the relative strength of the electric field vectors of the laser beams and the relative *z*-axis coordinates, we may realize the following family of tight-binding models for cold atoms on the displaced dice lattice\n \n$$\\begin{aligned}\n\\hat{H}&=&\\sum\\limits\\_{<i,j>,\\sigma}(t\\_{ba}b^{\\dagger}\\_{i\\sigma}a\\_{j\\sigma}+\\text{H.c.})\n+\\sum\\limits\\_{<i,j>,\\sigma}(t\\_{bc}b^{\\dagger}\\_{i\\sigma}c\\_{j\\sigma}+\\text{H.c.}) \\notag \\\\\n&&+\\sum\\limits\\_{i,\\sigma}(\\varepsilon\\_{a}a^{\\dagger}\\_{i\\sigma}a\\_{i\\sigma}\n+\\varepsilon\\_{b}b^{\\dagger}\\_{i\\sigma}b\\_{i\\sigma}\n+\\varepsilon\\_{c}c^{\\dagger}\\_{i\\sigma}c\\_{i\\sigma}).\\end{aligned}$$ \nBy taking *ɛ**b* as the reference energy, and defining *ɛ**a**b* = *ɛ**a* − *ɛ**b* and *ɛ**c**b* = *ɛ**c* − *ɛ**b*, it is the same as Eq.(5) of the main text. The parameters in the model follow the same definitions for Eq.(5) of the main text. To validate this model, the intralayer (i.e., intrasublattice) nearest-neighboring (NN) hopping amplitudes and the NN hopping amplitudes between the A layer and the C layer should be negligible as compared to the BA interlayer and BC interlayer NN hopping amplitudes. Put it another way, the retained BA interlayer NN hopping amplitudes and the BC interlayer NN hopping amplitudes should dominate among the six kinds of NN hopping amplitudes.\nTo justify the above approximation, we have to make a reasonable estimation over the magnitudes of these relevant NN hopping amplitudes (6 in total). We suppose all three layers of the displaced dice lattice have reached the 2D limit and the on-site interactions have been tuned to zero by the Feshbach resonance. We also assume that fermionic cold atoms occupy only the lowest-energy bound states of the potential wells. In this case, a reasonable order of magnitude estimation over the relevant hopping amplitudes follows by taking the ground state eigenvectors of the approximate harmonic models for each quantum well as the local Wannier orbitals and substitute them into the definition of the hopping amplitudes. A merit of this approach, despite less accurate, is that all hopping amplitudes may be evaluated in a fully analytical manner. Having an explicit analytical expression for the hopping amplitudes in terms of the parameters of the optical lattice greatly facilitates a qualitative understanding over the model parameters of the tight-binding model.\nFor the displaced dice lattice defined in the main text, the single-body Hamiltonian is\n \n$$\\hat{H}\\_{0}=-\\frac{\\hbar^{2}}{2m}\\boldsymbol{\\nabla}^{2}+V(x,y,z),$$ \nwhere *m* is the mass of the cold fermionic alkali-metal element such as 40K or 6Li. *V* is the sum of the optical dipole potentials of all three layers\n \n*V*(*x*, *y*, *z*) = *U**A*(*x*, *y*, *z*) + *U**B*(*x*, *y*, *z*) + *U**C*(*x*, *y*, *z*), \nwhere the subindices A, B, and C indicate the three triangular layers. Here, following the discussions of the main text, we do not include the interlayer interference terms in the optical potential. Analyses for their effects are deferred to future more quantitative calculations of the band structures. Taking the minimum of one potential well of the B layer as the origin of coordinate, (*x*, *y*, *z*) = (0, 0, 0), we have\n \n$$U\\_{B}(x,y,z)=-V\\_{B}[1+4\\cos\\frac{bx}{2}(\\cos\\frac{bx}{2}+\\cos\\frac{\\sqrt{3}by}{2})]g^{2}(z),$$ \n \n$$U\\_{A}(x,y,z)=U\\_{B}(x,y,z)|\\_{x\\rightarrow x-\\frac{a}{\\sqrt{3}},y\\rightarrow y,z\\rightarrow z-z\\_{A};V\\_{B}\\rightarrow V\\_{A}},$$ \n \n$$U\\_{C}(x,y,z)=U\\_{B}(x,y,z)|\\_{x\\rightarrow x+\\frac{a}{\\sqrt{3}},y\\rightarrow y,z\\rightarrow z-z\\_{C};V\\_{B}\\rightarrow V\\_{C}},$$ \nwhere the Gaussian factor *g*(*z*) = exp( − *z*2/*ρ*0*z*2), the length of the reciprocal lattice vector $b=4\\pi/(\\sqrt{3}a)$.\nGenerally, the hopping amplitude between two NN sites (i.e., two NN minima of the optical potential) located at **r**0 = (*x*0, *y*0, *z*0) and **r**1 = (*x*1, *y*1, *z*1) is \n \n*t**N**N*(**r**0**r**1) = ∫*d*3**r***ψ*0\\*(**r** − **r**0)*Ĥ*0*ψ*0(**r** − **r**1). \nIn this definition, the layer (sublattice) index is implicitly included in the reference coordinates **r**0 and **r**1. *ψ*0(**r** − **r**0) is the Wannier function for the local orbital at **r**0.\nFor a qualitative estimation of the hopping amplitudes, we take *ψ*0 as the ground-state wave function for a quantum well in the harmonic approximation. In Section II analyzing the condition for attaining the 2D limit, we focused on the optical dipole potential for a single layer. In the present displaced dice lattice containing three layers, the bottom of the potential well in one layer is also influenced by the optical potentials of the other two layers. Therefore, the approximate harmonic oscillator model for the quantum well in each layer should also be supplemented with the contribution from the other two layers. For the B-layer, we consider the potential well whose minima is at (0, 0, 0). For the A-layer, we consider the potential well whose minima is at $(a/\\sqrt{3},0,z\\_{A})$. For the C-layer, we consider the potential well whose minima is at $(-a/\\sqrt{3},0,z\\_{C})$.\nFirst, we consider the potential well of the A-layer at $(a/\\sqrt{3},0,z\\_{A})$. It is easy to see that $U\\_{A}(a/\\sqrt{3},0,z\\_{A})=-9V\\_{A}$ and $U\\_{B}(a/\\sqrt{3},0,z\\_{A})=U\\_{C}(a/\\sqrt{3},0,z\\_{A})=0$. Therefore, in contrast to *U**A* which attains its minimum, *U**B* and *U**C* both attain their maxima at $(a/\\sqrt{3},0,z\\_{A})$. To get an approximate model close to $(a/\\sqrt{3},0,z\\_{A})$, we expand the optical dipole potentials of all three layers into the polynomials of the relative coordinates $\\tilde{x}=x-a/\\sqrt{3}$, *ỹ* = *y*, and *z̃* = *z* − *z**A*. To the leading order of these relative coordinates, we have\n \n$$\\begin{cases}\nU\\_A(\\tilde{x},\\tilde{y},\\tilde{z})\\simeq-9V\\_{A}+\\frac{8\\pi^{2}V\\_{A}}{a^{2}}(\\tilde{x}^{2}+\\tilde{y}^{2}) +\\frac{18V\\_{A}}{\\rho\\_{0z}^{2}}\\tilde{z}^{2}, \\\\\nU\\_B(\\tilde{x},\\tilde{y},\\tilde{z})\\simeq -\\frac{4\\pi^{2}V\\_{B}}{a^{2}}e^{-2z\\_{A}^{2}/\\rho\\_{0z}^{2}}(\\tilde{x}^{2}+\\tilde{y}^{2}), \\\\\nU\\_C(\\tilde{x},\\tilde{y},\\tilde{z})\\simeq -\\frac{4\\pi^{2}V\\_{C}}{a^{2}}e^{-2(z\\_{A}-z\\_{C})^{2}/\\rho\\_{0z}^{2}}(\\tilde{x}^{2}+\\tilde{y}^{2}). \\\\\n\\end{cases}$$ \nTherefore, for a potential well of *U**A*, the presence of *U**B* and *U**C* reduces the strength of the harmonic oscillator in the *x**y* plane from $\\frac{8\\pi^{2}V\\_{A}}{a^{2}}$ to\n \n$$\\frac{8\\pi^{2}V\\_{A}}{a^{2}}[1-\\frac{V\\_{B}}{2V\\_{A}}e^{-2z\\_{A}^{2}/\\rho\\_{0z}^{2}} -\\frac{V\\_{C}}{2V\\_{A}}e^{-2(z\\_{A}-z\\_{C})^{2}/\\rho\\_{0z}^{2}}]\\equiv\\frac{8\\pi^{2}\\bar{V}\\_{A}}{a^{2}},$$ \nwhere we have defined the reduced parameter *V̄**A*. To the leading order of *z̃*, the harmonic oscillator along the *z* direction is unchanged. Effectively, the presence of *U**B* and *U**C* enhances the relative strength of the approximate harmonic confinement along the *z* direction. The ground state energy of this anisotropic harmonic oscillator model is\n \n$$\\varepsilon\\_{0A}=-9V\\_{A}+\\hbar\\omega\\_{xyA}+\\frac{1}{2}\\hbar\\omega\\_{zA},$$ \nwhere the angular frequencies are\n \n$$\\omega\\_{xyA}=\\frac{4\\pi}{a}\\sqrt{\\frac{\\bar{V}\\_{A}}{m}}, \\hspace{1cm} \\omega\\_{zA}=\\frac{6}{\\rho\\_{0z}}\\sqrt{\\frac{V\\_{A}}{m}}.$$ \nThe analysis for the *B* layer and the *C* layer are the same. For the potential well of the B layer centering at (0, 0, 0), the effective anisotropic harmonic oscillator potential is\n \n$$U\\_B(\\tilde{x},\\tilde{y},\\tilde{z})\\simeq-9V\\_{B}+\\frac{8\\pi^{2}\\bar{V}\\_{B}}{a^{2}}(\\tilde{x}^{2}+\\tilde{y}^{2}) +\\frac{18V\\_{B}}{\\rho\\_{0z}^{2}}\\tilde{z}^{2},$$ \nwhere the relative coordinates are simply *x̃* = *x*, *ỹ* = *y*, and *z̃* = *z*. The reduced oscillator strength in the *x**y* plane is characterized by\n \n$$\\bar{V}\\_{B}=V\\_{B}[1-\\frac{V\\_{A}}{2V\\_{B}}e^{-2z\\_{A}^{2}/\\rho\\_{0z}^{2}}-\\frac{V\\_{C}}{2V\\_{B}}e^{-2z\\_{C}^{2}/\\rho\\_{0z}^{2}}].$$ \nThe ground state energy of this anisotropic harmonic oscillator model is\n \n$$\\varepsilon\\_{0B}=-9V\\_{B}+\\hbar\\omega\\_{xyB}+\\frac{1}{2}\\hbar\\omega\\_{zB},$$ \nwhere the angular frequencies are\n \n$$\\omega\\_{xyB}=\\frac{4\\pi}{a}\\sqrt{\\frac{\\bar{V}\\_{B}}{m}}, \\hspace{1cm} \\omega\\_{zB}=\\frac{6}{\\rho\\_{0z}}\\sqrt{\\frac{V\\_{B}}{m}}.$$ \nFor the potential well of the C layer centering at $(-a/\\sqrt{3},0,z\\_{C})$, the effective anisotropic harmonic oscillator potential is\n \n$$U\\_C(\\tilde{x},\\tilde{y},\\tilde{z})\\simeq-9V\\_{C}+\\frac{8\\pi^{2}\\bar{V}\\_{C}}{a^{2}}(\\tilde{x}^{2}+\\tilde{y}^{2}) +\\frac{18V\\_{C}}{\\rho\\_{0z}^{2}}\\tilde{z}^{2},$$ \nwhere the relative coordinates are $\\tilde{x}=x+a/\\sqrt{3}$, *ỹ* = *y*, and *z̃* = *z* − *z**C*. The reduced oscillator strength in the *x**y* plane is characterized by\n \n$$\\bar{V}\\_{C}=V\\_{C}[1-\\frac{V\\_{A}}{2V\\_{C}}e^{-2(z\\_{C}-z\\_{A})^{2}/\\rho\\_{0z}^{2}}-\\frac{V\\_{B}}{2V\\_{C}}e^{-2z\\_{C}^{2}/\\rho\\_{0z}^{2}}].$$ \nThe ground state energy of this anisotropic harmonic oscillator model is\n \n$$\\varepsilon\\_{0C}=-9V\\_{C}+\\hbar\\omega\\_{xyC}+\\frac{1}{2}\\hbar\\omega\\_{zC},$$ \nwhere the angular frequencies are\n \n$$\\omega\\_{xyC}=\\frac{4\\pi}{a}\\sqrt{\\frac{\\bar{V}\\_{C}}{m}}, \\hspace{1cm} \\omega\\_{zC}=\\frac{6}{\\rho\\_{0z}}\\sqrt{\\frac{V\\_{C}}{m}}.$$ \nThe wave function of the lowest bound state is written generally as\n \n$$\\psi\\_{0}(\\mathbf{r}-\\mathbf{r}\\_{0})=N\\_{0}e^{-\\frac{\\alpha\\_{xy}^{2}}{2}[(x-x\\_{0})^{2}+(y-y\\_{0})^{2}] -\\frac{\\alpha\\_{z}^{2}}{2}(z-z\\_{0})^{2}},$$ \nwhere $\\alpha\\_{xy}=\\sqrt{m\\omega\\_{xy}/\\hbar}$, $\\alpha\\_{z}=\\sqrt{m\\omega\\_{z}/\\hbar}$, and the normalization factor $N\\_{0}=\\alpha\\_{xy}\\sqrt{\\alpha\\_{z}}/\\pi^{\\frac{3}{4}}$. *m*, $a=\\frac{2\\lambda}{3}$, and *ρ*0*z* are the same for all three triangular layers. The angular frequencies are however dependent on the layer index through the above definitions. Correspondingly, according to the position of **r**0 and **r**1, we should put the layer index (*A*, *B*, or *C*) to *ω**x**y*, *ω**z*, and other related quantities.\nWe are now ready to evaluate the hopping amplitudes. For the single-body Hamiltonian defined by Eq.(E2) and the local orbital defined by Eq.(E18), the NN hopping amplitudes defined by Eq.(E5) may be calculated analytically. For the displaced dice lattice, there are six independent NN hopping amplitudes. For the intralayer NN hopping amplitude in the B layer, we take **r**0 = (0, 0, 0) and $\\mathbf{r}\\_{1}=(\\frac{\\sqrt{3}}{2},\\frac{1}{2},0)a$. The result for the integral is\n \n$$\\begin{aligned}\n&&t\\_{NN}(BB)= \\notag \\\\\n&&-\\frac{\\hbar^{2}}{2m}(\\frac{1}{4}\\alpha\\_{xyB}^{4}a^{2}-\\alpha\\_{xyB}^{2}-\\frac{1}{2}\\alpha\\_{zB}^{2})\\exp(-\\frac{1}{4}\\alpha\\_{xyB}^{2}a^{2}) \\notag \\\\\n&&-3\\{[1-\\frac{2}{3}\\exp(-\\frac{b^{2}}{4\\alpha\\_{xyB}^{2}})]V\\_{B}+[1+\\frac{1}{3}\\exp(-\\frac{b^{2}}{4\\alpha\\_{xyB}^{2}})]\\cdot\n\\notag \\\\\n&&\\cdot[V\\_{A}\\exp(-\\frac{2\\alpha\\_{zB}^{2}\\rho\\_{0z}^{2}}{2+\\alpha\\_{zB}^{2}\\rho\\_{0z}^{2}}\\frac{z\\_{A}^{2}}{\\rho\\_{0z}^{2}})\n\\notag \\\\\n&&+V\\_{C}\\exp(-\\frac{2\\alpha\\_{zB}^{2}\\rho\\_{0z}^{2}}{2+\\alpha\\_{zB}^{2}\\rho\\_{0z}^{2}}\\frac{z\\_{C}^{2}}{\\rho\\_{0z}^{2}})]\\}\\cdot\n\\notag \\\\\n&&\\cdot\\sqrt{\\frac{\\alpha\\_{zB}^{2}\\rho\\_{0z}^{2}}{2+\\alpha\\_{zB}^{2}\\rho\\_{0z}^{2}}}\\exp(-\\frac{1}{4}\\alpha\\_{xyB}^{2}a^{2}).\\end{aligned}$$ \nA crucial qualitative feature of the result is the existence of a common exponential factor with the exponent proportional to the square of *a*, which is the distance between **r**0 and **r**1. Because the three layers of the displaced dice lattice are all triangular lattices and their relative positions are similar, we could get the NN hopping amplitudes within the A layer and those within the C layer by suitable substitutions of indices. Without writing out the explicit expressions, the relevant substitutions are simply\n \n*t**N**N*(*A**A*) = *t**N**N*(*B**B*)∣*B* → *A*, *A* → *C*, *C* → *B*, *z**A* → *z**C* − *z**A*, *z**C* → − *z**A*, \n \n*t**N**N*(*C**C*) = *t**N**N*(*B**B*)∣*B* → *C*, *C* → *A*, *A* → *B*, *z**A* → − *z**C*, *z**C* → *z**A* − *z**C*. \nTherefore, the NN intralayer hopping amplitudes in all three layers have an exponential factor, whose negative exponent is proportional to *a*2, the square of the distance between the NN potential wells.\nNow we calculate the three NN interlayer hopping amplitudes. For the NN hopping amplitude between the B layer and the A layer, we choose **r**0 = (0, 0, 0) on the B layer and $\\mathbf{r}\\_{1}=(\\frac{a}{\\sqrt{3}},0,z\\_{A})$ on the A layer. The complete expression for the hopping amplitude is too lengthy and so we split it into four parts\n \n*t**N**N*(*B**A*) = *t**B**A**K* + *t**B**A**A* + *t**B**A**B* + *t**B**A**C*, \nwhere the first term is the matrix element of the kinetic energy term of *Ĥ*0, the remaining three terms are related separately to the three components of the potential energy term of *Ĥ*0. In addition, we define the following abbreviations for the composite quantities\n \n$$\\begin{cases}\n\\eta\\_{21}=\\frac{2\\alpha\\_{xyB}\\alpha\\_{xyA}}{\\alpha\\_{xyB}^{2}+\\alpha\\_{xyA}^{2}}, \\\\\n\\zeta\\_{21}=\\frac{2\\alpha\\_{zB}\\alpha\\_{zA}}{\\alpha\\_{zB}^{2}+\\alpha\\_{zA}^{2}}, \\\\\n\\alpha\\_{xy21}=\\alpha\\_{xyB}\\alpha\\_{xyA}, \\\\\n\\alpha\\_{z21}=\\alpha\\_{zB}\\alpha\\_{zA}. \\\\\n\\end{cases}$$ \nWhen *V**A* and *V**B* are at most slightly different (i.e., *V**A* ≃ *V**B*), which is the case that we focus on, *η*21 ≃ 1, *ζ*21 ≃ 1, *α**x**y*21 ≃ *α**x**y**B*2, and *α**z*21 ≃ *α**z**B*2. The results for the four terms of *t**N**N*(*B**A*) are\n \n$$\\begin{aligned}\nt^{K}\\_{BA}&=&-\\frac{\\hbar^{2}}{2m}\\eta\\_{21}\\sqrt{\\zeta\\_{21}} \\{\\frac{(\\eta\\_{21}\\alpha\\_{xy21})^{2}}{4}(\\frac{a}{\\sqrt{3}})^{2}\n\\notag \\\\\n&&+\\frac{(\\zeta\\_{21}\\alpha\\_{z21})^{2}}{4}z\\_{A}^{2}-\\eta\\_{21}\\alpha\\_{xy21}-\\frac{\\zeta\\_{21}\\alpha\\_{z21}}{2}\\}\\cdot\n\\notag \\\\\n&&\\cdot\\exp[-\\frac{\\eta\\_{21}\\alpha\\_{xy21}}{4}(\\frac{a}{\\sqrt{3}})^{2}-\\frac{\\zeta\\_{21}\\alpha\\_{z21}}{4}z\\_{A}^{2}].\\end{aligned}$$ \n \n$$\\begin{aligned}\nt^{A}\\_{BA}&=&-3V\\_{A}\\eta\\_{21}\\sqrt{\\frac{2\\alpha\\_{z21}\\rho\\_{0z}^{2}}{4+(\\alpha\\_{zB}^{2}+\\alpha\\_{zA}^{2})\\rho\\_{0z}^{2}}}\\cdot\n\\notag \\\\\n&&\\cdot\\{1+\\frac{2}{3}e^{\\frac{-b^{2}}{2(\\alpha\\_{xyB}^{2}+\\alpha\\_{xyA}^{2})}} [\\cos\\frac{4\\pi\\alpha\\_{xyB}^{2}}{3(\\alpha\\_{xyB}^{2}+\\alpha\\_{xyA}^{2})}\n\\notag \\\\\n&&+2\\cos\\frac{2\\pi\\alpha\\_{xyB}^{2}}{3(\\alpha\\_{xyB}^{2}+\\alpha\\_{xyA}^{2})}]\\}\\cdot\n \\\\\n&&\\cdot\\exp[-\\frac{\\eta\\_{21}\\alpha\\_{xy21}}{4}(\\frac{a}{\\sqrt{3}})^{2} -\\frac{(4+\\alpha\\_{zA}^{2}\\rho\\_{0z}^{2})\\alpha\\_{zB}^{2}}{8+2(\\alpha\\_{zB}^{2}+\\alpha\\_{zA}^{2})\\rho\\_{0z}^{2}}z\\_{A}^{2}]. \\notag\\end{aligned}$$ \n \n*t**B**A**B* = *t**B**A**A*∣*V**A* → *V**B*, *α**x**y**A* ↔ *α**x**y**B*, *α**z**A* ↔ *α**z**B*. \n \n$$\\begin{aligned}\nt^{C}\\_{BA}&=&-3V\\_{C}\\eta\\_{21}\\sqrt{\\frac{2\\alpha\\_{z21}\\rho\\_{0z}^{2}}{4+(\\alpha\\_{zB}^{2}+\\alpha\\_{zA}^{2})\\rho\\_{0z}^{2}}}\\cdot\n\\notag \\\\\n&&\\cdot\\{1+\\frac{2}{3}e^{\\frac{-b^{2}}{2(\\alpha\\_{xyB}^{2}+\\alpha\\_{xyA}^{2})}} [\\cos\\frac{2\\pi(\\alpha\\_{xyB}^{2}-\\alpha\\_{xyA}^{2})}{3(\\alpha\\_{xyB}^{2}+\\alpha\\_{xyA}^{2})}\n\\notag \\\\\n&&+2\\cos\\frac{2\\pi(\\alpha\\_{xyB}^{2}+2\\alpha\\_{xyA}^{2})}{3(\\alpha\\_{xyB}^{2}+\\alpha\\_{xyA}^{2})}]\\}\\cdot\n\\notag \\\\\n&&\\cdot\\exp[-\\frac{\\eta\\_{21}\\alpha\\_{xy21}}{4}(\\frac{a}{\\sqrt{3}})^{2}\n\\\\\n&&-\\frac{(\\alpha\\_{z21}\\rho\\_{0z}z\\_{A})^{2}+4(\\alpha\\_{zB}z\\_{C})^{2}+4\\alpha\\_{zA}^{2}(z\\_{C}-z\\_{A})^{2}}{8+2(\\alpha\\_{zB}^{2}+\\alpha\\_{zA}^{2})\\rho\\_{0z}^{2}}]. \\notag\\end{aligned}$$ \nIn comparison to the three NN intralayer hopping amplitudes, the exponential factors have two changes. Firstly, the in-plane exponent is now proportional to $(\\frac{a}{\\sqrt{3}})^{2}$ instead of *a*2. Secondly, there is additional exponential decay associated with the difference in the *z*-coordinates (i.e., depending on *z**A* and *z**C*).\nFor the NN hopping amplitude between the B layer and the C layer, we choose **r**0 = (0, 0, 0) on the B layer and $\\mathbf{r}\\_{1}=(-\\frac{a}{\\sqrt{3}},0,z\\_{C})$ on the C layer. The complete expression for the hopping amplitude is again split into four parts\n \n*t**N**N*(*B**C*) = *t**B**C**K* + *t**B**C**A* + *t**B**C**B* + *t**B**C**C*, \nwhere the first term is the matrix element of the kinetic energy term of *Ĥ*0, the remaining three terms are related separately to the three components of the potential energy term of *Ĥ*0. The four terms of *t**N**N*(*B**C*) turn out to be very similar in expression to the four terms of *t**N**N*(*B**A*), and could be obtained from the expressions shown above by the following substitutions\n \n$$\\begin{cases}\nt^{K}\\_{BC}=t^{K}\\_{BA}|\\_{A\\rightarrow C}, \\\\\nt^{A}\\_{BC}=t^{C}\\_{BA}|\\_{A\\rightarrow C,C\\rightarrow A}, \\\\\nt^{B}\\_{BC}=t^{B}\\_{BA}|\\_{A\\rightarrow C}, \\\\\nt^{C}\\_{BC}=t^{A}\\_{BA}|\\_{A\\rightarrow C}. \\\\\n\\end{cases}$$ \nFrom this similarity, *t**N**N*(*B**C*) show similar dependencies as *t**N**N*(*B**A*) on the parameters of the optical dipole potential.\nFinally, for the NN hopping amplitude between the A layer and the C layer, we choose $\\mathbf{r}\\_{0}=(\\frac{a}{\\sqrt{3}},0,z\\_{A})$ on the A layer and $\\mathbf{r}\\_{1}=(\\frac{a}{2\\sqrt{3}},\\frac{a}{2},z\\_{C})$ on the C layer. The complete expression for the hopping amplitude is also split into four parts\n \n*t**N**N*(*A**C*) = *t**A**C**K* + *t**A**C**A* + *t**A**C**B* + *t**A**C**C*, \nwhere the first term is the matrix element of the kinetic energy term of *Ĥ*0, the remaining three terms are related separately to the three components of the potential energy term of *Ĥ*0. These four terms may also be obtained from the four terms for *t**N**N*(*B**A*) by substitution of parameters as follows\n \n$$\\begin{cases}\nt^{K}\\_{AC}=t^{K}\\_{BA}|\\_{B\\rightarrow C,z\\_{A}\\rightarrow z\\_{A}-z\\_{C}}, \\\\\nt^{A}\\_{AC}=t^{A}\\_{BA}|\\_{B\\rightarrow C,z\\_{A}\\rightarrow z\\_{A}-z\\_{C}}, \\\\\nt^{C}\\_{AC}=t^{A}\\_{AC}|\\_{A\\leftrightarrow C}=t^{A}\\_{BA}|\\_{B\\rightarrow A, A\\rightarrow C,z\\_{A}\\rightarrow z\\_{A}-z\\_{C}}, \\\\\nt^{B}\\_{AC}=t^{C}\\_{BA}|\\_{V\\_{C}\\rightarrow V\\_{B},\\alpha\\_{xyB}\\rightarrow\\alpha\\_{xyC},\\alpha\\_{zB}\\rightarrow\\alpha\\_{zC},z\\_{A}\\leftrightarrow z\\_{A}-z\\_{C}}. \\\\\n\\end{cases}$$ \nIn comparison with *t**N**N*(*B**A*) and *t**N**N*(*B**C*) whose dominant terms have an exponent proportional to − *z**A*2 and − *z**C*2, *t**N**N*(*A**C*) is exponentially smaller because the dominant terms have an exponent proportional to − (*z**A* − *z**C*)2. For example, (*z**A* − *z**C*)2 = 4*z**A*2 in the special case *z**A* = − *z**C*, which amount to a fourth-power decay in the hopping amplitude. On the other hand, we have shown that the interlayer NN hopping amplitudes have an exponent proportional to $(\\frac{a}{\\sqrt{3}})^{2}$ which is $\\frac{1}{3}$ of the corresponding exponent for the three intralayer NN hopping amplitudes. This shows that the intralayer NN hopping amplitudes have a component that is third power smaller than the corresponding factor for the interlayer NN hopping amplitudes. Combining these two trends, it is possible to tune *z**A* and − *z**C* to certain medium values, so that all the four kinds of hopping amplitudes that are neglected in the tight-binding model are much smaller (e.g., smaller by about one to two orders of magnitude) than *t**N**N*(*B**A*) and *t**N**N*(*B**C*) that are retained in the model.\nAs an order-of-magnitude comparison among the six NN hopping amplitudes, let us set *V**A* = *V**B* = *V**C* and *z**A* = − *z**C* = *z*0. For simplicity, we also suppress the layer indices on *α**x**y* and *α**z*. The magnitudes of the various hopping amplitudes are determined by the dominant exponential factors therein. From the above explicit expressions, the dominant exponential factors are exp[ − (*α**x**y**a*)2/4] for *t**N**N*(*α**α*) (*α* = *A*, *B*, *C*), are exp[ − (*α**x**y**a*)2/12 − (*α**z**z*0)2/4] for *t**N**N*(*B**A*) and *t**N**N*(*B**C*), and is exp[ − (*α**x**y**a*)2/12 − (*α**z**z*0)2] for *t**N**N*(*A**C*). To justify the tight-binding model of Eq.(E1), we require the exponents for *t**N**N*(*B**A*) and *t**N**N*(*B**C*) to be at least two orders of magnitude larger than those for the other four NN hopping amplitudes. This amounts to\n \n$$\\exp[-\\frac{(\\alpha\\_{xy}a)^{2}}{6}+\\frac{(\\alpha\\_{z}z\\_{0})^{2}}{4}]<10^{-2},$$ \nand\n \n$$\\exp[-\\frac{3(\\alpha\\_{z}z\\_{0})^{2}}{4}]<10^{-2}.$$ \nEq.(E33) leads to $\\exp[-\\frac{(\\alpha\\_{z}z\\_{0})^{2}}{4}]<(0.01)^{1/3}\\simeq0.2154$, and correspondingly $\\exp[\\frac{(\\alpha\\_{z}z\\_{0})^{2}}{4}]>(100)^{1/3}\\simeq4.642$. Substituting $\\exp[\\frac{(\\alpha\\_{z}z\\_{0})^{2}}{4}]=5$ into Eq.(E32), we get $\\exp[-\\frac{(\\alpha\\_{xy}a)^{2}}{12}]<\\sqrt{0.002}\\simeq0.0447$. This inequality sets a lower bound to the strength of the optical dipole potential, which turns out to be *V̄**α* > 1.004*E**R* or equivalently ℏ*ω**x**y**α* > 4.251*E**R* (*α* = *A*, *B*, *C*). This constraint is weaker than that inferred in Sec.II for the 2D limit. It should always be satisfied for the considered deep lattices. For known strengths of the optical dipole potential and the *ρ*0*z* parameter, $\\exp[\\frac{(\\alpha\\_{z}z\\_{0})^{2}}{4}]\\simeq5$ or larger may easily be fulfilled by tuning *z*0.\nSuppose we have tuned the parameters so that the BA and BC NN interlayer hopping amplitudes dominate among the six NN hopping amplitudes. The tight-binding model has four parameters [*ɛ**a**b*, *ɛ**c**b*, *t**b**a* = *t**N**N*(*B**A*), and *t**b**c* = *t**N**N*(*B**C*)]. Correspondingly, we also have control over four parameters of the optical dipole potential (*V**A*/*V**B*, *V**C*/*V**B*, *z**A* − *z**B* = *z**A*, *z**C* − *z**B* = *z**C*). Therefore, we may tune the four parameters in the tight-binding model freely within the scope of applicability of the model.\nWe mention in passing that, all the tight-binding parameters obtained above may be reexpressed in terms of strengths of the optical potential (i.e., *V**i*, *i* = *A*, *B*, *C*), the geometric parameters (i.e., *ρ*0*z* and *z**i*, *i* = *A*, *B*, *C*) measured in units of the wavelength *λ*, and the recoil energy *E**R*. In particular, it is conventional to express all the energies in units of the recoil energy. The conversion is easy to make through the following relations\n \n$$\\begin{cases}\n\\hbar\\omega\\_{xyi}=3\\sqrt{2}E\\_{R}\\sqrt{\\frac{\\bar{V}\\_{i}}{E\\_{R}}}, \\\\\n\\hbar\\omega\\_{zi}=\\frac{3\\sqrt{2}}{\\pi\\gamma\\_{0}}E\\_{R}\\sqrt{\\frac{V\\_{i}}{E\\_{R}}}, \\\\\n\\alpha\\_{xyi}^{2}\\lambda^{2}=6\\sqrt{2}\\pi^{2}\\sqrt{\\frac{\\bar{V}\\_{i}}{E\\_{R}}}, \\\\\n\\alpha\\_{zi}^{2}\\lambda^{2}=\\frac{6\\sqrt{2}\\pi}{\\gamma\\_{0}}\\sqrt{\\frac{V\\_{i}}{E\\_{R}}}, \\\\\n\\end{cases}$$ \nwhere *γ*0 = *ρ*0*z*/*λ*, *i* = *A*, *B*, *C*. We will not write out the explicit expressions of the tight-binding parameters in terms of this alternative parameter set.\nImplementation of the lattice phase-modulation spectroscopy for the interband transitions\n=========================================================================================\nThe conventional stimulus for the interband transitions of electronic systems in a crystal is the optical field, the electric field component of which causes parity-changing interband excitations. Because the wave vector of the optical field triggering the excitation is much smaller than the scale of the wave vectors in the Brillouin zone (BZ) of the material, the optical excitations are usually taken as vertical. That is, the initial and final electronic states of the transition have the same wave vector. For a light field with the electric field component **E**(**r**, *t*), its interaction with the electronic system is written as\n \n − *q***E**(**r**, *t*) ⋅ **r** = − **F***E**M*(**r**, *t*) ⋅ **r**, \nwhere *q* is the charge of the electron and **F***E**M*(**r**, *t*) is the force acting on the electrons at the position **r** and time *t*. Only the electric field component is retained because the effect of the magnetic field component is negligible. For optical excitations of a 2D or q-2D material, it is usually reasonable to neglect the spatial dependence of the electric field and the force in all directions, and thus **F***E**M*(**r**, *t*) = **F***E**M*(*t*).\nFor the neutral ultracold atoms on an optical lattice, a similar interaction may be constructed by virtue of the analogy between the electromagnetic Lorentz force and the inertial force for the center-of-mass motion. By shaking the optical lattice periodically, the optical dipole potential of the optical lattice oscillate in space in a periodic manner. We assume the modulation **l**(*t*) of the optical lattice (the optical potential) in the laboratory frame is uniform and therefore independent of **r**. Turning to the reference frame comoving with the optical lattice, all ultracold atoms in the optical lattice are subjected to the following inertial force\n \n$$\\mathbf{F}\\_{inertial}(\\mathbf{r},t)=\\mathbf{F}\\_{inertial}(t)=-m\\frac{d^{2}\\mathbf{l}(t)}{dt^{2}}\\equiv-m\\ddot{\\mathbf{l}}.$$ \nThis inertial force in turn amounts to adding the following potential energy to the model for the cold atoms in the comoving frame\n \n*Ĥ**d* = − **F***i**n**e**r**t**i**a**l*(*t*) ⋅ **r**. \nFor a sinusoidally shaken optical lattice, the above formula has the same form as Eq.(F1) for the electronic systems. When transformed to the tight-binding model, we replace **r** by the position operator of the cold atoms. Therefore, sinusoidally shaking the optical lattice may excite the parity-changing interband excitations in the cold atoms, just as the optical field excites the same kind of excitations in electronic systems.\nIn addition to the above heuristic justifications, the interaction term may be derived from a more formal approach. For this purpose, we specify the explicit expression of the sinusoidal modulation **l**(*t*). For generality, we consider an elliptical shaking in the *x**y* plane of the optical lattice. That is, in each period of revolution, **l**(*t*) traces out an ellipse centering at **l**(*t*) = **0**. We assume the major (minor) axis of the ellipse is along **e***a* (**e***b*), with **e***a* ⋅ **e***a* = **e***b* ⋅ **e***b* = 1, **e***a* ⋅ **e***b* = 0, and $\\mathbf{e}\\_{a}\\times\\mathbf{e}\\_{b}=\\hat{\\mathbf{z}}$. We have\n \n**l**(*t*) = *a*cos(*ω*0*t*)**e***a* + *b**η*sin(*ω*0*t*)**e***b*, \nwhere *ω*0 is the angular frequency of the shaking. *a* ≥ *b* ≥ 0 characterize the amplitude of the driving. *η* = ± 1 characterizes the rotating direction (i.e., chirality) of **l**(*t*). Viewing against the $\\hat{\\mathbf{z}}$ direction, the rotation is counterclockwise (clockwise) for *η* = 1 (*η* = − 1). When *b* = 0, the modulation is linear. When *a* = *b*, the modulation is circular. In terms of the following basis set\n \n$$\\begin{cases}\n\\mathbf{e}\\_{1}=\\frac{1}{\\sqrt{2}}(\\mathbf{e}\\_{a}-\\mathbf{e}\\_{b}), \\\\\n\\mathbf{e}\\_{2}=\\frac{1}{\\sqrt{2}}(\\mathbf{e}\\_{a}+\\mathbf{e}\\_{b}), \\\\\n\\end{cases}$$ \nthe sinusoidal modulation is reformulated as \n \n**l**(*t*) = *c*[cos(*ω*0*t* + *φ*)**e**1 + cos(*ω*0*t* − *φ*)**e**2], \nwhere\n \n$$\\begin{cases}\nc=\\sqrt{(a^{2}+b^{2})/2}, \\\\\n\\sin\\varphi=b\\eta/(\\sqrt{2}c), \\\\\n\\cos\\varphi=a/(\\sqrt{2}c). \\\\\n\\end{cases}$$ \nFinally, we may of course express **l**(*t*) in terms of the basis vectors of the *O**x**y**z* Cartesian coordinate system defined referring to the underlying optical lattice. Suppose the major axis (minor axis) of the ellipse is rotated from the *x* axis (*y* axis) by an angle of *α*, we have\n \n$$\\begin{cases}\n\\mathbf{e}\\_{a}=\\hat{\\mathbf{x}}\\cos\\alpha+\\hat{\\mathbf{y}}\\sin\\alpha, \\\\\n\\mathbf{e}\\_{b}=-\\hat{\\mathbf{x}}\\sin\\alpha+\\hat{\\mathbf{y}}\\cos\\alpha. \\\\\n\\end{cases}$$ \nIn the coordinate basis, we express the modulation as\n \n$$\\mathbf{l}(t)=\\delta x(t)\\hat{\\mathbf{x}}+\\delta y(t)\\hat{\\mathbf{y}}.$$ \nIn the special cases with sin*α* = 0 or cos*α* = 0, *δ**x*(*t*) and *δ**y*(*t*) are very simply related to *l**a*(*t*) and *l**b*(*t*) defined in Eq.(F4). In other more general cases, we have\n \n$$\\begin{cases}\n\\delta x(t)=c\\_{1}\\cos(\\omega\\_{0}t+\\varphi\\_{1}), \\\\\n\\delta y(t)=c\\_{2}\\cos(\\omega\\_{0}t+\\varphi\\_{2}), \\\\\n\\end{cases}$$ \nwhere we have defined\n \n$$\\begin{cases}\nc\\_{1}=\\sqrt{a^{2}\\cos^{2}\\alpha+b^{2}\\sin^{2}\\alpha}, \\\\\n\\cos\\varphi\\_{1}=a\\cos\\alpha/c\\_{1}, \\\\\n\\sin\\varphi\\_{1}=b\\eta\\sin\\alpha/c\\_{1}, \\\\\n\\end{cases}$$ \nand\n \n$$\\begin{cases}\nc\\_{2}=\\sqrt{a^{2}\\sin^{2}\\alpha+b^{2}\\cos^{2}\\alpha}, \\\\\n\\cos\\varphi\\_{2}=a\\sin\\alpha/c\\_{2}, \\\\\n\\sin\\varphi\\_{2}=-b\\eta\\cos\\alpha/c\\_{2}. \\\\\n\\end{cases}$$ \nThis general expression in the basis of $\\hat{\\mathbf{x}}$ and $\\hat{\\mathbf{y}}$ is clearly more complex and less comprehensible than either Eq.(F4) or Eq.(F6). Physically, however, all three expressions for **l**(*t*) are completely equivalent.\nIn the presence of the above elliptical modulation **l**(*t*), the optical dipole potential *V*(**r**) becomes *Ṽ*(**r**, *t*) = *V*(**r** − **l**(*t*)). The single-body Hamiltonian for the ultracold atoms in the shaken optical lattice turns into\n \n$$\\hat{\\tilde{H}}=\\frac{\\mathbf{p}^{2}}{2m}+\\tilde{V}(\\mathbf{r},t).$$ \nIn terms of two successive unitary transformations to the Schrödinger equation, the time dependence of the model may be relegated from the potential energy term to a new term of the same form as Eq.(F3). We start from the Schrödinger equation,\n \n$$i\\hbar\\frac{\\partial\\tilde{\\psi}(\\mathbf{r},t)}{\\partial t}=\\hat{\\tilde{H}}\\tilde{\\psi}(\\mathbf{r},t).$$ \nIn the first transformation, we take\n \n$$\\tilde{\\psi}(\\mathbf{r},t)= e^{\\frac{i}{\\hbar}(-\\mathbf{l}\\cdot\\mathbf{p}+\\frac{m}{4}\\mathbf{l}\\cdot\\dot{\\mathbf{l}})}\\tilde{\\psi}\\_{1}(\\mathbf{r},t) \\equiv U\\_{1}(\\mathbf{r},t)\\tilde{\\psi}\\_{1}(\\mathbf{r},t).$$ \nThe factor exp( − *i***l** ⋅ **p**/ℏ) of *U*1 implements a spatial translation of distance **l**. The Hamiltonian becomes\n \n$$\\begin{aligned}\n&&\\hat{\\tilde{H}}\\_{1}=U\\_{1}^{\\dagger}\\hat{\\tilde{H}}U\\_{1}-i\\hbar U\\_{1}^{\\dagger}\\frac{\\partial}{\\partial t}U\\_{1} \\notag \\\\\n&&\\hspace{0.5cm} =\\frac{(\\mathbf{p}-m\\dot{\\mathbf{l}})^{2}}{2m}+V(\\mathbf{r})-\\frac{1}{2}m\\omega\\_{0}^{2}c^{2},\\end{aligned}$$ \nwhere $c=\\sqrt{(a^{2}+b^{2})/2}$ is defined in Eq.(F7). In the second transformation, we take\n \n$$\\tilde{\\psi}\\_{1}(\\mathbf{r},t)= e^{\\frac{i}{\\hbar}m\\dot{\\mathbf{l}}\\cdot\\mathbf{r}}\\tilde{\\psi}\\_{2}(\\mathbf{r},t) \\equiv U\\_{2}(\\mathbf{r},t)\\tilde{\\psi}\\_{2}(\\mathbf{r},t).$$ \nThe Hamiltonian transforms into\n \n$$\\begin{aligned}\n&&\\hat{\\tilde{H}}\\_{2}=U\\_{2}^{\\dagger}\\hat{\\tilde{H}}\\_{1}U\\_{2}-i\\hbar U\\_{2}^{\\dagger}\\frac{\\partial}{\\partial t}U\\_{2} \\notag \\\\\n&&\\hspace{0.5cm} =\\frac{\\mathbf{p}^{2}}{2m}+V(\\mathbf{r})+m\\ddot{\\mathbf{l}}\\cdot\\mathbf{r}-\\frac{1}{2}m\\omega\\_{0}^{2}c^{2}.\\end{aligned}$$ \nIt is easy to see that, the third term in the final expression of $\\hat{\\tilde{H}}\\_{2}$ is exactly the same as Eq.(F3).\nHow could we realize the above sinusoidal modulation in our displaced dice lattice, in experiments? Firstly, we express the general elliptical modulation defined above in terms of the parameters defining our optical lattice. From Eq.(4) of the main text, the positions of the potential minima of the optical lattice are controlled by the phase parameters *ϕ**i*(*j*) (*i*, *j* = 1, 2, 3). Therefore, **l**(*t*) is expressed in terms of the modulations of these phases (i.e., *δ**ϕ**i*(*j*)). Since all three layers are modulated in the same manner, the modulations in these phases are independent of the layer index *i*. We therefore define (*i*, *j* = 1, 2, 3)\n \n*δ**ϕ**i*(*j*)(*t*) = *δ**ϕ*(*j*)(*t*). \nFrom Eq.(4) of the main text, the displacement of the optical lattice in the *x**y* plane is related to these phase modulations through\n \n$$\\begin{cases}\n\\delta x(t)=-\\frac{\\sqrt{3}a}{4\\pi}[\\delta\\phi^{(1)}(t)-\\delta\\phi^{(3)}(t)], \\\\\n\\delta y(t)=\\frac{a}{4\\pi}[\\delta\\phi^{(1)}(t)+\\delta\\phi^{(3)}(t)-2\\delta\\phi^{(2)}(t)], \\\\\n\\end{cases}$$ \nwhere *a* = 2*λ*/3 is the lattice constant of the displaced dice lattice. We assume *δ**ϕ*(*j*)(*t*) ≪ 2*π* (*j* = 1, 2, 3). Therefore, *δ**x*(*t*) and *δ**y*(*t*) are much smaller than the lattice constant *a*, corresponding to a weak stimulus to the system. There are infinitely many approaches of realizing the two displacements *δ**x*(*t*) and *δ**y*(*t*) in terms of three phase modulations *δ**ϕ*(*j*)(*t*) (*j* = 1, 2, 3). One implementation is by setting\n \n$$\\begin{cases}\n\\delta\\phi^{(1)}(t)=\\frac{4\\pi}{\\sqrt{3}a}[-\\frac{1}{2}\\delta x(t)+\\frac{\\sqrt{3}}{2}\\delta y(t)], \\\\\n\\delta\\phi^{(2)}(t)=0, \\\\\n\\delta\\phi^{(3)}(t)=\\frac{4\\pi}{\\sqrt{3}a}[\\frac{1}{2}\\delta x(t)+\\frac{\\sqrt{3}}{2}\\delta y(t)]. \\\\\n\\end{cases}$$ \nThe coefficient $\\frac{4\\pi}{\\sqrt{3}a}$ is the magnitude of the reciprocal lattice vector. In terms of Eqs.(F9)-(F12) and defining\n \n$$\\begin{cases}\nC\\_{\\zeta}=\\frac{1}{2}\\sqrt{c\\_{1}^{2}+3c\\_{2}^{2}+2\\sqrt{3}\\zeta c\\_{1}c\\_{2}\\cos(\\varphi\\_{1}-\\varphi\\_{2})}, \\\\\n\\cos\\varphi\\_{\\zeta}=\\frac{1}{2C\\_{\\zeta}}(\\zeta c\\_{1}\\cos\\varphi\\_{1}+\\sqrt{3}c\\_{2}\\cos\\varphi\\_{2}), \\\\\n\\sin\\varphi\\_{\\zeta}=\\frac{1}{2C\\_{\\zeta}}(\\zeta c\\_{1}\\sin\\varphi\\_{1}+\\sqrt{3}c\\_{2}\\sin\\varphi\\_{2}), \\\\\n\\end{cases}$$ \nwhere *ζ* = ± , we can reformulate Eq.(F21) as\n \n$$\\begin{cases}\n\\delta\\phi^{(1)}(t)=\\frac{4\\pi C\\_{-}}{\\sqrt{3}a}\\cos(\\omega\\_{0}t+\\varphi\\_{-}), \\\\\n\\delta\\phi^{(2)}(t)=0, \\\\\n\\delta\\phi^{(3)}(t)=\\frac{4\\pi C\\_{+}}{\\sqrt{3}a}\\cos(\\omega\\_{0}t+\\varphi\\_{+}). \\\\\n\\end{cases}$$ \nAn efficient and successful method of implementing such phase modulations is to attach piezoelectric actuators to certain optical elements related to the phase factors. According to Fig.3(a) of the main text, the *δ**ϕ*(1)(*t*) [*δ**ϕ*(3)(*t*)] modulation may be realized by mounting the two mirrors M3 and M4 (M1 and M2) on piezoelectric actuators, and setting the motion of the two in step with each other.\nFinally, we list the physical quantities relevant to the interband transitions of fermionic cold atoms confined in the symmetrically biased displaced dice lattice. For the circular modulation that we considered in the main text, we have *a* = *b* and\n \n$$\\mathbf{l}(t)=a[\\cos(\\omega\\_{0}t)\\hat{\\mathbf{x}}+\\eta\\sin(\\omega\\_{0}t)\\hat{\\mathbf{y}}].$$ \nSimilar to the polarized light field, it is more convenient to express the harmonically varying field in the complex representation, which brings the above expression to\n \n$$\\tilde{\\mathbf{l}}(t)=a(\\hat{\\mathbf{x}}+\\eta e^{i\\frac{\\pi}{2}}\\hat{\\mathbf{y}})e^{-i\\omega\\_{0}t}=a(\\hat{\\mathbf{x}}+i\\eta \\hat{\\mathbf{y}})e^{-i\\omega\\_{0}t}.$$ \nIn the basis of $\\hat{\\mathbf{x}}$ and $\\hat{\\mathbf{y}}$, this gives the standard polarization vector (i.e., the Jones vector) $\\mathbf{n}\\_{\\eta}=(1,i\\eta)/\\sqrt{2}$ for the circularly polarized modulation. This modulation exerts a parity-changing perturbation that may trigger an interband transition. The strength of this transition is characterized by the matrix element of the component of the current operator (for the center-of-mass motion of the ultracold atoms) along $\\dot{\\mathbf{l}}(t)$ [$\\dot{\\tilde{\\mathbf{l}}}(t)$]. According to Eq.(F16), $\\dot{\\mathbf{l}}(t)$ [$\\dot{\\tilde{\\mathbf{l}}}(t)$] plays the role of the artificial vector potential. By transforming the Hamiltonian to the wave vector space, the **k**-resolved current operator is represented as\n \n$$\\mathbf{J}(\\mathbf{k})=\\boldsymbol{\\nabla}\\_{\\mathbf{k}}h(\\mathbf{k}).$$ \nTherefore, the intensity of the vertical transition at **k** to a circular modulation characterized by $\\mathbf{n}\\_{\\eta}=(1,i\\eta)/\\sqrt{2}$ is proportional to the matrix element of\n \n$$J\\_{\\eta}(\\mathbf{k})=\\mathbf{J}(\\mathbf{k})\\cdot\\mathbf{n}\\_{\\eta}=\\frac{1}{\\sqrt{2}}[\\frac{\\partial h(\\mathbf{k})}{\\partial k\\_{x}}+i\\eta\\frac{\\partial h(\\mathbf{k})}{\\partial k\\_{y}}].$$ \nIn the basis *ψ***k***σ*† = [*a***k***σ*†, *b***k***σ*†, *c***k***σ*†], the tight-binding model in the **k** space is\n \n$$\\hat{H}=\\sum\\limits\\_{\\mathbf{k},\\sigma}\\psi^{\\dagger}\\_{\\mathbf{k}\\sigma}h(\\mathbf{k})\\psi\\_{\\mathbf{k}\\sigma}.$$ \nThe Hamiltonian matrix that is independent of the spin label is\n \n$$h(\\mathbf{k})=\\begin{pmatrix} \\varepsilon\\_{ab} & \\xi\\_{ab}(\\mathbf{k}) & 0 \\\\\n \\xi\\_{ab}^{\\ast}(\\mathbf{k}) & 0 & \\xi\\_{bc}(\\mathbf{k}) \\\\\n 0 & \\xi\\_{bc}^{\\ast}(\\mathbf{k}) & \\varepsilon\\_{cb}\n \\end{pmatrix},$$ \nwhere\n \n$$\\begin{cases}\n\\xi\\_{ab}(\\mathbf{k})=t\\_{ba}(1+e^{i\\mathbf{k}\\cdot\\mathbf{a}\\_{1}}+e^{i\\mathbf{k}\\cdot\\mathbf{a}\\_{2}}), \\\\\n\\xi\\_{bc}(\\mathbf{k})=t\\_{bc}(1+e^{i\\mathbf{k}\\cdot\\mathbf{a}\\_{1}}+e^{i\\mathbf{k}\\cdot\\mathbf{a}\\_{2}}). \\\\\n\\end{cases}$$ \nFor *ɛ**a**b* = − *ɛ**c**b* = Δ and *t**b**a* = *t**b**c* = *t*0, the above model reduces to the biased and displaced dice lattice that we studied in the main text. As was shown in the main text, the band structure of this model has three bands. Each band is twofold degenerate by spin. We suppress the spin label hereafter since it is irrelevant to the present discussions and will only give a factor of two to the final results.\nWe take the normalized eigenvector for the flat band at energy *E*0(**k**) = 0 as\n \n$$\\langle\\psi\\_{0}(\\mathbf{k})|=\\frac{1}{E(\\mathbf{k})}\\begin{pmatrix} -\\xi^{\\ast}(\\mathbf{k}), & \\Delta, & \\xi(\\mathbf{k}) \\end{pmatrix},$$ \nwhere *ξ*(**k**) = *t*0(1 + *e**i***k** ⋅ **a**1 + *e**i***k** ⋅ **a**2). For the two dispersive bands with energies *E**ν*(**k**) = *ν**E*(**k**) (*ν* = ± ), we take\n \n$$\\langle\\psi\\_{\\nu}(\\mathbf{k})|=\\frac{|\\xi(\\mathbf{k})|}{E(\\mathbf{k})}\n\\begin{pmatrix} \\frac{\\xi^{\\ast}(\\mathbf{k})}{E(\\mathbf{k})-\\nu\\Delta"},"data_id":{"kind":"string","value":"arxiv_0002853"}}},{"rowIdx":699,"cells":{"content":{"kind":"string","value":"% of that in the parent compound, where the 23Na spectrum is split by 540 kHz.\nThe NMR study of Oh *et al.* reports two species of antiferromagnetism in a sample with *x* ≈ 0.017, one with a large ordered moment (*s*-AFM) and the other one with a small moment (*w*-AFM). Our data from lower dopings are consistent with the finding of a small volume fraction of a *w*-AFM phase, but we find [inset, Fig, [tn1](b)] that this volume fraction does not change with doping. This result indicates that the appearance of the *w*-AFM phase is not an intrinsic consequence of Co doping; if it is a disorder effect then it must be of a different type, perhaps with its origin in a strain or chemical inhomogeneity. In Fig. [tn1](d), we present the spin-lattice relaxation rate, 1/23*T*1*T*, measured at the peak frequencies of both the *s*-AFM and the *w*-AFM signals. As noted in Sec. II the magnetization recoveries at the two separate frequencies each follow the single-component function expected for *I* = 3/2 nuclei. The relaxation rates for both signals fall at the same ordering temperature *T**N*. For the *w*-AFM component, a phase having such a small ordered moment but a high onset temperature *T**N* is generically very unlikely, and we suggest that the consistent explanation is a microscopic phase separation into AFM and PM regimes, but with weak magnetic order (appearing as the *w*-AFM phase) induced in the PM phase by its proximity to the *s*-AFM one. We note in addition that the significantly faster 23*T*1 below *T**N* in the *w*-AFM phase [Fig. [tn1](d)] is also consistent with spin fluctuations being only partially suppressed by a weak proximity effect.\nThe concept of “nanoscale phase separation” in iron-based superconductors is known from the depleted iron selenide materials *A*2Fe4Se5 (A = K, Rb, Cs, Tl; “245”), which appear to show a robust AFM phase accompanied by an equally robust but quite separate PM phase; the latter is the only part of the system to turn SC at *T**c*, forming a percolating SC phase despite having a volume fraction below 10%. However, it is generally thought that this phase separation is primarily a consequence of vacancy-induced structural inhomogeneity, causing a clear doping inhomogeneity, whereas our results (previous paragraph) appear to exclude this in NaFe1 − *x*Co*x*As. Here we observe for *x* = 0.013 [Fig. [tn1](d)] that 1/23*T*1*T* shows a similar fall in both the *s*- and *w*-AFM signals at *T**c*, which is approximately 10 K in a field of 11.5 T. Although this result implies that the same type of SC state sets in at all lattice sites, we caution that the drop in 1/23*T*1*T* is not at all sharp and the superconductivity is weak at best. We find only a very modest decrease in 1/23*T*1*T*, by a factor of two from 10 K down to 2 K, whereas the data at *x* = 0.019, which we present in Sec. IV, show a much larger drop of 1/23*T*1*T* below *T**c* for the PM phase. For completeness, we comment here that we also did not find an appreciable decrease in the Knight shifts (23*K* or 75*K*) below *T**c* in the *w*-AFM phase (data not shown). However, it is worth noting that NMR results showing similar drops in 1/*T*1*T* far below *T**N* for undoped NaFeAs and CaFe2As2 systems have been interpreted as a type of activated behavior of magnetic domain-wall motion. We leave to a future study the investigation of whether the *w*-AFM phase may in fact arise from magnetic domain walls, whose characteristic width gives the small but doping-insensitive volume fraction we observe.\nIn this regime we can only report that our current data are not sufficient to differentiate between a scenario of microscopic coexistence, which would be expected to show a far clearer signal, and a scenario where the PM regime is a domain-boundary phase whose SC coherence length, unlike 245, exceeds the domain size (the length scale of the nanoscopic phase separation), causing proximity superconductivity to pervade the entire magnetic regime. We can state that our results are fully consistent with STM data for a similarly underdoped system (*x* = 0.014), which show phase inhomogeneity, a SC gap on all sites, and a strong anticorrelation (competition) between the AFM and SC order, a topic we discuss next (Sec. IV). To summarize our analysis of the underdoped regime, our data for the constant *w*-AFM volume fraction and the common magnetic onset temperatures for *s-* and *w*-AFM are strong evidence in support of proximity magnetism in the *w*-AFM/PM phase. We return in Sec. VI to a detailed discussion of the phase separation between the PM and *s*-AFM regions, and of its implications for iron-based superconductivity.\n[spec2](Color online) 23Na central line for the NaCo175 crystal at selected temperatures in fields of (a) 1 T and (b) 11.5 T applied along the *c*-axis. Solid curves are fits to the spectrum with one or more Gaussian functions. (c) Temperature dependence of the magnetic (MVF) and superconducting (SVF) volume fractions at each field, deduced from the Gaussian fits. (d) Field dependence of the onset temperatures of static antiferromagnetism, *T**N**o**n*, and superconductivity, *T**c**o**n*. \nField-Controlled Volume Competition\n===================================\nWe turn next to our results for crystals with slightly higher doping levels, *x* = 0.0175 (which we label NaCo175) and *x* = 0.0190 (NaCo190). The NaCo175 sample has a lower onset Néel temperature and smaller ordered moment, while NaCo190 shows no long-ranged magnetic order, and so these samples represent the evolution of the system to optimal doping.\nFigure [spec2](a) shows the 23Na spectra at selected temperatures in a field of 1 T oriented along the crystal *c*-axis. The spectrum at 30 K shows a narrow line with a FWHM of approximately 5 kHz. On cooling to 25 K, a shoulder feature develops at both sides of the peak, which on further cooling grows in weight, whereas the weight of the center peak decreases. This feature indicates the development of two phases at low temperature, with PM sites giving the sharp center peak and magnetic sites giving the broad shoulders, which in contrast to undoped NaFeAs show a wide distribution of local fields. Here *T**N**o**n* = 25 K is the onset temperature of antiferromagnetism. By measuring the RF inductance, we find that the onset temperature of superconductivity is *T**c**o**n* = 20 K, and below 16 K the spectrum becomes too small to detect because of strong RF screening in the SC phase.\nAs shown in Fig. [spec2](a), the spectra below *T**N* can be fitted by two superposed Gaussian functions. We deduce the magnetic volume fraction (MVF) from the ratio of the spectral weight, taken from the Gaussian fit, of the magnetic (shoulder) feature to the total weight. The MVF at a field of 1 T is shown as a function of temperature in Fig. [spec2](c), where it clearly starts to develop at *T**N**o**n* = 25 K, and increases with cooling. At *T* = 16 K, the MVF reaches 90%, indicating that the sample is almost entirely magnetic. The average ordered moment for the magnetic part can also be estimated from the NMR spectrum, as the extension of the shoulder away from the central peak reflects the increase of internal static field (ordered moment). At *T* = 16 K, the FWHM of the magnetic part of the spectrum is approximately 150 kHz, which corresponds to 28% of the moment in the parent compound NaFeAs, or in other words an average moment of *μ* ≃ 0.09*μ**B*/Fe with a spatially inhomogeneous distribution. This behavior suggests that AFM order develops in islands below *T**N**o**n* and enlarges on cooling both in moment size and especially in volume fraction.\nAt higher magnetic fields, both *T**N**o**n* and the MVF are suppressed. Figure [spec2](b) shows 23Na spectra in a field of 11.5 T applied along ${\\hat c}$. The spectrum is single-peaked and sharp above 20 K, with no shoulder feature and hence no static magnetism. Below this, the shoulder appears and we fit the spectrum with three Gaussian functions to account for the center (PM) and shoulder (magnetic) components. In Fig. [spec2](c), the MVF at 11.5 T is seen to increase on cooling, similar to the low-field data, but with a lower onset temperature (*T**N**o**n* = 20 K) and a lower MVF ≃ 50% at 12 K. Similar results for intermediate fields, also shown in Fig. [spec2](c), demonstrate the continuous nature of these effects.\nThe most striking feature of Fig. [spec2](c) occurs at the onset of superconductivity (*T**c**o**n*). Above *T**c**o**n*, the MVF increases monotonically on cooling at a fixed field, but below *T**c* it falls away; at 11.5 T (*T**c* = 18 K), the MVF decreases from 50% at 12 K to 18% at 2 K. This behavior demonstrates a direct competition for volume fraction between antiferromagnetism and superconductivity, which is also visible in the spectra shown in Fig. [spec2](b). In Fig. [spec2](d), we show both *T**N**o**n* and *T**c**o**n* as functions of field. *T**N* shows a quite significant decrease with field, which can be fitted by the functional form $T\\_N = T\\_N (0) \\sqrt{1 - (H/H\\_c)^2}$, producing an estimate of the critical field for *T**N**o**n* = 0 to occur at *H**c* ≈ 19 ± 1 T. By contrast, the field-induced decrease of *T**c**o**n* is slower, consistent with the critical field for superconductivity being located at *H**c*2 ≈ 50 T in this system, and suggesting that antiferromagnetism is suppressed at far lower fields than superconductivity. We comment here that such strong field effects on *T**N* are highly unusual in iron-based SC materials, where the in-plane magnetic interactions are normally many tens of meV, and we stress that this result is obtained only for our samples close to optimal doping. For the underdoped samples discussed in Sec. III, we found no significant field-induced changes either to *T**N* or to the *s*-AFM volume fraction up to 12 T (data not shown). We return to this issue in Sec. VI.\nIndeed our MVF data demonstrate that superconductivity is more stable than antiferromagnetism, replacing it at low temperatures for all fields. We will show later that for *x* = 0.0175, superconductivity occupies the PM phase, but not the AFM phase, during the replacement process. If the RF screening is non-uniform and strong in the SC regions, then our data provide an upper bound for the MVF below *T**c*. The drop of MVF on cooling indicates that the SC volume fraction, meaning the fraction of a percolating SC state, increases. However, the average moment in the magnetic regions remains large, *μ* ≃ 0.09*μ**B*/Fe, when *T* ≪ *T**c*, and therefore we observe that AFM and SC order compete over the system volume, excluding each other in a first-order manner rather than coexisting with a reduced order parameter. Although competitive behavior of antiferromagnetism and superconductivty has been reported by neutron scattering studies of the Ba(Fe1 − *x*Co*x*)2As2 system, these cannot distinguish whether the phases compete by order-parameter suppression (second order) or volume suppression (first order). Our unambiguous demonstration of volume competition in NaFe1 − *x*Co*x*As is a key result whose origin and implications we discuss in Sec. VI.\n[invt1t3](Color online) (a) Spin-lattice relaxation rate 1/23*T*1*T* measured on 23Na for NaCo175, shown as a function of temperature with a field of 11.5 T applied along ${\\hat c}$. Dotted lines denote *T**N* and *T**c*, circles are data measured at the peak of the spectrum (PM phase), and diamonds are measured in the broad shoulder (AFM phase). Solid lines are guides to the eye. (b) 1/75*T*1*T* measured on 75As for NaCo175 as a function of temperature with the same field. Inset: 75As NMR spectra at selected temperatures. Solid lines are Gaussian fits to the data. \nTurning now to further details of magnetism in the NaCo175 sample, we note that the observed shoulder spectrum of 23Na is consistent with the *s*-AFM phase reported in Sec. III, but with the lower *T**N* value expected at a higher doping. However, there is no longer any evidence for ordered moments in the 23Na PM signal, suggesting that the *w*-AFM phase is absent at this doping. We confirm that the absence of ordered moments is not a resolution issue by comparing with the 75As NMR spectrum. As noted in Sec. II, 75As has a much stronger hyperfine coupling than 23Na, and therefore its strong sensitivity to any weak magnetic order makes it the optimal probe for excluding a *w*-AFM component in the PM signal. As shown in the inset of Fig. [invt1t3](b), the 75As spectrum is single-peaked at all temperatures. A narrow line with FWHM ≃ 14 kHz is observed above *T**N**o**n* = 20 K, and below this its spectral weight begins to decrease due to the increasing AFM volume fraction. The AFM signal lies outside our 75As measurement window, because its FWHM is very large (it can be estimated from the 23Na data to be around 2.4 MHz). The single-peaked form of the 75As spectrum below *T**N**o**n* is consistent with the PM 23Na signal, which remains sharp on cooling. Below *T**c**o**n* = 16 K, the spectra shift downward, as expected for singlet superconductivity, and at this point the 23Na spectrum does become broadened; at *T* = 6 K, far below *T**c*, we observe a FWHM ≃ 24 kHz as a consequence of the vortex structure in the SC phase. This value of the FWHM sets a strict limit on the ordered moment of the PM phase, which should be less than 0.6% of the moment in NaFeAs (0.32*μ**B*/Fe), and thus effectively excludes any possibility of *w*-AFM character at *x* = 0.0175.\nNext we focus on the SC state of the NaCo175 sample. At low temperatures, the PM phase is found to be purely superconducting by inspection of the spin-lattice relaxation rates for both 23Na and 75As. Figure [invt1t3](a) shows 1/23*T*1*T* in a field applied along the *c*-axis; above *T**N**o**n* we observe a decrease on cooling down to 40 K, followed by an increase on further cooling below 40 K. The high-temperature behavior is consistent with local-moment fluctuations and the low-temperature upturn with the spin fluctuations of itinerant electrons at the Fermi surface. Below *T**N**o**n*, 1/23*T*1*T* is no longer uniform, showing a different form if taken at different parts of the spectrum. For the shoulder, 1/23*T*1*T* drops quickly to a small value below *T**N**o**n*, as a consequence of the onset of static AFM order. For the peak, 1/23*T*1*T* continues to increase on cooling, falling only when the SC state is reached. Figure [invt1t3](b) shows the corresponding results taken from 75As, which are naturally uniform because the spectra [inset, Fig. [invt1t3](b)] have only a PM peak and no shoulders. From *T* = 220 K down to 80 K, 1/75*T*1*T* decreases linearly with temperature due to thermal excitation of local spin fluctuations in 2D. Below 80 K, the relaxation rate increases strongly with the 1/75*T*1*T* = *A*/(*T* − Θ) form characteristic of low-energy itinerant spin fluctuations. The uniform sharp drop of 1/*T*1*T* at *T**c* for the PM signal of both nuclei indicates that the PM state becomes fully SC and it is believed from ARPES measurements that a full gap opens at all points on the Fermi surface; however, we comment that NMR data have not been able to verify this second point directly (the apparent linear form of 1/23*T*1*T* visible around *T* = *T**c*/2 in Fig. [invt1t3](a), which may be of extrinsic or intrinsic origin).\n[spec4](Color online) (a) 23Na and (b) 75As NMR spectra for NaCo190, shown at selected temperatures a field of 11.5 T applied along ${\\hat c}$.\nThe spin-lattice relaxation rates in Fig. [invt1t3](a) suggest further that the *s*-AFM phase is not strongly coupled to superconductivity. Measurements of 1/23*T*1*T* for magnetic sites (in the shoulder of the spectrum) show a drop at *T**N**o**n*, where magnetic order sets in, but there is no discernible drop at *T**c*. Thus there is no evidence that the *s*-AFM phase supports even weak or proximity superconductivity. We stress that 1/23*T*1*T* at 12 K and 11.5 T, deep within the ordered phases, reaches a similar value for sites in both the SC and the *s*-AFM regions. This appears to be a clear statement that electrons on the Fermi surface are gapped by either type of order, and, taken together with Fig. [spec2](c), that relaxation contributions become dominated by SC electrons at low temperatures. This again reflects the fact that the competition between antiferromagnetism and superconductivity for electrons on the Fermi surface (i.e. in reciprocal space), and its apparent first-order nature, results in the volume competition (in real space) we observe in the NMR spectra.\nFinally, we comment that further information concerning the volume-competition effect can be gained by investigating samples with higher doping, namely *x* ≃ 0.019. Our NaCo190 sample shows a structural transition at *T**s* ≃ 35 K, which we discuss in detail in Sec. V. Figures [spec4](a) and (b) show respectively the 23Na and the 75As spectra for different temperatures. For 23Na, the spectrum is single-peaked and no shoulder feature develops on cooling, even down to 2 K. This observation excludes the existence not only of a possible *w*-AFM component but also of the *s*-AFM phase. For 75As, the spectrum also has a single peak, with FWHM ≃ 40 kHz at the lowest temperatures, which also excludes any type of AFM order. Below 18 K, the spectra shift to lower frequencies and a line broadening is clearly visible for 75As, which is the hallmark of the onset of singlet-pairing superconductivity. Thus the effect of doping on volume competition is to terminate the battle in favor of superconductivity at *x* ≃ 0.019, where one finds a single, uniform phase with only structural and SC transitions, but a complete absence of AFM order.\nPhase Diagram\n=============\nWe now compile all of our results, from samples across the full range of doping, to prepare a definitive (*x*, *T*) phase diagram. First, the structural transition can be detected from the frequency of a chosen satellite line in the 75As spectrum, which is shown in Fig. [sus5](a) as a frequency shift relative to the center line. When the field is applied in the *a**b*-plane, cooling from the tetragonal to the orthorhombic phase causes each satellite to shift and to split into two due to sample twinning. The sudden change of the satellite frequency as a function of temperature, clearly visible in Fig. [sus5](a), determines *T**s* for the structural transition at each doping. We comment here that above *T**s* this satellite frequency, which for perfect field alignment is the quadrupole frequency *ν**Q*, is generally expected to show a systematic increase with sample doping ; however, such a dependence is barely discernible in our data [Fig. [sus5](a)] due to the fact that the maximum doping we studied in the NaFe1 − *x*Co*x*As system is so low (*x* = 0.023). The variation in our measured values of *f* − *f*0 above *T**s* for the different samples is in fact dominated by the small but finite misalignment of the magnetic field, whose exact orientation with respect to the crystalline *a*-and *b*-axes was not determined. NMR provides an accurate measurement of *T**s* up to *x* = 0.019, beyond which the sample remains tetragonal at all temperatures.\n[sus5](Color online) (a) Determination of the structural transition temperature *T**s* from the frequency shift *f* − *f*0 of the satellite line in the 75As spectrum measured with an in-plane field, shown for a range of sample dopings. (b) Determination of the SC onset temperature *T**c* from measurements of the inductance change Δ*Z*/*Z*0 of the NMR coil at zero field, also for a range of dopings. (c) Phase diagram showing the values of *T**N* determined from the NMR spectra of Figs. [tn1](a) and [tn1](b), *T**s* determined from the satellite line shift, and *T**c* determined from the RF inductance. \nNext, the transition to static magnetic order can be detected at all dopings by the decrease of the PM spectral weight, as shown in Figs. [tn1](a) and (b). Finally, the SC transition is detected by the relative inductance change of the NMR coil, as noted in Sec. II and shown in Fig. [sus5](b); these measurements can also be benchmarked from the drop of the NMR Knight shift and the spin-lattice relaxation rate, at least for *x* ≥ 0.0175 (see Sec. III). In Fig. [sus5](c) we present the hierarchy of deduced transition temperatures, which show the clear evolution characteristic of iron-based superconductors. In the parent compound, NaFeAs, the system becomes fully magnetic below *T**N* ≃ 41.5 K, but as electron doping is induced by Co substitution, the antiferromagnetism is gradually suppressed and superconductivity develops.\n[pd6](Color online) (a) Temperature dependence of 1/75*T*1*T* for NaFe1 − *x*Co*x*As crystals under a field of 11.5 T applied in the *a**b*-plane of the crystal. Solid lines are fits to the function 1/75*T*1*T* = *A*/(*T* − Θ) + *B**T* + *C**T*2. Inset: 75*T*1*T* as a function of temperature. *T**s* and Θ*o* denote respectively the temperatures where 75*T*1*T* changes slope and where it goes to zero. (b) Phase diagram of NaFe1 − *x*Co*x*As established by NMR: *T**s*, *T**c*, Θ*o*, and *T**N* denote respectively the structural transition, the SC transition temperature, and the Curie-Weiss temperature, all obtained from the fit to 1/75*T*1*T* above *T**c*, and the Néel temperature obtained from the PM spectral weight [Figs. [tn1](a) and (b)]. Data for *T**c* below *x* = 0.0175 are taken only from inductance measurements [Fig. [sus5](b)]. *x*1 denotes the approximate doping where the *s*-AFM+*w*-AFM region and the *s*-AFM+PM region meet and *x*2 denotes the doping where *T**N* = 0 and beyond which AFM order is absent.\nHowever, in NaFe1 − *x*Co*x*As there is a complementary approach to obtaining the phase diagram, which is that all three phase transitions can be detected directly and accurately in the spin-lattice relaxation rate. This is not possible in other pnictide materials, and as we explain below it is also a consequence of the strongly 2D fluctuations acting in the 111 system. In Fig. [pd6](a) we show 1/75*T*1*T*, obtained over the full range of available dopings with the field applied in the *a**b*-plane, a geometry known to maximize the sensitivity of the measurement to the dominant in-plane spin fluctuations in pnictides as the (*π*, 0) magnetically ordered state is approached. At high temperatures, the 1/75*T*1*T* values are similar for all dopings other than the heavily doped *x* = 0.07 sample, and fall slowly on cooling down to 100 K. Below 100 K, a Curie-type upturn develops for all dopings (other than *x* = 0.07), becoming progressively stronger for lower doping values. Clearly the low-energy spin fluctuations responsible for this behavior, which have an itinerant origin, are suppressed strongly by increasing doping, reflecting a very high sensitivity to details of the Fermi surface. By contrast, the very small change above 100 K suggests an origin in local-moment (valence-electron) fluctuations, consistent with a general two-component interpretation of the spin response in pnictides.\nIf the low-temperature 1/75*T*1*T* data are fitted with the function 1/75*T*1*T* = *A*/(*T* − Θ), one may extract a Curie temperature Θ. The structural transition can be deduced by considering the quantity 75*T*1*T*, as shown in the inset of Fig. [pd6](a) for different underdoped and near-optimal dopings. To a good approximation, 75*T*1*T* = (*T* − Θ)/*A* can be fitted with two straight lines of different gradients, one (which we denote Θ*t*) corresponding to the tetragonal structure and the other, Θ*o*, to the orthorhombic one. The change in this gradient gives the structural transition temperature *T**s*. The values of Θ*t* and Θ*o* can be obtained from the intercept of the fitting line with 75*T*1*T* = 0.\nWe find that the values of Θ*t* (obtained by extrapolation from above *T**s*) are all negative and decrease with doping, but these are of limited physical meaning because the tetragonal structure is replaced at *T**s*, perhaps precisely because it does not allow the magnetic transition from which the orthorhombic phase can profit. By contrast, Θ*o*, which is determined from the structure and data below *T**s*, is positive and decreases with doping up to *x* = 0.019. The fact that Θ*o* is far larger than Θ*t* reinforces the evidence that the low-energy spin fluctuations are strongly enhanced below *T**s*, suggesting a clear role for the magnetic sector in driving the structural transition. Although there is already significant evidence for a coupling between the lattice structure and the magnetism in 111 materials, this coupling is manifestly not strong enough that *T**N* coincides with *T**s*, as it does in the 122 system. As noted in Sec. I, some authors have taken the lack of coincidence between *T**N* and *T**s* as a hallmark of strong two-dimensionality in some iron pnictide materials.\nAs noted above, the Néel temperature *T**N* is determined directly from the decrease of the PM spectral weight [Figs. [tn1](a), [tn1](b), [sus5](c)]. For NaFe1 − *x*Co*x*As, the fitting parameter Θ*o*, which we obtain from our data below *T**s*, has exactly the same value as *T**N* for *x* < 0.0175. This extremely significant result has not been reported in any other iron pnictide systems. *T**N* is a true measure of static AFM order, setting in due to all of the couplings in the system. By contrast, Θ*o* is a measure of the 2D spin fluctuations of the FeAs planes. For quasi-2D systems, the energy scale of in-plane fluctuations (*J*) sets a characteristic temperature $T\\_{\\rm BKT}$ in the spin response, but true long-range order is forbidden by the Mermin-Wagner theorem. However, a weak coupling *J**c* between these planes is sufficient to create long-range order in three dimensions and the transition temperature will be $T\\_N \\approx\nT\\_{\\rm BKT}$. Only when *J**c* becomes a significant fraction of *J* will the planar physics be supplemented by conventional three-dimensional fluctuations and *T**N* will exceed $T\\_{\\rm BKT}$ by an amount depending on *J**c*. The magnetic interactions in the parent compound NaFeAs have been measured directly in a very recent study of spin-wave dispersion relations by inelastic neutron scattering. These authors find in-plane couplings *S**J*1*a* ≃ 40 meV, *S**J*1*b* ≃ 16 meV, and *S**J*2 ≃ 19 meV, respectively for superexchange processes in the *a*, *b*, and diagonal (*a* ± *b*) directions in the FeAs plane, where *S* is the effective spin (moment) on the Fe ions, but an interplanar coupling of only *S**J**c* ≃ 1.8 meV. Thus our observation of perfect coincidence between Θ*o* and *T**N* is completely consistent with the measured value *J**c*/*J* ∼ 0.1, reflecting the minimal contributions from interplanar coupling, and these results form the best evidence yet available for the extremely 2D nature of the NaFeAs system. We defer a discussion of the microscopic implications of this result to Sec. VI.\nWe are now in a position to present the complete phase diagram, which is shown in Fig. [pd6](b). We begin by drawing attention to the error bars on the doping (*x*) axis, which given the extreme sensitivity of the system to small changes in *x* is critical information. As noted in Sec. II, we do not possess probes capable of determining the doping level to the 0.1% accuracies mandated by the physics of NaFe1 − *x*Co*x*As, and so we have taken the nominal doping in every case. However, we are able to benchmark our samples against each other and the remarkably smooth evolution in properties, as well as the perfect match between the *T**s* datasets, shown in Figs. [sus5](c) and [pd6](b), indicates that our conservative estimate of the doping error as ± 0.2% is reasonable. This smooth evolution over such a fine range of doping also supports the suggestion that NaFe1 − *x*Co*x*As provides some of the most homogeneous and highest-quality crystals of any iron-based SC material. Returning to the matching of datasets between Figs. [sus5](c) and [pd6](b), the agreement in *T**s* benchmarks the value of the analysis of Fig. [pd6](a); the agreement in *T**c* is perfect for *x* ≥ 0.0175, but the values extracted from 1/75*T*1*T* at lower dopings are inaccurate for the reasons discussed in Sec. III, and so the data shown in Fig. [pd6](b) for the underdoped samples are those of Fig. [sus5](c); the agreement between *T**N* and Θ*o* is remarkable for the reasons discussed above.\nOur NMR data [Figs. [sus5](c) and [pd6](b)] provide direct evidence for a complete separation of *T**s* and *T**N* at all dopings in NaFe1 − *x*Co*x*As, with the structural transition always occurring some 12 − 15 K above the magnetic one. Thus, in common with the majority of iron-based SC systems, the coupling between the lattice and the magnetism is “subcritical” in the sense of a combined transition. The *T**s* line terminates at a slightly higher doping than *T**N*, and in fact at a value coinciding with optimal doping, as indicated also in transport studies. The *T**c* line is quite flat in the doping range 0.017 < *x* < 0.022, whereas all of *T**s*, Θ*o*, and *T**N* change dramatically. In the zero-temperature limit, antiferromagnetism and superconductivity cohabit for 0 < *x* < 0.0175, as shown in Sec. IV [Fig. [spec2](c)], but within the phase-separated *s*-AFM and PM (*w*-AFM) regions (Sec. III). We comment again here that the onset of the *w*-AFM phase at the same temperature, *T**N*, as the *s*-AFM one suggests that the former is actually a set of narrow PM regions ordered by a proximity effect. Above *T**c*, the *s*-AFM and *w*-AFM phases “coexist,” by which we mean “cohabit as phase-separated regions,” in the doping range *x* < *x*1 and the *s*-AFM and PM phases cohabit over the range *x*1 < *x* < *x*2, as represented in Fig. [pd6](b). Our data fix the values *x*1 ≃ 0.0175 and *x*2 ≃ 0.018.\nTo determine the detailed structure of the phase diagram close to optimal doping, we show in Fig. [pd6](b) all of the values of Θ*o* determined from the Curie-Weiss fit below *T**s*. As noted above, Θ*o* coincides with *T**N* for *x* ≤ 0.0175. However, for doping *x* = 0.019, Θ*o* determined from the 1/75*T*1*T* curve above *T**c* predicts a finite Néel temperature (13.5 K) below *T**c* (22 K). In a scenario where antiferromagnetism and superconductivity coexist, one might expect a magnetic quantum critical point (Θ*o* = 0) to occur at *x* ≈ 0.02. However, we demonstrated at the end of Sec. IV that AFM order is completely absent for *x* = 0.019. Thus we conclude that in NaFe1 − *x*Co*x*As, the appropriate scenario for the regime *T**c* > Θ*o* is a complete suppression of long-range AFM order by volume competition (Sec. IV).\nGiven that Θ*o* (*T**N*) = 25 K at *x* = 0.0175 and Θ*o* = 13.5 K at *x* = 0.019, it is reasonable to estimate that the AFM transition line touches the SC dome at *x* ≈ 0.018, although we do not have samples with this doping. Because superconductivity suppresses the MVF below *T**c*, we expect that the *s*-AFM transition line is pushed to lower dopings below *T**c*, as also represented in the slope of the line to (*x*2, 0) in Fig. [pd6]. For all dopings (*x*) beyond this line, our experiments show unequivocally that the system is single-phased with only one transition, the onset of superconductivity in a structurally tetragonal system.\nDiscussion\n==========\nOur NMR measurements across the phase diagram in NaFe1 − *x*Co*x*As reveal a number of key features, including the first-order volume competition, the microscopic phase separation, and the dominance of superconductivity at low temperatures. We discuss these points in turn, finding that their common denominator is the two-dimensional nature of the NaFeAs system.\nBeginning with the volume competition effect, the large changes of the MVF we observe (Sec. IV) as functions of both temperature and field suggest a mutual exclusion of AFM and SC order in real space. Such a real-space competition implies a first-order transition between two states with finite order parameters, which would appear to reflect a strong competition in reciprocal space, meaning for electrons at the Fermi surface. Certainly the fact that the low-energy spin fluctuations, which are due to itinerant (Fermi-surface) contributions, are fully gapped below both *T**N* and *T**c* for the *x* = 0.0175 sample suggests that both types of order, stabilized by their own particular Fermi-surface electronic order parameter ($\\langle\nc\\_{\\bf k \\uparrow}^{\\dag} c\\_{{\\bf k} + {\\bf Q} \\downarrow} \\rangle$ or $\\langle\nc\\_{{\\bf k}\\uparrow}^{\\dag} c\\_{- {\\bf k} \\downarrow}^{\\dag} \\rangle$), compete for the same electrons.\nSuch a temperature- and field-controlled magnetic and SC volume fraction has not been reported in any other iron pnictides. We suggest that there are two reasons why this highly unconventional phenomenon has been observed (to date) only in NaFe1 − *x*Co*x*As. One is the extreme 2D nature of the Fermi surfaces and the other is the very fine control of the doping level, which is obtained in NaFeAs systems with no loss of chemical homogeneity.\nAddressing first the two-dimensionality, the question of whether electronic correlation effects result in competition or coexistence between antiferromagnetism and superconductivity has been fraught with contradictions in the iron-based SC systems. Microscopic coexistence of AFM and SC order has been reported in several compounds with the 122 structure, including by some of us. AFM and SC states in iron-based SC materials depend rather sensitively on the interactions of quasiparticles at the Fermi surface, and as a result it was suggested in Ref. that the ability of an iron-based SC system to host both types of order may be dictated by the variety of projected 2D Fermi surfaces available. The *c*-axis band dispersion in BaFe2As2 is quite significant, making it possible that the 122 structure may allow the coexistence of the two types of order for electrons on different parts of the Fermi surface. In Ba(Fe1 − *x*Ru*x*)2As2, the fact that 1/75*T*1*T* drops due to gap formation at *T**N* and again at *T**c* demonstrates that additional SC electrons are present in the AFM phase. By contrast, this dispersion is much weaker in the NaFe1 − *x*Co*x*As system, as shown both by ARPES studies and by the present results (Sec. V), indicating a highly 2D system with little flexibility in Fermi-surface sizes and spanning wave vectors. This leaves little option for the AFM and SC order but to “fight it out” for the available electrons, leading to the strong competition we observe (Sec. IV): in NaFe1 − *x*Co*x*As, 1/23*T*1*T* drops only at *T**N* or at *T**c*, and to the same low-temperature value, showing directly that antiferromagnetism and superconductivity compete for the same electrons. Our results suggest strongly that the microscopic coexistence of AFM and SC order is not possible in NaFe1 − *x*Co*x*As, which among other things should exclude any intrinsic superconductivity on sites with *s*-AFM order in the underdoped region (Sec. III).\nOn this note, we turn next to the issue of phase separation, for which we find evidence throughout the phase diagram. Clearly distinguishable *s*-AFM and *w*-AFM regions are present in the underdoped system (Sec. III), although the volume fraction of the *w*-AFM phase barely changes with either doping or temperature and there is only one magnetic transition. At near-optimal underdopings, there is clear phase separation between the AFM and PM/SC regions, demonstrated very explicitly by the large changes of AFM and SC volume fractions we observe in our NaCo175 sample as functions of both temperature and field (Sec. IV), which strongly suggest a first-order transition between the two phases. The phase diagram of Fig. [pd6](b) shows directly that the optimal *T**c* is achieved when both the AFM and orthorhombic phases are suppressed. Although this type of phase diagram has been interpreted as an incipient AFM quantum critical point of the orthorhombic system, in reality its suppression by the onset of the mutually exclusive (volume-competitive) SC phase is the dominant physics.\nSuch a complex phase diagram may also exist in other iron-based SC systems, although, for reasons of the fine doping control mentioned above, none has yet been resolved in the kind of detail possible in NaFe1 − *x*Co*x*As. Within this intricate phase structure, the inevitable presence of weak disorder could certainly provide an extrinsic origin for the phase separation of *s*-AFM and *w*-AFM regions in the underdoped regime. However, we appeal again to the evidence for remarkably high sample homogeneity in NaFe1 − *x*Co*x*As and to the obvious fact of very low carrier densities. If the origin of phase separation between the *s*-AFM and *w*-AFM regions is the same as that between the *s*-AFM and PM/SC regions near optimal doping, then such behavior may in fact be intrinsic in systems with strong electronic correlations. In cuprate materials, the stripe phase may be considered as an atomic-scale phase separation of AFM and PM/SC regions. Similar phase separation has been reported recently close to the phase boundary both in an organic superconductor and in heavy-fermion superconductors tuned by pressure and magnetic field.\nComplete nanoscale phase separation is familiar in iron-based SC materials from the case of the 245 iron selenides. However, as noted in Sec. III, the situation in NaFe1 − *x*Co*x*As does not appear to be the same, first in that structural and hence doping inhomogeneity is significant in 245 materials and second in that weak superconductivity can be observed throughout the sample below *T**c*. In Sec. III we outlined two scenarios for the present result, a microscopic coexistence or a microscopic phase separation, and, as noted above, the results of Sec. IV make a coexistence appear extremely unlikely. Although we cannot claim evidence for a nanoscale phase separation from our data, all of our results are consistent with such a scenario, under the proviso that the length scale of the phase-separation phenomenon be extremely short. We have observed a phase susceptible to an apparent bulk magnetic order by proximity effects, which is a definite statement that the phase-separation length scale should be short compared to the magnetic correlation length. The weakness and feeble onset of the apparent bulk SC regime could be the fingerprints of proximity superconductivity originating in narrow PM regions but spreading throughout the very small magnetic domains. Microscopically, in the absence of doping inhomogeneity effects, the scale of the phenomenon is expected to result from a subtle interplay between electronic correlations and lattice or charge inhomogeneities (independent of Co doping), and could indeed be on the nanometer scale.\nFirst-order phase separation close to the AFM quantum phase transition has certainly been suggested in other iron-based SC materials, most notably BaFe2 − *x*Ni*x*As2, although the fact that *T**s* and *T**N* merge in the 122 systems may cause qualitative differences in the phase diagram. One key proposal from these neutron scattering studies is an incommensurate nature of the resulting AFM phase. If the direct volume competition of the AFM and SC phases we observe below *T**c**o**n* takes place in a nanoscale lamellar structure, then an incommensurate AFM signal may indeed be observed. A different interpretation, namely a double quantum critical point, has been offered from combined transport and NMR studies in BaFe2 − *x*Ni*x*As2, but the magnetic structure below *T**c* was not resolved. NMR measurements on Ba(Fe1 − *x*Co*x*)2As2 suggest a cluster spin-glass phase close to optimal doping ; in NaFe1 − *x*Co*x*As this type of physics can be excluded explicitly from our data, which do not contain the stretched spin recovery of a spin glass, indicating again the high quality of our samples.\nOne of the fundamental questions in formulating a microscopic model for the iron-based SC materials is the equitable treatment of local-moment (valence-electron) and itinerant (conduction-electron, or Fermi-surface) contributions to the macroscopic properties of magnetism and pairing. Returning again to the key observation (Sec. IV) of large changes in the MVF with both temperature and field, these suggest a microscopic phase separation, which one expects to be driven by a first-order magnetic quantum phase transition. Because the field is applied along the *c*-axis, which is perpendicular to the direction of the ordered moments, a spin-flop transition cannot account for the observed field effect. Rather, the strong suppression of *T**N**o**n* by an applied field near optimal doping (Sec. III; the rate is approximately − 0.5 K/T close to *H* = 0), the classical critical behavior with increasing field [Fig. [spec2](d)], and the extreme field sensitivity to the competing SC phase close to the putative magnetic quantum critical point, all suggest that the interactions causing AFM order are largely local in nature.\nIn this context it is worth remarking that for overdoped systems up to *x* = 0.07, where low-energy spin fluctuations are entirely absent [Fig. [pd6](a)] but *T**c* is still approximately 10 K, it is local-moment spin fluctuations originating in the valence electrons that provide the pairing interactions for superconductivity. High-pressure NMR studies in overdoped NaFe1 − *x*Co*x*As samples demonstrate direct contributions to superconductivity from both itinerant-electron spin fluctuations (which are pressure-dependent) and local-moment fluctuations (which are largely pressure-independent). Thus it is clear that both local-moment magnetism and itinerant electrons are required for a complete understanding of the nature of antiferromagnetism and superconductivity, and in this light we revisit the key result of Sec. V that Θ*o* = *T**N*. The low-energy spin fluctuations causing the Curie-Weiss upturn in 1/*T*1*T*, and hence determining Θ*o*, are due only to Fermi-surface electrons, so the equality with *T**N* indicates only very weak contributions to long-range order from an inter-plane interaction *J**c* (Sec. V), which from the previous paragraph we conclude is mediated by local-moment fluctuations. This result, demonstrating clearly the extremely 2D nature of the NaFeAs system, suggests that the cornerstone of a microscopic model should be the itinerant contribution to (planar) magnetic order, without which the local-moment interactions appear quite unable to order alone.\nFinally, we comment that the suppression of antiferromagnetism by superconductivity at low temperatures for the *x* = 0.0175 sample, even though *T**N* exceeds *T**c*, indicates that the SC phase possesses additional electronic or magnetic channels with which it can “win” against AFM order. A full microscopic model of the band structure, Fermi surfaces, and correlation effects due to valence-electron contributions is required to account for this effect, and here we can only present the possibilities suggested by our data. One may be that superconductivity competes only with the AFM tendencies of the itinerant electrons, which in other iron-based superconductors are reinforced by sizeable local-moment contributions, but that (previous paragraph) the extremely 2D nature of NaFeAs weakens this link to the point that the AFM phase is disrupted completely.\nThe microscopic physics underlying such behavior will be found naturally in an orbital-specific model. Even in a fully 2D system, the electronic bands of iron pnictides contain five different *d*-orbitals, and hence many degrees of freedom in orbital symmetries and admixtures. Antiferromagnetism and superconductivity are favored by itinerant electrons in bands of different orbital content, and the unique feature of NaFe1 − *x*Co*x*As is that the band occupations, and thus their Fermi surfaces, are inordinately sensitive to the doping because of the highly 2D nature of the system. A recent ARPES study of NaFe1 − *x*Co*x*As samples with small and large *x* has provided some indications for orbital-selective connections between the competing AFM and SC phases, specifically concerning the relative *d**x**y* and *d**x**z*/*d**y**z* content of the bands near the Fermi surface. We suggest that similarly detailed studies of samples near optimal doping have the potential to reveal the underlying physics of NaFeAs.\nIn summary, by using NMR as a local probe sensitive to both antiferromagnetism and superconductivity, we observe a strong volume competition between the two phases at the boundary of the antiferromagnetic phase transition in NaFe1 − *x*Co*x*As. The volume fractions of the two phases can be controlled by varying both the temperature and the applied magnetic field, and show a complete mutual exclusion in real space. Thus our NMR data support a first-order phase transition between antiferromagnetism and superconductivity, which is driven by the competition between their electronic order parameters in reciprocal space. As striking as the volume competition effect is the exquisite sensitivity of the competition to doping, with optimal doping and all of the phase-separation effects occuring between 0 and 2%. These phenomena have their origin in the extremely weak interplane coupling in the NaFeAs materials, resulting in a very two-dimensional nature of the electronic band structure, and hence of the Fermi surfaces. One key anomaly compared to other iron-based superconductors is the winning of superconductivity over antiferromagnetism in real space at moderate underdopings, even where the magnetic transition temperature is higher, suggesting that the generic behavior of a two-dimensional iron pnictide may be for the electronic or magnetic channels of the Fermi-surface electrons to favor superconductivity. Further, because the very weak Co doping also appears to be remarkably homogeneous, NaFe1 − *x*Co*x*As is an excellent system in which to seek evidence of unconventional phases arising purely due to intrinsic electronic correlations. For this we obtain additional information concerning the *w*-AFM minority phase, which may be a paramagnetic regime occurring as thin lamellae due to nanoscale phase separation, but appears antiferromagnetic by proximity for underdoped samples and is the first region to turn superconducting on the approach to optimal doping.\nWe are grateful to T. Giamarchi, S. P. 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Dioguardi, J. Crocker, A. C. Shockley, C. H. Lin, K. R. Shirer, D. M. Nisson, M. M. Lawson, N. apRoberts-Warren, P. C. Canfield, S. L. Bud’ko, S. Ran, and N. J. Curro, Phys. Rev. Lett. **111**, 207201 (2013).\nZ. R. Ye, Y. Zhang, M. Xu, Q. Q. Ge, Q. Fan, F. Chen, J. Jiang, P. S. Wang, J. Dai, W. Yu, B. P. Xie, and D. L. Feng, unpublished (arxiv:1303.0682).\n\nPhase separation, competition, and volume-fraction control in NaFe1 − *x*Co*x*As\n================================================================================\nWe report a detailed nuclear magnetic resonance (NMR) study by combined 23Na and 75As measurements over a broad range of doping to map the phase diagram of NaFe1 − *x*Co*x*As. In the underdoped regime (*x* ≤ 0.017), we find a magnetic phase with robust antiferromagnetic (AFM) order, which we denote the *s*-AFM phase, cohabiting with a phase of weak and possibly proximity-induced AFM order (*w*-AFM) whose volume fraction *V* ≃ 8% is approximately constant. Near optimal doping, at *x* = 0.0175, we observe a phase separation between static antiferromagnetism related to the *s*-AFM phase and a paramagnetic (PM) phase related to *w*-AFM. The volume fraction of AFM phase increases upon cooling, but both the Néel temperature and the volume fraction can be suppressed systematically by applying a *c*-axis magnetic field. On cooling below *T**c*, superconductivity occupies the PM region and its volume fraction grows at the expense of the AFM phase, demonstrating a phase separation of the two types of order based on volume exclusion. At higher dopings, static antiferromagnetism and even critical AFM fluctuations are completely suppressed by superconductivity. Thus the phase diagram we establish contains two distinct types of phase separation and reflects a strong competition between AFM and superconducting phases both in real space and in momentum space. We suggest that both this strict mutual exclusion and the robustness of superconductivity against magnetism are consequences of the extreme two-dimensionality of NaFeAs.\nIntroduction\n============\nCompeting electronic phases underlie a number of the most unconventional phenomena in condensed matter systems. When this competition is sufficiently strong, the usual outcome is a phase separation. One outstanding example of this situation is provided by materials showing colossal magnetoresistance, where competing magnetic interactions lead to phase separation between conducting ferromagnetic and insulating antiferromagnetic (AFM) regions. As a consequence, an external magnetic field can be used to control the resistance over many orders of magnitude, offering possible applications in electronic devices. In cuprate superconductors, the competition between antiferromagnetism and superconductivity forms the basis for the majority of the observed phenomena and for several classes of materials the debate can be phrased in terms of the extent to which phase separation is the outcome. The stripe phase, which has been the object of heated research interest for two decades, can be considered as a form of atomic-scale phase separation between AFM and superconducting (SC) regions, and such self-organizing heterostructures are a direct reflection of the electronic correlations whose effects are essential to understanding the mechanism of high-temperature superconductivity.\nThe competition between AFM and SC phases also forms the foundation for the physics of iron-based superconductors, where it is manifest in the emergence of a tetragonal SC phase upon doping- or pressure-induced suppression of an orthorhombic AFM phase. Iron-based superconductors have in common a quasi-two-dimensional atomic structure of weakly coupled FeAs or FeSe planes, although the exact crystal structure varies somewhat among the 1111, 122, 111, and 11 families of materials ; in fact the phenomena we report here will highlight some of the important differences arising between families as a consequence of the strength of their interplane coupling. Currently, the detailed phase diagram close to optimal doping (the concentration giving the maximum SC transition temperature, *T**c*) remains hotly debated, with evidence cited in favor of phase coexistence, of a possible AFM quantum critical point, or of heterostructures of AFM and SC phases. The levels of doping and disorder, and their impact on the phases and their competition, seem to vary between structural families, defying any search for universal properties. However, this variety does open additional avenues in the search for novel forms of phase separation or heterostructure formation, and with them the scope for obtaining further clues to the mechanism of high-*T**c* superconductivity.\nThe 111 family is based on the materials LiFeAs and NaFeAs, with doping effected most easily as NaFe1 − *x*Co*x*As. The parent compound NaFeAs has a separate structural transition (*T**s* ≃ 55 K) and magnetic transition (*T**N* ≃ 41 K), the latter to an AFM phase with small ordered moments (*μ* ≃ 0.32*μ**B*/Fe). Here we choose to use the notation *T**N*, rather than *T**S**D**W*, to reflect the strong local-moment character of the magnetic phase, an issue to which we return in Sec. VI. The separation of *T**s* and *T**N*, and also the relatively low *T**N* values, count among the initial pieces of evidence for a rather weak interlayer coupling between FeAs planes in the NaFeAs system. The crystal quality, particularly the homogeneity of dopant distribution, is thought to be among the best in any iron-based superconductors, as measured in transition widths and observed by scanning tunneling microscopy (STM). Nevertheless, for underdoped 111 compounds a coexistence of inhomogeneous antiferromagnetism and superconductivity has been suggested by transport, angle-resolved photoemission spectroscopy (ARPES), and STM measurements. By contrast, the “coexistence” of a strongly ordered AFM phase (*s*-AFM) and a weakly ordered one (*w*-AFM) on different spatial sites (this situation may be denoted more specifically as a “cohabitation”) has been reported from NMR measurements. Clearly a phase inhomogeneity is observed in all of these studies. However, a detailed analysis of the intrinsic properties of the primary phases, of the exact phase diagram around optimal doping, and of the different phase volume fractions, is still required.\nIn this paper, we exploit the power of NMR as a completely local probe to resolve the appearance and properties of the different AFM and SC phases in NaFe1 − *x*Co*x*As. For low dopings, we confirm the cohabitation of two regimes, *s*-AFM and *w*-AFM, finding that the *w*-AFM phase has a constant volume fraction of order 8%; this indicates an intrinsic effect unrelated to the doping concentration and we suggest that the *w*-AFM phenomenon is actually a proximity-induced moment distribution in a paramagnetic (PM) phase. For dopings around optimal, we find at *x* ≃ 0.0175 (*T**c* ≃ 20 K) the onset of regions of antiferromagnetism below 25 K, where the AFM volume grows with cooling but the application of a magnetic field suppresses both *T**N* and the magnetic volume fraction. At lower temperatures, superconductivity enters in the PM phase and its volume fraction increases at the expense of the AFM region both on cooling and (somewhat paradoxically) with increasing field. For *x* ≃ 0.019 (*T**c* ≃ 22 K), superconductivity suppresses not only AFM order but also the critical AFM fluctuations below *T**c*, forming in the terminology of some authors the mechanism by which the AFM quantum critical point is “avoided.”\nOur results present direct evidence for the mutual exclusion of antiferromagnetism and superconductivity, which leads to a “volume competition” between regions of established (finite-order-parameter) phases replacing each other in space in a first-order manner. This volume competition can be controlled systematically by the temperature and magnetic field, and we suggest that it exists in many other iron-based superconductors. Our detailed studies of the spin-lattice relaxation rate across the phase diagram indicate the importance of both itinerant (conduction-electron) and local-moment (valence-electron) contributions to both types of order. A theoretical interpretation of the strong competition points to the key role of the very two-dimensional (2D) Fermi surfaces in 111 systems and to orbital-selective phenomena depending on the specific bands involved at the different Fermi surfaces.\nThe structure of this paper is as follows. In Sec. II we summarize our basic sample properties and measurement procedures. We begin the presentation of our results in Sec. III by considering the nature of the AFM phases in the underdoped regime. In Sec. IV we focus on our samples close to optimal doping to elucidate the nature of phase cohabitation and volume competition. With these results in hand, in Sec. V we complete a detailed phase diagram for the NaFe1 − *x*Co*x*As system. Section VI contains an interpretation of our results and a discussion of their implications for the understanding of superconductivity in iron-based materials, concluded by a short summary.\nMaterials and Methods\n=====================\nOur NaFe1 − *x*Co*x*As single crystals are synthesized by the flux-grown method with NaAs as the flux. The Co doping levels are monitored by the inductively coupled plasma (ICP) technique. However, ICP measurements are subject to significant inaccuracies and are by no means appropriate to establish the doping concentrations to the degree of precision required to study the NaFe1 − *x*Co*x*As system, where all doping levels are anomalously low (optimal doping $x\\_{\\rm opt} = 1.9$%). Here we report the nominal stoichiometries of the different crystals and establish their relative doping values from our physical measurements by seeking continuity and possible irregularities (Sec. V).\nThe SC transition temperature *T**c* was determined *in situ* by the sudden decrease in inductance of the NMR coil. The zero-field *T**c* values at different dopings are consistent with earlier reports. We have performed NMR measurements on both the 23Na and 75As nuclei, with the field applied both within the crystalline *a**b*-plane and along the *c*-axis. We use a TecMag spectrometer and obtain the NMR spectra from the Fourier transform of the spin-echo signal. The spin-lattice relaxation rates 1/23*T*1 and 1/75*T*1 were measured by the inversion method and all magnetization recovery rates could be fitted well with the function 1 − *I*(*t*)/*I*0 = *A*(0.1*e*− *t*/*T*1 + 0.9*e*− 6*t*/*T*1) (appropriate for *I* = 3/2 nuclei).\nFor detecting the magnitude of the different ordered magnetic moments in the NaFe1 − *x*Co*x*As system, it is important to be able to use both the 23Na and 75As spectra, in order to exploit their very different relative hyperfine coupling strengths, 75*A**h**f*/23*A**h**f* ≃ 23. Because 23Na has a rather weak hyperfine coupling, the resulting narrow line width makes it a very accurate probe of magnetically ordered states with inhomogeneous moment distributions. By contrast, 75As has a strong hyperfine coupling, making it sensitive to very weak ordered moments and ideal for proving the absence of magnetic order in a true PM phase.\n[tn1](Color online) (a) Spectral weight of the 75As PM signals of NaFe1 − *x*Co*x*As samples with five different dopings *x*, shown as a function of temperature and measured with a field of 11.5 T applied along the *c*-axis. The sharp loss of spectral weight indicates the onset of AFM ordering. (b) Spectral weight of the 23Na signal at the center frequency as a function of temperature for four sample dopings. The inset shows the residual spectral weight of the center line at low temperatures. (c) 23Na spectra for the parent compound (*x* = 0) at temperatures above and far below *T**N*. (d) Spin-lattice relaxation rate 1/23*T*1*T* for *x* = 0.013; dotted lines mark the onset temperatures *T**N* and *T**c* for AFM and SC order at the measurement field of 11.5 T.\nNature of Underdoped Phase Separation\n=====================================\nWe begin by considering our underdoped samples to investigate the potentially inhomogeneous AFM phases reported previously. The onset temperature for the transition to magnetic order can be determined from the 75As and 23Na NMR spectra. Figure [tn1](a) shows the spectral weight of the center peak of the 75As spectrum as a function of temperature for five underdoped samples. This quantity is the PM signal and it drops sharply at the onset of magnetic order, as the character of the magnetic environment is altered and spectral weight is transferred away from the center. We determine the Néel temperature *T**N* at each doping and we note that, far below *T**N*, the spectral weight appears to decrease to zero, indicating no residual PM phase at any doping. In Fig. [tn1](b) we show the spectral weight of 23Na at the center frequency for four dopings up to *x* = 0.013. Again the sharp drop of spectral weight indicates the onset of AFM order. However, a residual 8% spectral weight persists far below *T**N* for all dopings below *x* = 0.017 [inset, Fig. [tn1](b)]. The spectra for the parent phase (*x* = 0), shown in Fig. [tn1](c), contain a PM signal with one center peak and two satellites above *T**N*. Far below *T**N*, there is a clear spectral splitting due to strong magnetic order, accompanied by a residual peak in the center. This result provides a good example of the sensitivity of 23Na measurements: our data demonstrate that a very weak AFM order must be present to account for the residual spectral weight and line-width broadening. This is consistent with the absence of the 75As PM signal, although the strong hyperfine coupling of 75As makes it difficult to discern the nature of the magnetic state. Below *T**N*, the line width of the 23Na spectrum is approximately 35 kHz, and therefore the upper bound on the ordered moments is only 6.5% of that in the parent compound, where the 23Na spectrum is split by 540 kHz.\nThe NMR study of Oh *et al.* reports two species of antiferromagnetism in a sample with *x* ≈ 0.017, one with a large ordered moment (*s*-AFM) and the other one with a small moment (*w*-AFM). Our data from lower dopings are consistent with the finding of a small volume fraction of a *w*-AFM phase, but we find [inset, Fig, [tn1](b)] that this volume fraction does not change with doping. This result indicates that the appearance of the *w*-AFM phase is not an intrinsic consequence of Co doping; if it is a disorder effect then it must be of a different type, perhaps with its origin in a strain or chemical inhomogeneity. In Fig. [tn1](d), we present the spin-lattice relaxation rate, 1/23*T*1*T*, measured at the peak frequencies of both the *s*-AFM and the *w*-AFM signals. As noted in Sec. II the magnetization recoveries at the two separate frequencies each follow the single-component function expected for *I* = 3/2 nuclei. The relaxation rates for both signals fall at the same ordering temperature *T**N*. For the *w*-AFM component, a phase having such a small ordered moment but a high onset temperature *T**N* is generically very unlikely, and we suggest that the consistent explanation is a microscopic phase separation into AFM and PM regimes, but with weak magnetic order (appearing as the *w*-AFM phase) induced in the PM phase by its proximity to the *s*-AFM one. We note in addition that the significantly faster 23*T*1 below *T**N* in the *w*-AFM phase [Fig. [tn1](d)] is also consistent with spin fluctuations being only partially suppressed by a weak proximity effect.\nThe concept of “nanoscale phase separation” in iron-based superconductors is known from the depleted iron selenide materials *A*2Fe4Se5 (A = K, Rb, Cs, Tl; “245”), which appear to show a robust AFM phase accompanied by an equally robust but quite separate PM phase; the latter is the only part of the system to turn SC at *T**c*, forming a percolating SC phase despite having a volume fraction below 10%. However, it is generally thought that this phase separation is primarily a consequence of vacancy-induced structural inhomogeneity, causing a clear doping inhomogeneity, whereas our results (previous paragraph) appear to exclude this in NaFe1 − *x*Co*x*As. Here we observe for *x* = 0.013 [Fig. [tn1](d)] that 1/23*T*1*T* shows a similar fall in both the *s*- and *w*-AFM signals at *T**c*, which is approximately 10 K in a field of 11.5 T. Although this result implies that the same type of SC state sets in at all lattice sites, we caution that the drop in 1/23*T*1*T* is not at all sharp and the superconductivity is weak at best. We find only a very modest decrease in 1/23*T*1*T*, by a factor of two from 10 K down to 2 K, whereas the data at *x* = 0.019, which we present in Sec. IV, show a much larger drop of 1/23*T*1*T* below *T**c* for the PM phase. For completeness, we comment here that we also did not find an appreciable decrease in the Knight shifts (23*K* or 75*K*) below *T**c* in the *w*-AFM phase (data not shown). However, it is worth noting that NMR results showing similar drops in 1/*T*1*T* far below *T**N* for undoped NaFeAs and CaFe2As2 systems have been interpreted as a type of activated behavior of magnetic domain-wall motion. We leave to a future study the investigation of whether the *w*-AFM phase may in fact arise from magnetic domain walls, whose characteristic width gives the small but doping-insensitive volume fraction we observe.\nIn this regime we can only report that our current data are not sufficient to differentiate between a scenario of microscopic coexistence, which would be expected to show a far clearer signal, and a scenario where the PM regime is a domain-boundary phase whose SC coherence length, unlike 245, exceeds the domain size (the length scale of the nanoscopic phase separation), causing proximity superconductivity to pervade the entire magnetic regime. We can state that our results are fully consistent with STM data for a similarly underdoped system (*x* = 0.014), which show phase inhomogeneity, a SC gap on all sites, and a strong anticorrelation (competition) between the AFM and SC order, a topic we discuss next (Sec. IV). To summarize our analysis of the underdoped regime, our data for the constant *w*-AFM volume fraction and the common magnetic onset temperatures for *s-* and *w*-AFM are strong evidence in support of proximity magnetism in the *w*-AFM/PM phase. We return in Sec. VI to a detailed discussion of the phase separation between the PM and *s*-AFM regions, and of its implications for iron-based superconductivity.\n[spec2](Color online) 23Na central line for the NaCo175 crystal at selected temperatures in fields of (a) 1 T and (b) 11.5 T applied along the *c*-axis. Solid curves are fits to the spectrum with one or more Gaussian functions. (c) Temperature dependence of the magnetic (MVF) and superconducting (SVF) volume fractions at each field, deduced from the Gaussian fits. (d) Field dependence of the onset temperatures of static antiferromagnetism, *T**N**o**n*, and superconductivity, *T**c**o**n*. \nField-Controlled Volume Competition\n===================================\nWe turn next to our results for crystals with slightly higher doping levels, *x* = 0.0175 (which we label NaCo175) and *x* = 0.0190 (NaCo190). The NaCo175 sample has a lower onset Néel temperature and smaller ordered moment, while NaCo190 shows no long-ranged magnetic order, and so these samples represent the evolution of the system to optimal doping.\nFigure [spec2](a) shows the 23Na spectra at selected temperatures in a field of 1 T oriented along the crystal *c*-axis. The spectrum at 30 K shows a narrow line with a FWHM of approximately 5 kHz. On cooling to 25 K, a shoulder feature develops at both sides of the peak, which on further cooling grows in weight, whereas the weight of the center peak decreases. This feature indicates the development of two phases at low temperature, with PM sites giving the sharp center peak and magnetic sites giving the broad shoulders, which in contrast to undoped NaFeAs show a wide distribution of local fields. Here *T**N**o**n* = 25 K is the onset temperature of antiferromagnetism. By measuring the RF inductance, we find that the onset temperature of superconductivity is *T**c**o**n* = 20 K, and below 16 K the spectrum becomes too small to detect because of strong RF screening in the SC phase.\nAs shown in Fig. [spec2](a), the spectra below *T**N* can be fitted by two superposed Gaussian functions. We deduce the magnetic volume fraction (MVF) from the ratio of the spectral weight, taken from the Gaussian fit, of the magnetic (shoulder) feature to the total weight. The MVF at a field of 1 T is shown as a function of temperature in Fig. [spec2](c), where it clearly starts to develop at *T**N**o**n* = 25 K, and increases with cooling. At *T* = 16 K, the MVF reaches 90%, indicating that the sample is almost entirely magnetic. The average ordered moment for the magnetic part can also be estimated from the NMR spectrum, as the extension of the shoulder away from the central peak reflects the increase of internal static field (ordered moment). At *T* = 16 K, the FWHM of the magnetic part of the spectrum is approximately 150 kHz, which corresponds to 28% of the moment in the parent compound NaFeAs, or in other words an average moment of *μ* ≃ 0.09*μ**B*/Fe with a spatially inhomogeneous distribution. This behavior suggests that AFM order develops in islands below *T**N**o**n* and enlarges on cooling both in moment size and especially in volume fraction.\nAt higher magnetic fields, both *T**N**o**n* and the MVF are suppressed. Figure [spec2](b) shows 23Na spectra in a field of 11.5 T applied along ${\\hat c}$. The spectrum is single-peaked and sharp above 20 K, with no shoulder feature and hence no static magnetism. Below this, the shoulder appears and we fit the spectrum with three Gaussian functions to account for the center (PM) and shoulder (magnetic) components. In Fig. [spec2](c), the MVF at 11.5 T is seen to increase on cooling, similar to the low-field data, but with a lower onset temperature (*T**N**o**n* = 20 K) and a lower MVF ≃ 50% at 12 K. Similar results for intermediate fields, also shown in Fig. [spec2](c), demonstrate the continuous nature of these effects.\nThe most striking feature of Fig. [spec2](c) occurs at the onset of superconductivity (*T**c**o**n*). Above *T**c**o**n*, the MVF increases monotonically on cooling at a fixed field, but below *T**c* it falls away; at 11.5 T (*T**c* = 18 K), the MVF decreases from 50% at 12 K to 18% at 2 K. This behavior demonstrates a direct competition for volume fraction between antiferromagnetism and superconductivity, which is also visible in the spectra shown in Fig. [spec2](b). In Fig. [spec2](d), we show both *T**N**o**n* and *T**c**o**n* as functions of field. *T**N* shows a quite significant decrease with field, which can be fitted by the functional form $T\\_N = T\\_N (0) \\sqrt{1 - (H/H\\_c)^2}$, producing an estimate of the critical field for *T**N**o**n* = 0 to occur at *H**c* ≈ 19 ± 1 T. By contrast, the field-induced decrease of *T**c**o**n* is slower, consistent with the critical field for superconductivity being located at *H**c*2 ≈ 50 T in this system, and suggesting that antiferromagnetism is suppressed at far lower fields than superconductivity. We comment here that such strong field effects on *T**N* are highly unusual in iron-based SC materials, where the in-plane magnetic interactions are normally many tens of meV, and we stress that this result is obtained only for our samples close to optimal doping. For the underdoped samples discussed in Sec. III, we found no significant field-induced changes either to *T**N* or to the *s*-AFM volume fraction up to 12 T (data not shown). We return to this issue in Sec. VI.\nIndeed our MVF data demonstrate that superconductivity is more stable than antiferromagnetism, replacing it at low temperatures for all fields. We will show later that for *x* = 0.0175, superconductivity occupies the PM phase, but not the AFM phase, during the replacement process. If the RF screening is non-uniform and strong in the SC regions, then our data provide an upper bound for the MVF below *T**c*. The drop of MVF on cooling indicates that the SC volume fraction, meaning the fraction of a percolating SC state, increases. However, the average moment in the magnetic regions remains large, *μ* ≃ 0.09*μ**B*/Fe, when *T* ≪ *T**c*, and therefore we observe that AFM and SC order compete over the system volume, excluding each other in a first-order manner rather than coexisting with a reduced order parameter. Although competitive behavior of antiferromagnetism and superconductivty has been reported by neutron scattering studies of the Ba(Fe1 − *x*Co*x*)2As2 system, these cannot distinguish whether the phases compete by order-parameter suppression (second order) or volume suppression (first order). Our unambiguous demonstration of volume competition in NaFe1 − *x*Co*x*As is a key result whose origin and implications we discuss in Sec. VI.\n[invt1t3](Color online) (a) Spin-lattice relaxation rate 1/23*T*1*T* measured on 23Na for NaCo175, shown as a function of temperature with a field of 11.5 T applied along ${\\hat c}$. Dotted lines denote *T**N* and *T**c*, circles are data measured at the peak of the spectrum (PM phase), and diamonds are measured in the broad shoulder (AFM phase). Solid lines are guides to the eye. (b) 1/75*T*1*T* measured on 75As for NaCo175 as a function of temperature with the same field. Inset: 75As NMR spectra at selected temperatures. Solid lines are Gaussian fits to the data. \nTurning now to further details of magnetism in the NaCo175 sample, we note that the observed shoulder spectrum of 23Na is consistent with the *s*-AFM phase reported in Sec. III, but with the lower *T**N* value expected at a higher doping. However, there is no longer any evidence for ordered moments in the 23Na PM signal, suggesting that the *w*-AFM phase is absent at this doping. We confirm that the absence of ordered moments is not a resolution issue by comparing with the 75As NMR spectrum. As noted in Sec. II, 75As has a much stronger hyperfine coupling than 23Na, and therefore its strong sensitivity to any weak magnetic order makes it the optimal probe for excluding a *w*-AFM component in the PM signal. As shown in the inset of Fig. [invt1t3](b), the 75As spectrum is single-peaked at all temperatures. A narrow line with FWHM ≃ 14 kHz is observed above *T**N**o**n* = 20 K, and below this its spectral weight begins to decrease due to the increasing AFM volume fraction. The AFM signal lies outside our 75As measurement window, because its FWHM is very large (it can be estimated from the 23Na data to be around 2.4 MHz). The single-peaked form of the 75As spectrum below *T**N**o**n* is consistent with the PM 23Na signal, which remains sharp on cooling. Below *T**c**o**n* = 16 K, the spectra shift downward, as expected for singlet superconductivity, and at this point the 23Na spectrum does become broadened; at *T* = 6 K, far below *T**c*, we observe a FWHM ≃ 24 kHz as a consequence of the vortex structure in the SC phase. This value of the FWHM sets a strict limit on the ordered moment of the PM phase, which should be less than 0.6% of the moment in NaFeAs (0.32*μ**B*/Fe), and thus effectively excludes any possibility of *w*-AFM character at *x* = 0.0175.\nNext we focus on the SC state of the NaCo175 sample. At low temperatures, the PM phase is found to be purely superconducting by inspection of the spin-lattice relaxation rates for both 23Na and 75As. Figure [invt1t3](a) shows 1/23*T*1*T* in a field applied along the *c*-axis; above *T**N**o**n* we observe a decrease on cooling down to 40 K, followed by an increase on further cooling below 40 K. The high-temperature behavior is consistent with local-moment fluctuations and the low-temperature upturn with the spin fluctuations of itinerant electrons at the Fermi surface. Below *T**N**o**n*, 1/23*T*1*T* is no longer uniform, showing a different form if taken at different parts of the spectrum. For the shoulder, 1/23*T*1*T* drops quickly to a small value below *T**N**o**n*, as a consequence of the onset of static AFM order. For the peak, 1/23*T*1*T* continues to increase on cooling, falling only when the SC state is reached. Figure [invt1t3](b) shows the corresponding results taken from 75As, which are naturally uniform because the spectra [inset, Fig. [invt1t3](b)] have only a PM peak and no shoulders. From *T* = 220 K down to 80 K, 1/75*T*1*T* decreases linearly with temperature due to thermal excitation of local spin fluctuations in 2D. Below 80 K, the relaxation rate increases strongly with the 1/75*T*1*T* = *A*/(*T* − Θ) form characteristic of low-energy itinerant spin fluctuations. The uniform sharp drop of 1/*T*1*T* at *T**c* for the PM signal of both nuclei indicates that the PM state becomes fully SC and it is believed from ARPES measurements that a full gap opens at all points on the Fermi surface; however, we comment that NMR data have not been able to verify this second point directly (the apparent linear form of 1/23*T*1*T* visible around *T* = *T**c*/2 in Fig. [invt1t3](a), which may be of extrinsic or intrinsic origin).\n[spec4](Color online) (a) 23Na and (b) 75As NMR spectra for NaCo190, shown at selected temperatures a field of 11.5 T applied along ${\\hat c}$.\nThe spin-lattice relaxation rates in Fig. [invt1t3](a) suggest further that the *s*-AFM phase is not strongly coupled to superconductivity. Measurements of 1/23*T*1*T* for magnetic sites (in the shoulder of the spectrum) show a drop at *T**N**o**n*, where magnetic order sets in, but there is no discernible drop at *T**c*. Thus there is no evidence that the *s*-AFM phase supports even weak or proximity superconductivity. We stress that 1/23*T*1*T* at 12 K and 11.5 T, deep within the ordered phases, reaches a similar value for sites in both the SC and the *s*-AFM regions. This appears to be a clear statement that electrons on the Fermi surface are gapped by either type of order, and, taken together with Fig. [spec2](c), that relaxation contributions become dominated by SC electrons at low temperatures. This again reflects the fact that the competition between antiferromagnetism and superconductivity for electrons on the Fermi surface (i.e. in reciprocal space), and its apparent first-order nature, results in the volume competition (in real space) we observe in the NMR spectra.\nFinally, we comment that further information concerning the volume-competition effect can be gained by investigating samples with higher doping, namely *x* ≃ 0.019. Our NaCo190 sample shows a structural transition at *T**s* ≃ 35 K, which we discuss in detail in Sec. V. Figures [spec4](a) and (b) show respectively the 23Na and the 75As spectra for different temperatures. For 23Na, the spectrum is single-peaked and no shoulder feature develops on cooling, even down to 2 K. This observation excludes the existence not only of a possible *w*-AFM component but also of the *s*-AFM phase. For 75As, the spectrum also has a single peak, with FWHM ≃ 40 kHz at the lowest temperatures, which also excludes any type of AFM order. Below 18 K, the spectra shift to lower frequencies and a line broadening is clearly visible for 75As, which is the hallmark of the onset of singlet-pairing superconductivity. Thus the effect of doping on volume competition is to terminate the battle in favor of superconductivity at *x* ≃ 0.019, where one finds a single, uniform phase with only structural and SC transitions, but a complete absence of AFM order.\nPhase Diagram\n=============\nWe now compile all of our results, from samples across the full range of doping, to prepare a definitive (*x*, *T*) phase diagram. First, the structural transition can be detected from the frequency of a chosen satellite line in the 75As spectrum, which is shown in Fig. [sus5](a) as a frequency shift relative to the center line. When the field is applied in the *a**b*-plane, cooling from the tetragonal to the orthorhombic phase causes each satellite to shift and to split into two due to sample twinning. The sudden change of the satellite frequency as a function of temperature, clearly visible in Fig. [sus5](a), determines *T**s* for the structural transition at each doping. We comment here that above *T**s* this satellite frequency, which for perfect field alignment is the quadrupole frequency *ν**Q*, is generally expected to show a systematic increase with sample doping ; however, such a dependence is barely discernible in our data [Fig. [sus5](a)] due to the fact that the maximum doping we studied in the NaFe1 − *x*Co*x*As system is so low (*x* = 0.023). The variation in our measured values of *f* − *f*0 above *T**s* for the different samples is in fact dominated by the small but finite misalignment of the magnetic field, whose exact orientation with respect to the crystalline *a*-and *b*-axes was not determined. NMR provides an accurate measurement of *T**s* up to *x* = 0.019, beyond which the sample remains tetragonal at all temperatures.\n[sus5](Color online) (a) Determination of the structural transition temperature *T**s* from the frequency shift *f* − *f*0 of the satellite line in the 75As spectrum measured with an in-plane field, shown for a range of sample dopings. (b) Determination of the SC onset temperature *T**c* from measurements of the inductance change Δ*Z*/*Z*0 of the NMR coil at zero field, also for a range of dopings. (c) Phase diagram showing the values of *T**N* determined from the NMR spectra of Figs. [tn1](a) and [tn1](b), *T**s* determined from the satellite line shift, and *T**c* determined from the RF inductance. \nNext, the transition to static magnetic order can be detected at all dopings by the decrease of the PM spectral weight, as shown in Figs. [tn1](a) and (b). Finally, the SC transition is detected by the relative inductance change of the NMR coil, as noted in Sec. II and shown in Fig. [sus5](b); these measurements can also be benchmarked from the drop of the NMR Knight shift and the spin-lattice relaxation rate, at least for *x* ≥ 0.0175 (see Sec. III). In Fig. [sus5](c) we present the hierarchy of deduced transition temperatures, which show the clear evolution characteristic of iron-based superconductors. In the parent compound, NaFeAs, the system becomes fully magnetic below *T**N* ≃ 41.5 K, but as electron doping is induced by Co substitution, the antiferromagnetism is gradually suppressed and superconductivity develops.\n[pd6](Color online) (a) Temperature dependence of 1/75*T*1*T* for NaFe1 − *x*Co*x*As crystals under a field of 11.5 T applied in the *a**b*-plane of the crystal. Solid lines are fits to the function 1/75*T*1*T* = *A*/(*T* − Θ) + *B**T* + *C**T*2. Inset: 75*T*1*T* as a function of temperature. *T**s* and Θ*o* denote respectively the temperatures where 75*T*1*T* changes slope and where it goes to zero. (b) Phase diagram of NaFe1 − *x*Co*x*As established by NMR: *T**s*, *T**c*, Θ*o*, and *T**N* denote respectively the structural transition, the SC transition temperature, and the Curie-Weiss temperature, all obtained from the fit to 1/75*T*1*T* above *T**c*, and the Néel temperature obtained from the PM spectral weight [Figs. [tn1](a) and (b)]. Data for *T**c* below *x* = 0.0175 are taken only from inductance measurements [Fig. [sus5](b)]. *x*1 denotes the approximate doping where the *s*-AFM+*w*-AFM region and the *s*-AFM+PM region meet and *x*2 denotes the doping where *T**N* = 0 and beyond which AFM order is absent.\nHowever, in NaFe1 − *x*Co*x*As there is a complementary approach to obtaining the phase diagram, which is that all three phase transitions can be detected directly and accurately in the spin-lattice relaxation rate. This is not possible in other pnictide materials, and as we explain below it is also a consequence of the strongly 2D fluctuations acting in the 111 system. In Fig. [pd6](a) we show 1/75*T*1*T*, obtained over the full range of available dopings with the field applied in the *a**b*-plane, a geometry known to maximize the sensitivity of the measurement to the dominant in-plane spin fluctuations in pnictides as the (*π*, 0) magnetically ordered state is approached. At high temperatures, the 1/75*T*1*T* values are similar for all dopings other than the heavily doped *x* = 0.07 sample, and fall slowly on cooling down to 100 K. Below 100 K, a Curie-type upturn develops for all dopings (other than *x* = 0.07), becoming progressively stronger for lower doping values. Clearly the low-energy spin fluctuations responsible for this behavior, which have an itinerant origin, are suppressed strongly by increasing doping, reflecting a very high sensitivity to details of the Fermi surface. By contrast, the very small change above 100 K suggests an origin in local-moment (valence-electron) fluctuations, consistent with a general two-component interpretation of the spin response in pnictides.\nIf the low-temperature 1/75*T*1*T* data are fitted with the function 1/75*T*1*T* = *A*/(*T* − Θ), one may extract a Curie temperature Θ. The structural transition can be deduced by considering the quantity 75*T*1*T*, as shown in the inset of Fig. [pd6](a) for different underdoped and near-optimal dopings. To a good approximation, 75*T*1*T* = (*T* − Θ)/*A* can be fitted with two straight lines of different gradients, one (which we denote Θ*t*) corresponding to the tetragonal structure and the other, Θ*o*, to the orthorhombic one. The change in this gradient gives the structural transition temperature *T**s*. The values of Θ*t* and Θ*o* can be obtained from the intercept of the fitting line with 75*T*1*T* = 0.\nWe find that the values of Θ*t* (obtained by extrapolation from above *T**s*) are all negative and decrease with doping, but these are of limited physical meaning because the tetragonal structure is replaced at *T**s*, perhaps precisely because it does not allow the magnetic transition from which the orthorhombic phase can profit. By contrast, Θ*o*, which is determined from the structure and data below *T**s*, is positive and decreases with doping up to *x* = 0.019. The fact that Θ*o* is far larger than Θ*t* reinforces the evidence that the low-energy spin fluctuations are strongly enhanced below *T**s*, suggesting a clear role for the magnetic sector in driving the structural transition. Although there is already significant evidence for a coupling between the lattice structure and the magnetism in 111 materials, this coupling is manifestly not strong enough that *T**N* coincides with *T**s*, as it does in the 122 system. As noted in Sec. I, some authors have taken the lack of coincidence between *T**N* and *T**s* as a hallmark of strong two-dimensionality in some iron pnictide materials.\nAs noted above, the Néel temperature *T**N* is determined directly from the decrease of the PM spectral weight [Figs. [tn1](a), [tn1](b), [sus5](c)]. For NaFe1 − *x*Co*x*As, the fitting parameter Θ*o*, which we obtain from our data below *T**s*, has exactly the same value as *T**N* for *x* < 0.0175. This extremely significant result has not been reported in any other iron pnictide systems. *T**N* is a true measure of static AFM order, setting in due to all of the couplings in the system. By contrast, Θ*o* is a measure of the 2D spin fluctuations of the FeAs planes. For quasi-2D systems, the energy scale of in-plane fluctuations (*J*) sets a characteristic temperature $T\\_{\\rm BKT}$ in the spin response, but true long-range order is forbidden by the Mermin-Wagner theorem. However, a weak coupling *J**c* between these planes is sufficient to create long-range order in three dimensions and the transition temperature will be $T\\_N \\approx\nT\\_{\\rm BKT}$. Only when *J**c* becomes a significant fraction of *J* will the planar physics be supplemented by conventional three-dimensional fluctuations and *T**N* will exceed $T\\_{\\rm BKT}$ by an amount depending on *J**c*. The magnetic interactions in the parent compound NaFeAs have been measured directly in a very recent study of spin-wave dispersion relations by inelastic neutron scattering. These authors find in-plane couplings *S**J*1*a* ≃ 40 meV, *S**J*1*b* ≃ 16 meV, and *S**J*2 ≃ 19 meV, respectively for superexchange processes in the *a*, *b*, and diagonal (*a* ± *b*) directions in the FeAs plane, where *S* is the effective spin (moment) on the Fe ions, but an interplanar coupling of only *S**J**c* ≃ 1.8 meV. Thus our observation of perfect coincidence between Θ*o* and *T**N* is completely consistent with the measured value *J**c*/*J* ∼ 0.1, reflecting the minimal contributions from interplanar coupling, and these results form the best evidence yet available for the extremely 2D nature of the NaFeAs system. We defer a discussion of the microscopic implications of this result to Sec. VI.\nWe are now in a position to present the complete phase diagram, which is shown in Fig. [pd6](b). We begin by drawing attention to the error bars on the doping (*x*) axis, which given the extreme sensitivity of the system to small changes in *x* is critical information. As noted in Sec. II, we do not possess probes capable of determining the doping level to the 0.1% accuracies mandated by the physics of NaFe1 − *x*Co*x*As, and so we have taken the nominal doping in every case. However, we are able to benchmark our samples against each other and the remarkably smooth evolution in properties, as well as the perfect match between the *T**s* datasets, shown in Figs. [sus5](c) and [pd6](b), indicates that our conservative estimate of the doping error as ± 0.2% is reasonable. This smooth evolution over such a fine range of doping also supports the suggestion that NaFe1 − *x*Co*x*As provides some of the most homogeneous and highest-quality crystals of any iron-based SC material. Returning to the matching of datasets between Figs. [sus5](c) and [pd6](b), the agreement in *T**s* benchmarks the value of the analysis of Fig. [pd6](a); the agreement in *T**c* is perfect for *x* ≥ 0.0175, but the values extracted from 1/75*T*1*T* at lower dopings are inaccurate for the reasons discussed in Sec. III, and so the data shown in Fig. [pd6](b) for the underdoped samples are those of Fig. [sus5](c); the agreement between *T**N* and Θ*o* is remarkable for the reasons discussed above.\nOur NMR data [Figs. [sus5](c) and [pd6](b)] provide direct evidence for a complete separation of *T**s* and *T**N* at all dopings in NaFe1 − *x*Co*x*As, with the structural transition always occurring some 12 − 15 K above the magnetic one. Thus, in common with the majority of iron-based SC systems, the coupling between the lattice and the magnetism is “subcritical” in the sense of a combined transition. The *T**s* line terminates at a slightly higher doping than *T**N*, and in fact at a value coinciding with optimal doping, as indicated also in transport studies. The *T**c* line is quite flat in the doping range 0.017 < *x* < 0.022, whereas all of *T**s*, Θ*o*, and *T**N* change dramatically. In the zero-temperature limit, antiferromagnetism and superconductivity cohabit for 0 < *x* < 0.0175, as shown in Sec. IV [Fig. [spec2](c)], but within the phase-separated *s*-AFM and PM (*w*-AFM) regions (Sec. III). We comment again here that the onset of the *w*-AFM phase at the same temperature, *T**N*, as the *s*-AFM one suggests that the former is actually a set of narrow PM regions ordered by a proximity effect. Above *T**c*, the *s*-AFM and *w*-AFM phases “coexist,” by which we mean “cohabit as phase-separated regions,” in the doping range *x* < *x*1 and the *s*-AFM and PM phases cohabit over the range *x*1 < *x* < *x*2, as represented in Fig. [pd6](b). Our data fix the values *x*1 ≃ 0.0175 and *x*2 ≃ 0.018.\nTo determine the detailed structure of the phase diagram close to optimal doping, we show in Fig. [pd6](b) all of the values of Θ*o* determined from the Curie-Weiss fit below *T**s*. As noted above, Θ*o* coincides with *T**N* for *x* ≤ 0.0175. However, for doping *x* = 0.019, Θ*o* determined from the 1/75*T*1*T* curve above *T**c* predicts a finite Néel temperature (13.5 K) below *T**c* (22 K). In a scenario where antiferromagnetism and superconductivity coexist, one might expect a magnetic quantum critical point (Θ*o* = 0) to occur at *x* ≈ 0.02. However, we demonstrated at the end of Sec. IV that AFM order is completely absent for *x* = 0.019. Thus we conclude that in NaFe1 − *x*Co*x*As, the appropriate scenario for the regime *T**c* > Θ*o* is a complete suppression of long-range AFM order by volume competition (Sec. IV).\nGiven that Θ*o* (*T**N*) = 25 K at *x* = 0.0175 and Θ*o* = 13.5 K at *x* = 0.019, it is reasonable to estimate that the AFM transition line touches the SC dome at *x* ≈ 0.018, although we do not have samples with this doping. Because superconductivity suppresses the MVF below *T**c*, we expect that the *s*-AFM transition line is pushed to lower dopings below *T**c*, as also represented in the slope of the line to (*x*2, 0) in Fig. [pd6]. For all dopings (*x*) beyond this line, our experiments show unequivocally that the system is single-phased with only one transition, the onset of superconductivity in a structurally tetragonal system.\nDiscussion\n==========\nOur NMR measurements across the phase diagram in NaFe1 − *x*Co*x*As reveal a number of key features, including the first-order volume competition, the microscopic phase separation, and the dominance of superconductivity at low temperatures. We discuss these points in turn, finding that their common denominator is the two-dimensional nature of the NaFeAs system.\nBeginning with the volume competition effect, the large changes of the MVF we observe (Sec. IV) as functions of both temperature and field suggest a mutual exclusion of AFM and SC order in real space. Such a real-space competition implies a first-order transition between two states with finite order parameters, which would appear to reflect a strong competition in reciprocal space, meaning for electrons at the Fermi surface. Certainly the fact that the low-energy spin fluctuations, which are due to itinerant (Fermi-surface) contributions, are fully gapped below both *T**N* and *T**c* for the *x* = 0.0175 sample suggests that both types of order, stabilized by their own particular Fermi-surface electronic order parameter ($\\langle\nc\\_{\\bf k \\uparrow}^{\\dag} c\\_{{\\bf k} + {\\bf Q} \\downarrow} \\rangle$ or $\\langle\nc\\_{{\\bf k}\\uparrow}^{\\dag} c\\_{- {\\bf k} \\downarrow}^{\\dag} \\rangle$), compete for the same electrons.\nSuch a temperature- and field-controlled magnetic and SC volume fraction has not been reported in any other iron pnictides. We suggest that there are two reasons why this highly unconventional phenomenon has been observed (to date) only in NaFe1 − *x*Co*x*As. One is the extreme 2D nature of the Fermi surfaces and the other is the very fine control of the doping level, which is obtained in NaFeAs systems with no loss of chemical homogeneity.\nAddressing first the two-dimensionality, the question of whether electronic correlation effects result in competition or coexistence between antiferromagnetism and superconductivity has been fraught with contradictions in the iron-based SC systems. Microscopic coexistence of AFM and SC order has been reported in several compounds with the 122 structure, including by some of us. AFM and SC states in iron-based SC materials depend rather sensitively on the interactions of quasiparticles at the Fermi surface, and as a result it was suggested in Ref. that the ability of an iron-based SC system to host both types of order may be dictated by the variety of projected 2D Fermi surfaces available. The *c*-axis band dispersion in BaFe2As2 is quite significant, making it possible that the 122 structure may allow the coexistence of the two types of order for electrons on different parts of the Fermi surface. In Ba(Fe1 − *x*Ru*x*)2As2, the fact that 1/75*T*1*T* drops due to gap formation at *T**N* and again at *T**c* demonstrates that additional SC electrons are present in the AFM phase. By contrast, this dispersion is much weaker in the NaFe1 − *x*Co*x*As system, as shown both by ARPES studies and by the present results (Sec. V), indicating a highly 2D system with little flexibility in Fermi-surface sizes and spanning wave vectors. This leaves little option for the AFM and SC order but to “fight it out” for the available electrons, leading to the strong competition we observe (Sec. IV): in NaFe1 − *x*Co*x*As, 1/23*T*1*T* drops only at *T**N* or at *T**c*, and to the same low-temperature value, showing directly that antiferromagnetism and superconductivity compete for the same electrons. Our results suggest strongly that the microscopic coexistence of AFM and SC order is not possible in NaFe1 − *x*Co*x*As, which among other things should exclude any intrinsic superconductivity on sites with *s*-AFM order in the underdoped region (Sec. III).\nOn this note, we turn next to the issue of phase separation, for which we find evidence throughout the phase diagram. Clearly distinguishable *s*-AFM and *w*-AFM regions are present in the underdoped system (Sec. III), although the volume fraction of the *w*-AFM phase barely changes with either doping or temperature and there is only one magnetic transition. At near-optimal underdopings, there is clear phase separation between the AFM and PM/SC regions, demonstrated very explicitly by the large changes of AFM and SC volume fractions we observe in our NaCo175 sample as functions of both temperature and field (Sec. IV), which strongly suggest a first-order transition between the two phases. The phase diagram of Fig. [pd6](b) shows directly that the optimal *T**c* is achieved when both the AFM and orthorhombic phases are suppressed. Although this type of phase diagram has been interpreted as an incipient AFM quantum critical point of the orthorhombic system, in reality its suppression by the onset of the mutually exclusive (volume-competitive) SC phase is the dominant physics.\nSuch a complex phase diagram may also exist in other iron-based SC systems, although, for reasons of the fine doping control mentioned above, none has yet been resolved in the kind of detail possible in NaFe1 − *x*Co*x*As. Within this intricate phase structure, the inevitable presence of weak disorder could certainly provide an extrinsic origin for the phase separation of *s*-AFM and *w*-AFM regions in the underdoped regime. However, we appeal again to the evidence for remarkably high sample homogeneity in NaFe1 − *x*Co*x*As and to the obvious fact of very low carrier densities. If the origin of phase separation between the *s*-AFM and *w*-AFM regions is the same as that between the *s*-AFM and PM/SC regions near optimal doping, then such behavior may in fact be intrinsic in systems with strong electronic correlations. In cuprate materials, the stripe phase may be considered as an atomic-scale phase separation of AFM and PM/SC regions. Similar phase separation has been reported recently close to the phase boundary both in an organic superconductor and in heavy-fermion superconductors tuned by pressure and magnetic field.\nComplete nanoscale phase separation is familiar in iron-based SC materials from the case of the 245 iron selenides. However, as noted in Sec. III, the situation in NaFe1 − *x*Co*x*As does not appear to be the same, first in that structural and hence doping inhomogeneity is significant in 245 materials and second in that weak superconductivity can be observed throughout the sample below *T**c*. In Sec. III we outlined two scenarios for the present result, a microscopic coexistence or a microscopic phase separation, and, as noted above, the results of Sec. IV make a coexistence appear extremely unlikely. Although we cannot claim evidence for a nanoscale phase separation from our data, all of our results are consistent with such a scenario, under the proviso that the length scale of the phase-separation phenomenon be extremely short. We have observed a phase susceptible to an apparent bulk magnetic order by proximity effects, which is a definite statement that the phase-separation length scale should be short compared to the magnetic correlation length. The weakness and feeble onset of the apparent bulk SC regime could be the fingerprints of proximity superconductivity originating in narrow PM regions but spreading throughout the very small magnetic domains. Microscopically, in the absence of doping inhomogeneity effects, the scale of the phenomenon is expected to result from a subtle interplay between electronic correlations and lattice or charge inhomogeneities (independent of Co doping), and could indeed be on the nanometer scale.\nFirst-order phase separation close to the AFM quantum phase transition has certainly been suggested in other iron-based SC materials, most notably BaFe2 − *x*Ni*x*As2, although the fact that *T**s* and *T**N* merge in the 122 systems may cause qualitative differences in the phase diagram. One key proposal from these neutron scattering studies is an incommensurate nature of the resulting AFM phase. If the direct volume competition of the AFM and SC phases we observe below *T**c**o**n* takes place in a nanoscale lamellar structure, then an incommensurate AFM signal may indeed be observed. A different interpretation, namely a double quantum critical point, has been offered from combined transport and NMR studies in BaFe2 − *x*Ni*x*As2, but the magnetic structure below *T**c* was not resolved. NMR measurements on Ba(Fe1 − *x*Co*x*)2As2 suggest a cluster spin-glass phase close to optimal doping ; in NaFe1 − *x*Co*x*As this type of physics can be excluded explicitly from our data, which do not contain the stretched spin recovery of a spin glass, indicating again the high quality of our samples.\nOne of the fundamental questions in formulating a microscopic model for the iron-based SC materials is the equitable treatment of local-moment (valence-electron) and itinerant (conduction-electron, or Fermi-surface) contributions to the macroscopic properties of magnetism and pairing. Returning again to the key observation (Sec. IV) of large changes in the MVF with both temperature and field, these suggest a microscopic phase separation, which one expects to be driven by a first-order magnetic quantum phase transition. Because the field is applied along the *c*-axis, which is perpendicular to the direction of the ordered moments, a spin-flop transition cannot account for the observed field effect. Rather, the strong suppression of *T**N**o**n* by an applied field near optimal doping (Sec. III; the rate is approximately − 0.5 K/T close to *H* = 0), the classical critical behavior with increasing field [Fig. [spec2](d)], and the extreme field sensitivity to the competing SC phase close to the putative magnetic quantum critical point, all suggest that the interactions causing AFM order are largely local in nature.\nIn this context it is worth remarking that for overdoped systems up to *x* = 0.07, where low-energy spin fluctuations are entirely absent [Fig. [pd6](a)] but *T**c* is still approximately 10 K, it is local-moment spin fluctuations originating in the valence electrons that provide the pairing interactions for superconductivity. High-pressure NMR studies in overdoped NaFe1 − *x*Co*x*As samples demonstrate direct contributions to superconductivity from both itinerant-electron spin fluctuations (which are pressure-dependent) and local-moment fluctuations (which are largely pressure-independent). Thus it is clear that both local-moment magnetism and itinerant electrons are required for a complete understanding of the nature of antiferromagnetism and superconductivity, and in this light we revisit the key result of Sec. V that Θ*o* = *T**N*. The low-energy spin fluctuations causing the Curie-Weiss upturn in 1/*T*1*T*, and hence determining Θ*o*, are due only to Fermi-surface electrons, so the equality with *T**N* indicates only very weak contributions to long-range order from an inter-plane interaction *J**c* (Sec. V), which from the previous paragraph we conclude is mediated by local-moment fluctuations. This result, demonstrating clearly the extremely 2D nature of the NaFeAs system, suggests that the cornerstone of a microscopic model should be the itinerant contribution to (planar) magnetic order, without which the local-moment interactions appear quite unable to order alone.\nFinally, we comment that the suppression of antiferromagnetism by superconductivity at low temperatures for the *x* = 0.0175 sample, even though *T**N* exceeds *T**c*, indicates that the SC phase possesses additional electronic or magnetic channels with which it can “win” against AFM order. A full microscopic model of the band structure, Fermi surfaces, and correlation effects due to valence-electron contributions is required to account for this effect, and here we can only present the possibilities suggested by our data. One may be that superconductivity competes only with the AFM tendencies of the itinerant electrons, which in other iron-based superconductors are reinforced by sizeable local-moment contributions, but that (previous paragraph) the extremely 2D nature of NaFeAs weakens this link to the point that the AFM phase is disrupted completely.\nThe microscopic physics underlying such behavior will be found naturally in an orbital-specific model. Even in a fully 2D system, the electronic bands of iron pnictides contain five different *d*-orbitals, and hence many degrees of freedom in orbital symmetries and admixtures. Antiferromagnetism and superconductivity are favored by itinerant electrons in bands of different orbital content, and the unique feature of NaFe1 − *x*Co*x*As is that the band occupations, and thus their Fermi surfaces, are inordinately sensitive to the doping because of the highly 2D nature of the system. A recent ARPES study of NaFe1 − *x*Co*x*As samples with small and large *x* has provided some indications for orbital-selective connections between the competing AFM and SC phases, specifically concerning the relative *d**x**y* and *d**x**z*/*d**y**z* content of the bands near the Fermi surface. We suggest that similarly detailed studies of samples near optimal doping have the potential to reveal the underlying physics of NaFeAs.\nIn summary, by using NMR as a local probe sensitive to both antiferromagnetism and superconductivity, we observe a strong volume competition between the two phases at the boundary of the antiferromagnetic phase transition in NaFe1 − *x*Co*x*As. The volume fractions of the two phases can be controlled by varying both the temperature and the applied magnetic field, and show a complete mutual exclusion in real space. Thus our NMR data support a first-order phase transition between antiferromagnetism and superconductivity, which is driven by the competition between their electronic order parameters in reciprocal space. As striking as the volume competition effect is the exquisite sensitivity of the competition to doping, with optimal doping and all of the phase-separation effects occuring between 0 and 2%. These phenomena have their origin in the extremely weak interplane coupling in the NaFeAs materials, resulting in a very two-dimensional nature of the electronic band structure, and hence of the Fermi surfaces. One key anomaly compared to other iron-based superconductors is the winning of superconductivity over antiferromagnetism in real space at moderate underdopings, even where the magnetic transition temperature is higher, suggesting that the generic behavior of a two-dimensional iron pnictide may be for the electronic or magnetic channels of the Fermi-surface electrons to favor superconductivity. Further, because the very weak Co doping also appears to be remarkably homogeneous, NaFe1 − *x*Co*x*As is an excellent system in which to seek evidence of unconventional phases arising purely due to intrinsic electronic correlations. For this we obtain additional information concerning the *w*-AFM minority phase, which may be a paramagnetic regime occurring as thin lamellae due to nanoscale phase separation, but appears antiferromagnetic by proximity for underdoped samples and is the first region to turn superconducting on the approach to optimal doping.\nWe are grateful to T. Giamarchi, S. P. Kou, H. H. Wen, Z. Y. Weng, and R. Yu for helpful discussions. Work at Renmin University of China was supported by the National Basic Research Program of China under Grant Nos. 2010CB923004, 2011CBA00112, and 2012CB921704 and by the NSF of China under Grant Nos. 11174365, 11222433, and 11374364. Work at Rice University and the University of Tennessee in Knoxville was supported by the US DOE Office of Basic Energy Sciences through contracts DE-SC0012311 (P.D.) and DE-FG02-08ER46528 (C.L.Z).\n99\nM. Uehara, S. Mori, C. H. Chen, and S.-W. Cheong, Nature **399**, 560 (1999).\nS. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada, A. Kapitulnik, and C. Howald, Rev. Mod. Phys. **75**, 1201 (2003).\nY. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. **130**, 3296 (2008).\nG. F. Chen, Z. Li, D. Wu, G. Li, W. Z. Hu, J. Dong, P. Zheng, J. L. Luo, and N. L. Wang, Phys. Rev. Lett. **100**, 247002 (2008).\nX. H. Chen, T. Wu, G. Wu, R. H. Liu, H. Chen, and D. 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Dioguardi, J. Crocker, A. C. Shockley, C. H. Lin, K. R. Shirer, D. M. Nisson, M. M. Lawson, N. apRoberts-Warren, P. C. Canfield, S. L. Bud’ko, S. Ran, and N. J. Curro, Phys. Rev. Lett. **111**, 207201 (2013).\nZ. R. Ye, Y. Zhang, M. Xu, Q. Q. Ge, Q. Fan, F. Chen, J. Jiang, P. S. Wang, J. Dai, W. Yu, B. P. Xie, and D. L. Feng, unpublished (arxiv:1303.0682).\n"},"data_id":{"kind":"string","value":"arxiv_0002854"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":6,"numItemsPerPage":100,"numTotalItems":46503,"offset":600,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6Im1sR1pFYktIVHRVSGRhX1RRdEczQ0N1S3k5ME9Qd25DNHY5elRjM3FVYzgifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4Mzg2MjA0MiwianRpIjoiNTEwOTRiYTctMjkzMC00MmIyLWJlM2EtYmZhNTFhODJhMTg3Iiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODM4NjU2NDIsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.mWacFbv_zC6aTXAOO4jtK_TGHUCtimPaUL-LBZ1lwrGs8kBsNZtJTwI53zGOnXfo1quk5_qkn1B566sAV20WAA","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
content stringlengths 34.5k 188k | data_id stringlengths 13 13 |
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as a diagnostic for M31 membership in field D. Instead, we employed a less conservative radial velocity cut of $v\_{\rm{helio}} < -100$ km s− 1 to identify potential M31 RGB stars. This cut likely recovers the majority of M31 members in this field, but increases the MW contamination fraction (in the velocity range of ... | arxiv_0002474 |
consider a class of dynamical vacuum models which incorporate into a single vacuum structure ${\bar{{\Lambda}}}(H)$ the rapid stage of inflation, followed by the radiation and cold matter epochs, until achieving our dark energy Universe. The early behavior of such “running vacuum model” ($\bar{\Lambda}$CDM) bares rese... | arxiv_0002477 |
}^{-1}$ is furthermore in agreement with previous calculations and well within the range of measured values ( ∼ 1900 ± 500 $\Omega^{-1}{\rm cm}^{-1}$ ). Next, we compute the longitudinal Kerr rotation $\theta\_{\rm K}$ spectrum for s-polarized light as a function of the induced *y*-magnetization (*M**y*) in Pt and W. ... | arxiv_0002484 |
\_{bio}}=0.75$, 75) and spaced closely/widely (*D**b* = 50, 400). For the stiffer biofilm, the flow over the widest-spaced colony in Fig. [Fig12b-pathlinesup75](d) clearly exhibits flow separation both upstream and downstream of the colony. As the spacing between the stiff biofilm colonies is reduced, the up/downstream... | arxiv_0002490 |
Computing Approximate Equilibria in Weighted Congestion Games via Best-Responses[3](#fn3)
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### November 24, 2020
We present a deterministic polynomial-time algorithm for computing *d**d* + *o*(*d*)-approximate (pure) Nash equilibri... | arxiv_0002491 |
\|\_{L^2(0,1)} =
\left \| \frac{1}{A(s)}\int\_{\Gamma(s)}{ E \cdot \left ( \Delta\phi - \overline{\overline{\Delta \phi}} \right ) \, dA} \right \|\_{L^2(0,1)} \\
& = {\|{\kappa}\|}\_{L^\infty(0,1)} \left \|
\mathcal{A} \left ( \frac{E}{\kappa} \cdot \left ( \Delta\phi - \overline{\overline{\Delta \phi}} \right ) \ri... | arxiv_0002495 |
Uncovering Circumbinary Planetary Architectural Properties from Selection Biases
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The new discoveries of circumbinary planetary systems shed light on the understanding of planetary system formation. Learning the architectural properties of ... | arxiv_0002496 |
evaluated as young in the *N**U**V* youth criterion [*Youth(NUV)*=1] are almost certainly young NYMG members. Table [tab:newnuvlx] shows the number of candidates which satisfy the supplementary youth criterion. About 26 per cent (*N*=522) of the new cool dwarf candidates satisfy the *Youth(NUV)* criterion. For members... | arxiv_0002499 |
Parallel Index-based Stream Join on a Multicore CPU
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There is increasing interest in using multicore processors to accelerate stream processing. For example, indexing sliding window content to enhance the performance of streaming queries is greatly improved by utilizin... | arxiv_0002501 |
Compatibility of Hodge theory on Alexander modules
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Let *U* be a smooth connected complex algebraic variety, and let $f\colon U\to {{\mathbb C}}^\*$ be an algebraic map. To the pair (*U*, *f*) one can associate an infinite cyclic cover *U**f*, and (homology) Alexander m... | arxiv_0002505 |
Elections, Protest, and Alternation of Power[4](#fn4)
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### January 2013
Despite many examples to the contrary, most models of elections assume that rules determining the winner will be followed. We present a model where elections are solely a public signal of the i... | arxiv_0002506 |
study of quantum algorithms. In the quantum query model, a quantum algorithm is given query access to the bits of an input *x*, and the goal is to compute some function *f* of *x* while minimizing the number of queried bits. Quantum query complexity captures much of the power of quantum computing, and most quantum alg... | arxiv_0002508 |
mathcal{S}^{\rm c}$ unambiguously.[↩](#fnref3)
4. Note that although notions of causal correlations and causal Bayesian networks both share the “causal” qualifier, they are distinct concepts: a causal correlation is *not* simply one that can be obtained from any particular causal Bayesian network.[↩](#fnref4)
5. For CI... | arxiv_0002511 |
II1 factor.
The analogous C\*-theory involving actions of countable amenable groups on compact metrizable spaces is more complicated and, despite many significant advances, still incomplete. While a variety of different tools and techniques have been developed over the last four decades, in large part stimulated by ne... | arxiv_0002514 |
to any address that has not also been read. I.e., P is **clean** if each of P’s $\var{write}$ instructions is eventually followed by a $\var{read}$ to the same address.
Our considered PSAM program simulates PRAM by implementing two binary trees; to clean up after itself, the program simply traverses those two trees wi... | arxiv_0002515 |
A Conversation with David R. Brillinger
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David Ross Brillinger was born on the 27th of October 1937, in Toronto, Canada. In 1955, he entered the University of Toronto, graduating with a B.A. with Honours in Pure Mathematics in 1959, while also serving as a Lieutenant in the Royal ... | arxiv_0002519 |
history of MW-like DM halos seen in simulations. The concentration is determined by the *z* = 2 concentration-mass relation from.
Orbital parameters
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All our simulations begin with GSE at the MW’s virial radius. The initial velocity is set such that the total energy is the energy of a circular orbit ... | arxiv_0002532 |
A Census of the Carina Nebula – II. Energy Budget and Global Properties of the Nebulosity
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### Accepted 0000, Received 0000, in original form 0000
[firstpage]
The first paper in this series took a direct census of energy input from... | arxiv_0002534 |
`Patient` and `Location` relations to the action initiator, target and location, which are objects modeled in the S-AOG. An event that has a `Causal` relation with a fluent change in the C-AOG is itself modeled by the T-AOG. There may also be a `Trigger` relation between a fluent in the C-AOG and an event in the T-AOG... | arxiv_0002535 |
Cost optimization of data flowsbased on task re-ordering
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Analyzing big data in a highly dynamic environment becomes more and more critical because of the increasingly need for end-to-end processing of this data. Modern data flows are quite complex and there are n... | arxiv_0002543 |
Viscoelastic hydrodynamics and holography
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=1
Introduction
============
When undergoing deformations, most observable materials are known to exhibit both elastic and viscous responses. Due to the coupling between fluid and elastic behaviour, such materials are said to be *viscoe... | arxiv_0002545 |
Poisson brackets in Sobolev spaces: a mock holonomy-flux algebra
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The purpose of this paper is to discuss a number of issues that crop up in the computation of Poisson brackets in field theories. This is specially important for the canonical approaches to ... | arxiv_0002547 |
Constant Rank Bimatrix Games are PPAD-hard
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The rank of a bimatrix game (*A*, *B*) is defined as *r**a**n**k*(*A* + *B*). Computing a Nash equilibrium (NE) of a rank-0, i.e., zero-sum game is equivalent to linear programming (von Neumann’28, Dantzig’51). In 2005, Kannan and The... | arxiv_0002548 |
**A magnetic instability ofthe non-Abelian Sakai-Sugimoto model**
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In this follow-up paper of we further discuss the occurrence of a magnetically induced tachyonic instability of the rho meson in the two-flavour Sakai-Sugimoto model, uplifting two remain... | arxiv_0002551 |
Compact spin qubits using the common gate structure of fin field-effect transistors
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The sizes of commercial transistors are of nanometer order, and there have already been many proposals of spin qubits using conventional complementary m... | arxiv_0002553 |
⟩ in the resulting, e.g., Hartree-Fock single-particle basis. Build the matrix *M**i**j*(Ω) as the *N**h* × *N**h* reduction of *R**α**β*(Ω) to the subspace of hole states of ∣Φ⟩ and compute its inverse *M*− 1(Ω).
2. Built the *N**p* × *N**h* rectangular matrix *ρ**a**i**p**h*(Ω) ≡ ∑*i* = 1*N**h**R**a**j*(Ω)*M**j**i*−... | arxiv_0002555 |
\mbox{and}\; \; \mu = k +\alpha)$$
and
$$(\mathcal{P}\_2g) (t) = g + 2\dot{g} + \ddot{g}.$$
The results proven in Section [sec:VariationalSolutions]—specifically the bounds obtained in terms of the Laplace parameter *s* and its real part *σ*—will now allow us to show that the operators involved belong precisely... | arxiv_0002557 |
Machine Learning Portfolio Allocation
=====================================
### March 2, 2020
We find economically and statistically significant gains when using machine learning for portfolio allocation between the market index and risk-free asset. Optimal portfolio rules for time-varying expected returns and volatili... | arxiv_0002560 |
Self-torque and angular momentum balance for a spinning charged sphere
======================================================================
### August 27, 2018
Angular momentum balance is examined in the context of the electrodynamics of a spinning charged sphere, which is allowed to possess any variable angular velo... | arxiv_0002561 |
Including a Luminous Central Remnant in Radiative Transfer Simulations for Type Iax Supernovae
===============================================================================================
### Accepted XXX. Received YYY; in original form ZZZ
[firstpage]
Type Iax supernovae (SNe Iax) are proposed to arise from deflag... | arxiv_0002570 |
{eqGaloisConcLatticeCut}
\operatorname{cl}\colon 2^V\setminus\{\varnothing,V\}\rightleftarrows \mathfrak{B}({\mathcal{C}})\setminus\{\hat{0},\hat{1}\}\colon \operatorname{inc},$$
proving the homotopy equivalence $|K\_{\mathcal{C}}|\simeq |\mathfrak{B}({\mathcal{C}})\setminus\{\hat{0},\hat{1}\}|$. The second homotopy ... | arxiv_0002571 |
On a Stokes-type system arising in fluid vesicle dynamics
=========================================================
This article is the first in a series of papers on the analysis of a basic model for fluid vesicle dynamics. There are two variants of this model, a parabolic one decribing purely relaxational dynamics an... | arxiv_0002575 |
± 0.08 & 3.30 ± 0.12 & 5.58 ± 0.25 & 4.50 ± 0.86 & 0.23 ± 0.02 & 1.36 ± 0.15 & 0.51
1.148 & − 0.63 ± 0.13 & 1.56 ± 0.09 & 3.75 ± 0.17 & 8.17 ± 0.47 & 4.40 ± 1.02 & 0.12 ± 0.02 & 1.05 ± 0.14 & 0.42
1.279 & − 0.50 ± 0.16 & 1.23 ± 0.12 & 3.43 ± 0.31 & 7.88 ± 0.88 & 3.62 ± 1.07 & 0.14 ± 0.03 & 0.91 ± 0.25 & 0.32
1... | arxiv_0002581 |
mathrm{F}}}\,{\mathrm{d}}\theta\otimes {\mathrm{d}}\theta\otimes{\mathrm{d}}\theta,\qquad
\mbox{where}\qquad
C\_{\,{\mathrm{Ising}},0}^{\,{\mathrm{F}}}
=\frac{{\mathrm{d}}^{\,3}\,\Psi\_{\,{\mathrm{Ising}}}^{\,{\mathrm{eq}}}}{{\mathrm{d}}\theta^{\,3}}
=-\,2\,{\mathrm{sech}}^{\,2}\theta\tanh\theta.$$
The *α*-connectio... | arxiv_0002585 |
Higgs boson, renormalization group, and naturalness in cosmology
================================================================
### Received: date / Accepted: date
We consider the renormalization group improvement in the theory of the Standard Model (SM) Higgs boson playing the role of an inflaton with a strong non-m... | arxiv_0002597 |
+ Θ*I**α*†*ν**L**α* in ([LightMassEigenstates]). In addition, the *N**I* directly couple to the Higgs boson via their Yukawa coupling to the physical Higgs field ${\rm h}$ in ([Lseesaw]) in unitary gauge. The full *N**I* interaction term at leading order in *θ* can be expressed as
$$\begin{aligned}
\label{NWW}
\math... | arxiv_0002602 |
Codimension-Two Spiral Spin-Liquid in the Effective Honeycomb-Lattice Compound Cs3Fe2Cl9
========================================================================================
[1](#fn1)
A codimension-two spiral spin-liquid is a correlated paramagnetic state with one-dimensional ground state degeneracy hosted within a... | arxiv_0002604 |
THEORETICAL AND EXPERIMENTAL STATUSOF RARE CHARM DECAYS
=======================================================
Rare charm decays offer the unique possibility to explore flavor-changing neutral-currents in the up-sector within the Standard Model and beyond. Due to the lack of effective methods to reliably describe its ... | arxiv_0002605 |
wedge}}^2 {{\mathcal Q}}{\longrightarrow}\left({{\textstyle\bigwedge}}^2 {{\mathcal Q}}\right)^{\otimes 2}$ given by $D^2 F^{{\scriptscriptstyle \vee}}$; and
5. $\operatorname{Sym}^2 {{\mathcal Q}}{\longrightarrow}\left({{\textstyle\bigwedge}}^2 {{\mathcal Q}}\right)^{\otimes 2}$ given by ${{\textstyle\bigwedge}}^2 F^{... | arxiv_0002606 |
Quantum Dynamics of the Avian Compass
=====================================
The ability of migratory birds to orient relative to the Earth’s magnetic field is believed to involve a coherent superposition of two spin states of a radical electron pair. However, the mechanism by which this coherence can be maintained in t... | arxiv_0002610 |
Type 1 AGN at low *z*. I. Emission properties
=============================================
We analyze the emission properties of a new sample of 3579 type 1 AGN, selected from the SDSS DR7 based on the detection of broad emission. The sample extends over a broad luminosity ${L\_{\rmn{bH\alpha}}}$ of $10^{40} - 10^{4... | arxiv_0002620 |
Homomorphism Tensors and Linear Equations[1](#fn1)
==================================================
Lovász (1967) showed that two graphs *G* and *H* are isomorphic if and only if they are *homomorphism indistinguishable* over the class of all graphs, i.e. for every graph *F*, the number of homomorphisms from *F* to *... | arxiv_0002624 |
Psi\_1[\ff](\bx,\bar{\bx})\D\_{\bar{\bx}}[\p\_{\bx}\ff](\bx)|\leq C(\e)$. Finally, working as we did to get inequality we get the required bound for each of the above terms. Therefore, combining all we have proved our goal.
**Acknowledgments:** AC is supported by the Spanish Ministry of Science and Innovation, through ... | arxiv_0002633 |
Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces
============================================================================
We consider a sequence, {*M**k*}*k* ∈ N, of bubble converging minimal hypersurfaces, or *H*-CMC hypersurfaces, in compact Riemannian manifolds without boundary, of di... | arxiv_0002635 |
Unshared Secret Key Cryptography
================================
Current security techniques can be implemented with either secret key exchange or physical layer wiretap codes. In this work, we investigate an alternative solution for MIMO wiretap channels. Inspired by the *artificial noise* (AN) technique, we propose ... | arxiv_0002638 |
0)–(9,2)–(6,2)–(6,0); (6,1)–(9,1); (7,0)–(7,2); (8,0)–(8,2); (7,1)–(8,2); (7,2)–(8,1); (12,0)–(15,0)–(15,2)–(12,2)–(12,0); (12,1)–(15,1); (13,0)–(13,2); (14,0)–(14,2); (14,1)–(15,2); (14,2)–(15,1); (18,0)–(21,0)–(21,2)–(18,2)–(18,0); (18,1)–(21,1); (19,0)–(19,2); (20,0)–(20,2); (18,0)–(19,1); (18,1)–(19,0); (24,0)–(27,... | arxiv_0002639 |
=1
Evolution of star formation and gas
===================================
Star-forming molecular clouds
-----------------------------
Here, I provide an overview of the observations and physics of star-forming molecular clouds in the Galaxy and nearby normal galaxies. The motivation is to develop the intuitive backgro... | arxiv_0002642 |
Asymptotic Shear and the Intrinsic Conformal Geometry of Null-Infinity
======================================================================
In this article we propose a new geometrization of the radiative phase space of asymptotically flat space-times: we show that the geometry induced on null-infinity by the presenc... | arxiv_0002646 |
that the manifestly doubly-periodic analogue of ([merov12]) can be found in ([doubperv12]).
### Reformulation of the meromorphic single-cycle formula
The F-IBP formula ([rhorho]) to break cycles of Ω*i**j*(*η*) was reformulated in such as to single out an arbitrary *z**a* with *a* = 2, 3, …, *m* instead of *z*1. We ca... | arxiv_0002648 |
dynamics , and opinion dynamics on multiplex coevolutionary networks.
Despite some skepticism over the multiplex framework in favor of simpler and more-established single-layer one, the compelling viewpoint of this paper is that the multiplex framework is not only useful but crucial in elevating the level of our under... | arxiv_0002650 |
Comparison of Methods for modelling Coronal Magnetic Fields
===========================================================
Four different approximate approaches used to model the stressing of coronal magnetic fields due to an imposed photospheric motion are compared with each other and the results from a full time-depend... | arxiv_0002652 |
Loops and polarization in strong-field QED
==========================================
In a previous paper we showed how higher-order strong-field-QED processes in long laser pulses can be approximated by multiplying sequences of “strong-field Mueller matrices”. We obtained expressions that are valid for arbitrary field... | arxiv_0002656 |
the real line is the Killip-Simon sum rule (see also Section 3.5). In this case, the reference measure is the semicircle distribution. In, we gave a probabilistic interpretation of the Killip-Simon sum rule (KS-SR) and a general strategy to construct and prove new sum rules. The starting point is a *N* × *N* random ... | arxiv_0002666 |
by Z2*P* reflection symmetry and SPT phases protected by Z2*T* time reversal. For bosonic systems with Z2*P* only or Z2*T* only, the classification of SPT phases is Z2 in *d* = 1, and Z2 × Z2 in *d* = 3. For fermions in *d* = 3 with Z2*P* and *σ*2 = 1, or with Z2*T* and *T*2 = ( − 1)*F*, the classification is Z16. Sim... | arxiv_0002668 |
). Naturally, as *ω*0 → ∞, we recover the behaviour of the impurity model, where a true bound state occurs .
Thus, for large phonon frequency, or whenever *γ* is close to unity, we can understand most properties starting from the non-interacting case, if we take the interaction into account by renormalisation of the ap... | arxiv_0002671 |
Automated Design of Pulse Sequences for Magnetic Resonance Fingerprinting using Physics-Inspired Optimization
=============================================================================================================
### Physical Sciences: Engineering. Keywords: Magnetic Resonance Imaging, optimization, pulse sequen... | arxiv_0002677 |
Positive Maps From Irreducibly Covariant Operators
==================================================
In this paper, we discuss positive maps induced by (irreducibly) covariant linear operators for finite groups. The application of group theory methods allows deriving some new results of a different kind. In particular... | arxiv_0002679 |
], [Sect:isoZsurfTracks], and [Sect:isochrones].
Basic tracks
------------
[Fig:ZamsDef]
Basic evolutionary tracks are constructed from the raw evolutionary tracks by first identifying reference points on each track in the log*R* − log*L* diagram, and then defining ‘equivalent’ models by regularly distributing them be... | arxiv_0002681 |
Posterior Belief Assessment: Extracting Meaningful Subjective Judgements from Bayesian Analyses with Complex Statistical Models
===============================================================================================================================
./style/arxiv-ba.cfg
In this paper, we are concerned with attrib... | arxiv_0002688 |
The disk of the eruptive protostar V900 Mon[1](#fn1)
====================================================
a MATISSE/VLTI and MUSE/VLT perspective
=======================================
### Received Dec 20, 2022; accepted Nov XX, 2023
In this work, we study the silicate dust content in the disk of one of the youngest ... | arxiv_0002689 |
Actively Learning Concepts and Conjunctive Queries under -Ontologies
=====================================================================
We consider the problem to learn a concept or a query in the presence of an ontology formulated in the description logic, in Angluin’s framework of active learning that allows the ... | arxiv_0002700 |
On latent position inference from doubly stochastic messaging activities
========================================================================
=1
We model messaging activities as a hierarchical doubly stochastic point process with three main levels, and develop an iterative algorithm for inferring actors’ relative l... | arxiv_0002701 |
both is an unexplored problem. Our goal is to design SGD algorithms that easily lend themselves to distributed implementations, and are robust to fluctuations in computation and network delays as well as unpredictable node failures. This work improves the true convergence speed of distributed SGD with respect to wallc... | arxiv_0002702 |
A fast algorithm with minimax optimal guarantees for topic models with an unknown number of topics.
===================================================================================================
Topic models have become popular for the analysis of data that consists in a collection of n independent multinomial obs... | arxiv_0002709 |
The global curvature estimate for the *n* − 2 Hessian equation
==============================================================
[1](#fn1)
This paper establishes the global curvature estimate for the *n* − 2 curvature equation with the general right hand side which partially solves this longstanding problem.
introduction
... | arxiv_0002712 |
An Online Learning Theory of Brokerage
======================================
We investigate brokerage between traders from an online learning perspective. At any round *t*, two traders arrive with their private valuations, and the broker proposes a trading price. Unlike other bilateral trade problems already studied i... | arxiv_0002718 |
relation than the cosmo-OWLS agn models apart at the high-temperature end and that the bolometric X-ray luminosity–X-ray spectroscopic temperature relation evolves slightly faster in the *WMAP*7 cosmology, especially for the highest temperature bin.
[lastpage]
---
1. E-mail: amandine.le-brun@cea.fr[↩](#fnref1)
2. We a... | arxiv_0002720 |
, matter inside compact general relativistic objects can also form a Bose-Einstein Condensate (BEC). This possibility has been intensively investigated in the literature (see for a detailed discussion of the condensation processes in astrophysics), and the existence of stars with majority matter content in the form of... | arxiv_0002722 |
Towards guarantees for parameter isolation in continual learning
================================================================
Deep learning has proved to be a successful paradigm for solving many challenges in machine learning. However, deep neural networks fail when trained sequentially on multiple tasks, a shortc... | arxiv_0002724 |
4119).
H. C. Burridge and P. F. Linden, *Questioning the Mpemba effect: hot water does not cool more quickly than cold*, [Sci. Rep. **6**, 37665 (2016)](https://doi.org/10.1038/srep37665).
J. Bechhoefer, A. Kumar, and R. Chétrite, *A fresh understanding of the Mpemba effect*, [Nat. Rev. Phys. **3**, 534 (2021)](https:/... | arxiv_0002728 |
Asymptotic structure of the gravitational field in five spacetime dimensions: Hamiltonian analysis
==================================================================================================
Introduction
============
The Bondi-Metzner-Sachs algebra BMS4 is the algebra of asymptotic symmetries of four-dimensional... | arxiv_0002729 |
Interactive Proofs For Quantum Computations
===========================================
The widely held belief that strictly contains raises fundamental questions: upcoming generations of quantum computers might already be too large to be simulated classically. Gottesman asked (): Is it possible to experimentally tes... | arxiv_0002748 |
Sparsity-Aware Intelligent Massive Random Access Control in Open RAN: A Reinforcement Learning Based Approach
=============================================================================================================
GBKsong
1
Massive random access of devices in the emerging Open Radio Access Network (O-RAN) brings ... | arxiv_0002752 |
all of the inputs of our energy deposition model. Section [sec:ionosphericmodel] describes the energy deposition model itself, and in Sect. [sec:results] we present and interpret our results. In Sect. [sec:conc], we highlight the main findings and discuss their implications.
Key model inputs
================
There are... | arxiv_0002753 |
[
]
The Stochastic Extragradient (SEG) method is one of the most popular algorithms for solving finite-sum min-max optimization and variational inequality problems (VIPs) appearing in various machine learning tasks. However, existing convergence analyses of SEG focus on its with-replacement variants, while practical im... | arxiv_0002757 |
0.18 & 2.99 & 102 & 1.62 & 1.95 & 18.18 & 365
g7.08e11 & 0.46 & 1.43 & 4.20 & 120 & 1.09 & 3.14 & 9.30 & 320
g6.96e11 & 0.49 & 0.24 & 5.80 & 122 & 1.10 & 3.40 & 8.16 & 321
g5.55e11 & 0.33 & 0.16 & 3.71 & 108 & 0.68 & 1.76 & 4.29 & 273
g5.38e11 & 0.52 & 0.83 & 4.71 & 125 & 0.82 & 1.87 & 5.66 & 292
g5.36e11 & 0... | arxiv_0002760 |
===========================================
As mentioned in the previous section, there is a close relation between boundary conditions for invertible TQFTs and anomalous TQFTs. In the present section we will exploit this relation in detail.
Let *Z̃* be a boundary condition for an (*n* + 1)-dimensional invertible TQ... | arxiv_0002763 |
from 104.0–105.5mJy, significantly larger than those of the GLIMPSE II PNs in Figure [compar2] (about 103.2–103.9mJy). The consistent results obtained from PNs and regions, which have remarkably different spatial sizes and integrated fluxes, strongly validate the reliabilities of the absolute diffuse calibration of MI... | arxiv_0002764 |
a.\end{aligned}$$
The proof can be simply obtained by the definition of *i*, the choice of a helpful set of Gaussian null coordinates, and following the steps explained above.
Let us consider now an ${r}\choose{s}$ tensor *T**b*1⋯*b**s**a*1⋯*a**r* at a point *p* ∈ *B* and expand this in terms of the basis vectors. Fr... | arxiv_0002766 |
arising in applications, rather than contrived examples that may not be representative of broader classes of functions.
In , the authors observe, given a depth 2 lower bound for neural network approximation, that the Lipschitz constant of the target function and the target accuracy parameters, can be traded off by res... | arxiv_0002780 |
Optimal finite difference grids and rational approximations of the square root: I. Elliptic problems., 53:1039–1066, 2000.
A. D. Kent and D. C. Worledge. A new spin on magnetic memories., 10:187–191, 2015.
M. Kläui, C. A. F. Vaz, L. Lopez-Diaz, and J. A. C. Bland. Vortex formation in narrow ferromagnetic rings., 15:R9... | arxiv_0002781 |
agnostic about the CFT, Fig. [dsvsdp] would provide evidence that the variable *p* is proportional to the *U*(1) charge of the operator *V**p*, since the scalar appearing in the OPE *V**p**V**p* has twice the *U*(1) charge of *V**p* (the *U*(1) charge is additive) and the scaling dimension Δ*s* is found to be Δ*s* = (... | arxiv_0002791 |
Priced Timed Petri Nets
=======================
[1](#fn1)
We consider priced timed Petri nets, i.e., unbounded Petri nets where each token carries a real-valued clock. Transition arcs are labeled with time intervals, which specify constraints on the ages of tokens. Furthermore, our cost model assigns token storage cost... | arxiv_0002798 |
a central position in this project. This is mostly because all the deep theorems (Deng lemma, Hennessy lemma, Coarsest congruence contained in weak equivalence) that we have formally proved in this project, were all talking about the relationship between weak bisimulation equivalence and rooted weak bisimulation equiv... | arxiv_0002799 |
N*2 in : The background is confined to a single scaling on the boundary of phase space.
The *D*2(1, 2) observable has a similar phase space structure to *M*2, and will therefore behave similarly, so we do not show it explicitly in this section. Note that *τ*2, 1 has the same phase space structure as *N*2, so it exhibit... | arxiv_0002801 |
Detecting Novel Processes with CANDIES – An Holistic Novelty Detection Technique based on Probabilistic Models
==============================================================================================================
### Received: 15. April 2016
In this article, we propose CANDIES (Combined Approach for Novelty De... | arxiv_0002805 |
The Implicit Bias of Batch Normalization in Linear Models and Two-layer Linear Convolutional Neural Networks
============================================================================================================
We study the implicit bias of batch normalization trained by gradient descent. We show that when learn... | arxiv_0002809 |
Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces
===========================================================================
We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjam... | arxiv_0002811 |
a variety of ways as shown in software packages FOLDX, SAAMBE, mCSM-AB, mCSM-PPI2, BindProfX, etc. AI approaches take the advantage of existing data and often outperform other methods when experimental data become available. Due to the structural complexity and high dimensionality of of protein-protein interactions (P... | arxiv_0002813 |
On the commitment value and commitment optimal strategies in bimatrix games
===========================================================================
Given a bimatrix game, the associated leadership or commitment games are defined as the games at which one player, the leader, commits to a (possibly mixed) strategy an... | arxiv_0002816 |
characteristics would require addressing in the near future, as this saturation region would affect the performance of devices.
[bipolar] The ambipolar characteristics of the electric field effect in graphene were first observed by Geim *et al.*. It is one of the key important properties that the conductivity *σ* incr... | arxiv_0002825 |
Saturn ring defect around a spherical particle immersed in nematic liquid crystal
=================================================================================
We consider a nematic liquid crystal occupying the three-dimensional domain in the exterior of a spherical colloid particle. The nematic is subject to Diric... | arxiv_0002826 |
On the impact of galaxy bias uncertainties on primordial non-Gaussianity constraints
=====================================================================================
Introduction
============
One of the main goals of modern cosmology is to determine the statistical properties of the density fluctuations of the pr... | arxiv_0002839 |
Ultra-Fast Converging Path-Integral Approach for RotatingIdeal Bose-Einstein Condensates
========================================================================================
A recently developed efficient recursive approach for analytically calculating the short-time evolution of the one-particle propagator to extr... | arxiv_0002840 |
The Magellanic Edges Survey I. Description and First Results
============================================================
### Accepted XXX. Received YYY; in original form ZZZ
[firstpage]
We present an overview of, and first science results from, the Magellanic Edges Survey (MagES), an ongoing spectroscopic survey mappi... | arxiv_0002841 |
STATISTICAL INFERENCE FOR THE PENALIZED EM ALGORITHM TO TEST DIFFERENTIAL ITEM FUNCTIONING
==========================================================================================
Recent advancements in testing differential item functioning (DIF) have greatly relaxed restrictions made by the conventional multiple gro... | arxiv_0002844 |
rich representation of each input word: this is a dense, *H*-dimensional real vector. Obviously, a very large data set is required to properly train such a model, and in Word2Vec’s case that was GoogleNews (100 billion words).
Other word embedding algorithms followed-up and improved upon the Word2Vec concept, such as ... | arxiv_0002846 |
+, 0, *F*2(*x*+)) stable in the competition plane in region **(k)**. We selected three parameter values from this region and the first one is *x*⋆ = .1 and *a*2 = 3.5 (marked by a top blue \* in region **(k)** of Figure [bifurcationdiagram]). The situation is depicted in Figure [Simubild11](a): The coexistence cycles t... | arxiv_0002849 |
Layer-by-layer assembly of multilayer optical lattices: Application to displaced dice lattice
=============================================================================================
We propose methods for synthesizing multilayer optical lattices of cold atoms in a layer-by-layer manner, to unlock the potential of... | arxiv_0002853 |
% of that in the parent compound, where the 23Na spectrum is split by 540 kHz.
The NMR study of Oh *et al.* reports two species of antiferromagnetism in a sample with *x* ≈ 0.017, one with a large ordered moment (*s*-AFM) and the other one with a small moment (*w*-AFM). Our data from lower dopings are consistent with... | arxiv_0002854 |
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