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[EQUATION] where the extremum is over [MATH] and [MATH] Except for [MATH] and [MATH] the saddle-point equations are algebraic and may be used to eliminate the respective variables. From the saddle point equation for [MATH] we immediately get [MATH] , those with respect to [MATH] and [MATH] give |
[EQUATION] Plugging these expressions in ( 31 ) yields [EQUATION] The extremum in [MATH] is somewhat unusual, it is at the lower boundary of the integration region, i.e. at [MATH] . Intuitively this means that we satisfy the inequality [MATH] in ( ) by [MATH] . It is not unreasonable that the solution volume is maximiz... |
[EQUATION] and we therefore finally find [EQUATION] where the extremum remains to be taken only over [MATH] and [MATH] . This is the expression given in the main text. |
The phase transition At the transition the solution volume [MATH] shrinks to zero and the overlap [MATH] between different solutions [MATH] has to tend to one. Then to leading order in [MATH] we have |
[EQUATION] and therefore [EQUATION] Keeping only the most divergent terms hence gives near the transition [EQUATION] The saddle-point equation with respect to [MATH] gives |
[EQUATION] resulting in [EQUATION] The saddle-point equation with respect to [MATH] gives [EQUATION] which together with ( 39 ) yields |
[EQUATION] Eqs. ( 40 ) and ( 42 ) give a parametric description of the transition line in the [MATH] [MATH] -plane. They are identical to the equations found in for the critical line. |
10 The competitive exclusion principle and the limit [MATH] We are interested in the stationary states of the model given by [EQUATION] |
This equation implies that in the stationary states for each component [MATH] either the population has to vanish [MATH] or the equation [MATH] has to be fulfilled. For a general vector of thresholds [MATH] and a full-rank matrix of metabolic strategies [MATH] , the overdetermined linear equation system |
[EQUATION] will have no solution [MATH] . This means that not all species can survive at the same time. For large system sizes a random [MATH] has full rank with overwhelming probability. It is then always possible to find a vector of resource availabilities [MATH] such that [MATH] equations are fulfilled. Therefore, t... |
In the model the following choice of thresholds is made [EQUATION] where [MATH] is a Gaussian random variable with zero mean and variance one and [MATH] is a small positive number quantifying the scatter of the thresholds. With this choice, Eq. ( 44 ) takes the form: |
[EQUATION] For [MATH] the vector [MATH] is a linear combination of the columns of [MATH] and one observes a highly degenerated situation. Then, [MATH] for all [MATH] is a solution of the system which makes the bracket in Eq. ( 43 ) zero for all [MATH] and hence all population sizes can be non-zero in the stationary sta... |
Assume that the system is in the S-phase and [MATH] species survive in the stationary state. Then the equation [EQUATION] with the matrix of the surviving metabolic strategies [MATH] and the scatter of thresholds of the surviving species [MATH] is fulfilled. For large system sizes the matrix [MATH] will have full rank ... |
[EQUATION] In the limit of vanishing scattering [MATH] the components of [MATH] equal one. This shows that the survival of [MATH] species implies that all resource availabilities are equal to one. |
Now consider the case that the system is in the V-phase and [MATH] species survive. Then, [MATH] and [MATH] . This means that Eq. ( 47 ) becomes an underdetermined equation system which does not have a unique solution and the components [MATH] do not necessarily approach one for [MATH] |
11 Finite size analysis The data in Figure 3 was obtained in the following way: For each choice of [MATH] and [MATH] we created 400 random realizations of the linear equation system |
[EQUATION] Using the non-negative least squares solver nnls of the scipy.optimize package in Python it was checked whether the system possesses a non-negative solution [MATH] . The data points show the average number of cases in which such a solution was found. |
To fit the data for the respective system sizes [MATH] we used the hyperbolic tangent function [EQUATION] where the parameters [MATH] and [MATH] were fitted. |
The finite-size approximation of the critical [MATH] is given by the value of [MATH] where the hyperbolic tangent vanishes and half of the systems posses a solution, i.e. [MATH] . The fits in the insets were done using a power law |
[EQUATION] where [MATH] and [MATH] are fit parameters. References Engel A. and Van den Broeck C., Statistical mechanics of learning (Cambridge University Press) 2001. Tikhonov M. and Monasson R., Phys. Rev. Lett., 118 (2017) 048103. |
# Source: arxiv 1806.11421 # Title: States-conserving density of states for Altshuler-Aronov effect: Heuristic derivation # Sections: all # Downloaded: 2026-03-03T05:15:59.438396+00:00 |
States-conserving density of states for Altshuler-Aronov effect: Heuristic derivation Abstract Altshuler and Aronov (AA) have shown that the electron-electron interaction in a weakly-disordered metal suppresses the single-particle density of states (DOS) in the vicinity of the Fermi level ( [MATH] ). According to the A... |
disordered systems, correlations, density of states, Altshuler-Aronov effect, conservation of states Introduction Altshuler and Aronov (AA) have shown Altshuler1 Altshuler3 Altshuler4 that the electron-electron (e-e) interaction in a weakly-disordered three-dimensional ( [MATH] ) metal suppresses the single-particle de... |
where [MATH] is a characteristic correlation energy. The DOS [MATH] at energies [MATH] was observed by tunneling spectroscopy Abeles Dynes McMillan2 ImryOvadyahu Schmitz1 Schmitz2 Escudero Teizer Mazur Luna2014 Luna2015 and by photoemission spectroscopy Kobayashi Some experiments Schmitz1 Schmitz2 Escudero Mazur Kobaya... |
was to show that the DOS in presence of the AA effect exhibits a states-conserving dependence on energy. It has been found Mazur that all states removed from near the Fermi level by the AA effect are found at energies above [MATH] |
in the energy range of [MATH] to [MATH] times [MATH] However, the observed states-conserving DOS Mazur was not compared with theory, because the relevant theories Altshuler1 Altshuler3 Altshuler4 LeeRamakrishnan Imry |
studied the AA effect in the low energy limit. In this work we study the AA effect beyond the low energy limit theoretically. We consider the model Altshuler4 Imry |
in which the electrons interact via the statically screened Coulomb interaction and the modification of the DOS is due to the Fock part of the self-energy. We derive the states-conserving DOS which is in qualitative accord with experimental observations at energies above [MATH] |
and which reproduces the AA theory at low energies. We show that, besides the direct experimental study of the states-conserving DOS Mazur such DOS was present (but not noticed) also in other experiments Schmitz1 Schmitz2 Escudero |
In our model Altshuler4 Imry electrons in the disordered metal interact via the static finite-ranged potential [MATH] . If [MATH] the electrons interact only with the random potential [MATH] , produced by disorder. In such case the electron energies [MATH] and wave functions [MATH] obey the Schrodinger equation [MATH] ... |
[MATH] is modified to [MATH] as Imry [EQUATION] where [MATH] is the Fock first-order self-energy correction: [EQUATION] Here [MATH] is the Fourier transform of [MATH] [MATH] is the Fermi function, and [MATH] is the sum over [MATH] with spin parallel to that of [MATH] Equations ( ) and ( ) hold if [MATH] |
Equations ( ) and ( ) describe a specific disordered sample. When averaged over many disordered samples, they remain unchanged except that [MATH] and [MATH] are the mean values. Most important, the disorder-averaged [MATH] |
can be calculated explicitly. For a diffusing electron Altshuler4 Imry LeeRamakrishnan [EQUATION] where [MATH] is the diffusion coefficient, [MATH] is the volume, and [MATH] is the DOS for a single spin orientation [MATH] is often replaced by [MATH] which is justified for [MATH] close to [MATH] ]. If we average [MATH] ... |
over all states [MATH] with energies [MATH] , equation ( ) can be rewritten as [EQUATION] where [EQUATION] In the last equation and in all following calculations we assume zero temperature for simplicity. |
Due to averaging over disorder the unperturbed DOS (per spin) reads [MATH] , as for the free electrons. The perturbed DOS versus [MATH] [MATH] , can be expressed |
Altshuler4 Imry LeeRamakrishnan from equation ( ) as [EQUATION] where the right hand side holds for [MATH] Note that the perturbed DOS, [MATH] , is expressed as a function of [MATH] rather than of [MATH] This approximation is valid within the first order perturbation theory Altshuler4 Imry LeeRamakrishnan |
It is customary to change the integral [MATH] in equation ( ) as [MATH] This infinite band approximation is justified for weak interaction. Then, substituting [MATH] by variable [MATH] , one can rewrite equation ( ) as LeeRamakrishnan |
[EQUATION] where Altshuler4 Imry LeeRamakrishnan [EQUATION] The AA effect was studied Altshuler4 Imry LeeRamakrishnan for [MATH] so small that [MATH] . Then |
[MATH] , or [EQUATION] In the simplest model Altshuler4 Imry LeeRamakrishnan with static screening [EQUATION] where [MATH] is the reciprocal screening length (the factor of [MATH] is due to the spin degeneracy), and [MATH] is the high-frequency permittivity of the metal. |
From equations ( ) and ( ) one obtains the result of Altshuler and Aronov, Altshuler1 Imry [EQUATION] where [MATH] is the DOS at the Fermi level and the second term on the right hand side is the AA interaction correction. In Refs. Altshuler1 Imry the integral in equation ( ) was calculated assuming [MATH] For [MATH] th... |
In the following text we present an alternative derivation which is not restricted to the low-energy limit. At low energies our derivation will reproduce equation ( 11 ) and also determine explicitly the term [MATH] However, our major goal is to go beyond the low energy limit and to derive the DOS which conserves the s... |
In Sect. 2 we show that the DOS given by equations ( ), ( ), and ( 10 ) does not conserve the states. In Sect. 3. we identify why this is so and present a heuristic derivation of the states-conserving DOS. Comparison with experiment is presented in Sects. 3 and 4. Finally, in Sect. 5 we interpret the AA effect with con... |
II The states conservation problem Figure shows schematically the typical experimental output Mazur . At energies below [MATH] the data show the AA singularity described by the [MATH] law. All states repelled from the AA singularity are found at energies above [MATH] in the range of about [MATH] This local conservation... |
In accord with figure and Ref. Mazur , the conservation of states for the AA model reviewed in Sect. 1 reads [EQUATION] Inserting equation ( ) into the conservation law ( 12 ) we find that the conservation of states is fulfilled only if |
[EQUATION] However, equation ( 13 ) is not fulfilled because [MATH] is positive for any [MATH] [see equation ( )]. This means that the model of Sect. 1 does not conserve the states. |
Furthermore, integral [MATH] not only fails to fulfill equation ( 13 but even diverges in the upper limit (see the next section). This means that also the self-energy ( ) diverges which is another problem, in addition to the states conservation problem. In principle, the divergence could be eliminated by considering th... |
[MATH] independently on the band width). We will see that the divergent self-energy is closely related to the conservation of states problem. |
To conserve the states and to obtain a finite self-energy, one has to identify the limitations of the model and to modify it properly. First, the diffusive approximation ( ) holds only at small energies. Second, the interaction ( 10 ) and diffusive matrix element ( ) are mutually independent while in reality they shoul... |
Concerning the first point, the diffusive approximation ( ) holds for the eigen-states which are correlated in time and space Altshuler4 Imry LeeRamakrishnan Therefore, equation ( ) is valid only if |
[MATH] and [MATH] , where [MATH] is the elastic scattering time and [MATH] is the Fermi velocity. In addition, the e-e interaction with spatial range [MATH] introduces the correlation time [MATH] which is usually longer than [MATH] and gives rise to the correlation energy [MATH] This restricts the validity of equation ... |
Second, in a realistic self-consistent model the interaction ( 10 ) has to be modified to the form which depends on the energy difference |
[MATH] and diffusion coefficient [MATH] . From figure it follows that the conservation of states (equation 12 ) can be fulfilled only if |
[MATH] in equation ( 13 ) changes the sign at energy [MATH] Indeed, one can see from equation ( ) that such sign change can arise only from the sign change of the interaction [MATH] |
III The states-conserving DOS and self-energy The question is how to modify equation ( to be valid also beyond the low energy limit. A direct self-consistent solution of the problem would be difficult. Therefore, as a first approach, we develop a simple heuristic theory. Inserting for [MATH] the equation ( 10 ), using ... |
[EQUATION] The right hand side of the last equation is composed of three terms. We will show that the only modification we need is the omission of the third term. |
All three integrals in equation ( 14 ) can be calculated analytically. Introducing notations [MATH] and [MATH] we obtain [EQUATION] |
where [EQUATION] corresponds to the first two terms in equation ( 14 ) and [EQUATION] corresponds to the third term. Using equations ( 16 ) and ( 17 ) we find that |
[EQUATION] where the second integral diverges because in the upper limit [MATH] We can now discuss the self-energy. Inserting equation ( 15 ) into the equation ( ) we obtain |
[EQUATION] where [MATH] and [MATH] are given by equations ( 16 ) and ( 17 ). Due to one of equations ( 18 ) the self-energy ( 19 ) diverges which is not a sound result. A physically sound self-energy should be finite and of the order of [MATH] |
(in our weak-interaction case). This can only be achieved if we omit in equation ( 19 ) the integral [MATH] or in other words, if we set [MATH] . In fact, we will see soon that the choice [MATH] |
is the only way how to ensure both the finite self-energy and conservation of states. Due to this choice equation ( 19 ) reduces to [MATH] where the remaining integral is finite and tractable analytically. We get |
[EQUATION] As expected, the self-energy ( 20 ) is finite and smaller than [MATH] It starts from zero value at [MATH] , reaches the peak value |
[MATH] at [MATH] , and eventually shows logarithmic decay to zero for [MATH] Now we discuss the DOS. From equations ( 15 ) and ( |
[EQUATION] We have already mentioned the conservation of states problem. To conserve the states, equation ( 21 ) has to fulfill equation ( 12 ). So it conserves the states only if |
[EQUATION] which is not the case due to the second of equations ( 18 ). Evidently, the term [MATH] is not sound due to its behavior at large energies, in accord with the fact that equation ( ) is not valid at large energies. On the other hand, the first of equations ( 18 ) implies that the term [MATH] is sound in the s... |
[MATH] and we obtain the DOS [MATH] which conserves the states. In final form [EQUATION] The DOS expression 23 ) is plotted in a full line in panel of figure . The full curve exhibits the AA singularity at energies [MATH] Due to the conservation of states, the states removed from the AA singularity are found at energie... |
For comparison, panels , and show the experimental data for various disordered metals, the experimental data of panel are shown also in panel (the dotted-dashed line). The origin of all these data is specified in the next section. Here we point out the following. |
First, all experimental curves exhibit the AA singularity accompanied by pile up of states at energies above [MATH] . This is a sign of the local conservation of states (Fig. ). Second, the states-conserving DOS ( 23 ) (the full curve in panel ) captures the main experimental features in the sense that it describes the... |
and even turn to zero due to the Anderson localization. Finally, experimental data are affected by finite [MATH] while we assume [MATH] |
Equation ( 23 ) not only conserves the states but in the limit [MATH] gives the formula [EQUATION] which reproduces the AA result ( 11 ) and in addition expresses [MATH] as |
[MATH] . In figure equation ( 24 is plotted in a dashed line. Finally, we express explicitly equation ( 21 ): [EQUATION] In figure the DOS expression ( 25 ) is shown in a dotted line, a similar dependence was found in work Rabatin It shows no pile up of states as it does not conserve them. It also deviates too early fr... |
IV Insight into experiment The experimental data in figure originate from the tunneling spectroscopy experiments Escudero Schmitz1 Schmitz2 Mazur In these experiments the electron current ( [MATH] ) is driven through the metal-insulator-metal (MIM) tunnel junction composed of the clean metal and disordered metal of int... |
in dependence on voltage [MATH] At low temperatures [MATH] can be expressed as [MATH] , where [MATH] and [MATH] is the differential conductance that is independent on the interaction Mazur In reality |
[MATH] is a parabolic function which has to be specified Mazur if one studies [MATH] at large energies. To our knowledge, experiment Mazur is the only experiment which studied the AA effect with intention to observe the pile up of states. In that work [MATH] was measured for the disordered [MATH] metal at [MATH] K and ... |
was determined from equation [MATH] after specifying [MATH] The resulting [MATH] data are those presented in panel of figure these data represent the first observation Mazur of the local conservation of states in the weakly disordered metal. Now we show that such local conservation of states was present (unnoticed) als... |
The [MATH] dependence for the [MATH] alloy, shown in panel of figure has been determined (by us) from the [MATH] data measured in work Escudero . Figure shows the [MATH] curves measured Escudero at [MATH] K and [MATH] K, normalized by [MATH] at [MATH] K. The only difference between these curves is a pronounced AA singu... |
The [MATH] dependencies for [MATH] and [MATH] , shown in panels and of figure were extracted by us from the [MATH] spectra measured in works Schmitz1 Schmitz2 The spectra are presented in figure together with our extraction procedure. The [MATH] spectra exhibit the AA singularity together with features due to the pile ... |
Discussion, alternative view on theory Our major result is the states-conserving DOS given by equation ( 23 ). At low energies equation ( 23 ) reproduces the original AA result ( 11 and determines the term [MATH] Most important, at large energies it provides the conservation of states, manifested by pile up of states a... |
Our derivation was heuristic. We have found that the AA self-energy consists of a diverging part and of the small part of size [MATH] (equation 20 ). We have removed the diverging part (on physical grounds) by omitting the term [MATH] |
and the remaining part has produced the states-conserving DOS which we have searched for. In this sense the conservation of states is inherent to the AA model. |
We want to finish by providing an alternative view on our derivation. In the AA model the matrix element ( ) and interaction ( 10 ) were independent, however, in reality they should affect each other. We show that just this happens in our model due to the omission of term [MATH] |
We first repeat in a slightly different form our major results. We take equation ( 14 ) and omit the third term on the right hand side (the third term is [MATH] ). We get |
[EQUATION] where the right hand side recalls the final result ( 16 ). If we compare equation ( 26 ) with the original equation ( ), we see that the diffusive approximation ( ) is modified as |
[EQUATION] and the interaction ( 10 ) as [EQUATION] Thus, the matrix element and interaction are no longer independent. First, equation ( 27 ) contains the factor [MATH] |
which has the Lorentzian shape with spread [MATH] . Due to this factor, expression ( 27 ) decreases with increase of [MATH] faster than the diffusive approximation ( ). This is in accord with expectation (Sect. 2) that the interaction will suppress correlation between the states with [MATH] Second, interaction [MATH] i... |
The energy dependence in equation ( 28 ) should not be confused with dynamic screening. If we replace [MATH] by the real part of [MATH] where [MATH] |
LeeRamakrishnan , the interaction remains positive. Finally, note that the interaction ( 28 ) is closely related to the AA effect [equations ( 11 ) and ( 24 )]. Indeed, the AA equation ( 11 ) is obtained as it stands if we skip in equation ( 26 ) the factor [MATH] but keep the factor [MATH] . Owing to this factor it is... |
Acknowledgement This work was supported by the APVV project No. APVV-0560-14 and by grant VEGA 2/0200/14. We thank Richard Hlubina for useful discussions and for help with equations ( 14 ), ( 16 ), and ( 17 ). |
# Source: arxiv 1806.11457 # Title: Simulating Ising and Potts models and external fields with non-equilibrium condensates # Sections: all # Downloaded: 2026-03-03T05:16:01.440754+00:00 |
Simulating Ising and Potts models and external fields with non-equilibrium condensates (June 2, 2018) Abstract Classical spin models with discrete or continuous degrees of freedom arise in many studies of complex physical systems. A wide class of hard real-life optimisation problems can be formulated as a minimisation ... |
Understanding of the phase transitions, structures of the ground states, dynamical behaviour of classical spin systems lies at the core of studies of complex physical systems. The interplay of different types of excitations, couplings, continuous and discrete degrees of freedom can lead to complex behaviour that is mos... |
Beyond the investigation of new physical regimes, not realisable in condensed matter systems lies another important area of research where finding the ground state configuration of spin models is associated with classical optimization (NP-hard) problems in vastly different areas such as vehicle routing and scheduling p... |
Among the various platforms that aim to simulate the classical spin Hamiltonians a new subclass of simulators has recently emerged – the gain-dissipative simulators – non-equilibrium systems that use a gain process to raise the system above the threshold for a phase transition to a coherent state (via a supercritical H... |
The nature of the couplings varies between the physical systems, so in this Letter we focus on non-equilibrium condensates and coupled lasers. So far in any realisation of a lattice of condensates and lasers the ‘spins’ – phases of CCs could take any value between [MATH] and [MATH] . There were various attempts to map ... |
To realise the XY model using the gain-dissipative simulators one needs to establish a feedback connection between the gain mechanism, coupling strength and the densities of the CCs as we recently established GD-polariton . The time evolution of the system of [MATH] CCs in the gain-dissipative simulator is described by... |
[EQUATION] where [MATH] is a complex amplitude of the [MATH] th CC with number density [MATH] and phase [MATH] [MATH] is the effective gain with [MATH] being the pumping rate, [MATH] and [MATH] are rates of linear and nonlinear dissipation respectively, [MATH] is the specified threshold number density, [MATH] is the co... |
To implement the external fields and the discrete versions of the XY model such as the Ising and the Potts models we need to break the symmetry of Eq. ( ) to phase rotations which can be done by forcing the system parametrically at a frequency [MATH] resonant with the basic frequency of the system at the Hopf bifurcati... |
[EQUATION] where [MATH] is the pumping strength of [MATH] th CC at the resonant frequency [MATH] . In Eq. ( ) we use the Madelung transformation [MATH] and separate real and imaginary parts dropping the noise term for simplicity to get |
[EQUATION] where [MATH] . The fixed point of the dynamical system given by the Eqs. ( ) satisfies [EQUATION] Since for each CC we choose the smallest [MATH] by raising it from below then at the threshold the global minimum of |
[EQUATION] is achieved while Eq. ( ) describes the gradient decent to that minimum. By taking the resonance [MATH] we introduce the effective external ”magnetic” field [MATH] into the model. For [MATH] the forcing term in Eq. ( ) reduces the invariance to a global phase shift to a discrete symmetry [MATH] and for a suf... |
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