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[EQUATION] Setting [EQUATION] where we have defined [EQUATION] the long-time average can be written as [EQUATION] where we have defined, for each [MATH] , the matrix |
[EQUATION] II Classical and quantum Fluctuations In this section we study the statistical properties of the stationary distribution of single-particle occupation numbers. In particular we analyze both “classical” and “quantum” fluctuations. Concerning the former they can be obtained from the study of the time fluctuati... |
In Fig. the long-time dynamics of the average occupation numbers [EQUATION] are shown for different [MATH] values. Let us first concentrate on the statistical properties of this “classical signal”, [MATH] . The distributions [MATH] , taken from the values in Fig. are shown in Fig. (a) (for two values of [MATH] [MATH] a... |
[MATH] as a function of the number of principal components of the stationary wave-packet (after relaxation) for the correspondent initial states (essentially what is shown in Fig. (b).) As one can see there is a very good agreement with the dependence [MATH] |
which is a strong result in view of the requirement of statistical mechanics. Let us stress that the decrease of relative fluctuations occurs not with respect to the number [MATH] of particles, but with the number of principal components contained in the stationary distribution [MATH] |
Concerning quantum fluctuations, they are defined by [EQUATION] for different initial states [MATH] and from them, the relative fluctuations [MATH] . In the canonical ensemble, for non-interacting bosons the following relation holds pathria |
[EQUATION] We have numerically checked this relation, see data in Fig. (a) from which one can see a good correspondence to the above relation in the case when the eigenstates are strongly chaotic. In Fig. (b), the same quantity has been plotted for a non-chaotic case. As one can see quantum fluctuations deviate strongl... |
# Source: arxiv 1806.00613 # Title: Anomalous cumulative inertia in human behaviour # Sections: all # Downloaded: 2026-03-03T05:16:46.625901+00:00 |
Anomalous cumulative inertia in human behaviour Human behaviour is dictated by past experiences via cumulative inertia (CI): the longer a certain behaviour has been going on, the less likely changes becomes. This is a well-known sociological phenomenon observed in employment, residence, addiction, criminal activity, wa... |
Cumulative inertia (CI) is a fundamental property of human behaviour, whether we are conscious of it or not . It is the statement that the longer we retain a job, own a home, or belong to a certain political group, the less likely we are to change those circumstances Consider for example tenured academics: with time, t... |
. CI is also historically well-documented in human conflicts. A war or strike which has gone on for 3 months is far more likely to continue for another three months than a confrontation which only started 2 weeks ago (both parties have had greater opportunity to become entrenched in their beliefs, reducing chances of a... |
CI is prevalent in online streaming; the longer our streaming session, the more likely we are to continue ; an effect also seen in television consumption |
. The same is true for correspondence where the likelihood of responding decreases with the time since receiving the message , and in political activity where inveterate members are less likely to leave than new members |
CI was popularised during studies of human movement and migration , where local attachments were observed to lead to reluctant movers. It has recently experienced a resurgence with new empirical support for postulated psychological interpretations of the phenomenon |
. In the seminal work of Kahnemann and Tversky, the endowment effect was put forth as an explanation of this aversion to change. As we experience our home and neighbourhood, a set of positives and negatives is acquired (amenities, commute, neighbours, etc.) which forms an endowment. Changing means the loss of this endo... |
CI can be expressed in terms of a rate of change [MATH] which decreases with time [MATH] . The expression [MATH] is the conditional probability of e.g. being employed for a duration in the interval [MATH] , given the employment has already lasted a duration [MATH] . On dimensional grounds, let the rate of change take t... |
[MATH] exhibits inertia because it increases with [MATH] (see and SI, part 1) This expression holds for [MATH] , a key property of CI. |
In this work we uncover a hitherto untreated trend in human behaviour which we call anomalous cumulative inertia (ACI), supported by empirical evidence. Crucially, the predictions from standard cumulative inertia regarding the mean remaining time break down due to a smaller rate parameter [MATH] . Instead, anomalous sy... |
, leaving a stable well-paying job, or our activities on the internet , including ‘clickbait’ websites which do their utmost to keep a user on the site for as long as possible |
A large body of literature examines bursty activity followed by long periods of inactivity for related examples, but do not address anomalous effects |
A hallmark of anomalous cumulative inertia is that classical treatments of cumulative inertia break down due to the long-term memory effects in place. Consequently, predictions pertaining to the mean remaining time [MATH] no longer apply; the mean has ceased to exist. The consequences of ACI are pervasive beyond the br... |
Before addressing this point in greater detail, we first motivate the rate of change [MATH] , which describes both CI and ACI depending on the value of rate parameter [MATH] . Figure collates data for rates of change observed in a variety of human behaviours. These include the rates at which wars (1820-1949) or British... |
In Figure we provide further empirical support for the presence of ACI. We consider the survival probabilities [MATH] of convicted individuals in the U.S.A. and Spain not committing another crime for a given duration. The American data distinguishes between individuals sent to prison or given parole, as well as if the ... |
The tails of [MATH] plotted in Figure are equivalent to those arising from the solutions to fractional equations in ( ) and ( ). The resulting Mittag-Leffler function [MATH] is an ideal fitting candidates for empirical cases when the tails dominate the dynamics, capable of describing systems with strong long-time effec... |
Fractional Calculus and ACI Let us consider the survival probability of employees [MATH] in a company which exhibits ACI. The associated strong inertia leads to the fractional equation for [MATH] |
[EQUATION] with the Riemann-Liouville fractional derivative defined as [MATH] . This clearly has a strong memory dependence. An other formulation of the form |
[EQUATION] using the fractional Caputo derivative also exists. The solution to ( ) and ( ) is given by the Mittag-Leffler function [MATH] , which is illustrated in Figure |
The empirical support for ACI and fractional calculus is provided in Figure and . The importance of fractional calculus becomes clear when one considers the competition across a network of companies or universities |
Highly reputable or prestigious companies can reasonably be modelled by ACI, and out-compete locations with classical CI. Worker allocation will then be dominated by companies with ACI. A heuristic interpretation is the well-established notion of ‘brain-drain’ between different countries, or international migration bet... |
. Instead, most workers will be found in the state with ACI, which we emphasize is different from an absorbing state. Individuals may still change, but overall are trapped in the anomalous state. In Figure we illustrate this by the trapping of arriving Syrian refugees in the EU. |
We now demonstrate the breakdown of standard techniques for modelling and data analysis concerning behaviours in systems subject to ACI. In what follows we illustrate this with the example of employees in a company founded at time [MATH] . If each employee leaving their job is replaced immediately, every job generates ... |
. Such a renewal process involving ( ), is known as a fractional Poisson process This can be gauged via the structural population density [MATH] : we consider not only the number of individuals in the company at time [MATH] , but also the time [MATH] they have held their current job. A similar analysis can be done for ... |
. However, the strong memory of ACI invalidates the assumptions of these techniques, resulting in fundamentally altered, time-dependent employment or residence time distributions (see ( )). For CI, the equilibrium density [MATH] is a decaying function of duration time [MATH] . However, a hallmark of ACI is the presence... |
Properties and Predictions of ACI If employment at a company follows CI, then the distribution of employment durations is a decreasing function of duration [MATH] |
[EQUATION] plotted here for [MATH] For ACI, the distribution is radically changed to a U-shape: [EQUATION] with high probabilities of short [MATH] and very long [MATH] durations (see SI, part 3 and |
). The constant [MATH] , as is plotted for [MATH] . Crucially, the presence of [MATH] in ( ) implies a strong memory dependence. The population density [MATH] of workers remaining for an additional duration [MATH] is |
[EQUATION] which is also time-dependent (see ). The constant [MATH] takes the same value as above. Conversely, for CI [MATH] takes the form of ( ). |
The high probability of durations [MATH] in ( ) are indicative of the long-time persistence of systems with ACI. The presence of this U-shape is thus indicative of the high likelihood of continuation of the current persistent behaviour. Consequently, ACI has profound implications for the long-term expectations of human... |
If recidivism , the likelihood of a criminal to reoffend, is subject to inertia , we predict qualitatively different outcomes if the inertia is anomalous. If the inertia is ‘classical’ with [MATH] , the average individual may attempt behavioural change for some time, but will almost inevitably return to crime |
. This can be seen from the finite mean remaining time However, if the recidivism inertia is anomalous with [MATH] , the average individual is unlikely to succeed in changing behaviour. In other words, the expected recidivism time exceeds the human life span, though the median time may still be evaluated (see SI, part ... |
A successful strategy for criminal intervention would thus identify populations with [MATH] -values close to unity, where small changes in prevention or deterrent policies would thus move this individuals into the category of ACI. Expressions ( ), ( ) can be generalised to an alternating renewal process accounting for ... |
Altogether, the predictions of ACI have strong implications across all manner of human behaviours. When applied to job durations |
, we encounter attractive companies with high worker retention as employees prefer not to leave. Of the same persistence is encountered in residence patterns |
, this is in indication of popular areas with primarily a sellers’ market and rich opportunities for property speculation. Finally, when concerned with criminal activity |
, ACI in the recidivism rate indicates individuals or crime types wherein rehabilitation is more likely than re-offending. It is expected that similar patterns in behavioural change may be of relevance in marketing for the development of brand loyalty |
, or elite polarisation in political models One might question the robustness of ACI as described by ( ) and ( ). It can be shown that a time-decreasing rate [MATH] arises from population heterogeneity where individuals change behaviour with constant rate, whereupon ACI is a long-term transient phenomenon |
Consequently, each individual need not be subject to (A)CI in order for a company, real estate market or otherwise to reflect that overall trend |
Furthermore, anomalous effects can self-organise by the growing popularity of certain positions, neighbourhoods, etc. If the average number of individuals working for a company is [MATH] , then an increase in [MATH] may be indicative of good working conditions there, thus further increasing the number of applicants and... |
In this work we have exposed the ubiquitous nature of anomalous cumulative inertia in human activity, supported by empirical evidence from employment, residence, recidivism, and a plethora of other examples. A key contribution is the paradigm shift introduced by recognising the power in anomalous cumulative inertia to ... |
We believe that the link established in this paper between fractional calculus and the universal phenomenon of ACI provides hitherto unexplored understanding of the human experience as we change homes, jobs and affiliations. This data-driven application of fractional equations for behavioural changes provides an inclus... |
Supplementary Information is available in the online version of this paper. Author Contributions HS carried out data analysis. All authors wrote the manuscript. |
Author Information Reprints and permissions information is available at . Correspondence should be addressed to HS ( helena.stage@manchester.ac.uk ). |
# Source: arxiv 1806.00829 # Title: Machine learning of quantum phase transitions # Sections: all # Downloaded: 2026-03-03T05:15:51.938265+00:00 |
Machine learning of quantum phase transitions Abstract Machine learning algorithms provide a new perspective on the study of physical phenomena. In this paper, we explore the nature of quantum phase transitions using multi-color convolutional neural-network (CNN) in combination with quantum Monte Carlo simulations. We ... |
Machine learning, especially deep learning, has recently shown to be a very powerful tool in the fields of image classification, speech recognition, video activity recognition, machine translation, game playing and so on Krizhevsky et al. 2012 ); LeCun et al. 2015 ); Goodfellow et al. 2016 The basic idea is to train a ... |
An important task in condensed matter physics is to characterize different phases of matter and transitions between them Sachdev ( 2007 ); Wen ( 2004 . Phases can for example be characterized by local order parameters in Landau’s theory of spontaneously symmetry breaking Landau ( 1937 , by topological invariants in top... |
A natural way to use machine learning to identify different phases of matter is with the aid of Monte Carlo method Landau and Binder ( 2014 . By stochastically moving through configuration space according to a partition function, a large number of samples can be obtained and labelled by different phases. These can then... |
Quantum Monte Carlo (QMC) methods operate in at least [MATH] -dimensional configuration spaces (determinant QMC needs more) Pollet ( 2012 , where [MATH] is the spatial dimension and the extra dimension refers to the imaginary time [MATH] [MATH] is the temperature) direction. To detect quantum phase transitions, thermal... |
In this paper, we propose a systematic way to compress the [MATH] -dimensional configurations such that they can be fed into multi-color conventional neural-networks (CNNs). This approach is inspired by the similarity of the data structure in QMC simulations to video data. Firstly, we investigate the efficiency of our ... |
In order to simulate quantum systems, QMC algorithms usually sample the partition function based on an expansion ansatz. For example, in the stochastic series expansion (SEE) for a Hamiltonian of the form [MATH] (with [MATH] are defined on the bonds), the partition function is expanded as |
[EQUATION] where [MATH] is the basis in occupation number representations and [MATH] is the operator-index sequence Sandvik ( 1999 ); Syljuåsen and Sandvik ( 2002 ); Louis and Gros ( 2004 ); Sengupta et al. 2002 . Then we can define a discrete imaginary time [MATH] such that the state after [MATH] steps propagation is ... |
While diagonal order (e.g., density waves) can be directly detected by the density distribution in each slice [MATH] , off-diagonal order (e.g. superfluid order or boson condensation) is characterized by a change between slices. Thus the machine learning algorithm should not only consider the information of density dis... |
After compression, the reduced data will be used to train a deep learning model for classification. The structure we used is demonstrated in Fig. The model has [MATH] input channels, which are called color channels. The [MATH] matrices obtained by compressing the QMC data are fed into these channels one-by-one in seque... |
Firstly, we consider a continuous quantum phase transition between a Mott-insulator and superfluid phase of the Bose-Hubbard model on the triangular lattice. The Hamiltonian reads |
[EQUATION] where [MATH] [MATH] ) is the creation (annihilation) operator a boson, [MATH] represent nearest-neighbour sites, [MATH] is the hopping strength, and [MATH] describes the on-site repulsion. This model is relevant in the context of ultra-cold atoms on a triangular optical lattices Becker et al. 2010 . At small... |
Hu et al. Using QMC, we produce 20000 samples for each [MATH] in the region [MATH] with step [MATH] , the length of imaginary time is around [MATH] . After compressing these samples with different [MATH] (which reflect the compression strength), we separate the samples at each point into two sets. One set is used for t... |
The above example shows that the deep learning model can well predict the quantum phases and related continuous phase transition. However, the quantum phase transition could also be first order or an intermediate phase could emerge between two phases. A question we address now is whether deep learning can predict the e... |
[EQUATION] where [MATH] denotes the repulsive interaction between nearest-neighbor sites, hard-core implies that only occupancies [MATH] are allowed, and [MATH] is the chemical potential. The phase diagram is shown in Fig. (a). The solid phase breaks the translational symmetry and the superfluid phase breaks the U(1) s... |
Following the same strategy as above, we collect the samples deep in phases from regions marked with color blocks in Fig. (a), and the length of imaginary time is around [MATH] . After data compression, we feed them into the deep learning model for training. Next we run the prediction in the whole parameter region. As ... |
Using an enlarged training set, we find a phase diagram identical to the one identified by the “confusion” approach discussed above (see supplemental material for details). |
In brief, the relation between prediction of probability and compression strength can be used to distinguish direct quantum phase transition and intermediate phases. The reason such compression can keep the key information in imaginary time is its equivalence to the high frequency truncation. When we increase the value... |
At last, we try to directly check to which extend the winding in imaginary time can be extracted. In QMC, the bosons can form a net current flowing around the periodic system in real space. Due to the periodic boundary condition in imaginary time (trace of partition function), such current can only wind around the syst... |
In order to count the winding number using deep learning techinques, the neural network with one output neuron is used. We then randomly select the samples with small winding numbers from the samples of the previously discussed Bose-Hubbard model and divide them into a new training and testing set. Meanwhile, the label... |
In conclusion, we proposed a systematic way of generating and compressing training samples to be used for machine learning in combination with quantum Monte Carlo methods. The neural networks for deep learning are composed of multi-color CNN following with fully connected neuron layers. By implementing this method for ... |
Related work: While completing this manuscript, we became aware of a related work Hsu et al. 2018 which shows that supervised machine learning can be used to detect novel phases that have not been trained in the context of many-body localization. |
Acknowledgements We thanks Chen Zhang, Zhi-Yuan Xie for helpful discussions, and Hubert Scherrer-Paulus for technical supports on Google Tensorflow and GPU. F. Pollmann acknowledges support from DFG through Research Unit FOR 1807 with grant no. PO 1370/2-1 and from the Nanosystems Initiative Munich (NIM) by the German ... |
Appendix A Prediction of the intermediate supersolid phase In this supplementary material, we explain the details about the calculations which give the probability of the supersolid phase of the extended hard-core Bose-Hubbard model on the triangular lattice. From the results of the main text, we get the signature of t... |
# Source: arxiv 1806.00883 # Title: The (he)art of gluing # Sections: all # Downloaded: 2026-03-03T02:40:06.065350+00:00 The (he)art of gluing |
Abstract We introduce a notion of gluability for poset-indexed Bridgeland slicings on triangulated categories and show how a gluing abelian slicing on the heart of a bounded [MATH] -structure naturally induces a family of perverse [MATH] -structures. Our setup generalises the one of Collins and Polishchuk. As a corolla... |
Prologue A common feature shared by several constructions involving [MATH] -structures on triangulated categories is the following. One starts with a (possibly infinite) semiorthogonal decomposition [MATH] whose triangulated subcategories [MATH] are endowed with distinguished [MATH] -structures and, thanks to the vanis... |
For instance, the ‘perverse motives’ considered in SW18 and the ‘perverse coherent sheves’ considered in AB10 arise this way. A closely related example is the construction of the exotic [MATH] -structure on the derived category of [MATH] -modules, for [MATH] a Koszul algebra, obtained by Koszul duality BGS96 |
In this note we assume he reader is familiar with the language of Bridgeland slicings Bri07 , in its generalization for an arbitrary poset [MATH] endowed with a [MATH] -action considered in GKR04 and surveyed in FLM15 . We refer to FLM15 for the notation and the definition used here. In particular we use the language o... |
The gluing procedure Let [MATH] a stable [MATH] -category or, if one prefers a more classical setting, a triangulated category. If one considers Bridgeland slicings indexed by arbitrary partially orderes sets (endowed with a compatible [MATH] -action) as in GKR04 , then one can think of associating with any [MATH] -pos... |
2.1 [MATH] -compatible slicings We begin by recalling the construction of the slice functor; see FLM15 for details. Let [MATH] be a [MATH] -toset, i.e., a totally ordered set together with a monotone action of [MATH] , that we will denote by [MATH] . A [MATH] -slicing on a stable [MATH] -category [MATH] is a morphism o... |
Clearly, if [MATH] is a morphism of [MATH] -tosets, then [EQUATION] induces a [MATH] -equivariant morphism of [MATH] -posets [MATH] , and so composition with [MATH] gives a morphism |
[EQUATION] The slices of [MATH] are given by [MATH] , for any [MATH] . Notice that, as [MATH] is monotone, the subset [MATH] is an interval in [MATH] . It is immediate to see that [MATH] restricts to a map |
[EQUATION] Namely, if [MATH] for every [MATH] then [MATH] for every [MATH] in [MATH] . As we are assuming the [MATH] -slicing [MATH] is a Bridgeland slicing, this implies that [MATH] for every [MATH] in [MATH] , for every [MATH] . Therefore [MATH] for every [MATH] in [MATH] and so, again by definition of Bridgeland sli... |
Notice that, if [MATH] is a Bridgeland slicing of [MATH] , and [MATH] is a morphism of [MATH] -tosets, then for any slicing [MATH] of [MATH] , the lower and the upper categories |
[MATH] and [MATH] can be equivalently defined as [EQUATION] where [MATH] denotes the extension-closed subcategory of [MATH] [MATH] . In particular the slices of are given by |
[MATH] Remark 2.1 The right hand sides of the above two expressions can clearly be defined for every morphism [MATH] from [MATH] to [MATH] (i.e., not necessarily monotone nor [MATH] -equivariant), and as soon as [MATH] is [MATH] -equivariant, the assignment |
[EQUATION] is an equivariant morphism from [MATH] to pairs of subcategories of [MATH] . Clearly, when [MATH] is not monotone there is no reason to expect that the pair [MATH] forms a [MATH] -structure on [MATH] |
Yet, it interesting to notice that the condition that [MATH] be monotone is only only sufficient in order to have this, and can indeed be relaxed. |
Definition 2.2 Let [MATH] and [MATH] be [MATH] -tosets, and let [MATH] a map of [MATH] -sets (i.e., a [MATH] -equivariant map, not necessarily nondecreasing). A Bridgeland [MATH] -slicing [MATH] of [MATH] is said to be [MATH] -compatible |
if the condition ‘ [MATH] with [MATH] ’ implies [MATH] and [MATH] Remark 2.3 Clearly, if [MATH] is monotone, then every [MATH] -slicing [MATH] is [MATH] -compatible as the condition ‘ [MATH] with [MATH] ’ is empty. |
Remark 2.4 Let [MATH] and [MATH] be [MATH] -tosets, let [MATH] a map of [MATH] -sets, and let [MATH] be an isomorphism of [MATH] -posets. Then a Bridgeland [MATH] -slicing [MATH] of [MATH] is [MATH] -compatible if and only if it is [MATH] -compatible. Similarly, if [MATH] is an isomorphism of [MATH] -posets, then [MATH... |
Lemma 2.5 Let [MATH] and [MATH] be [MATH] -tosets, and let [MATH] be a [MATH] -equivariant morphism of [MATH] -sets (i.e., not necessarily a monotone map) and let [MATH] be a Bridgeland slicing of [MATH] which is [MATH] -compatible. Then, for any slicing [MATH] of [MATH] , the pair of subcategories |
[MATH] is a [MATH] -structure on [MATH] Proof. As [MATH] is [MATH] -equivariant and [MATH] , we have [EQUATION] and similarly for the lower subcategory [MATH] . To show that |
[EQUATION] it suffices to show that, if [MATH] and [MATH] then [MATH] . As [MATH] we cannot have [MATH] , so either [MATH] or vice versa. In the first case, [MATH] by definition of Bridgeland [MATH] -slicing. In the second case, we have [MATH] and [MATH] as [MATH] and [MATH] . Therefore, since [MATH] is [MATH] -compati... |
[EQUATION] with [MATH] and [MATH] . As [MATH] is a Bridgeland slicing, we have a factorization of the initial morphism [MATH] of the form |
[EQUATION] with [MATH] for all [MATH] , with [MATH] . Let us now consider the sequence of symbols [MATH] and [MATH] obtained putting in the [MATH] -th place [MATH] if [MATH] and [MATH] if [MATH] . If this sequence is of the form [MATH] , then there exists an index [MATH] such that |
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