arXiv subjects

Enej Ilievski

Publications and source records attributed to Enej Ilievski.

At least 19 recordsLinked to original sources

arXiv2026

Anomalous hydrodynamic fluctuations in the quantum XXZ spin chain

The quantum XXZ spin-1/2 chain features non-Gaussian spin current fluctuations in the regime of easy-axis anisotropy. Using ballistic macroscopic fluctuation theory, we derive the exact probability distribution of typical spin-current fluctuations in thermal equilibrium. The obtained nested Gaussian distribution is fully characterized by its variance which we analytically relate to the spin diffusion constant and static spin susceptibility, and compare with numerical simulations. By unveiling how the same mechanism which leads to anomalous charge current fluctuations in single-file systems manifests itself in the XXZ chain, our approach establishes the universal hydrodynamic origin of the observed anomalous fluctuations.

cond-mat.stat-mech↗

arXiv2026

Effective field theories of nonlinear fluctuating hydrodynamics in one dimension

Describing emergent macroscopic phenomena in low spatial dimensions is known to be notoriously challenging, primarily due to strong interactions that render perturbative approaches inapplicable. On the other hand, low-dimensional systems host a wealth of unorthodox phenomena. A prominent example is the emergence of superdiffusive transport in one-dimensional interacting systems, traditionally studied in the framework of nonlinear fluctuating hydrodynamics. After identifying and discussing internal inconsistencies in the previous formulations, in this work we develop a general and systematic approach for constructing effective field theories of one-dimensional hydrodynamic systems in the form of coupled stochastic Langevin-type equations compatible with the physical requirements of local equilibrium states such as the thermodynamic Maxwell relation and fluctuation-dissipation symmetry. We implemented a general numerical integration scheme and exemplified our construction on a simple model of two interacting hydrodynamic modes with a non-Gaussian stationary equilibrium measure.

cond-mat.stat-mech↗

arXiv2026

Multiscale Structure of Eigenstate Thermalization

The eigenstate thermalization hypothesis provides a framework for understanding thermalization in isolated quantum many-body systems by characterizing statistical properties of local observables in energy eigenstates. Here we demonstrate that distributions of matrix elements in macroscopic systems may depend not only on the macrostate parameters, such as the densities of local conserved charges, but generally also on the properties of ensembles used in sampling eigenstates. To this end, we depart from the conventional analysis of microcanonical windows and consider statistical ensembles with an adjustable scale parameter prescribing the magnitude of charge fluctuations. We specifically consider an integrable field theory that permits efficient numerical sampling of matrix elements and reliable extrapolation to the thermodynamic limit. Moreover, in this system, we identify a class of states that enables explicit closed-form computation of the suppression rate of matrix elements. Our findings reveal an underlying multiscale structure of matrix elements captured by a non-analytic fluctuation-scale dependence of algebraic exponents governing their statistical properties.

cond-mat.stat-mech↗

arXiv2026

Thermalization in classical systems with discrete phase space

We study the emergence of statistical mechanics in isolated classical systems with local interactions and discrete phase spaces. We establish that thermalization in such systems does not require global ergodicity; instead, it arises from effective local ergodicity, where dynamics in a subsystem may appear pseudorandom. To corroborate that, we analyze the spectrum of the unitary evolution operator and propose an ansatz to describe statistical properties of local observables expanded in the eigenfunction basis - the classical counterpart of the Eigenstate Thermalization Hypothesis. Our framework provides a unified perspective on thermalization in classical and quantum systems with discrete spectra.

cond-mat.stat-mech↗

arXiv2026

Anisotropic Landau-Lifshitz Model in Discrete Space-Time

We construct an integrable lattice model of classical interacting spins in discrete space-time, representing a discrete-time analogue of the lattice Landau-Lifshitz ferromagnet with uniaxial anisotropy. As an application we use this explicit discrete symplectic integration scheme to compute the spin Drude weight and diffusion constant as functions of anisotropy and chemical potential. We demonstrate qualitatively different behavior in the easy-axis and the easy-plane regimes in the non-magnetized sector. Upon approaching the isotropic point we also find an algebraic divergence of the diffusion constant, signaling a crossover to spin superdiffusion.

cond-mat.stat-mech↗

arXiv2025

Quantum many-body spin ratchets

Introducing a class of SU(2) invariant quantum unitary circuits generating chiral transport, we examine the role of broken space-reflection and time-reversal symmetries on spin transport properties. Upon adjusting parameters of local unitary gates, the dynamics can be either chaotic or integrable. The latter corresponds to a generalization of the space-time discretized (Trotterized) higher-spin quantum Heisenberg chain. We demonstrate that breaking of space-reflection symmetry results in a drift in the dynamical spin susceptibility. Remarkably, we find a universal drift velocity given by a simple formula which, at zero average magnetization, depends only on the values of SU(2) Casimir invariants associated with local spins. In the integrable case, the drift velocity formula is confirmed analytically based on the exact solution of thermodynamic Bethe ansatz equations. Finally, by inspecting the large fluctuations of the time-integrated current between two halves of the system in stationary maximum-entropy states, we demonstrate violation of the Gallavotti-Cohen symmetry, implying that such states cannot be regarded as equilibrium ones. We show that the scaled cumulant generating function of the time-integrated current instead obeys a generalized fluctuation relation.

cond-mat.stat-mech↗

arXiv2025

Integrable fishnet circuits and Brownian solitons

We introduce classical many-body dynamics on a one-dimensional lattice comprising local two-body maps arranged on discrete space-time mesh that serve as discretizations of Hamiltonian dynamics with arbitrarily time-varying coupling constants. Time evolution is generated by passing an auxiliary degree of freedom along the lattice, resulting in a `fishnet' circuit structure. We construct integrable circuits consisting of Yang-Baxter maps and demonstrate their general properties, using the Toda and anisotropic Landau-Lifschitz models as examples. Upon stochastically rescaling time, the dynamics is dominated by fluctuations and we observe solitons undergoing Brownian motion.

cond-mat.stat-mech↗

arXiv2024

Landau-Lifschitz magnets: exact thermodynamics and transport

The classical Landau--Lifshitz equation -- the simplest model of a ferromagnet -- provides an archetypal example for studying transport phenomena. In one-spatial dimension, integrability enables the classification of the spectrum of linear and nonlinear modes. An exact characterization of finite-temperature thermodynamics and transport has nonetheless remained elusive. We present an exact description of thermodynamic equilibrium states in terms of interacting modes. This is achieved by retrieving the classical Landau--Lifschitz model through the semiclassical limit of the integrable quantum spin-$S$ anisotropic Heisenberg chain at the level of the thermodynamic Bethe ansatz description. In the axial regime, the mode spectrum comprises solitons with unconventional statistics, whereas in the planar regime we additionally find two special types of modes of radiative and solitonic type. The obtained framework paves the way for analytical study of unconventional transport properties: as an example we study the finite-temperature spin Drude weight, finding excellent agreement with Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2024

Universal anomalous fluctuations in charged single-file systems

Introducing a general class of one-dimensional single-file systems (meaning that particle crossings are prohibited) of interacting hardcore particles with internal degrees of freedom (called charge), we exhibit a novel type of dynamical universality reflected in anomalous statistical properties of macroscopic fluctuating observables such as charge transfer. We find that stringent dynamical constraints lead to universal anomalous statistics of cumulative charge currents manifested both on the timescale characteristic of typical fluctuations and also in the rate function describing rare events. By computing the full counting statistics of net transferred charge between two extended subsystems, we establish a number of unorthodox dynamical properties in an analytic fashion. Most prominently, typical fluctuations in equilibrium are governed by a universal distribution that markedly deviates from the expected Gaussian statistics, whereas large fluctuations are described by an exotic large-deviation rate function featuring an exceptional triple critical point. Far from equilibrium, competition between dynamical phases leads to dynamical phase transitions of first and second order and spontaneous breaking of fluctuation symmetry of the univariate charge large-deviation function. The rich phenomenology of the outlined dynamical universality is exemplified on an exactly solvable classical cellular automaton of charged hardcore particles. We determine the dynamical phase diagram in the framework of Lee-Yang's theory of phase transitions and exhibit a hyper-dimensional diagram of distinct dynamical regimes. Our findings lead us to conclude that the conventional classification of dynamical universality classes based on the algebraic dynamical exponents and asymptotic scaling functions that characterize hydrodynamic relaxation of the dynamical structure factor is incomplete and calls for refinement.

cond-mat.stat-mech↗

arXiv2023

Dynamical criticality of magnetization transfer in integrable spin chains

Recent studies have found that fluctuations of magnetization transfer in integrable spin chains violate the central limit property. Here we revisit the problem of anomalous counting statistics in the Landau-Lifshitz field theory by specializing to two distinct anomalous regimes featuring a dynamical critical point. By performing optimized numerical simulations using an integrable space-time discretization we extract the algebraic growth exponents of time-dependent cumulants which attain their threshold values. The distinctly non-Gaussian statistics of magnetization transfer in the easy-axis regime is found to converge towards the universal distribution of charged single-file systems. At the isotropic point we infer a weakly non-Gaussian distribution, corroborating the view that superdiffusive spin transport in integrable spin chains does not belong to any known dynamical universality class.

cond-mat.stat-mech↗

arXiv2023

Squeezed ensembles and anomalous dynamic roughening in interacting integrable chains

It is widely accepted that local subsystems in isolated integrable quantum systems equilibrate to generalized Gibbs ensembles. Here, we demonstrate the failure of canonical generalized thermalization for a particular class of initial states in certain types of interacting integrable models. Particularly, we show that in the easy-axis regime of the quantum XXZ chain, pure non-equilibrium initial states with no magnetic fluctuations instead locally relax to squeezed generalized Gibbs ensembles, referring to exotic equilibrium states governed by non-local equilibrium Hamiltonians with sub-extensive charge fluctuations that violate the self-affine scaling. The behaviour at the isotropic point is exceptional and depends on the initial state. We find that relaxation from the Néel state is governed by extensive fluctuations and a super-diffusive dynamical exponent compatible with the Kardar-Parisi-Zhang universality. On the other hand, there are other non-fluctuating initial states that display diffusive scaling. Our predictions can be directly tested in state-of-the-art cold atomic experimental settings.

cond-mat.quant-gas↗

arXiv2022

Grokking phase transitions in learning local rules with gradient descent

We discuss two solvable grokking (generalisation beyond overfitting) models in a rule learning scenario. We show that grokking is a phase transition and find exact analytic expressions for the critical exponents, grokking probability, and grokking time distribution. Further, we introduce a tensor-network map that connects the proposed grokking setup with the standard (perceptron) statistical learning theory and show that grokking is a consequence of the locality of the teacher model. As an example, we analyse the cellular automata learning task, numerically determine the critical exponent and the grokking time distributions and compare them with the prediction of the proposed grokking model. Finally, we numerically analyse the connection between structure formation and grokking.

cond-mat.stat-mech↗

arXiv2022

Popcorn Drude weights from quantum symmetry

Integrable models provide emblematic examples of non-ergodic phenomena. One of their most distinguished properties are divergent zero-frequency conductivities signalled by finite Drude weights. Singular conductivities owe to long-lived quasiparticle excitations that propagate ballistically through the system without any diffraction. The case of the celebrated quantum Heisenberg chain, one of the best-studied many-body paradigms, turns out to be particularly mysterious. About a decade ago, it was found that the spin Drude weight in the critical phase of the model assumes an extraordinary, nowhere continuous, dependence on the anisotropy parameter in the shape of a `popcorn function'. This unprecedented discovery has been afterwards resolved at the level of the underlying deformed quantum symmetry algebra which helps explaining the erratic nature of the quasiparticle spectrum at commensurate values of interaction anisotropy. This work is devoted to the captivating phenomenon of discontinuous Drude weights, with the aim to give a broader perspective on the topic by revisiting and reconciling various perspectives from the previous studies. Moreover, it is argued that such an anomalous non-ergodic feature is not exclusive to the integrable spin chain but can be instead expected in a number of other integrable systems that arise from realizations of the quantum group $\mathcal{U}_{q}(\mathfrak{sl}(2))$, specialized to unimodular values of the quantum deformation parameter $q$. Our discussion is framed in the context of gapless anisotropic quantum chains of higher spin and the sine-Gordon quantum field theory in two space-time dimensions.

cond-mat.stat-mech↗

arXiv2022

Exact anomalous current fluctuations in a deterministic interacting model

We analytically compute the full counting statistics of charge transfer in a classical automaton of interacting charged particles. Deriving a closed-form expression for the moment generating function with respect to a stationary equilibrium state, we employ asymptotic analysis to infer the structure of charge current fluctuations for a continuous range of timescales. The solution exhibits several unorthodox features. Most prominently, on the timescale of typical fluctuations the probability distribution of the integrated charge current in a stationary ensemble without bias is distinctly non-Gaussian despite diffusive behavior of dynamical charge susceptibility. While inducing a charge imbalance is enough to recover Gaussian fluctuations, we find that higher cumulants grow indefinitely in time with different exponents, implying singular scaled cumulants. We associate this phenomenon with the lack of a regularity condition on moment generating functions and the onset of a dynamical critical point. In effect, the scaled cumulant generating function does not, irrespectively of charge bias, represent a faithful generating function of the scaled cumulants, yet the associated Legendre dual yields the correct large-deviation rate function. Our findings hint at novel types of dynamical universality classes in deterministic many-body systems.

cond-mat.stat-mech↗

arXiv2021

Absence of Normal Fluctuations in an Integrable Magnet

We investigate dynamical fluctuations of transferred magnetization in the one-dimensional lattice Landau--Lifshitz magnet with uniaxial anisotropy, representing an emblematic model of interacting spins. We demonstrate that the structure of fluctuations in thermal equilibrium depends radically on the characteristic dynamical scale. In the ballistic regime, typical fluctuations are found to follow a normal distribution and scaled cumulants are finite. In stark contrast, on the diffusive and superdiffusive timescales, relevant respectively for the easy-axis and isotropic magnet at vanishing total magnetization, typical fluctuations are no longer Gaussian and, remarkably, scaled cumulants are divergent. The observed anomalous features disappear upon breaking integrability, suggesting that the absence of normal fluctuations is intimately tied to the presence of soliton modes. In a nonequilibrium setting of the isotropic magnet with weakly polarized step-profile initial state we find a slow drift of dynamical exponent from the superdiffusive towards the diffusive value.

cond-mat.stat-mech↗

arXiv2021

Superdiffusion in spin chains

This review summarizes recent advances in our understanding of anomalous transport in spin chains, viewed through the lens of integrability. Numerical advances, based on tensor-network methods, have shown that transport in many canonical integrable spin chains -- most famously the Heisenberg model -- is anomalous. Concurrently, the framework of generalized hydrodynamics has been extended to explain some of the mechanisms underlying anomalous transport. We present what is currently understood about these mechanisms, and discuss how they resemble (and differ from) the mechanisms for anomalous transport in other contexts. We also briefly review potential transport anomalies in systems where integrability is an emergent or approximate property. We survey instances of anomalous transport and dynamics that remain to be understood.

cond-mat.stat-mech↗

arXiv2021

Semi-classical quantisation of magnetic solitons in the anisotropic Heisenberg quantum chain

Using the algebro-geometric approach, we study the structure of semi-classical eigenstates in a weakly-anisotropic quantum Heisenberg spin chain. We outline how classical nonlinear spin waves governed by the anisotropic Landau-Lifshitz equation arise as coherent macroscopic low-energy fluctuations of the ferromagnetic ground state. Special emphasis is devoted to the simplest types of solutions, describing precessional motion and elliptic magnetisation waves. The internal magnon structure of classical spin waves is resolved by performing the semi-classical quantisation using the Riemann-Hilbert problem approach. We present an expression for the overlap of two semi-classical eigenstates and discuss how correlation functions at the semi-classical level arise from classical phase-space averaging.

cond-mat.stat-mech↗

arXiv2020

Undular diffusion in nonlinear sigma models

We discuss general features of charge transport in non-relativistic classical field theories invariant under non-abelian unitary Lie groups by examining the full structure of two-point dynamical correlation functions in grand-canonical ensembles at finite charge densities (polarized ensembles). Upon explicit breaking of non-abelian symmetry, two distinct transport laws characterized by dynamical exponent $z=2$ arise. While in the unbroken symmetry sector the Cartan fields exhibit normal diffusion, the transversal sectors governed by the nonlinear analogues of Goldstone modes disclose an unconventional law of diffusion characterized by a complex diffusion constant and undulating patterns in the spatiotemporal correlation profiles. In the limit of strong polarization, one retrieves the imaginary-time diffusion for uncoupled linear Goldstone modes, whereas for weak polarizations the imaginary component of the diffusion constant becomes small. In models of higher rank symmetry, we prove absence of dynamical correlations among distinct transversal sectors.

cond-mat.stat-mech↗