Integrable Matrix Models in Discrete Space-Time

We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models provide an efficient integrable Trotterization of non-relativistic $\sigma$-models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau-Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar-Parisi-Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.Comment: v2, 60 pages, 10 figures, 1 tabl

Universal distributions 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 dynamical criticality. By performing optimized numerical simulations using an integrable space-time discretization we extract the algebraic growth exponents of time-dependent cumulants. 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 critical isotropic point we instead infer a weakly non-Gaussian distribution, indicating that superdiffusive spin transport found in integrable spin chains does not belong to any known dynamical universality class.Comment: 4.5 page

Universal anomalous fluctuations in charged single-file systems

Conventional classification of dynamical phenomena is based on universal hydrodynamic relaxation characterized by algebraic dynamical exponents and asymptotic scaling of the dynamical structure factor. This work uncovers a novel type of dynamical universality reflected in statistical properties of macroscopic fluctuating observables such as the transmitted charge. By considering a general class of one-dimensional single-file systems ({meaning that particle crossings are prohibited}) of interacting hardcore charged particles, we demonstrate that stringent dynamical constraints give rise 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. Despite dynamical criticality, we find the large-deviation rate function of the joint particle-charge transfer obeys the fluctuation relation. Curiously, the univariate charge-current rate function experiences a spontaneous breaking of fluctuation symmetry upon varying the particle and charge densities in a nonequilibrium initial state. The rich phenomenology of the outlined dynamical universality is exemplified on an exactly solvable classical cellular automaton of charged hardcore particles.Comment: 39+5 page

Kardar-Parisi-Zhang physics in integrable rotationally symmetric dynamics on discrete space-time lattice

We introduce a deterministic SO(3) invariant dynamics of classical spins on a discrete space–time lattice and prove its complete integrability by explicitly finding a related non-constant (baxterized) solution of the set-theoretic Yang–Baxter equation over the 2-sphere. Equipping the algebraic structure with the corresponding Lax operator we derive an infinite sequence of conserved quantities with local densities. The dynamics depend on a single continuous spectral parameter and reduce to a (lattice) Landau–Lifshitz model in the limit of a small parameter which corresponds to the continuous time limit. Using quasi-exact numerical simulations of deterministic dynamics and Monte Carlo sampling of initial conditions corresponding to a maximum entropy equilibrium state we determine spin-spin spatio-temporal (dynamical) correlation functions with relative accuracy of three orders of magnitude. We demonstrate that in the equilibrium state with a vanishing total magnetization the correlation function precisely follows Kardar–Parisi–Zhang scaling hence the spin transport belongs to the universality class with dynamical exponent z=3/2, in accordance to recent related simulations in discrete and continuous time quantum Heisenberg spin 1/2 chains

Neravnovesna statistična fizika v diskretnem prostor-času

We study the equilibrium and non-equilibrium statistical properties of interacting many-body systems, focusing on classical integrable models in one spatial dimensions. While integrability allows one to solve the initial value problem for a nonlinear system, the averaging over ensembles of initial conditions, implicit in a statistical description, is analytically intractable. Even numerical simulations of integrable systems are delicate since direct discretization invariably break integrability. By embedding an integrable system as a compatibility condition of a pair of linear problems, we instead define families of classical integrable system on a discrete space-time lattice, whose limits are Hamiltonian lattice/field-theory integrable models and facilitate their efficient numerical simulations. We solve the initial value problem of a model in discrete space-time by using the inverse scattering transform and formulate its thermodynamics within the soliton gas approximation. By using the defined integrable discretization, we study spin transport in the anisotropic lattice Landau–Lifshitz model. In integrable spin chains with non-abelian symmetry we find spin superdiffusion with the scaling function of the Kardar-Parisi-Zhang universality class. A refined view of dynamics is given by full-counting statistics of conserved quantities. We introduce the class of charged single-file systems and demonstrate their dynamical universality which we study in detail. We detect robust signs of dynamical criticality in the anisotropic lattice Landau–Lifshitz model and find an unexpected connection with charged single-file systems.Študiramo ravnovesne in neravnovesne statistične lastnosti sklopljenih mnogo-delčnih sistemov, kjer se osredotočimo na klasične integrabilne sisteme v eni prostorski dimenziji. Kljub temu, da integrabilnost omogoča ekzaktno rešitev začetnega problema za nelinearen sistem je povprečenje po začetnih pogojih analitično nedostopno. Celo numerične simulaticje integrabilnih sistemov so delikatne, saj direktne diskretizacije zlomijo integrabilnost. Postopamo drugače in integrabilne sisteme zapišemo kot kompatibilnostni pogoj za par linearnih problemov. S tem definiramo družine klasičnih integrabilnih sistemov na diskretni prostorsko-časovni mreži, katerih limite so Hamiltonski integrabilini modeli, ki jih lahko učinkovito numerično simuliramo. Z uporabo metode inverznega sipanja rešimo začetni problem modela v diskretnem prostor-času in formuliramo termodinamiko modela v približku solitonskega plina. S pomočjo definiranih integrabilnih diskretizacij študiramo spinski transport v anizotropnem Landau–Lifshitzovem modelu na mreži. V integrabilnih spinskih verigah z neabelsko simetrijo je spinska dinamika superdifuzvna s skalirno funkcijo Kardar-Parisi-Zhangovega univerzalnostnega razreda. Podrobnejši opis dinamike nam omogoča statistika polnega štetja ohranjenih količin. Vpeljemo razred nabitih enovrstičnih sistemov in pokažemo, da je njihova dinamika univerzalna. V anizotropnem Landau–Lifshitzovem modelu na mreži zaznamo robustne znake dinamične kritičnosti in najdemo nepričakovano povezavo z nabiti enovrstičnimi sistemi

Transport in an integrable classical spin space-time lattice

V delu študiramo transport v Landu-Lifshitzovem modelu na mreži v eni prostorski dimenziji, ki predstavlja klasičen analog kvantne spinske verige. Model motiviramo s kvantnomehansko obravnavo izmenjalne interakcije. Vpeljemo koncept integrabilnosti v klasični mehaniki ter zapišemo enačbe gibanja modela z uporabo Laxovih operatorjev in pogoja ničelne ukrivljenosti. Integrabilnost modela sledi iz involutivne lastnosti para matrik monodromij, ki jo dokažemo z uporabo R-matrike. Predstavimo Trotterjev razcep Liouvillove enačbe, ki nam omogoči učinkovito simulacijo dinamike modela. Naiven razcep modela se izkaže za neintegrabilnega, kar potrdimo z izračunom Lyapunovega spektra. Uvedemo novo integrabilno posplošitev Landau-Lifshitzovega modela na diskretno krajevno-časovno mrežo ter poiščemo dvodelčni hamiltonian, ki generira želeno dinamiko ter novo R-matriko modela. Pokažemo, da je takšen model sam svoj dual. V nemagnetiziranem stanju se model izkaže za superdifuzivnega, z analizo korelacijskih funkcij pokažemo, da spada v Kardar-Parisi-Zhangov univerzalnostni razred. V magnetiziranem stanju prevlada balistično obnašanje, opazimo bogato odvisnost korelacijske funkcije magnetizacije od celotne magnetizacije.In the present work we study the transport properties of the lattice Landau-Lifshitz model in one spatial dimensions, which represents a classical analogue of a quantum spin chain. The model is motivated by a quantum mechanical treatment of the exchange interaction. We introduce the concept of integrability in classical mechanics and write down the model’s equations of motion using Lax operators and the zero-curvature condition. The integrability of the models follows from an involutive property of a pair of monodromy matrices which we prove using an R-matrix. We present the Trotter decomposition of the Liouville equation, which allows for an efficient simulation of the model dynamics. The naive decomposition of the model turns out to be nonintegrable, which we verify by computing the Lyapunov spec- trum. We introduce a novel integrable generalization of the Landau-Lifshitz model on the discrete time lattice and compute the two-body Hamiltonian that generates the requisite dynamics and find a novel R-matrix. We show that the model is self-dual. The model turns out to be superdiffusive in a non-magnetized state, by analyzing its correlation functions we further show it belongs into the Kardar-Parisi-Zhang universality class. In a magnetized state ballistic trasnport predominates, a rich dependance of the correlation function of magnetization upon net magnetization is observed

Undular diffusion in nonlinear sigma models

We discuss general features of charge transport in nonrelativistic 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 analogs 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

Anisotropic Landau-Lifshitz model in discrete space-time

Weconstruct an integrable lattice model of classical interacting spins in discrete spacetime, 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

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.Comment: v2: 4.5 + 22 pages, minor corrections and fixed typos, new section on phase transition in SM; v3: slightly updated notatio