日本物理学会誌及びJournal of the Physical Society of Japanの月刊について

Journal of Statistical Mechanics: theory and experiment

Proceedings of the American Mathematical Society

Over the last decade impressive progress has been made in the theoretical understanding of transport properties of clean, one-dimensional quantum lattice systems. Many physically relevant models in one dimension are Bethe-ansatz integrable, including the anisotropic spin-1/2 Heisenberg (also called the spin-1/2 XXZ chain) and the Fermi-Hubbard model. Nevertheless, practical computations of correlation functions and transport coefficients pose hard problems from both the conceptual and technical points of view. Only because of recent progress in the theory of integrable systems, on the one hand, and the development of numerical methods, on the other hand, has it become possible to compute their finite-temperature and nonequilibrium transport properties quantitatively. Owing to the discovery of a novel class of quasilocal conserved quantities, there is now a qualitative understanding of the origin of ballistic finite-temperature transport, and even diffusive or superdiffusive subleading corrections, in integrable lattice models. The current understanding of transport in one-dimensional lattice models, in particular, in the paradigmatic example of the spin-1/2 XXZ and Fermi-Hubbard models, is reviewed, as well as state-of-the-art theoretical methods, including both analytical and computational approaches. Among other novel techniques, matrix-product-state-based simulation methods, dynamical typicality, and, in particular, generalized hydrodynamics are covered. The close and fruitful connection between theoretical models and recent experiments is discussed, with examples given from the realms of both quantum magnets and ultracold quantum gases in optical lattices.

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Finite-temperature transport in one-dimensional quantum lattice models

Transformations of soliton equations into the bilinear forms involving four dependent variables are discussed. It is found that both nonlinear Schrodinger equation and classical Heisenberg ferromagnet equation are transformed into the same bilinear from, while the equation \begin{aligned} \phi_{xt}{=}\phi(1-|\phi_{t}|^{2})^{1/2} \end{aligned} shares the same bilinear form with the Pohlmeyer-Lund-Regge-Getmanov equation. Transformations of the complex sine-Gordon equation and the Landau-Lifshitz equation into the bilinear forms are also described.

Bilinearization of Soliton Equations (非線形波動現象の数理物理学(研究会報告))

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.

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Kardar–Parisi–Zhang Physics in Integrable Rotationally Symmetric Dynamics on Discrete Space–Time Lattice

We introduce a class of integrable dynamical systems of interacting classical matrixvalued 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 σ-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.

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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.

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Integrable Matrix Models in Discrete Space-Time

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.

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Undular Diffusion in Nonlinear Sigma Models

We perform energy spectrum analysis of the active turbulence in 3D bulk active nematic using continuum numerical modelling. Specifically, we calculate the spectra of two main energy contributions---kinetic energy and nematic elastic energy---and combine this with the geometrical analysis of the nematic order and flow fields, based on direct defect tracking and calculation of autocorrelations. We show that the active nematic elastic energy is concentrated at scales corresponding to the effective defect-to-defect separation, scaling with activity as $\sim\zeta^{0.5}$, whereas the kinetic energy is largest at somewhat larger scales of typically several 100 nematic correlation lengths. Nematic biaxiallity is shown to have no role in active turbulence at most of length scales, but can affect the nematic elastic energy by an order of magnitude at scales of active defect core size. The effect of an external aligning field on the 3D active turbulence is explored, showing a transition from an effective active turbulent to an aligned regime. The work is aimed to provide a contribution towards understanding active turbulence in general three-dimensions, from the perspective of main energy-relevant mechanisms at different length scales of the system.

Žiga Krajnik

Papers

Kardar–Parisi–Zhang Physics in Integrable Rotationally Symmetric Dynamics on Discrete Space–Time Lattice

Integrable matrix models in discrete space-time

Integrable Matrix Models in Discrete Space-Time

Undular Diffusion in Nonlinear Sigma Models

Spectral energy analysis of bulk three-dimensional active nematic turbulence