Abstract
We review recent progress in understanding the notion of locality in integrable quantum lattice systems. The central concept concerns the so-called quasilocal conserved quantities, which go beyond the standard perception of locality. Two systematic procedures to rigorously construct families of quasilocal conserved operators based on quantum transfer matrices are outlined, specializing on anisotropic Heisenberg XXZ spin-1/2 chain. Quasilocal conserved operators stem from two distinct classes of representations of the auxiliary space algebra, comprised of unitary (compact) representations, which can be naturally linked to the fusion algebra and quasiparticle content of the model, and non-unitary (non-compact) representations giving rise to charges, manifestly orthogonal to the unitary ones. Various condensed matter applications in which quasilocal conservation laws play an essential role are presented, with special emphasis on their implications for anomalous transport properties (finite Drude weight) and relaxation to non-thermal steady states in the quantum quench scenario.
Highlights
Local conservation laws are amongst the most important fundamental concepts in theoretical physics. In generic systems these usually comprise of energy, momentum, particle number, etc., and correspond to Noether charges connected to rather obvious physical symmetries
The simplest extensions should involve quantum lattice models associated with the so-called fundamental solutions to Yang–Baxter equation, with underlying symmetry algebras based on Lie algebras of higher rank and their quantizations
The notion of locality indisputably plays a monumental role in the foundations of statistical mechanics, both on the classical and quantum level
Summary
Local conservation laws are amongst the most important fundamental concepts in theoretical physics. Interacting quantum systems, where local degrees of freedom (quantum spins, fermions, or bosons) are arranged in a regular 1D lattice, are typically considered integrable in one of the following cases: Firstly, there may exist a canonical (Bogoliubov) transformation which maps the local degrees of freedom to non-interacting quasiparticles. Such is, for example, the situation with quantum transverse field Ising model, or XY spin-1/2 chain [1]. Besides defining and describing integrability in closed quantum many-body systems in 1D [7], it covers 2D equilibrium classical statistical systems [8], nonequilibrium classical driven diffusive 1D systems [9], as well as classical Hamiltonian systems [10,11], and since more recently, integrable nonequilibrium steady states of open quantum interacting systems [12]
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