We discuss a general procedure to construct an integrable real-time Trotterization of interacting lattice models. As an illustrative example, we consider a spin-1/2 chain, with continuous time dynamics described by the isotropic (XXX) Heisenberg Hamiltonian. For periodic boundary conditions, local conservation laws are derived from an inhomogeneous transfer matrix, and a boost operator is constructed. In the continuous time limit, these local charges reduce to the known integrals of motion of the Heisenberg chain. In a simple Kraus representation, we also examine the nonequilibrium setting, where our integrable cellular automaton is driven by stochastic processes at the boundaries. We show explicitly how an exact nonequilibrium steady-state density matrix can be written in terms of a staggered matrix product ansatz, and we propose quasilocal conservation laws for the model with periodic boundary conditions. This simple Trotterization scheme, in particular in the open system framework, could prove to be a useful tool for experimental simulations of the lattice models in terms of trapped ion and atom optics setups.
We discuss a general procedure to construct an integrable real-time Trotterization of interacting lattice models. As an illustrative example, we consider a spin-1/2 chain, with continuous time dynamics described by the isotropic (XXX) Heisenberg Hamiltonian. For periodic boundary conditions, local conservation laws are derived from an inhomogeneous transfer matrix, and a boost operator is constructed. In the continuous time limit, these local charges reduce to the known integrals of motion of the Heisenberg chain. In a simple Kraus representation, we also examine the nonequilibrium setting, where our integrable cellular automaton is driven by stochastic processes at the boundaries. We show explicitly how an exact nonequilibrium steady-state density matrix can be written in terms of a staggered matrix product ansatz, and we propose quasilocal conservation laws for the model with periodic boundary conditions. This simple Trotterization scheme, in particular in the open system framework, could prove to be a useful tool for experimental simulations of the lattice models in terms of trapped ion and atom optics setups.
We present two directions of research in statistical mechanics of nonequilibrium onedimensional quantum systems. One is related to isolated integrable models, the other one to exactly solvable dissipatively driven spin chains. The device enabling their analysis is identified as the quantum group symmetry.
In the framework of isolated systems we focus on the concept of integrable Floquet driven systems. In particular, we show how to build such systems out of the basic constituents of integrability structure. This allows substantiated and conclusive statements,for instance, about the transport phenomena. Despite their inherent relation, the presented integrable periodically driven models originate in two different ideas. On the one hand we have Trotterisations of integrable
spin chains, concieved in an effort to better understand the dynamics of spin models. Then, there are systems originating in the attempts to solve quantum field theory in an algebraically closed form, for example, the quantum Hirota equation. The bulk of our consideration of isolated quantum systems consists of the construction of extensive conservation laws that either (i) constitute the generalised Gibbs ensemble and the hydrodynamic description of thermalisation after a quantum quench, or (ii) prevent the decay of current autocorrelations and thus characterise ideal transport in the system. The exact way in which these charges determine the dynamics stems from the symmetries of the quantum group representations.
The second part of the exposition is dedicated to open quantum systems. Again we distinguish two formally related settings. The first one is that of a boundary driven quantum cellular automaton, our particular example being the integrable Trotterisation of the Heisenberg magnet. Starting from a repeated interaction protocol, in which the system under scrutiny is repeatedly coupled to the environment, we introduce the Kraus map as a general form of a dissipative time evolution of the density matrix. We then solve for its unique nonequilibrium steady state, using integrability structure of the model. The other setting is that of a dissipatively boundary driven spin chain in the continuous time. Here we present the recently developed formalism of inhomogeneous Lax structure. Using it we demonstrate the solvability of the XXZ and XYZ spin chains, acted upon by the Lindblad operators that polarise the boundary spins in arbitrary directions. The ansatz for the steady state is particularly interesting, since it exhibits a
previously unknown integrability structure, differing from site to site in the spin chain. This structure can independently produce nontrivial conservation laws in an isolated spin
chain with arbitrary boundary fields.
We review recent progress on constructing non-equilibrium steady state density operators of boundary driven locally interacting quantum chains, where driving is implemented via Markovian dissipation channels attached to the chain's ends. We discuss explicit solutions in three different classes of quantum chains, specifically, the paradigmatic (anisotropic) Heisenberg spin- chain, the Fermi–Hubbard chain, and the Lai–Sutherland spin-1 chain, and discuss universal concepts which characterize these solutions, such as matrix product ansatz and a more structured walking graph state ansatz. The central theme is the connection between the matrix product form of nonequilibrium states and the integrability structures of the bulk Hamiltonian, such as the Lax operators and the Yang–Baxter equation. However, there is a remarkable distinction with respect to the conventional quantum inverse scattering method, namely addressing nonequilibrium steady state density operators requires non-unitary irreducible representations of Yang–Baxter algebra which are typically of infinite dimensionality. Such constructions result in non-Hermitian, and often also non-diagonalisable families of commuting transfer operators which in turn result in novel conservation laws of the integrable bulk Hamiltonians. For example, in the case of the anisotropic Heisenberg model, quasi-local conserved operators which are odd under spin reversal (or spin flip) can be constructed, whereas the conserved operators stemming from orthodox Hermitian transfer operators (via logarithmic differentiation) are all even under spin reversal.
A continuous family of quasilocal exact conservation laws is constructed in the anisotropic Heisenberg (XXZ) spin-1/2 chain for periodic (or twisted) boundary conditions and for a set of commensurate anisotropies densely covering the entire easy plane interaction regime. All local conserved operators follow from the standard (Hermitian) transfer operator in fundamental representation (with auxiliary spin s=1/2), and are all even with respect to a spin flip operation. However, the quasilocal family is generated by differentiation of a non-Hermitian highest weight transfer operator with respect to a complex auxiliary spin representation parameter s and includes also operators of odd parity. For a finite chain with open boundaries the time derivatives of quasilocal operators are not strictly vanishing but result in operators localized near the boundaries of the chain. We show that a simple modification of the non-Hermitian transfer operator results in exactly conserved, but still quasilocal operators for periodic or generally twisted boundary conditions. As an application, we demonstrate that implementing the new exactly conserved operator family for estimating the high-temperature spin Drude weight results, in the thermodynamic limit, in exactly the same lower bound as for almost conserved family and open boundaries. Under the assumption that the bound is saturating (suggested by agreement with previous thermodynamic Bethe ansatz calculations) we propose a simple explicit construction of infinite time averages of local operators such as the spin current.
In the thesis we present an analytic approach towards exact description for steady state density operators of nonequilibrium quantum dynamics in the framework of open systems. We employ the so-called quantum Markovian semi-group evolution, i.e. a general form of time-autonomous positivity and trace-preserving dynamical equation for reduced density operators, by only allowing unitarity-breaking dissipative terms acting at the boundaries of a system. Such setup enables to simulate macroscopic reservoirs for different values of effective thermodynamic potentials, causing incoherent transitions between quantum states which are modeled with aid of the Lindblad operators. This serves as a simple minimalistic model for studying quantum transport properties, either in the linear response domain or in more general regimes far from canonical equilibrium. We are mainly exploring possibilities of identifying nonequilibrium situations which are amenable to exact description within matrix product state representation, by exclusively focusing on steady states, i.e. fixed points of the Lindblad equation, of certain prototypic interacting integrable spin chains driven by incoherent polarizing processes. Finally, we define a concept of pseudo-local extensive almost-conserved quantities by allowing a violation of time-invariance up to boundary-localized terms. We elucidate the role of such quantities on non-ergodic behaviour of temporal correlation functions rendering anomalous transport properties. It turns out that such conservation laws can be generated by means of boundary universal quantum transfer operators of the fundamental integrable models.
We review recent progress in understanding the notion of locality in integrable quantum lattice systems. The central concept are 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.
We extend T. Prosen's construction of quasilocal conserved quantities for the XXZ model [Phys. Rev. Lett. 106, 217206 (2011)] to the case of periodic boundary conditions. These quasilocal operators stem from a two-parameter transfer matrix which employs a highest-weight representation of the quantum group algebra inherent in the Yang-Baxter algebra. In contrast with the open chain, where the conservation law is weakly violated by boundary terms, the quasilocal operators in the periodic chain exactly commute with the Hamiltonian and other local conserved quantities.