Infinite Quantum Spin Chains Research Articles

On the set of reduced states of translation invariant, infinite quantum systems

The set of two-body reduced states of translation invariant, infinite quantum spin chains can be approximated from inside and outside using matrix product states and marginals of finite systems, respectively. These lead to hierarchies of algebraic approximations that become tight only in the limit of infinitely many auxiliary variables. We show that this is necessarily so for any algebraic ansatz by proving that the set of reduced states is not semialgebraic. We also provide evidence that additional elementary transcendental functions cannot lead to a finitary description.

Groupoid and algebra of the infinite quantum spin chain

It is well known that certain features of a quantum theory cannot be described in the standard picture on a Hilbert space. In particular, this happens when we try to formally frame a quantum field theory, or a thermodynamic system with finite density. This forces us to introduce different types of algebras, more general than the ones we usually encounter in a standard course of quantum mechanics. We show how these algebras naturally arise in the Schwinger description of the quantum mechanics of an infinite spin chain. In particular, we use the machinery of Dirac-Feynman-Schwinger (DFS) states developed in recent works to introduce a dynamics based on the modular theory by Tomita-Takesaki, and consequently we apply this approach to describe the Ising model.

Tensor-network approach to compute genuine multisite entanglement in infinite quantum spin chains

We devise a method based on the tensor-network formalism to calculate genuine multisite entanglement in ground states of infinite spin chains containing spin-1/2 or spin-1 quantum particles. The ground state is obtained by employing an infinite time-evolving block decimation method acting upon an initial matrix product state for the infinite spin system. We explicitly show how such infinite matrix product states with translational invariance provide a natural framework to derive the generalized geometric measure, a computable measure of genuine multisite entanglement, in the thermodynamic limit of quantum many-body systems with both spin-1/2 and higher-spin particles.

Dissipative entanglement of quantum spin fluctuations

We consider two non-interacting infinite quantum spin chains immersed in a common thermal environment and undergoing a local dissipative dynamics of Lindblad type. We study the time evolution of collective mesoscopic quantum spin fluctuations that, unlike macroscopic mean-field observables, retain a quantum character in the thermodynamical limit. We show that the microscopic dissipative dynamics is able to entangle these mesoscopic degrees of freedom, through a purely mixing mechanism. Further, the behaviour of the dissipatively generated quantum correlations between the two chains is studied as a function of temperature and dissipation strength.

Dissipative dynamics of quantum fluctuations

One way to look for complex behaviours in many‐body quantum systems is to let the number N of degrees of freedom become large and focus upon collective observables. Mean‐field quantities scaling as tend to commute, whence complexity at the quantum level can only be inherited from complexity at the classical level. Instead, fluctuations of microscopic observables scale as and exhibit collective Bosonic features, typical of a mesoscopic regime half‐way between the quantum one at the microscopic level and the classical one at the level of macroscopic averages. Here, we consider the mesoscopic behaviour emerging from an infinite quantum spin chain undergoing a microscopic dissipative, irreversible dynamics and from global states without long‐range correlations and invariant under lattice translations and dynamics. We show that, from the fluctuations of one site spin observables whose linear span is mapped into itself by the dynamics, there emerge bosonic operators obeying a mesoscopic dissipative dynamics mapping Gaussian states into Gaussian states. Instead of just depleting quantum correlations because of decoherence effects, these maps can generate entanglement at the collective, mesoscopic level, a phenomenon with no classical analogue that embodies a peculiar complex behaviour at the interface between micro and macro regimes.

Global quantum discord in matrix product states and the application

Global quantum discord (GQD) is a measure of quantum correlation for multi-site quantum states. In this paper, we propose an efficient procedure to calculate global quantum discord (Gn) for consecutive n-site subsystems in infinite-size matrix product states (MPSs). We apply the method to study the scaling behavior of Gn in several one-dimensional infinite quantum spin chains, i.e., (1) a three-site interaction model whose ground state can be exactly expressed as MPSs, and (2) a spin-12 XXZ chain whose ground state is approximately expressed as MPSs with the help of infinite time-evolving block decimation (iTEBD) algorithm. In both models, as the increase of n, Gn shows an approximately linear growth. Unambiguous clue for the convergence of the incremental ΔGn=Gn−Gn−1 is observed when n is large enough. Moreover, in non-critical (gapped) regions ΔGn converges very fast, while in critical (gapless) regions it converges relatively slow. The behaviors are explained in a model-independent physical picture with finite-range correlations. Based on these results, we propose to use “discord per site” (Gnn) to describe the global correlations in infinite-size spin chains. Moreover, we find that the global discord shows a size-independent maximum at the infinite-order quantum phase transition point of the XXZ model.

Tensor network techniques for the computation of dynamical observables in one-dimensional quantum spin systems

We analyze the recently developed folding algorithm (Bañuls et al 2009 Phys. Rev. Lett.102 240603) for simulating the dynamics of infinite quantum spin chains and we relate its performance to the kind of entanglement produced under the evolution of product states. We benchmark the accomplishments of this technique with respect to alternative strategies using Ising Hamiltonians with transverse and parallel fields, as well as XY models. Also, we evaluate its capability of finding ground and thermal equilibrium states.