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arXiv:cond-mat/0406440 · Physical Review Letters

Mixed-State Dynamics in One-Dimensional Quantum Lattice Systems: A Time-Dependent Superoperator Renormalization Algorithm

Michael P. Zwolak, Guifré Vidal

Published Nov 12, 2004• 655 citations
arXivHTMLPDF

Abstract

We present an algorithm to study mixed-state dynamics in one-dimensional quantum lattice systems. The algorithm can be used, e.g., to construct thermal states or to simulate real time evolution given by a generic master equation. Its two main ingredients are (i) a superoperator renormalization scheme to efficiently describe the state of the system and (ii) the time evolving block decimation technique to efficiently update the state during a time evolution. The computational cost of a simulation increases significantly with the amount of correlations between subsystems, but it otherwise depends only linearly on the system size. We present simulations involving quantum spins and fermions in one spatial dimension.

DOI
10.1103/physrevlett.93.207205

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We present an algorithm to study mixed-state dynamics in one-dimensional quantum lattice systems. The algorithm can be used, e.g., to construct thermal states or to simulate real time evolution given by a generic master equation. Its two main ingredients are (i) a superoperator renormalization scheme to efficiently describe the state of the system and (ii) the time evolving block decimation technique to efficiently update the state during a time evolution. The computational cost of a simulation increases significantly with the amount of correlations between subsystems, but it otherwise depends only linearly on the system size. We present simulations involving quantum spins and fermions in one spatial dimension.
The infinite time-evolving block decimation algorithm [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)] allows to simulate unitary evolution and to compute the ground state of one-dimensional (1D) quantum lattice systems in the thermodynamic limit. Here we extend the algorithm to tackle a much broader class of problems, namely, the simulation of arbitrary one-dimensional evolution operators that can be expressed as a (translationally invariant) tensor network. Relatedly, we also address the problem of finding the dominant eigenvalue and eigenvector of a one-dimensional transfer matrix that can be expressed in the same way. New applications include the simulation, in the thermodynamic limit, of open (i.e., master equation) dynamics and thermal states in 1D quantum systems, as well as calculations with partition functions in two-dimensional (2D) classical systems, on which we elaborate. The present extension of the algorithm also plays a prominent role in the infinite projected entangled-pair states approach to infinite 2D quantum lattice systems.
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All systems are open to an essentially uncontrollable environment that acts as a source of decoherence and dissipation. In some cases the environment's only effect is to add a weak relaxation mechanism and thus can be ignored for short timescales. In others, however, the presence of the environment can fundamentally alter the behavior of the system. Such is the case in mesoscopic superconductors where the environment can stabilize superconductivity and in spin-boson systems where the environment induces a localization transition. Likewise, in technological applications we are often interested in systems operating far from equilibrium. Here the environment might act as a particle reservoir or strong driving force. In all these examples, we need accurate methods to describe the influence of the environment on the system and to solve for the resulting dynamics or equilibrium states. In this thesis, we develop computational and conceptual approaches to efficiently simulate quantum systems in contact with an environment. Our starting point is the use of numerical renormalization techniques. Thus, we restrict our attention to one-dimensional lattices or small quantum systems coupled to an environment. We have developed several complementary algorithms: a superoperator renormalization algorithm for simulating real-time Markovian dynamics and for calculating states in thermal equilibrium; a blocking algorithm for simulating integro-differential equations with long-time memory; and a tensor network algorithm for branched lattices, which can be used to simulate strongly dissipative systems. Further, we provide support for an idea that to generically and accurately simulate the real-time dynamics of strongly dissipative systems, one has to include all or part of the environment within the simulation. In addition, we discuss applications and open questions.
We present a numerical method to simulate the time evolution, according to a generic Hamiltonian made of local interactions, of quantum spin chains and systems alike. The efficiency of the scheme depends on the amount of entanglement involved in the simulated evolution. Numerical analysis indicates that this method can be used, for instance, to efficiently compute time-dependent properties of low-energy dynamics in sufficiently regular but otherwise arbitrary one-dimensional quantum many-body systems. As by-products, we describe two alternatives to the density matrix renormalization group method.
In this introductory review, we present and compare four algorithms for the numerical integration of the Lindblad equation for one-dimensional quantum lattice systems. All four methods are based on Matrix Product State representations and can be viewed as extensions of the Time-Evolving Block Decimation (TEBD) algorithm to open quantum systems. Two approaches directly integrate the vectorized Lindblad equation, one of them explicitly enforcing the positivity of the density matrix. The other two rely on stochastic unravelings of the Lindblad equation, namely the quantum trajectory and quantum state diffusion approaches. We discuss the principles, numerical implementation, accuracy, and computational efficiency of the different methods, and benchmark them against an exactly solvable free fermion model.
We study the evolution of one-dimensional quantum lattice systems when the ground state is perturbed by altering one site in the middle of the chain. For a large class of models, we observe a similar pattern of entanglement growth during the evolution, characterized by a moderate increase of significant Schmidt coefficients in all relevant bipartite decompositions of the state. As a result, the evolution can be accurately described by a matrix product state and efficiently simulated using the time-evolving block decimation algorithm.