We develop a new algorithm based on the time-dependent variational principle applied to matrix product states to efficiently simulate the real-and imaginary-time dynamics for infinite one-dimensional quantum lattices. This procedure (i) is argued to be optimal, (ii) does not rely on the Trotter decomposition and thus has no Trotter error, (iii) preserves all symmetries and conservation laws, and (iv) has low computational complexity. The algorithm is illustrated by using both an imaginary-time and a real-time example. DOI: 10.1103/PhysRevLett.107.070601 PACS numbers: 05.10.Cc, 02.70.Àc, 03.67.Àa, 75.40.Gb The density-matrix renormalization group (DMRG) is arguably the most powerful tool available for the study of one-dimensional strongly interacting quantum lattice systems [1]. The DMRG-now understood as an application of the variational principle to matrix product states (MPSs) [2]-was originally conceived as a method to calculate ground-state properties. However, there has been a recent explosion of activity, spurred by insights from quantum information theory, in developing powerful extensions allowing the study of finite-temperature properties and nonequilibrium physics via time evolution [3]. The simulation of nonequilibrium properties with the DMRG was first attempted in Ref.[4], but modern implementations are based on the time-evolving block decimation algorithm (TEBD) [5] or the variational matrix product state approach [6].At the core of a TEBD algorithm lies the Lie-Trotter decomposition for the propagator expðidtĤÞ, which splits it into a product of local unitaries. This product can then be dealt with in a parallelized and efficient way: When applied to an MPS, one obtains another MPS with a larger bond dimension. To proceed, one then truncates the MPS description by discarding irrelevant variational parameters. This is such a flexible idea that it has allowed even the study of the dynamics of infinite translation-invariant lattice systems via the infinite TEBD [7]. Despite its success, the TEBD has some drawbacks: (i) The truncation step may not be optimal; (ii) conservation laws, e.g., energy conservation, may be broken; and (iii) symmetries, e.g., translation invariance, are broken (although translation invariance by two-site shifts is retained for nearestneighbor Hamiltonians). The problem is that when the Lie-Trotter step is applied to the state-stored as an MPS-it leaves the variational manifold and a representative from the manifold must be found that best approximates the new time-evolved state. There are a variety of ways to do this based on diverse distance measures for quantum states, but implementations become awkward when symmetries are brought into account.In this Letter, we introduce a new algorithm to solve the aforementioned problems-intrinsic to the TEBD-without an appreciable increase in computational cost. The resulting imaginary-time algorithm quickly converges towards the globally best uniform MPS (uMPS) approximation for translational-invariant ground states of strongly corr...

Using quantization in the Fock space of operators we compute the non-equilibrium steady state in an open Heisenberg XY spin 1/2 chain of finite but large size coupled to Markovian baths at its ends. Numerical and theoretical evidence is given for a far from equilibrium quantum phase transition with spontaneous emergence of long-range order in spin-spin correlation functions, characterized by a transition from saturation to linear growth with the size of the entanglement entropy in operator space.PACS numbers: 02.30. Ik, 05.70.Fh, 75.10.Pq, 03.67.Mn Non-perturbative physics of many-body open quantum systems far from equilibrium is largely an unexplored field. In one-dimensional locally interacting quantum systems equilibrium phase transitions -quantum phase transitions (QPT) -can occur at zero temperature only and are by now well understood [1]. QPT are typically characterized by vanishing of the Hamiltonian's spectral gap in the thermodynamic limit at the critical point, and (logarithmic) enhancement of the entanglement entropy and other measures of quantum correlations in the ground state [2]. Much less is known about the physics of QPT out of equilibrium, studies of which have been usually limited to near equilibrium regimes or using involved and approximate analytical techniques (e.g. [3,4]).There exist two general theoretical approaches to a description of non-equilibrium open quantum systems, namely the non-equilibrium Green's function method [5], and the quantum master equation [6,7]. In this Letter we adpot the latter and present a quasi-exactly solvable example of an open Heisenberg XY spin 1/2 chain exhibiting a novel type of phase transition far from equilibrium; characterized by a sudden appearance of long-range magnetic order in non-equilibrium steady state (NESS) as the magnetic field is reduced, and the transition from saturation to linear growth with size of the operator space entanglement entropy (OSEE) of NESS.The Hamiltonian of the quantum XY chain reads(1) where σ x,y,z m , m = 1, . . . , n are Pauli operators acting on a string of n spins. We may assume that parameters γ (anisotropy) and h (magnetic field) are non-negative. It is known that XY model (1) exhibits (equilibrium) critical behavior in the thermodynamic limit n → ∞ along the lines: γ = 0, h ≤ 1, and h = 1. Here we consider an open XY chain whose density matrix evolution ρ(t) is governed by the Lindblad master equation [6] (we set = 1)and study a phase transition in NESS. The simplest nontrivial bath (Lindblad) operators acting only on the first and the last spin are chosen (M = 4)where σ Note that Lindblad equation (2) can be rigorously derived within the so-called Markov approximation [7] which is justified for macroscopic baths with fast internal relaxation times. As shown in [9], Eq. (2) with (1,3) can be solved exactly in terms of normal master modes (NMM) which are obtained from diagonalization of 4n × 4n matrix A written in terms of 4 × 4 blockswhere Following [9], the key concept is 4 n dimensional Fock space of operators...

The efficiency of time-dependent density matrix renormalization group methods is intrinsically connected to the rate of entanglement growth. We introduce a measure of entanglement in the space of operators and show, for a transverse Ising spin-1 / 2 chain, that the simulation of observables, contrary to the simulation of typical pure quantum states, is efficient for initial local operators. For initial operators with a finite index in Majorana representation, the operator space entanglement entropy saturates with time to a level which is calculated analytically, while for initial operators with infinite index the growth of operator space entanglement entropy is shown to be logarithmic.

We demonstrate that Monte Carlo sampling can be used to efficiently extract the expectation value of projected entangled pair states with a large virtual bond dimension. We use the simple update rule introduced by Xiang et al. in Phys. Rev. Lett 101, 090603 (2008) to obtain the tensors describing the ground state wavefunction of the Antiferromagnetic Heisenberg model and evaluate the finite size energy and staggered magnetization for square lattices with periodic boundary conditions of linear sizes up to L = 16 and virtual bond dimensions up to D = 16. The finite size magnetization errors are 0.003(2) and 0.013(2) at D = 16 for a system of size L = 8, 16 respectively. Finite D extrapolation provides exact finite size magnetization for L = 8, and reduces the magnetization error to 0.005(3) for L = 16, significantly improving the previous state of the art results.The efficient simulation of strongly correlated quantum many body systems presents one of the major open problems and challenges in condensed matter physics. A major step forward was made by Steven White 2 in the case of 1 dimensional quantum spin chains by introducing the density matrix renormalization group (DMRG), which soon became the method of choice for simulating 1 dimensional manybody systems at zero temperature. By reformulating DMRG as a variational method within the class of matrix product states (MPS) 3-5 , it has become clear how DMRG can be generalized to deal with systems in two dimensions 6,7 ; the quantum states of the corresponding variational class are known as projected entangled pair states (PEPS) and are part of the class called tensor product states which also includes the multiscale entanglement renormalization ansatz 8 and infinite PEPS 9 . More recently, it has also been demonstrated how the PEPS class can take into account fermionic anti-commutation relations 10-16 . Numerical algorithms based on these ansatze, such as variational minimization of the ground state energy and imaginary time evolution are also developing fast 1,7,9,17,18 , and a wide range of applications has been studied 19-27 .The computational complexity of algorithms based on the PEPS ansatz with virtual bond dimension D scales as D 12 for the finite PEPS algorithm with open boundary condition 17 , χ 3 D 4 for the infinite PEPS (iPEPS) algorithm 9 , χ 6 for the tensor entanglement renormalization (TERG) algorithm for square lattices 18 and χ 5 for honeycomb lattices 1,18 , where χ is the number of Schmidt coefficients kept in the various approximations. The large scaling power presents the main bottleneck in scaling up the number of variational parameters, which is necessary near second order phase transitions 28 . The common characteristics of all these algorithms is that the tensor network is always contracted over the physical indices, which effectively squares the computational cost of contracting the tensor network as compared to a tensor network corresponding to a classical spin system. As first shown in 29,30 for the case of matrix product states and str...

The complexity of representation of operators in quantum mechanics can be characterized by the operator space entanglement entropy ͑OSEE͒. We show that in the homogeneous Heisenberg XY spin 1/2 chains the OSEE for initial local operators grows at most logarithmically with time. The prefactor in front of the logarithm generally depends only on the number of stationary points of the quasiparticle dispersion relation and for the XY model changes from 1/3 to 2/3 exactly at the point of quantum phase transition to long-range magnetic correlations in the nonequilibrium steady state. In addition, we show that the presence of a small disorder triggers a saturation of the OSEE.