Time-Dependent Variational Principle for Quantum Lattices

Abstract

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...

Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

“…These are variational state vectors that are described by O (d N D 2 ) variational parameters, where D ∈ N is a refinement parameter and d the dimension of the local Hilbert space. There are several variants of this approach, based on either a Trotter-decomposition [64] or a time-dependent variational principle [25]. Such schemes are subsumed under the term time-dependent density matrix renormalization group method (t-DMRG).…”

Calculus of continuous matrix product states

“…18 An essential ingredient of this algorithm is the study of (infinitesimally) small variations of MPS, i.e., the set of MPS tangent vectors. Indeed, it was rigorously proven that the set of MPS can be given the structure of a variational manifold with a well-defined tangent space 36 by eliminating some singular points or regions.…”

“…These tangent states are relevant when studying time evolution or elementary excitations along the lines of analogous MPS algorithms. [18][19][20][21] Before the conclusion in Sec. X, we also discuss how several of the cMPS equations in this manuscript compare to their better known analogues of lattice matrix product states in Sec.…”

Dynamical windows for real-time evolution with matrix product states

“…However TEBD comes into its own for real-time evolution. More recently, a new algorithm, called the time-dependent variational principle (TDVP), 16,17 has also been introduced to study both the realand imaginary-time dynamics for infinite one-dimensional quantum lattices.…”