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...

We show that the time-dependent variational principle provides a unifying framework for time-evolution methods and optimisation methods in the context of matrix product states. In particular, we introduce a new integration scheme for studying time-evolution, which can cope with arbitrary Hamiltonians, including those with long-range interactions. Rather than a Suzuki-Trotter splitting of the Hamiltonian, which is the idea behind the adaptive time-dependent density matrix renormalization group method or time-evolving block decimation, our method is based on splitting the projector onto the matrix product state tangent space as it appears in the Dirac-Frenkel time-dependent variational principle. We discuss how the resulting algorithm resembles the density matrix renormalization group (DMRG) algorithm for finding ground states so closely that it can be implemented by changing just a few lines of code and it inherits the same stability and efficiency. In particular, our method is compatible with any Hamiltonian for which DMRG can be implemented efficiently and DMRG is obtained as a special case of imaginary time evolution with infinite time step.Comment: 5 pages + 5 pages supplementary material (6 figures) (updated example, small corrections

It is shown how to construct renormalization group flows of quantum field theories in real space, as opposed to the usual Wilsonian approach in momentum space. This is achieved by generalizing the multiscale entanglement renormalization ansatz to continuum theories. The variational class of wavefunctions arising from this RG flow are translation invariant and exhibit an entropy-area law. We illustrate the construction for a free non-relativistic boson model, and argue that the full power of the construction should emerge in the case of interacting theories.Classical statistical mechanics, quantum many-body systems, and relativistic quantum field theories all involve an extremely large number of degrees of freedom living at different length scales. The interactions between these degrees of freedom are the source of notorious difficulties in their study. However, much insight has been gained from the renormalization group (RG), which has proven to be the natural tool to deal with the different length scales in such systems [1]. In its original development, the RG acts as a fixed operation at the level of the classical partition function (or related quantities such as the effective action). This operation is typically formulated in momentum space and can only be implemented exactly for free theories. Perturbative expansions in a small parameter are required for interacting theories. In addition, this formulation of the RG is only applicable to quantum systems using the quantumto-classical mapping, which is known to fail in some cases [2].One exception is Wilson's numerical renormalization group [1], which can be interpreted as an implementation of the RG directly at the level of the quantum wave function. Together with White's more powerful density matrix renormalization group (DMRG) [3], these methods are now understood as a variational optimization over the class of matrix product states (MPS) [4]. Based on the observation of an entropy/area law [5] in locally interacting quantum lattices, quantum-information-theoretical considerations have resulted in the development of other sophisticated variational classes for quantum lattice systems. These are generally known as tensor network states and can be associated with RG schemes, allowing the classification of gapped phases of matter [6]. Unlike Wilson's fixed RG scheme, these are variable RG schemes that can be variationally optimized. They are formulated in realspace and deal equally well with free and interacting systems. One specific scheme, called entanglement renormalization [7], can also be applied to study critical phases and can be used to compute, e.g., scaling exponents [8]. The corresponding variational class, the multiscale entanglement renormalization ansatz (MERA) [9], is set apart by its unique properties, including, the ability to support algebraically decaying correlations and logarithmic corrections to the entropy/area law in (1 + 1) dimensions. This class has been successfully applied to study the physics of a wide variety of strongly interacting...

Quantum tensor network states and more particularly projected entangledpair states provide a natural framework for representing ground states of gapped, topologically ordered systems. The defining feature of these representations is that topological order is a consequence of the symmetry of the underlying tensors in terms of matrix product operators. In this paper, we present a systematic study of those matrix product operators, and show how this relates entanglement properties of projected entangled-pair states to the formalism of fusion tensor categories. From the matrix product operators we construct a C * -algebra and find that topological sectors can be identified with the central idempotents of this algebra. This allows us to construct projected entangled-pair states containing an arbitrary number of anyons. Properties such as topological spin, the S matrix, fusion and braiding relations can readily be extracted from the idempotents. As the matrix product operator symmetries are acting purely on the virtual level of the tensor network, the ensuing Wilson loops are not fattened when perturbing the system, and this opens up the possibility of simulating topological theories away from renormalization group fixed points. We illustrate the general formalism for the special cases of discrete gauge theories and string-net models.1 arXiv:1511.08090v2 [cond-mat.str-el]

The matrix product state formalism is used to simulate Hamiltonian lattice gauge theories. To this end, we define matrix product state manifolds which are manifestly gauge invariant. As an application, we study 1+1 dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model, and are able to determine very accurately the ground state properties and elementary one-particle excitations in the continuum limit. In particular, a novel particle excitation in the form of a heavy vector boson is uncovered, compatible with the strong coupling expansion in the continuum. We also study full quantum non-equilibrium dynamics by simulating the real-time evolution of the system induced by a quench in the form of a uniform background electric field. PACS numbers:Gauge theories hold a most prominent place in physics. They appear as effective low energy descriptions at different instances in condensed matter physics and nuclear physics. But far and foremost they lie at the root of our understanding of the four fundamental interactions that are each mediated by the gauge fields corresponding to a particular gauge symmetry. At the perturbative quantum level, this picture translates to the Feynman diagrammatic approach that has produced physical predictions with unlevelled precision, most famously in quantum electrodynamics (QED). However the perturbative approach miserably fails once the interactions become strong. This problem is most pressing for quantum chromodynamics (QCD), where all low energy features like quark confinement, chiral symmetry breaking and mass generation are essentially non-perturbative.Lattice QCD, which is based on Monte Carlo sampling of Wilson's Euclidean lattice version of gauge theories, has historically been by far the most successful method in tackling this strongly coupled regime. Using up a sizable fraction of the global supercomputer time, state of the art calculations have now reached impressive accuracy, for instance in the ab initio determination of the light hadron masses [1]. But in spite of its clear superiority, the lattice Monte Carlo sampling also suffers from a few drawbacks. There is the infamous sign problem that prevents application to systems with large fermionic densities. In addition, the use of Euclidean time, as opposed to real time, presents a serious barrier for the understanding of dynamical non-equilibrium phenomena. Over the last few years there has been a growing experimental and theoretical interest in precisely these elusive regimes, e.g. in the study of heavy ion collisions or early time cosmology.In this letter we study the application of tensor network states (TNS) as a possible complementary approach to the numerical simulation of gauge theories. This is highly relevant as this Hamiltonian method is free from the sign problem and allows for real-time dynamics. As a first application we focus on the massive Schwinger model. For this model the TNS approach has been studied before by Byrnes et al [2] and Bañuls et al [3]. By integrating out the ...