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arXiv:1207.0652 · Physical Review B

Infinite boundary conditions for matrix product state calculations

Ho N. Phien, Guifré Vidal, Ian P. McCulloch

Published Dec 10, 2012• 94 citations
arXivHTMLPDF

Abstract

We propose a formalism to study dynamical properties of a quantum many-body system in the thermodynamic limit by studying a finite system with ``infinite boundary conditions'' where both finite-size effects and boundary effects have been eliminated. For one-dimensional systems, infinite boundary conditions are obtained by attaching two boundary sites to a finite system, where each of these two sites effectively represents a semi-infinite extension of the system. One can then use standard finite-size matrix product state techniques to study a region of the system while avoiding many of the complications normally associated with finite-size calculations such as boundary Friedel oscillations. We illustrate the technique with an example of time evolution of a local perturbation applied to an infinite (translationally invariant) ground state, and use this to calculate the spectral function of the $S=1$ Heisenberg spin chain. This approach is more efficient and more accurate than conventional simulations based on finite-size matrix product state and density-matrix renormalization-group approaches.

DOI
10.1103/physrevb.86.245107

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We propose a formalism to study dynamical properties of a quantum many-body system in the thermodynamic limit by studying a finite system with ``infinite boundary conditions'' where both finite-size effects and boundary effects have been eliminated. For one-dimensional systems, infinite boundary conditions are obtained by attaching two boundary sites to a finite system, where each of these two sites effectively represents a semi-infinite extension of the system. One can then use standard finite-size matrix product state techniques to study a region of the system while avoiding many of the complications normally associated with finite-size calculations such as boundary Friedel oscillations. We illustrate the technique with an example of time evolution of a local perturbation applied to an infinite (translationally invariant) ground state, and use this to calculate the spectral function of the S=1S=1S=1 Heisenberg spin chain. This approach is more efficient and more accurate than conventional simulations based on finite-size matrix product state and density-matrix renormalization-group approaches.
We propose an alternative to the infinite density-matrix renormalization approach for accessing quantum many-body states within a finite-size calculation that faithfully mimics the thermodynamic limit. Our method constructs environment matrix product operators (MPOs) representing the Hamiltonian of semi-infinite regions surrounding the target system. Starting from the finite-size ground-state MPS, we contract its Hamiltonian representation to generate effective environment MPOs, which are then attached to a renewed finite system in a recursive manner. This iterative embedding drives the system toward a bulk-like state with negligible finite-size effects. The scheme requires no assumption of homogeneity and achieves unprecedentedly long real-time dynamics free from boundary reflections.
Response functions $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}_{x}(t){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}}_{y}(0)\ensuremath{\rangle}$ for one-dimensional strongly correlated quantum many-body systems can be computed with matrix product state (MPS) techniques. Especially, when one is interested in spectral functions or dynamic structure factors of translation-invariant systems, the response for some range $|x\ensuremath{-}y|<\ensuremath{\ell}$ is needed. We demonstrate how the number of required time-evolution runs can be reduced substantially: (a) If finite-system simulations are employed, the number of time-evolution runs can be reduced from $\ensuremath{\ell}$ to $2\sqrt{\ensuremath{\ell}}$. (b) To go beyond, one can employ infinite MPS (iMPS) such that two evolution runs suffice. To this purpose, iMPS that are heterogeneous only around the causal cone of the perturbation are evolved in time, i.e., the simulation is done with infinite boundary conditions. Computing overlaps of these states, spatially shifted relative to each other, yields the response functions for all distances $|x\ensuremath{-}y|$. As a specific application, we compute the dynamic structure factor for ground states of bilinear-biquadratic spin-1 chains with very high resolution and explain the underlying low-energy physics. To determine the initial uniform iMPS for such simulations, infinite-system density matrix renormalization group (iDMRG) can be employed. We discuss that, depending on the system and chosen bond dimension, iDMRG with a cell size nc{n}_{c}nc​ may converge to a nontrivial limit cycle of length mmm. This then corresponds to an iMPS with an enlarged unit cell of size mncm{n}_{c}mnc​.
We propose the use of a dynamical window to investigate the real-time evolution of quantum many-body systems in a one-dimensional lattice. In a recent paper [Phien et al., Phys. Rev. B 86, 245107 (2012)], we introduced infinite boundary conditions in order to investigate real-time evolution of an infinite system under a local perturbation. This was accomplished by restricting the update of the tensors in the matrix product state to a finite region known as a window, with left and right boundaries held at fixed positions. Here we consider instead the use of a dynamical window, where the positions of left and right boundaries are allowed to change in time. In this way, all computational efforts can be devoted to the space-time region of interest, which leads to a remarkable reduction in simulation costs. For illustrative purposes, we consider two applications in the context of the spin-1 antiferromagnetic Heisenberg model in an infinite spin chain: one is an expanding window, with boundaries that are adjusted to capture the expansion in time of a local perturbation of the system; the other is a moving window of fixed size, where the position of the window follows the front of a propagating wave.
We introduce an efficient method to calculate the ground state of one-dimensional lattice models with periodic boundary conditions. The method works in the representation of matrix product states (MPS), related to the density matrix renormalization group (DMRG) method. It improves on a previous approach by Verstraete et al. We introduce a factorization procedure for long products of MPS matrices, which reduces the computational effort from m5{m}^{5}m5 to m3{m}^{3}m3, where mmm is the matrix dimension, and $m\ensuremath{\simeq}100--1000$ in typical cases. We test the method on the S=12S=\frac{1}{2}S=21​ and S=1S=1S=1 Heisenberg chains. It is also applicable to nontranslationally invariant cases. The method makes ground-state calculations with periodic boundary conditions about as efficient as traditional DMRG calculations for systems with open boundaries.
Efficient MPS Algorithm for Periodic Boundary Conditions and Applications Section: Solid matter Original Author's Text: English Abstract: We present the implementation of an efficient algorithm for the calculation of the spectrum of one-dimensional quantum systems with periodic boundary conditions. This algorithm is based on a matrix product representation for quantum states (MPS) and a similar representation for Hamiltonians and other operators (MPO). It is significantly more efficient for systems of about 100 sites and more than for small quantum systems. We apply the formalism to calculate the ground state and the first excited state of a spin-1 Heisenberg ring and deduce the size of a Haldane gap. The results are compared to previous high-precision DMRG calculations. Furthermore, we study the spin-1 systems with a biquadratic nearest-neighbor interaction and show the first results of an application to a mesoscopic Hubbard ring of spinless fermions, which carries a persistent current. Key words: matrix product representation for quantum states (MPS) and Hamiltonians (MPO), spin-1 Heisenberg ring, density matrix renormalization group (DMRG). List Issues Configure
We present the implementation of an efficient algorithm for the calculation of the spectrum of one-dimensional quantum systems with periodic boundary conditions. This algorithm is based on a matrix product representation for quantum states (MPS) and a similar representation for Hamiltonians and other operators (MPO). It is significantly more efficient for systems of about 100 sites and more than for small quantum systems. We apply the formalism to calculate the ground state and the first excited state of a spin-1 Heisenberg ring and deduce the size of a Haldane gap. The results are compared to previous high-precision DMRG calculations. Furthermore, we study the spin-1 systems with a biquadratic nearest-neighbor interaction and show the first results of an application to a mesoscopic Hubbard ring of spinless fermions, which carries a persistent current.
We describe how to implement the time-dependent variational principle for matrix product states in the thermodynamic limit for nonuniform lattice systems. This is achieved by confining the nonuniformity to a (dynamically expandable) finite region with fixed boundary conditions. The suppression of nonphysical quasiparticle reflections from the boundary of the nonuniform region is also discussed. Using this algorithm we study the dynamics of localized excitations in infinite systems, which we illustrate in the case of the spin-1 antiferromagnetic Heisenberg model and the ${\ensuremath{\phi}}^{4}$ model.