Matrix product states represent ground states faithfully

Abstract

We quantify how well matrix product states approximate exact ground states of one-dimensional quantum spin systems as a function of the number of spins and the entropy of blocks of spins. We also investigate the convex set of local reduced density operators of translational invariant systems. The results give a theoretical justification for the high accuracy of renormalization group algorithms and justifies their use even in the case of critical systems.

Relaxation Dynamics in the Spin-1 Heisenberg Antiferromagnetic Chain after a Quantum Quench of the Uniaxial Anisotropy

“…The value of the plateau O α ≈ −0.3743 (α = x, y, z), the non-local order parameter for S = 1, was in accordance with earlier numerical simulations [29]. Figure 5 presents how the individual components of the non-local correlations change over time for various values of the uniaxial anisotropy.…”

“…In order to investigate the ground state and its dynamical properties after a sudden change of the Hamiltonian parameters the MPS formalism has been employed [28,29]. The observation that for physical systems only minor part of the Hilbert space is involved [30], resulted in the rapid development of numerical methods based on a variational method within the space of MPS.…”

“…Moreover, it is one of the most attractive features of the MPS representation that the time evolution can also be performed very efficiently. Therefore, discrete time as t = N ∆t can be used for the Hamiltonian, when a second-order Trotter decomposition is applied [29] the time-evolution operator can be presented as e − i H∆t = e − i Ho∆t/2 e − i He∆t e − i Ho∆t/2 +O(∆t 3 ), (8) where…”

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

“…More precisely, we say that a pure state satisfies an area law if for any region R ⊂ V the (Rényi) entropy of the reduced state on R can be bounded by the size of the boundary of R, up to a constant. States of 1D systems satisfying an area law can be provably well approximated by matrix product states [68]. Indeed, t-DMRG simulates time evolution for short times to essentially machine precision.…”

“…Approximating 1D ground states of gapped Hamiltonians with MPS: Since ground states of any 1D local Hamiltonian with a spectral gap ∆E > 0 satisfy an area law for Rényi entropies they can be approximated [68] by matrix product states (MPS) in polynomial time [36]. This is used by the static density-matrix renormalization group method (DMRG) [58] (see also the chapter of Ors Legeza, Thorsten Rohwedder and Reinhold Schneider) for simulating ground state properties [59], which has led to a wealth of novel insights in condensed matter physics.…”