Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling

Abstract

We investigate the use of matrix product states (MPS) to approximate ground states of critical quantum spin chains with periodic boundary conditions (PBC). We identify two regimes in the (N,D) parameter plane, where N is the size of the spin chain and D is the dimension of the MPS matrices. In the first regime MPS can be used to perform finite size scaling (FSS). In the complementary regime the MPS simulations show instead the clear signature of finite entanglement scaling (FES). In the thermodynamic limit (or large N limit), only MPS in the FSS regime maintain a finite overlap with the exact ground state. This observation has implications on how to correctly perform FSS with MPS, as well as on the performance of recent MPS algorithms for systems with PBC. It also gives clear evidence that critical models can actually be simulated very well with MPS by using the right scaling relations; in the appendix, we give an alternative derivation of the result of Pollmann et al. [Phys. Rev. Lett. 102, 255701 (2009)] relating the bond dimension of the MPS to an effective correlation length.

arXiv comment18 pages, 13 figures

References in corpus (22)

• The density-matrix renormalization group in the age of matrix product states
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Efficient simulation of one-dimensional quantum many-body systems
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Time-dependent variational principle for quantum lattices
• DMRG and periodic boundary conditions: a quantum information perspective
• Entanglement spectrum in one-dimensional systems
• Matrix product operator representations
• Theory of finite-entanglement scaling at one-dimensional quantum critical points
• Scaling of entanglement support for Matrix Product States
• Matrix-product-groundstates for one-dimensional spin-1 quantum antiferromagnets
• Numerical Renormalization Group at Criticality
• Efficient Matrix Product State Method for periodic boundary conditions
• Phase diagram of an anisotropic frustrated ferromagnetic spin-1/2 chain in a magnetic field: a density matrix renormalization group study
• Exploiting translational invariance in Matrix Product State simulations of spin chains with periodic boundary conditions
• Field-induced phase transitions of repulsive spin-1 bosons in optical lattices
• Boundary quantum critical phenomena with entanglement renormalization
• Spontaneous symmetry breaking and bifurcations in ground state fidelity for quantum lattice systems
• Symmetry breaking and criticality in tensor-product states
• Hyperbolic Deformation on Quantum Lattice Hamiltonians
• Continuous quantum phase transition between two topologically distinct valence bond solid states associated with the same spin value
• Stiffness in 1D Matrix Product States with periodic boundary conditions

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