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arXiv:1005.5195 · Physical Review B

Exploiting translational invariance in matrix product state simulations of spin chains with periodic boundary conditions

B. Pirvu, Frank Verstraete, Guifré Vidal

Published Mar 17, 2011• 52 citations
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Abstract

We present a matrix product state (MPS) algorithm to approximate ground states of translationally invariant systems with periodic boundary conditions. For a fixed value of the bond dimension $D$ of the MPS, we discuss how to minimize the computational cost to obtain a seemingly optimal MPS approximation to the ground state. In a chain with $N$ sites and correlation length $\ensuremath{\xi}$, the computational cost formally scales as $g(D,\ensuremath{\xi}/N){D}^{3}$, where $g(D,\ensuremath{\xi}/N)$ is a nontrivial function. For $\ensuremath{\xi}\ensuremath{\ll}N$, this scaling reduces to ${D}^{3}$, independent of the system size $N$, making our method $N$ times faster than previous proposals. We apply the algorithm to obtain MPS approximations for the ground states of the critical quantum Ising and Heisenberg spin-$1/2$ models as well as for the noncritical Heisenberg spin-$1$ model. In the critical case, for any chain length $N$, we find a model-dependent bond dimension $D(N)$ above which the polynomial decay of correlations is faithfully reproduced throughout the entire system.

DOI
10.1103/physrevb.83.125104

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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)(N,D)(N,D) parameter plane, where NNN is the size of the spin chain and DDD 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 NNN 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 appendixes, 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.
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We develop an efficient algorithm to find a matrix product state representation of the ground-state wavefunctions for translationally invariant finite-size periodic lattice systems in one spatial dimension. This is based on the observation that the efficient computation of the ground-state energy per site only needs to retain a certain number of the largest eigenvalues of the transfer matrix for a matrix product state, without any sacrifice of accuracy. The computational cost is independent of the system's size, and scales as χ 3 with χ being the truncation dimension. The algorithm is tested for the critical quantum Ising model in a transverse field on a finite-size lattice, with the size as large as 4800 for the truncation dimension 200.
We present some exact results for the optimal matrix product state (MPS) approximation to the ground state of the infinite isotropic Heisenberg spin-1/2 chain. Our approach is based on the systematic use of Schmidt decompositions to reduce the problem of approximating for the ground state of a spin chain to an analytical minimization. This allows one to show that results of standard simulations, e.g. density matrix renormalization group and infinite time evolving block decimation, do correspond to the result obtained by this minimization strategy and, thus, both methods deliver optimal MPS with the same energy but, otherwise, different properties. We also find that translational and rotational symmetries cannot be maintained simultaneously by the MPS ansatz of minimum energy and present explicit constructions for each case. Furthermore, we analyze symmetry restoration and quantify it to uncover new scaling relations. The method we propose can be extended to any translational invariant Hamiltonian.
On Translation-Invariant Matrix Product States and advances in MPS representations of the WWW-state
This work is devoted to the study Translation-Invariant (TI) Matrix Product State (MPS) representations of quantum states with periodic boundary conditions (PBC). We pursue two directions: we introduce new methods for constructing TI MPS representations of a certain class of TI states and study their optimality in terms of their bond dimension. We pay particular attention to the nnn-party WWW-state and construct a TI MPS representation of bond dimension ⌊n2⌋+1\left \lfloor \dfrac{n}{2} \right \rfloor +1⌊2n​⌋+1 for it. We further study properties of this class and show that we can can always achieve a bond dimension of nnn for TI MPS representation of states in this class. In the framework of studying optimality of TI MPS representations with PBC, we study the optimal bond dimension d(ψ)d(ψ)d(ψ) for a given state ψψψ. In particular we introduce a deterministic algorithm for the search of d(ψ)d(ψ)d(ψ) for an arbitary state. Using numerical methods, we verify the optimality of our previous construction for the nnn-party WWW-state for small nnn.