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

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 $\xi$, the computational cost formally scales as $g(D,\xi /N)D^3$, where $g(D,\xi /N)$ is a nontrivial function. For $\xi \ll N$, 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.

Figure 1

The properties of a block of size $l$ within a PBC system can be equivalent to those of a block with the same size in the bulk of an OBC system. Depending on $l$ and the correlation length $\xi$, the left and right boundary conditions of the OBC system are more or less correlated.

Figure 2

(a) Graphical representation of the TI PBC MPS $|\psi (\mathbf{A})\rangle$ of a TI spin chain with four sites. Note the identical tensors $\mathbf{A}$ at each site. (b) Small perturbation $\delta \mathbf{A}$ is added to the MPS tensor $\mathbf{A}$. (c) Norm of a state $\langle \psi (\mathbf{A})|\psi (\mathbf{A})\rangle$. (d) Expectation value of a two-site operator, e.g., $\langle \psi (\mathbf{A})\left|H_{s,s+1}\right|\psi (\mathbf{A})\rangle$. (e) The expectation value is expanded in powers of $\delta \mathbf{A}$.

Figure 3

Graphical representation of the tensor $H_{\mathrm{eff}}(\mathbf{A})$ and $N_{\mathrm{eff}}(\mathbf{A})$.

Figure 4

Critical quantum Ising chain with $N=100$: Relative precision of the MPS ground-state energy as compared to the analytical result as a function of the parameters $(m,n)$ for $D=16$ (left) and 32 (right).

Figure 5

Critical Heisenberg chain with $N=100$: Relative precision of the MPS ground-state energy as compared to the analytical result as a function of the parameters $(m,n)$ for $D=16$ (left) and 32 (right).

Figure 6

Critical quantum Ising chain with $N=100$ (left) and 400 (right): Relative precision of the MPS ground-state energy as a function of the parameter $n$ for different bond dimensions $D$. The scan has been performed along the line $m=5n$ up to the maximal value of $m$ and then along the line with constant $m=(N-2)/2$.

Figure 7

Critical quantum Ising model: relative precision of the MPS ground-state energy for different $N$ as a function of $D$.

Figure 8

Critical Heisenberg model: relative precision of the MPS ground-state energy for different $N$ as a function of $D$.

Figure 9

Correlation functions for a critical quantum Ising chain with $N=500$. Left: order parameter correlator ${\Gamma }^{\mathit{ZZ}}(\Delta r)$ and as inset the half-chain correlator as a function of $D$. Right: correlator ${\Gamma }^{\mathit{XX}}(\Delta r)$ and as inset the half-chain correlator as a function of $D$.

Figure 10

Absolute values of the correlation functions for a critical Heisenberg chain with $N=500$. Left: correlator ${\Gamma }^{\mathit{ZZ}}(\Delta r)$ and as inset the half-chain correlator as a function of $D$. Right: correlator ${\Gamma }^{\mathit{XX}}(\Delta r)$ and as inset the half-chain correlator as a function of $D$.

Figure 11

Relative precision of correlation functions in the MPS ground state of the quantum Ising model with $N=500$. Left: error of the order parameter correlation function ${\Gamma }^{\mathit{ZZ}}(\Delta r)$ for several different $D$ in the high-precision regime. Right: relative precision of the half-chain correlators ${\Gamma }_{N/2}^{\mathit{ZZ}}$ and ${\Gamma }_{N/2}^{\mathit{XX}}$ as a function of the MPS bond dimension $D$.

Figure 12

Spin-$1$ Heisenberg chain with $N=100$: relative precision of the MPS ground-state energy as compared to the best numerical approximation as a function of the parameters $(m,n)$ for $D=16$ (left) and 100 (right).

Figure 13

Spin-$1$ Heisenberg chain with $N=100$. Left: relative precision of the MPS ground-state energy as a function of $D$. Right: absolute values of the correlation functions ${\Gamma }^{\mathit{ZZ}}(\Delta r)$ and as inset the half-chain correlator as a function of $D$.

Figure 14

Graphical representation of $H_{\mathit{AA}}^{\mathit{AA}}$, $H_{\mathit{AA}}^A$, $H_{\mathit{AA}}^A$, $T=T_A^A$, and $T_A$.

Figure 15

Graphical representation of a term with small $s$ and its approximation within the subspace spanned by $n$ dominant eigenvectors of $T$.