Infinite boundary conditions for matrix product state calculations

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.

Figure 1

(a) Single-site translationally invariant canonical form iMPS. (b) One-site translationally invariant mixed canonical form iMPS. (c) Partition the whole chain into three parts: the left and right semi-infinite sublattices, and the middle part which is a window that contains $N$ sites. (d) Finite-size MPS effectively represent the iMPS with left- and right-effective sites representing the left and right semi-infinite sublattices.

Figure 2

The full Hamiltonian of the system is decomposed into the tensor product of local matrix product operators.

Figure 3

(a) Generalized transfer matrix for finding the left dominant eigenvector $\vec{E}$. (b) Generalized transfer matrix for finding the right dominant eigenvector $\vec{E}$. (c) Equation to find left dominant eigenvector. (d) Equation to find right dominant eigenvector.

Figure 4

The wave packet (local magnetization) is formed after flipping one spin in the middle of the chain. The result is obtained by using a two-site translationally invariant iMPS for the ground state with bond dimension $\chi =160$.

Figure 5

(a) Applying the operator $e^{-i\tilde{H}\delta t}$ to the effective finite MPS. (b) Update the left and right tensors by contracting tensors $\{U_L,U_{LW},\tilde{I},A^{s_1}\}$ and $\{U_R,U_{RW},\tilde{I},A^{s_6}\}$. (c) Update the new tensors when applying two-body gate $u$ on the odd or even link. Contracting all the tensors involved and take the SVD of that. The bond dimension will increase after taking the SVD, so we need to do the truncation to keep new tensors $A^{\prime s_i},A^{\prime s_{i+1}}$, and ${\lambda }^{\prime }$ in the desired bond dimension. (d) Shifting ${\lambda }^{\prime }$ to the right (or left) of the updated link if sweeping direction is from left to right (or right to left) by taking two successive SVDs.

Figure 6

Wave packet propagates in time with window size $N=60$ (blue dots). The window is expanded to $N=200$ (black solid lines) after simulation to see how the wave front propagates beyond the infinite boundaries.

Figure 7

Wave packet propagates in time with different window sizes which are fixed at the beginning of real-time evolution. The lines are distinguishable in the center of the plot.

Figure 8

Comparisons of the differences of local magnetization in time at a fixed site $x_i$ between window sizes $N=\{60,100,120,160\}$ and a highly accurate calculation using a 500-site finite chain. (a) Perturbed point of chain $x_i=0$. (b) Boundary point of the chain $x_i=30$.

Figure 9

Plots of the real and imaginary parts of the Green's function versus time and spin chain space.

Figure 10

(a) Spectral function versus momentum and frequency for the spin-1 isotropic Heisenberg model; the window is $N=60$. (b) Spectrum viewed from the top when it is projected on the $(\omega ,q)$ plane. (c) The dispersion relation is derived from the maximum of the spectrum.