Environment-matrix-product operator for boundary-free large-scale quantum many-body simulations

Abstract

We propose an alternative to the infinite density-matrix renormalization approach for accessing quantum many-body states within a finite-size calculation that faithfully mimics the thermodynamic limit. Our method constructs environment matrix product operators (MPOs) representing the Hamiltonian of semi-infinite regions surrounding the target system. Starting from the finite-size ground-state MPS, we contract its Hamiltonian representation to generate effective environment MPOs, which are then attached to a renewed finite system in a recursive manner. This iterative embedding drives the system toward a bulk-like state with negligible finite-size effects. The scheme requires no assumption of homogeneity and achieves unprecedentedly long real-time dynamics free from boundary reflections.

Figures (4)

• Figure 1: Graphical representation of (a) finite MPS with auxiliaries and MPO Hamiltonian, $\hat{h}$, with embedded environments, $H_L$ and $H_R$. (b) Matrix representation of the Hamiltonian (tensor) of the left subsystem of size $N_L$ in terms of the Schmidt state $|\psi_L\rangle$, which becomes $H^{(n+1)}_L$ after the contraction. Those of the right subsystem are also shown on the right. (c) Process of contracting the tensor in (b), which is repeated by increasing $n_l$.
• Figure 2: Static properties of the ground state of TFIM for a system of $N=60$ and for the bond dimension, $m_{L/R}=30$, of the environment MPO. (a) Energy density $e_i(N)$ at bond $i$ (with its both edge sites included by half) for several choices of the number of embedding $n=1,2,\cdots,10$ in the gapless critical point, $\Gamma/J=1$. The $n$-dependence of $|e_i(N)-e(\infty)|$ at the center site for $\chi=30$ is shown in the inset. (b) $e_i(N)$ for the gapped state $\Gamma/J=0.5$ with $n=1$, in comparison with OBC ($n=0$). (c) Spin-spin correlation function $\langle \sigma_i^z \sigma_j^z \rangle - \langle \sigma_i^z\rangle\langle\sigma_{j}^z \rangle$ with fixed values of $j$, for the gapped phase $\Gamma/J=0.5$. The top and bottom panels show the case of embedding $n=1$ and OBC, respectively.
• Figure 3: (a) Entanglement entropy $S(\ell)$ as a function of subsystem size $\ell$ of the TFIM at $\Gamma/J=1$ and $N=60$ and $\chi=30$ for several choices of embedding, $n=1,2,\cdots,5,10$, and for OBC. The solid line in the semilog plot indicates the function $\frac{c}{3}\log(\ell)$ with $c=1/2$. (b) Effective site length $N_{\rm eff}$, obtained by fitting the EE when we assume the form $S_{\rm OBC}(\ell)$ with $c=1/2$ to the environment-embedded data, where we compare the cases, $\chi=10,20,30$.
• Figure 4: Real-time dynamics of TFIM+L in a $E_8$ regime with $\Gamma=J=1$, $h=0.05$. (a) Density plot of $C_j(t)=|\langle\sigma_j(t)\sigma_{N/2}(0)\rangle|$ for $N=240$ with $n=1$ embedding (left panel) and OBC (right). (b) Density plot of the dynamical structure factor $S^{zz}(k, \omega)$ for $N=240$. The Fourier transform of $C_j(t)$ uses Hann window to position space, and Gaussian window $\sigma=\sqrt{T^2/(-2*\log(\alpha))}$ to time space drescher2023dynamical, with a linear extrapolationwhite2008spectralbarthel2009spectral, using the data $t=[0:85]$. (c) $\mathfrak{R}e\{S^{zz}(k=0, \omega)\}$ for the data in panel (b) with embedding (solid line) and OBC(broken line). Vertical lines denote the $E_8$ spectrum from the field theory.