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Fermionic Isometric Tensor Network States in Two Dimensions

Zhehao Dai, Yantao Wu, Taige Wang, Michael P. Zaletel

TL;DR

This work developed and benchmarked a time-evolution block-decimation (TEBD) algorithm for real-time and imaginary-time evolution, which produces ground-state energies for gapped systems, systems with a Dirac point, and systems with gapless edge mode to good accuracy.

Abstract

We generalize isometric tensor network states to fermionic systems, paving the way for efficient adaptations of 1D tensor network algorithms to 2D fermionic systems. As the first application of this formalism, we developed and benchmarked a time-evolution block-decimation (TEBD) algorithm for real-time and imaginary-time evolution. The imaginary-time evolution produces ground-state energies for gapped systems, systems with a Dirac point, and systems with gapless edge modes to good accuracy. The real-time TEBD captures the scattering of two fermions and the chiral edge dynamics on the boundary of a Chern insulator.

Fermionic Isometric Tensor Network States in Two Dimensions

TL;DR

This work developed and benchmarked a time-evolution block-decimation (TEBD) algorithm for real-time and imaginary-time evolution, which produces ground-state energies for gapped systems, systems with a Dirac point, and systems with gapless edge mode to good accuracy.

Abstract

We generalize isometric tensor network states to fermionic systems, paving the way for efficient adaptations of 1D tensor network algorithms to 2D fermionic systems. As the first application of this formalism, we developed and benchmarked a time-evolution block-decimation (TEBD) algorithm for real-time and imaginary-time evolution. The imaginary-time evolution produces ground-state energies for gapped systems, systems with a Dirac point, and systems with gapless edge modes to good accuracy. The real-time TEBD captures the scattering of two fermions and the chiral edge dynamics on the boundary of a Chern insulator.
Paper Structure (6 sections, 1 theorem, 13 equations, 11 figures)

This paper contains 6 sections, 1 theorem, 13 equations, 11 figures.

Table of Contents

  1. Virtual fermion ansatz
  2. Replacing multiple entangled fermion pairs by a single pair
  3. Equivalence of the swap gate ansatz and the virtual fermion ansatz
  4. Alternative interpretation of the isometric condition in the swap-gate convention
  5. Finite-depth-local-unitary circuits preserve fisoTNS
  6. Moses Move for symmetric TNS

Key Result

Theorem 1

For any two fisoTNS $|\psi\rangle$ and $|\psi'\rangle$ that have the same tensors everywhere except at the orthogonality center, with $\hat{\Lambda}_{(x_0,y_0)}$ and $\hat{\Lambda}'_{(x_0,y_0)}$ for each respectively, $\langle\psi|\psi'\rangle = \sum_{l,r,d,u,k}\Lambda^{*k}_{l,r,d,u}\Lambda'^{k}_{l,

Figures (11)

  • Figure 1: Isometric tensor network states in 1D and 2D. (a) A matrix product state in isometric form. The red dot represents the orthogonality center. (b) Isometric condition for tensors on the left (right) of the center $A$ ($B$). A line without a circle on it represents identity. (c) The expectation of any physical operator (blue square) on the orthogonality center equals the contraction of the operator with the central tensor and its conjugate. (d) 2D isometric tensor network. The red dot represents the orthogonality center. On the shaded region, the orthogonality hypersurface, all arrows point in. (e) Isometric condition for tensors in the lower-left and the upper-right quadrant. (f) The same as (c) but in 2D.
  • Figure 2: (a) Definition of the fermionic wavefunction in the swap gate convention. An extra sign (diamond) is inserted at every crossing. (b) In the upper-right quadrant of a fisoTNS, we require $(-1)^{P_kP_d} A$ to be an isometric tensor. The minus sign comes from putting the outgoing legs next to each other. (c) After the action of a brick wall circuit, we can write the resulting state as a new fisoTNS by splitting the product of two tensors, the unitary and the swap gate back into two isometric tensors. Here we choose a particular way of splitting the tensor where the original vertical legs are assigned to the second tensor and the left tensor has trivial up and down legs (dashed lines)
  • Figure 3: TEBD in fisoTNS. (a) Column $i$ is the OC. (b) Contraction of column $i$ and $i+1$ into one column. (c) Apply the TEBD gates on the OC using the MPS TEBD method. (d) After the MM algorithm, column $i+1$ is now the OC.
  • Figure 4: Error in ground state energy density. (a) is presented with the optimal $d\tau$. The system size is $9 \times 9$ for Chern and $10 \times 10$ for Dirac, Insulator, and $p+ip$ SC. $f = \chi$. The MM error is typically $10^{-4}$ or $10^{-5}$, comparable with the truncation error of the MPS TEBD within the OC.
  • Figure 5: Real-time evolution of two fermions scattering with nearest neighbor hopping on a $12 \times 12$ square lattice without (a) and with (b) interaction. The two fermions are initialized at the top-left and bottom-left corner. We show the ED, fisoTNS, and 1D TDVP results from top to bottom. From left to right, we show three time slices, before, during and after the collision. We show the error of various methods in (c-d), including fTNS with different configurations and 1D W$_{\mathrm{II}}$ and TDVP methods. The solid lines show the excitation error and the dashed lines show the background error (see main text for definition).
  • ...and 6 more figures

Theorems & Definitions (2)

  • Definition 1
  • Theorem 1