Investigating topological order using recurrent neural networks

Mohamed Hibat-Allah, Roger G. Melko, and Juan Carrasquilla

Phys. Rev. B 108, 075152 – Published 22 August 2023

Abstract

Recurrent neural networks (RNNs), originally developed for natural language processing, hold great promise for accurately describing strongly correlated quantum many-body systems. Here, we employ two-dimensional RNNs to investigate two prototypical quantum many-body Hamiltonians exhibiting topological order. Specifically, we demonstrate that RNN wave functions can effectively capture the topological order of the toric code and a Bose-Hubbard spin liquid on the kagome lattice by estimating their topological entanglement entropies. We also find that RNNs favor coherent superpositions of minimally entangled states over minimally entangled states themselves. Overall, our findings demonstrate that RNN wave functions constitute a powerful tool to study phases of matter beyond Landau's symmetry-breaking paradigm.

Received 28 March 2023
Accepted 4 August 2023

DOI:https://doi.org/10.1103/PhysRevB.108.075152

Physics Subject Headings (PhySH)

Topological order

Language models Strongly correlated systems

Artificial neural networks Bose-Hubbard model Machine learning Variational approach

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Mohamed Hibat-Allah ^1,2,3,*, Roger G. Melko^2,3, and Juan Carrasquilla^1,2,4

¹Vector Institute, MaRS Centre, Toronto, Ontario M5G 1M1, Canada
²Department of Physics and Astronomy, University of Waterloo, Ontario N2L 3G1, Canada
³Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, Ontario N2L 2Y5, Canada
⁴Department of Physics, University of Toronto, Ontario M5S 1A7, Canada

^*[email protected]

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Vol. 108, Iss. 7 — 15 August 2023

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Images

Figure 1
(a) An illustration of the positive RNN wave function. Each RNN cell (blue squares) receives an input $σ_{n - 1}$ and a hidden state $h_{n - 1}$ and outputs a new hidden state $h_{n}$ . The latter is fed to a Softmax layer (denoted by S, red circle) that outputs a conditional probability $P_{i}$ (orange circle). The conditional probabilities are multiplied after taking a square root to obtain the wave function $Ψ_{θ} (σ)$ (pink circle). (b) Illustration of the sampling scheme for an RNN wave function. After obtaining the probability vector $y_{i}$ from the Softmax layer (S) at step $i$ , we sample it to produce $σ_{i}$ which is fed again to the RNN with the hidden state $h_{i}$ to produce the next configuration $σ_{i + 1}$ .(c) A 2D RNN with periodic boundary conditions. A bulk RNN cell receives two hidden states $h_{i, j - 1}$ and $h_{i - {(- 1)}^{j}, j}$ , as well as two input vectors $σ_{i, j - 1}$ and $σ_{i - {(- 1)}^{j}, j}$ (not shown) illustrated by the black solid arrows. To handle periodic boundary conditions, RNN cells at the boundary receive an additional $h_{i, j + 1}$ and $h_{i + {(- 1)}^{j}, j}$ , as well as two input vectors $σ_{i, j + 1}$ and $σ_{i + {(- 1)}^{j}, j}$ (not shown) illustrated by green solid arrows. The sampling path is illustrated with curved dashed red arrows. The initial memory states and the initial inputs are taken as null vectors and are fed to the 2DRNN cells as indicated by the pink dashed arrows.
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Figure 2
(a) a sketch of the parts $A$ , $B$ , and $C$ that we use for Kitaev-Preskill construction to compute the TEE in a system of interest. (b)–(d) An illustration of the subregions $A, B, C$ chosen for the Kitaev-Preskill constructions. (b) For the 2D toric code, each subregion has three spins. We also do the same for the lattice lengths $L = 6, 8, 10$ in Fig. 4. For the hard-core Bose-Hubbard model on the kagome lattice, we target two different system sizes. (c) shows the construction for $L = 6$ and (d) provides the construction for $L = 8$ . In (c) and (d), each site corresponds to a block of three bosons.
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Figure 3
Mapping of 2D toric code lattice and kagome lattice to a square lattice that can be handled by a 2D RNN wave function. For the toric code lattice in (a), every two sites inside the dashed green ellipses are merged. For the kagome lattice in (c), every three sites in a unit cell enclosed by the dashed green circles are combined.
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Figure 4
Entanglement properties of the 2D toric code. (a) Second Renyi entropy $S_{2}$ scaling is computed using our RNN wave function on the 2D toric code for different lengths of the boundary $L_{A} = L$ , where $L$ is the system length and the total system size is given as $L \times L \times 2$ . Note that our toric code embedded in a torus is divided into two equal cylinders to compute $S_{2}$ . (b) TEE computed with the Kitaev-Preskill construction [see Fig. 2] versus the overall system length $L$ . The values found by the RNN are very close to $ln (2)$ . Error bars correspond to one standard deviation and are smaller than the symbol size.
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Figure 5
An illustration of the vertical and the horizontal loops used to compute the Wilson loop operators [see Eq. (8)] and the 't Hooft loop operators [see Eq. (9)].
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Figure 6
A plot of the topological entanglement entropy against the interaction strength $V$ (in units of $t$ ) of Hard-core Bose-Hubbard model on kagome lattice for system sizes $N = L \times L \times 3$ where $L = 6, 8$ . The calculations were performed using the Kitaev-Preskill construction [see Figs. 2 and 2]. The continuous black horizontal line corresponds to a zero TEE, and the dashed blue horizontal line for a $ln (2)$ TEE.
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