Validating Phase-Space Methods with Tensor Networks in Two-Dimensional Spin Models with Power-Law Interactions

Sean R. Muleady, Mingru Yang, Steven R. White, and Ana Maria Rey

Phys. Rev. Lett. 131, 150401 – Published 13 October 2023

Abstract

Using a recently developed extension of the time-dependent variational principle for matrix product states, we evaluate the dynamics of 2D power-law interacting $X X Z$ models, implementable in a variety of state-of-the-art experimental platforms. We compute the spin squeezing as a measure of correlations in the system, and compare to semiclassical phase-space calculations utilizing the discrete truncated Wigner approximation (DTWA). We find the latter efficiently and accurately captures the scaling of entanglement with system size in these systems, despite the comparatively resource-intensive tensor network representation of the dynamics. We also compare the steady-state behavior of DTWA to thermal ensemble calculations with tensor networks. Our results open a way to benchmark dynamical calculations for two-dimensional quantum systems, and allow us to rigorously validate recent predictions for the generation of scalable entangled resources for metrology in these systems.

Received 31 May 2023
Accepted 7 September 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.150401

Physics Subject Headings (PhySH)

Long-range interactions Quantum metrology

2-dimensional systems Quantum spin models

Phase space methods Tensor network methods

Statistical Physics & ThermodynamicsAtomic, Molecular & OpticalCondensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Sean R. Muleady ^1,2,*, Mingru Yang ^3,4,*, Steven R. White ³, and Ana Maria Rey ^1,2

¹JILA, National Institute of Standards and Technology and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
²Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA
³Department of Physics and Astronomy, University of California, Irvine, California 92697, USA
⁴University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Wien, Austria

^*These authors contributed equally to this work.

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Vol. 131, Iss. 15 — 13 October 2023

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Images

Figure 1
Dynamics of (a),(b) the spin squeezing $ξ^{2}$ (shown in decibels) and (c),(d) total spin $⟨ {\hat{S}}^{2} ⟩$ generated by Eq. (1), shown for select interaction ranges $α$ and $Δ = - 1.8$ , $- 0.2$ on a $6 \times 6$ lattice. The total spin is normalized by its initial value, $⟨ {\hat{S}}^{2} ⟩_{0} = (N / 2) (N / 2 + 1)$ , and the time axis is scaled by the average spin interaction $\bar{J}$ and $| Δ |$ . Solid lines are obtained by GSE-TDVP (longest evolved times denoted with vertical bars for visibility), while the dashed lines are obtained by DTWA. We show exact results for the collective case with $α = 0$ for comparison (black, dashed). (e)–(h) Analogous results for a $10 \times 10$ lattice. (i) We also plot $\bar{J} / J^{⊥}$ for various 2D square lattices of side length $L$ , with expected power laws shown for comparison.
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Figure 2
(a)–(c) Optimal spin squeezing $ξ^{2}$ (in decibels) computed via GSE-TDVP (squares) and DTWA (faded lines) for various 2D lattice sizes, interaction ranges $α$ , and $Δ$ . The dotted horizontal lines denote the expected results for the OAT model ( $α = 0$ ), while the vertical dashed lines correspond to $Δ = 0$ . Unfilled squares denote estimated values for parameters where the GSE-TDVP dynamics approach close to, but do not achieve, a local minimum for the longest evolved times, as shown in the Supplemental Material [32]. (d) Fits for the power-law scaling of the optimal spin squeezing with the particle number $N$ , where $ξ^{2} \sim N^{ν}$ . The associated fitting error is denoted by error bars for exact results, or the shaded region for DTWA results. We supplement our data with exact and DTWA results for a $4 \times 4$ lattice, and only provide fits when data for three or more system sizes are available.
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Figure 3
(a),(b) Comparison between the long-time values (lines) and thermal ensemble averages (squares, same in each panel) of the transverse magnetization $⟨ {\hat{S}}_{⊥}^{2} ⟩$ , normalized by its initial value of $⟨ {\hat{S}}_{⊥}^{2} ⟩_{0} = N (N + 1) / 4$ for a $6 \times 6$ lattice. Dynamical results are obtained via DTWA for various scaled times $t / t_{OAT}^{*}$ . The thermal averages are obtained via METTS for $Δ > 0$ and via a purification method for $Δ < 0$ . (c) We also show the temperature $T$ of the thermal state, scaled by $\bar{J}$ . (d),(e) Dynamical growth of the bipartite entanglement entropy, $S_{vN} = - Tr [{\hat{ρ}}_{A} \ln {\hat{ρ}}_{A}]$ , where ${\hat{ρ}}_{A}$ is the reduced density matrix for a bipartition of the lattice about the center. We show GSE-TDVP results on an $8 \times 8$ lattice for various values of $Δ$ and $α$ , scaling the time axis by $\bar{J} | Δ |$ . We also provide the results for the OAT model (black, dashed) for comparison. (f) From the available short-time data, we estimate the growth rate of the entropy by fitting $S_{vN} \approx r \times t$ , and plot the resulting coefficient $r$ , scaled by $\bar{J} | Δ |$ . The resulting fit for the OAT model over the range $t \bar{J} | Δ | ≲ 0.3$ is also shown (black, dotted).
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