Improved summations of $n$-point correlation functions of projected entangled-pair states

Improved summations of $n$ -point correlation functions of projected entangled-pair states

Boris Ponsioen, Juraj Hasik, and Philippe Corboz

Phys. Rev. B 108, 195111 – Published 7 November 2023

Abstract

Numerical treatment of two-dimensional strongly-correlated systems is both extremely challenging and of fundamental importance. Infinite projected entangled-pair states (PEPS), a class of tensor networks, have demonstrated cutting-edge performance for ground-state calculations, working directly in the thermodynamic limit. Furthermore, in recent years the application of PEPS has been extended to also low-lying excited states, using an ansatz that targets quasiparticle states above the ground state with high accuracy. A major technical challenge for those simulations is the accurate evaluation of summations of two- and three-point correlation functions with reasonable computational cost. In this paper, we show how a reformulation of $n$ -point functions in the context of PEPS leads to extra contributions to the results that prove to play an important role. Benchmarks for the frustrated $J_{1} - J_{2}$ Heisenberg model illustrate the improved precision, efficiency, and stability of the simulations compared to previous approaches. Leveraging automatic differentiation to generate the most tedious and error-prone parts of the computation, the straightforward implementation presented here is a step towards broader adoption of the PEPS excitation ansatz in future applications.

Received 3 July 2023
Accepted 20 September 2023

DOI:https://doi.org/10.1103/PhysRevB.108.195111

Physics Subject Headings (PhySH)

Frustrated magnetism Quasiparticles & collective excitations

Quantum spin models

Heisenberg model Projected entangled pair states Tensor network methods Variational approach

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Boris Ponsioen , Juraj Hasik, and Philippe Corboz

Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands

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Issue

Vol. 108, Iss. 19 — 15 November 2023

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Images

Figure 1
Cut of the dynamical structure factor of the $J_{1} - J_{2}$ Heisenberg model for $J_{2} / J_{1} = 0.3$ at $k = (π, 0)$ computed with the PEPS excitation ansatz at bond dimension $D = 4$ , compared with variational Monte Carlo (VMC) data of Ref. [43]. The VMC data was normalized in such a way that the sum of spectral weight is equal to the static structure factor [computed to be $s (π, 0) \approx 0.186]$ , while the PEPS data always exactly fulfills this sum rule by construction [for PEPS we obtain $s (π, 0) \approx 0.182]$ . Both sets of data were convoluted with a Lorentzian with broadening factor $η = 0.1$ .
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Figure 2
Static structure factors $s_{α} (π, π)$ of $D = 5$ ground states of the $J_{1} - J_{2}$ model obtained with and without projector derivatives. The ground states and the results without projector derivatives appeared in [46].
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Figure 3
Energies and norms of the single-mode approximation (SMA) $S_{k}^{+} | Ψ (A) 〉$ evaluated without (blue) and with projector derivatives (red). [(a),(b)] Energies of SMA at $J_{2} / J_{1} = 0, 0.25$ respectively, for bond dimensions $D = 3, 4$ . Note that due to the fast decay of singular values at these bond dimensions, obtaining accurate results without projector derivatives at very high values of $χ$ is not possible, as numerical noise becomes significant. (c),(d) Norms of the SMA, equal to the static structure factors in the transverse channel, again for $J_{2} / J_{1} = 0, 0.25$ .
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Figure 4
Basis size dependency for $D = 3, 4$ in the small- $χ$ regime at $J_{2} / J_{1} = 0.25$ . Results on the left (blue) were obtained without projector derivatives, showing that $D = 3, 4$ require respectively $χ = 48, 84$ in order to converge to reliable results. On the right hand side (red), the results obtained with projector derivatives show the substantial improvements: even at an extremely low $χ = 12$ we observe complete convergence of eigenvalues up to the full basis size. The opacity of each marker in the plots corresponds to the norm of the excited state, which can be used to identify the most stable and relevant solutions.
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Figure 5
Basis size dependency for $D = 4$ for various values of $J_{2} / J_{1} = 0.25$ . The left column (blue) shows results without projector derivatives at $χ = 72$ , which is an intermediate value for which reasonable selection of stable eigenvalues can be made. On the right-hand side (red) we show the effect of adding the projector derivatives at $χ = 36$ , demonstrating that even at this small value of $χ$ complete convergence of eigenvalues can be obtained. The opacity of each marker in the plots corresponds to the norm of the excited state, which can be used to identify the most stable and relevant solutions.
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