Quantum criticality and entanglement for the two-dimensional long-range Heisenberg bilayer

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Quantum criticality and entanglement for the two-dimensional long-range Heisenberg bilayer

Menghan Song, Jiarui Zhao, Yang Qi, Junchen Rong, and Zi Yang Meng

Phys. Rev. B 109, L081114 – Published 26 February 2024

Abstract

The study of quantum criticality and entanglement in systems with long-range (LR) interactions is still in its early stages, with many open questions remaining to be explored. In this work, we investigate critical exponents and scaling of entanglement entropy (EE) in the LR bilayer Heisenberg model using large-scale quantum Monte Carlo simulations. By applying modified (standard) finite-size scaling above (below) the upper critical dimension and field theory analysis, we obtain precise critical exponents in three regimes: the LR Gaussian regime with a Gaussian fixed point, the short-range (SR) regime with Wilson-Fisher exponents, and a LR non-Gaussian regime where the critical exponents vary continuously from LR Gaussian to SR values. We compute the Rényi EE both along the critical line and in the Néel phase, and we observe that as the LR interaction is enhanced, the area-law contribution in EE gradually vanishes both at quantum critical points (QCPs) and in the Néel phase. The log-correction in EE arising from sharp corners at the QCPs also decays to zero as the LR interaction grows, whereas that for Néel states, caused by the interplay of Goldstone modes and restoration of the symmetry in a finite system, is enhanced. Relevant experimental settings to detect these nontrivial properties for quantum many-body systems with LR interactions are discussed.

Received 12 June 2023
Revised 14 December 2023
Accepted 7 February 2024

DOI:https://doi.org/10.1103/PhysRevB.109.L081114

Physics Subject Headings (PhySH)

Entanglement entropy Long-range interactions

Bilayer films Quantum spin models

Conformal field theory Critical exponents Heisenberg model Quantum Monte Carlo

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Menghan Song ¹, Jiarui Zhao ^1,*, Yang Qi^2,3,4, Junchen Rong^5,†, and Zi Yang Meng ^1,‡

¹Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong SAR, China
²State Key Laboratory of Surface Physics, Fudan University, Shanghai 200438, China
³Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China
⁴Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
⁵Institut des Hautes Études Scientifiques, 91440 Bures-sur-Yvette, France

^*jrzhao@connect.hku.hk
^†junchenrong@gmail.com
^‡zymeng@hku.hk

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Issue

Vol. 109, Iss. 8 — 15 February 2024

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Images

Figure 1
Model and phase diagram of the 2D LR antiferromagnetic Heisenberg bilayer. (a) The bilayer antiferromagnetic model with interlayer interaction $J_{⊥}$ and intralayer LR interaction $J_{i j}$ . (b) The ground-state phase diagram of the model expanded by the axes of $α$ and $g = \frac{J_{⊥}}{J}$ , obtained from QMC simulation and finite-size analysis, as exemplified in Figs. 2 and 2 and in the Supplemental Material [90]. The Néel phase spontaneously breaks the spin $S U (2)$ symmetry and the dimer phase is a product state without symmetry breaking. The QCPs separating them belong to the $(2 + 1) D O (3)$ universality when $α > α_{c} = 3.9621$ , a non-Gaussian fixed point when $\frac{10}{3} < α < α_{c}$ , and a Gaussian fixed point when $α < \frac{10}{3}$ . QCPs in LR Gaussian, LR non-Gaussian, and SR regions are denoted by green, yellow, and white dots (where the simulations are performed) and lines, respectively.
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Figure 2
Data collapse and the critical exponents $ν$ and $β$ obtained in $α \in (2, 10]$ . (a) Data collapse at $α = 8$ (SR case) with critical exponents $ν = 0.706 (1)$ and $β = 0.366 (5)$ . (b) Data collapse at $α = 3$ (LR case with Gaussian fixed point above the upper critical dimension) with critical exponents $ν^{'} = 0.747 (4), β = 0.497 (7)$ , and $ν = ν^{'} \frac{d}{d_{uc}} = 0.996 (7)$ . The green shaded area in panels (c) and (d) denote the LR Gaussian regime ( $α < \frac{10}{3}$ ) where $d = 2$ is larger than the upper critical dimension $d_{uc}$ . In the region of $\frac{10}{3} < α < α_{c}$ (yellow shaded area), the system is in a non-Gaussian fixed point, and when $α > α_{c}$ (white area) the critical exponents become their SR $(2 + 1) D O (3)$ WF values [86].
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Figure 3
EE at the QCPs. $S_{A}^{(2)}$ and its scaling behavior at the QCPs shown in Fig. 1 with system sizes $L = 8, 10, 12, ..., 22$ . (a) Rényi entropies versus the system sizes for different interaction powers $α$ . The inset shows the square entanglement region $A$ with the boundary length $l = 2 L$ . (b) The area-law coefficient $a$ versus $α$ . (c) The log-coefficient $s_{c}$ versus $α$ .
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Figure 4
EE inside the Néel phase. $S_{A}^{(2)}$ and its scaling behavior for the single-layer LR Heisenberg model, i.e., $g = 0$ in Eq. (1) in the Néel phase with system sizes $L = 8, 12, 16, ..., 44$ . (a) Rényi entropies versus the system sizes for different interaction powers $α$ . The inset shows the cylinder entanglement region $A$ with the boundary length $l = 2 L$ . For clarity, $S_{A}^{(2)}$ is modified with a constant so that $S_{A}^{(2)} (l = 16)$ is the same for every $α$ . (b) The area-law coefficient $a$ versus $α$ . (c) The logarithmic coefficient $s_{G}$ versus $α$ . The red line shows the result $s_{G} = \frac{N_{G} [d - z (α)]}{2}$ , with $z (α)$ obtained from spin wave theory and QMC data ( $L = 64$ ) in Ref. [2].
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