Asymptotic Freedom at the Berezinskii-Kosterlitz-Thouless Transition without Fine-Tuning Using a Qubit Regularization

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Asymptotic Freedom at the Berezinskii-Kosterlitz-Thouless Transition without Fine-Tuning Using a Qubit Regularization

Sandip Maiti, Debasish Banerjee, Shailesh Chandrasekharan, and Marina K. Marinkovic

Phys. Rev. Lett. 132, 041601 – Published 22 January 2024

Abstract

We propose a two-dimensional hard-core loop-gas model as a way to regularize the asymptotically free massive continuum quantum field theory that emerges at the Berezinskii-Kosterlitz-Thouless transition. Without fine-tuning, our model can reproduce the universal step-scaling function of the classical lattice $X Y$ model in the massive phase as we approach the phase transition. This is achieved by lowering the fugacity of Fock-vacuum sites in the loop-gas configuration space to zero in the thermodynamic limit. Some of the universal quantities at the Berezinskii-Kosterlitz-Thouless transition show smaller finite size effects in our model as compared to the traditional $X Y$ model. Our model is a prime example of qubit regularization of an asymptotically free massive quantum field theory in Euclidean space-time and helps understand how asymptotic freedom can arise as a relevant perturbation at a decoupled fixed point without fine-tuning.

Received 16 August 2023
Revised 9 November 2023
Accepted 11 December 2023

DOI:https://doi.org/10.1103/PhysRevLett.132.041601

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Conformal field theory Lattice field theory Lower-dimensional field theories Renormalization group Sigma models XY model

Continuous symmetries

Monte Carlo methods

Particles & FieldsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Sandip Maiti ^1,2,*, Debasish Banerjee ^1,2,†, Shailesh Chandrasekharan ^3,‡, and Marina K. Marinkovic ^4,§

¹Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata 700064, India
²Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India
³Department of Physics, Box 90305, Duke University, Durham, North Carolina 27708, USA
⁴Institut für Theoretische Physik, Wolfgang-Pauli-Straße 27, ETH Zürich, 8093 Zürich, Switzerland

^*Corresponding author: sandip.maiti@saha.ac.in
^†Corresponding author: debasish.banerjee@saha.ac.in
^‡Corresponding author: sch27@duke.edu
^§Corresponding author: marinama@ethz.ch

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Issue

Vol. 132, Iss. 4 — 26 January 2024

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Images

Figure 1
Length scales in a lattice field theory that reproduces asymptotically free quantum field theories.
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Figure 2
Illustration of the renormalization group flow of a generic qubit regularized model that reproduces the physics of the asymptotically free QFTs. At the decoupled quantum critical point, qubit models describe the physics of a critical system containing two decoupled theories. However, when a small nonzero coupling is introduced between the theories, the long-distance physics flows toward the desired universal physics of the UV fixed point theory.
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Figure 3
The left figure shows an illustration of a dimer configuration that contributes to the partition function of the model arising from Eq. (3). Interlayer dimers (or instantons) have weight $λ$ while the intralayer dimers have weight 1. By giving the dimers an orientation as illustrated, each dimer configuration can also be viewed as a configuration of self-avoiding oriented loops. The configuration on the right is such a mapping of the configuration on the left. The instantons are mapped into Fock-vacuum sites, shown as blue circles. The dimer model shows that the loop model is critical when $λ = 0$ .
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Figure 4
The figure shows the universal step-scaling function [i.e., $ξ (2 L) / ξ (L)$ vs $ξ (L) / L$ ] obtained from the $X Y$ model Eq. (1) (solid line) [60] and compares it with data from the model Eq. (3) at $λ = 0.01$ (red), 0.2 (blue), 0.4 (purple), and 0.6 (green), for various lattice sizes shown with different symbols. For small values of $L$ , our data deviate from the solid line. We define $ℓ_{UV}$ as the minimum value of $L$ when the data begin to fall on the solid line. From the figure we estimate $ℓ_{UV} \approx 80$ for $λ = 0.6$ and $ℓ_{UV} \approx 160$ for $λ = 0.4$ . For very small $λ$ we expect the $ξ (L) / L$ to approach the universal UV prediction of $ξ (L) / L = 0.750 691 2 \dots$ (see Ref. [35]), when $L \sim ℓ_{UV}$ before beginning to follow the solid line. We see this at $λ = 0.2$ and 0.01. Since at these couplings $ℓ_{UV} > 1280$ , we predict that the data at these couplings will also eventually follow the solid line, but only for $L ≫ ℓ_{UV}$ , which we cannot access. To show this feature, in the inset we plot $ξ (L) / L$ as a function of $L$ at $λ = 0.01$ . Note that the data approaches $ξ (L) / L = 0.750 691 2 \dots$ when $L \sim ℓ_{UV}$ as expected. Based on our prediction above, this is only a plateau and that for $L ≫ ℓ_{UV}$ (which we cannot access) $ξ (L) / L$ will eventually approach zero. The inset also shows that the large $L$ behavior of $λ = 0$ is very different and stabilizes at $ξ (L) / L = 0.4889 (6)$ . In the inset we also show the data from [35] in the traditional $X Y$ model [Eq. (1)] at two values of $β$ close to the transition. These data are still far from the universal value due to logarithmic corrections as explained in [35].
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Figure 5
The figure shows the helicity modulus $ϒ$ as a function of $L$ for $λ = 0.6$ , 0.4, 0.2, 0.01, and 0.0. We expect $ϒ \to 0$ for $L ≫ ℓ_{UV}$ when $λ \neq 0$ . This is clearly seen at $λ = 0.6$ . At $λ = 0.4$ , since $ℓ_{UV}$ is larger, we only see the initial part of the decrease toward zero. At $λ = 0.2$ , $ℓ_{UV}$ is even larger, and so we only observe the flat part expected in the UV. At the UV fixed point we expect $ϒ \approx 2 / π$ . When $λ = 0.01$ we do observe the data converging well to this universal value (top solid line). Since $λ = 0.01$ is not really the critical point, this line, too, will eventually turn around at very large values of $L$ and go to zero. On the other hand when $λ = 0$ we find that $ϒ \approx 0.606$ in the large $L$ limit [60]. We demonstrate the difference between $λ = 0$ and $λ = 0.01$ in the inset, where we also show data from [35] for the helicity modulus in the traditional $X Y$ model Eq. (1) at two values of $β$ close to the transition. Note that the values from the traditional model are far from $2 / π$ , a well-known difficulty due to large logarithmic corrections. In contrast, our qubit model is able to recover the UV physics more easily.
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