The numerical value for a universal quantity of a two-dimensional dimerized quantum antiferromagnet

Spatial dimension two is special because of the famous Mermin–Wagner theorem Mer66 . Specifically, continuous symmetry cannot be broken spontaneously (at finite temperature $T$) in spatial dimension two. As a result, for two-dimensional (2D) quantum antiferromagnets, as $T$ rises from zero temperature, one encounters a crossover instead of a phase transition to a unique phase called the quantum critical regime (QCR).

Using the relevant field theory, properties of QCR are investigated Chu94 . In particular, several universal quantities are proposed. Among these quantities, we are especially interested in $\chi_{u}c^{2}/T$ due to the recently found discrepancy between the analytic prediction and the numerical calculations. Here $\chi_{u}$ and $c$ are the uniform susceptibility and the spin-wave velocity, respectively.

Analytically, it is predicted that the numerical value of $\chi_{u}c^{2}/T$ is given by (around) 0.27185. Although this prediction was confirmed by earlier Monte Carlo studies Chu94 ; Tro98 , recent investigations of 2D dimerized bilayer and plaquette spin-1/2 Heisenberg models conclude that $\chi_{u}c^{2}/T\sim 0.32(0.33)$ Sen15 ; Tan181 . Because of this discrepancy, it will be interesting to conduct a further examination on the numerical value of $\chi_{u}c^{2}/T$.

In this study, we perform large-scale Monte Carlo simulations to determine the $\chi_{u}c^{2}/T$ of a 2D dimerized quantum antiferromagnetic Heisenberg model. By simulating lattices as large as $L=512$ ($L$ is the linear system size), we obtain $\chi_{u}c^{2}/T\sim 0.32$ which matches quantitatively with recent outcomes claimed in Refs. Sen15 ; Tan181 . Our result suggests that a refinement of analytic calculation is needed.

The rest of the paper is organized as follows. After the introduction, the model and the measured observables are described in Sec. II. We then present the obtained results in Sec. III. In particular, the numerical evidence to support $\chi_{u}c^{2}/T\sim 0.32$ is demonstrated. We conclude our investigation in Sec. VI.

The Hamiltonian of the considered 2D spin-$\frac{1}{2}$ dimerized herringbone Heisenberg model takes the following expression

$\displaystyle H$

$\displaystyle=$

$\displaystyle\sum_{\langle ij\rangle}J\,\vec{S}_{i}\cdot\vec{S}_{j}+\sum_{\langle i^{\prime}j^{\prime}\rangle}J^{\prime}\,\vec{S}_{i^{\prime}}\cdot\vec{S}_{j^{\prime}},$

(1)

where $J$ and $J^{\prime}$ are the antiferromagnetic couplings connecting nearest neighbor spins $\langle ij\rangle$ and $\langle i^{\prime}j^{\prime}\rangle$, respectively, and $\vec{S}_{i}$ is the spin-$\frac{1}{2}$ operator at site $i$. A cartoon representation of the studied model is depicted in fig. 1. $J$ is set to 1 in our investigation. As the magnitude of $J^{\prime}$ increases, a phase transition will occur for a particular value of $J^{\prime}>J$. This special point $J^{\prime}/J$ in the parameter space is denoted by $(J^{\prime}/J)_{c}$ and is found to be $(J^{\prime}/J)_{c}=2.4981(2)$ in the literature Pen20 . The investigation presented in this study is conducted at the critical point $(J^{\prime}/J)_{c}$.

To examine the universal quantity $\chi_{u}c^{2}/T$ of QCR, the uniform susceptibility $\chi_{u}$ and the spin-wave velocity $c$ are measured. On a finite lattice of linear size $L$, the uniform susceptibility $\chi_{u}$ is defined by

$$\chi_{u}=\frac{\beta}{L^{2}}\left\langle\left(\sum_{i}S_{i}^{z}\right)^{2}\right\rangle.$$

(2)

The quantity $\beta$ appearing above is the inverse temperature. The spin-wave velocity $c$ for the investigated model is calculated through the temporal and spatial winding numbers squared ($\langle W_{t}^{2}\rangle$ and $\langle W_{i}^{2}\rangle$ with $i\in\{1,2\}$).

To conduct the proposed investigation, we have carried out large-scale quantum Monte Carlo calculations (QMC) using the stochastic series expansion algorithm (SSE) with efficent operate-loop update San99 . The obtained outcomes are described in the following subsections.

In this study, we calculate the numerical value of $\chi_{u}c^{2}/T$ corresponding to the 2D dimerized quantum herringbone Heisenberg model using QMC. By simulating lattice as large as $L=512$ at moderate low temperatures, we obtain $\chi_{u}c^{2}/T\sim 0.32$. Our result deviates from the associated analytic prediction $\chi_{u}c^{2}/T=0.27185$ but agrees quantitatively with recent numerical calculations of 2D spin-1/2 bilayer and plaquette Heisenberg models. Based on the obtained outcomes in this study, it will be interesting to refine the relevant theoretical calculations.

Partial support from National Science and Technology Council (NSTC) of Taiwan is acknowledged (MOST 110-2112-M-003-015 and MOST 111-2112-M-003-011).

F.J.J proposed the project, conducted the calculations, analyzed the data, and wrote up the manuscript.

The author declares no conflict of interest.