Quantum criticality at finite temperature for two-dimensional $JQ_{3}$ models on the square and the honeycomb lattices

Studying the characteristics of transitions between various phases of matters has long been an active research topic in the condensed matter physics Nig92 ; Car96 ; Car10 ; Sac11 . In particular, with the advances of experimental techniques and theoretical approaches, some exotic phases and (or) abnormal transitions between various states are observed and (or) proposed. One such a noticeable example is the transition between the Néel phase and the valence bond solid San07 .

The so-called $SU(2)$ $JQ$-type models of (quantum) spins in two-dimension (2D) San07 ; Lou09 ; Sen10 , which will be introduced later, are systems hosting the exotic phase transition between the Néel and the valence bond solid states. The models have two types of couplings, namely the $J$-term and the $Q$-term. In addition, when the magnitudes of the couplings increase, the $J$-term and the $Q$-term favor the formation of singlets and pairs of singlets (valence bond solid, VBS), respectively. On the one hand, when the magnitude of $Q$ is much smaller than that of $J$, the $SU(2)$ symmetry is broken spontaneously and the system is in the antiferromagnetic long-range order. On the other hand, the translation symmetry is broken in the case of $Q\gg J$. If one starts with a value of $Q/J$ ($Q>0$ and $J$ is fixed to be 1) so that the long-range order exists and the translation symmetry of the system is unbroken, then by increasing the magnitude of $Q$ one expects that the restoration of the broken $SU(2)$ symmetry and the breaking of the translation symmetry should occur at some couplings $Q/J$. In particular, these two distinct physical incidents may take place at different values of $Q/J$. If the two mentioned transitions occur at the same $\left(Q/J\right)_{c}$, which is the most interesting scenario, then without fine-tuning this phase transition should be first order according to the famous Landau-Ginsburg paradigm Gin50 .

Although the intriguing exotic phase transition introduced in the previous paragraph should be first order in general, it is argue theoretically that this phase transition, as well as those which belong to some other models possessing similar characteristics of the $JQ$-type models, can be second order Sen040 ; Sen041 . Such an exotic critical behavior is referred to as the deconfined quantum criticality (DQC) in the literature.

Many numerical investigations have been carried out in order to understand DQC, and various conclusions are obtained Kuk06 ; Mel07 ; Jia08 ; Kuk08 ; San10 ; Che13 ; Har13 . Although consistent critical exponents are obtained by simulating square lattices as large as 448² Sha16 ; San201 , one still cannot rule out the possibility that the considered phase transition of the $SU(2)$ $JQ$ model (on the square lattice) is a weakly first order one. Indeed, it is shown in Ref. Iin19 that while the phase transition of the ferromagnetic 5-state Potts model on the square lattice is a weakly first order phase transition, yet a high quality data collapse with certain values of critical exponents, which should be a feature of a second order phase transition, is reached. It is also unexpected that when more and more data of large lattices are included in the associated analyses, the determined values of $(Q/J)_{c}$ and the correlation length exponent $\nu$ change accordingly in a sizeable manner San07 ; Mel07 ; San10 ; Sha16 ; San201 . For instance, when the largest linear system sizes are $L=32$, 64 (128), 256, and 448, the calculated values of $\nu$ are given by 0.78, 0.68, 0.446, and 0.448 (or 0.455), respectively. Hence, it will extremely interesting to explore further the issue of the nature of these exotic phase transitions. In conclusion, at the moment it is a consensus that if these exotic phase transitions are second order, then they must receive abnormally large corrections (to the scaling at the corresponding criticalities).

Studies of the $SU(2)$ $JQ$-type models have been targeted at uncovering the nature of DQC. Moreover, due to the famous Mermin-Wagner theorem Mer66 ; Hoh67 ; Col73 ; Gel01 , investigating the finite-temperature properties of these models are focusing on the VBS perspective. Yet by exploring the quantum critical regime (QCR) and the associated universal quantities, which are features of the antiferromagnetic phase of the $JQ$-type models Chu93 ; Chu931 ; Chu94 ; San95 ; Tro96 ; Tro97 ; Tro98 ; Kim99 ; Kim00 ; San11 ; Sen15 ; Tan181 , one may gain certain aspects of the nature of DQC Kau08 ; Kau081 ; Puj13 . Indeed, studying the QCR of various $JQ$-type models which have either a first order or a second order phase transitions may shed some light on the DQC.

Due to the motivations described in the previous paragraphs, in this investigation we study three 2D $SU(2)$ $JQ_{3}$ models (Which will be introduced in detail later). In particular, two of these three systems are on the square lattice and the remaining one is on the honeycomb lattice Lou09 ; Sen10 ; Puj13 ; Har13 . Moreover, among these three considered $JQ_{3}$ models, one is shown to have a first order phase transition and the other two are argued to possess second order phase transitions Lou09 ; Sen10 ; Puj13 ; Har13 . As a result, studying these three systems may reveal discrepancies of the features of QCR that result from the nature of the phase transitions.

In our calculations, the associated universal quantities of QCR, namely $S(\pi,\pi)/\left(\chi_{s}T\right)$, $\chi_{u}c^{2}/T$ and $\rho_{s}L/c$ are obtained. Here $S(\pi,\pi)$, $\chi_{s}$, $T$, $\chi_{u}$, $c$, $\rho_{s}$, and $L$ are the staggered structure factor, the staggered susceptibility, the temperature, the uniform susceptibility, the spinwave velocity, the spin stiffness, and the linear system size, respectively. Remarkably, by simulating large systems ($448^{2}$ spins for the square lattice and more than $300^{2}$ spins for the honeycomb lattice), we find that there is a substantial difference in the temperature dependence of these considered universal quantities between models with first and second order phase transitions. Specifically, the features of QCR only show up for the two $JQ_{3}$ models which are shown to possess second order phase transitions in the literature. In addition, the $S(\pi,\pi)/\left(\chi_{s}T\right)$ data of both the models having second order phase transitions reach the same plateau when regarded as a function of the inverse temperature $\beta$. A similar scenario occurs for $\chi_{u}c^{2}/T$ as well.

Remarkably, by considering the $S(\pi,\pi)/\left(\chi_{s}T\right)$ data of both the models having the characteristics of QCR as a function of $\beta c$, a high quality data collapse shows up. The quantity $\chi_{u}c^{2}/T$ also demonstrates the same behavior when treated as a function of $\beta c$. The outcome associated with $\rho_{s}L/c$ for the model on the honeycomb lattice requires a careful interpretation and can be satisfactory accounted for by taking into account a logarithmic correction to the corresponding theoretical predictions.

Although one cannot completely rule out the possibility of weakly first order phase transitions, the obtained results shown here provide evidence to support the hypothesis that the targeted phase transitions, from the Néel to the valence bond solid states, are likely continuous for two of the three studied $JQ_{3}$ models.

Finally, we also make a comparison between the numerical results obtained here and the related theoretical predictions in Refs. Chu94 and Kau08 , and have concluded that the leading order large $N$-expansion calculations carried out in Refs. Chu94 and Kau08 are not sufficient to match the outcomes determined in our study.

The findings presented in our investigation are not only interesting in themselves, but also can be adopted as useful criterions to distinguish first order phase transitions from the second order ones associated with DQC.

The rest of this paper is organized as follows. In Sect. II the studied $JQ_{3}$ models and the relevant observables of QCR are introduced. We then demonstrate our numerical results, particularly the mentioned dramatical difference of the temperature dependence of the calculated universal quantities are demonstrated in Sec. III. Finally, in Sec. IV we present discussions and conclusions.

The Hamiltonians of the three investigated 2D $SU(2)$ $JQ_{3}$ models on the square and the honeycomb lattices have the same expression which is given by

where in Eq. (1) $J$ (which is set to be 1 here) and $Q$ are the couplings for the two-spin and the six-spin interactions, respectively, $\vec{S}_{i}$ is the spin-1/2 operator at site $i$, and $P_{ij}$ is the singlet pair projection operator between nearest neighbor sites $i$ and $j$. Figure 1 contains the cartoon representations of the studied models. In the following, the top, the middle, and the bottom models of fig. 1 will be named the ladder, the staggered, and the honeycomb $JQ_{3}$ models respectively, if no confusion arises. We would also like to emphasize the fact that although the boundary condition (BC) for the honeycomb lattice considered here is different from that of Ref. Puj13 , bulk properties such as the critical point should be independent of the implemented BC. Hence, whenever $(Q/J)_{c}$ for the $JQ_{3}$ model on the honeycomb lattice is needed, the one determined in Ref. Puj13 will be used.

To study the quantum critical regime associated with the investigated $JQ_{3}$ models, particularly to calculate the corresponding universal quantities, several relevant observables, such as the staggered structure factor $S(\pi,\pi)$, the staggered susceptibility $\chi_{s}$, the uniform susceptibility $\chi_{u}$, the spinwave velocity $c$, the spin stiffness $\rho_{s}$, and the temporal and spatial winding numbers squared ($\langle W_{t}^{2}\rangle$ and $\langle W_{i}^{2}\rangle$ for $i=1,2$) are measured in our calculations. The formal expressions of these observables (on a $L_{1}$ by $L_{2}$ lattice) are as follows

where $\beta$ and $L_{i}$ for $i=1,2$ are the inverse temperature and the linear box size (in the $i$-spatial direction) used in the simulations, respectively, and $S_{i}^{z}$ is the third-component of the spin-half operator $\vec{S}_{i}$ at site $i$.

To calculate the desired physical observables, we have carried out large-scale quantum Monte Carlo calculations (QMC) using the stochastic series expansion (SSE) algorithm with very efficient operator-loop update San99 ; San101 . In addition, the simulations are performed at (close to) the associated critical points established in the literature. Specifically, we conduct the simulations at $(Q/J)_{c}=1.500$, 1.1933 and 1.1936 for the ladder, the staggered, and the honeycomb $JQ_{3}$ models, respectively. Finally, outcomes of up to $448^{2}$ (over $300^{2}$) spins on the square (honeycomb) lattice are obtained.

The calculated numerical results regarding the universal quantities introduced previously will be described in detail in the following. Particularly, in the analyses we assume that the associated dynamic critical exponents $z$ are all given by $z=1$ Sen040 ; Sen041 .

Using the first principle quantum Monte Carlo simulations, we investigate the QCR of three 2D $SU(2)$ $JQ_{3}$ models on both the square and the honeycomb lattices. Particularly, three universal quantities associated with QCR, namely $S(\pi,\pi)/\left(\chi_{s}T\right)$, $\chi_{u}c^{2}/T$ and $\rho_{s}L/c$ are obtained and their $\beta$ ($\beta c$) dependence is studied.

Among the three studied models, the ladder and the honeycomb $JQ_{3}$ models are shown to possess second order phase transitions from the Néel phase to the VBS phase Lou09 ; Puj13 ; Har13 , and for the staggered $JQ_{3}$ model the associated transition is first order Sen10 . Our investigation related to the QCR leads to results consistent with these scenarios.

Specifically, while for the ladder and the honeycomb $JQ_{3}$ models, the quantity $S\left(\pi,\pi\right)/\left(\chi_{s}T\right)$ ($\chi_{u}c^{2}/T$) of both models reach the same plateau value and a good scaling emerges when considered as a function of $\beta c$, no such behavior is found for the staggered $JQ_{3}$ models. Moreover, $\rho_{s}L/c$ obtains a constant outcome in the $LT\rightarrow\infty$ limit for the ladder model. Similar situation applies to the honeycomb model when a logarithmic correction to the observable $\rho_{s}L/c$ is taken into account. For the staggered model both the quantities $\chi_{u}c^{2}/T$ and $\rho_{s}L/c$ grow with $L$ or $\beta$. This is in agreement with the fact that the targeted phase transition of the staggered model is first order.

It is surprising to observe that to obtain a good scaling, the universal quantity associated with $\rho_{s}$ on the honeycomb lattice requires a logarithmic correction. This is not the case for the same observable on the square lattice. It will be interesting to understand this from a theoretical perspective.

Apart from the three universal quantities considered here, one frequently studied observable associated with QCR is the Wilson ratio $W=\chi_{u}T/C_{V}$. Here $C_{V}$ is the specific heat and can be calculated directly or through the internal energy density $E$. However, due to the subtlety of both approaches for the determination of $C_{V}$ shown in Refs. Sen15 ; Pen20 , here we do not attempt to carry out a high precision calculation of the quantity $W$. Nevertheless, the investigation conducted here provide evidence to support the scenario that for the ladder and the honeycomb $JQ_{3}$ models, the associated quantum phase transitions from the Néel phase to the VBS phase are likely to be continuous.

The results presented here are not only interesting in themselves, but also can be used as criterions to distinguish second order phase transitions from first order ones for exotic phase transitions such as the DQC studied here.

This study is partially supported by the Ministry of Science and Technology of Taiwan.