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arXiv:2412.18892 · Physical review. B./Physical review. B

Emergent intermediate phase in the J 1 − J 2 XY model from tensor network approaches

Feng-Feng Song, Hanggai Nuomin, Naoki Kawashima

Published Mar 20, 2025• 1 citation
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Abstract

We investigate the finite-temperature phase diagram of the classical ${J}_{1}\ensuremath{-}{J}_{2}$ XY model on a square lattice using a tensor network approach designed for frustrated spin systems. This model, characterized by competing nearest-neighbor and next-to-nearest-neighbor interactions, exhibits a complex interplay between $U(1)$ and ${Z}_{2}$ symmetries. Our study reveals an emergent intermediate phase around ${J}_{2}/{J}_{1}\ensuremath{\sim}0.505$, which is characterized by a ${Z}_{2}$ long-range stripe order without phase coherence in the XY spins. The intermediate phase features two well-separated phase transitions: a higher-temperature Ising transition and a lower-temperature Berezinskii-Kosterlitz-Thouless (BKT) transition. The relative separation between these transitions is significantly larger than previously reported, enabling a clearer investigation of their distinct thermodynamic properties. For $0.5<{J}_{2}/{J}_{1}<0.501$, two transitions merge into a single first-order phase transition, a phenomenon that cannot be explained solely by mapping to the Ising-XY model. As ${J}_{2}/{J}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$, the transition evolves continuously into the BKT universality class. These findings advance the understanding of the mechanisms driving phase transitions in frustrated spin systems, and they suggest potential experimental realizations in platforms such as ultracold atoms, Josephson junction arrays, and optical lattices.

DOI
10.1103/physrevb.111.104428

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We present a comprehensive study on the frustrated J_{1}-J_{2} classical q-state clock model with even q>4 on a two-dimensional square lattice, revealing a rich ensemble of phases driven by competing interactions. In the unfrustrated regime (J_{1}>2J_{2}), the model reproduces the standard clock model phenomenology: a low-temperature Z_{q}-broken ferromagnet, an intermediate XY-like critical quasi-long-range-ordered phase with emergent U(1) symmetry, and a high-temperature paramagnet. For J_{1}<2J_{2}, frustration stabilizes five distinct regimes: the disordered paramagnet, a stripe-ordered phase breaking Z_{q}×Z_{2} symmetry, two Z_{2}-broken nematic phases (one with and one without quasi-long-range order), and an exotic stripe phase with emergent discrete Z_{q} spin degrees of freedom prohibited in the microscopic Hamiltonian. Remarkably, this seemingly forbidden Z_{q} order emerges via a relevant operator in the infrared long-wavelength limit, rather than from an irrelevant perturbation, highlighting a nonstandard route to emergence. Using large-scale corner transfer matrix renormalization group calculations, complemented by classical Monte Carlo simulations, we map the complete phase diagram and identify Berezinskii-Kosterlitz-Thouless, Ising, first-order, and unconventional Landau-incompatible transitions between different phases. Finally, we propose an effective field-theoretic framework that encompasses these emergent orders and their interwoven transitions.
By using extensive tensor network calculations, we map out the phase diagram of the frustrated J_{1}-J_{2} Ising model on the square lattice. In particular, we focus on the cases with controversy in the phase diagram, especially the stripe transition in the regime g=|J_{2}/J_{1}|>1/2 (J_{2}>0,J_{1}<0). While recent studies claimed that the phase transition is of first order when 1/2<g<g^{*} (with the smallest g^{*} being 0.67), our simulations suggest that if there is such a first-order region, it is smaller than those found in earlier studies by other methods. Combining with the analysis of critical properties, we provide evidence that the classical J_{1}-J_{2} model evolves continuously from two decoupled Ising models (g→∞ with central charge c=1) to a point belonging to the tricritical Ising universality class (with c=0.7) as g decreases to g^{*}≃0.54.
We consider the two-dimensional (2D) ${J}_{1}\ensuremath{-}{J}_{2}$ classical XY\mathrm{XY}XY model on a square lattice. In the frustrated phase corresponding to J2>J1/2,{J}_{2}>{J}_{1}/2,J2​>J1​/2, an Ising-like order parameter emerges by an ``order due to disorder'' effect. This leads to a discrete Z2{Z}_{2}Z2​ symmetry plus the U(1)U(1)U(1) global one. Using a powerful algorithm we show that the system undergoes two transitions at different but still very close temperatures, one of Kosterlitz-Thouless (KT) type and another one which does not belong to the expected Ising universality class. A new analysis of the KT transition has been developed in order to avoid the use of the nonuniversal helicity jump and to allow the computation of the exponents without a precise determination of the critical temperature. Moreover, our huge number of data enables us to exhibit the existence of large finite size effects explaining the dispersed results found in the literature concerning the more studied frustrated 2D, XY\mathrm{XY}XY models.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions J1<0{J}_{1}<0J1​<0 (nearest neighbor, ferromagnetic) and J2>0{J}_{2}>0J2​>0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4{Z}_{4}Z4​ symmetry and is stabilized at low temperatures for g=J2/∣J1∣>1/2g={J}_{2}/|{J}_{1}|>1/2g=J2​/∣J1​∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The J1{J}_{1}J1​-J2{J}_{2}J2​ model has a line of very weak first-order phase transitions in the whole region 1/2<g<g∗1/2<g<{g}^{*}1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from g=g∗g={g}^{*}g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point g∗{g}^{*}g∗ in the J1{J}_{1}J1​-J2{J}_{2}J2​ model. The continuous transitions near g∗{g}^{*}g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of g>1/2g>1/2g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.
Meandering stripes in the frustrated J1J_1J1​-J2J_2J2​ Ising model on the honeycomb lattice
We study the frustrated J1J_1J1​-J2J_2J2​ Ising model on the honeycomb lattice with ferromagnetic nearest-neighbor couplings fixed at J1=1J_1=1J1​=1 and strong antiferromagnetic next-nearest-neighbor interactions, i.e., J2≤−1/4J_2 \leq -1/4J2​≤−1/4. Little is known for this range of J2J_2J2​, whereas for less negative values of J2J_2J2​ the system orders ferromagnetically at low temperatures and appears to remain in the Ising universality class. In previous work it was shown that the model has a largely degenerate ground state, and it was conjectured that there is some kind of phase transition. We introduce a complex-valued nematic order parameter, which can differentiate between the high-temperature paramagnetic phase and the observed partially-disordered stripe phase at lower temperatures. Configurations in this phase consist of stripes of spins parallel with respect to one lattice direction, which collectively meander along the remaining two, producing partially disordered ground states. The sharp peaks in the specific heat observed in earlier work only appear when using periodic boundary conditions and are absent for free boundaries. Additionally, we reveal a striking dependence of the behavior on the aspect ratio of the considered samples. Ultimately, even a careful finite-size scaling analysis for J2=−0.5J_2 = -0.5J2​=−0.5 and J2=−1J_2 = -1J2​=−1 is unable to clearly discern between a crossover without any singularities and some form of continuous transition, including the possibility of an infinite-order transition of the Berezinskii-Kosterlitz-Thouless (BKT) type. For J2=−1/4J_2=-1/4J2​=−1/4 we find that the system remains disordered at all temperatures and that it exhibits a finite ground-state entropy per site, for which our simulations provide the accurate asymptotic estimate S(T=0)/N=0.230 960 93(14)S(T=0)/N = 0.230\,960\,93(14)S(T=0)/N=0.23096093(14) in the thermodynamic limit N→∞N\rightarrow\inftyN→∞.
A general framework is proposed to solve the two-dimensional (2D) fully frustrated XY model for the Josephson junction arrays in a perpendicular magnetic field. The essential idea is to encode the ground-state local rules induced by frustrations in the local tensors of the partition function. The partition function is then expressed in terms of a product of a 1D transfer matrix operator, whose eigenequation can be solved by an algorithm of matrix product states rigorously. The singularity of the entanglement entropy for the 1D quantum analog provides a stringent criterion to distinguish various phase transitions without identifying a priori order parameter. Two very close phase transitions are determined at ${T}_{c1}\ensuremath{\approx}0.4459$ and ${T}_{c2}\ensuremath{\approx}0.4532$, respectively. The former corresponds to a Berezinskii-Kosterlitz-Thouless phase transition describing the phase coherence of XY spins, and the latter is an Ising-like continuous phase transition below which a chirality order with spontaneously broken Z2{Z}_{2}Z2​ symmetry is established.