Two-dimensional $J_1$-$J_2$ clock model: Enhanced symmetries, emergent orders, and Landau-incompatible transitions

TL;DR

This work analyzes a 2D frustrated $q$-state clock model with competing $J_1$ and $J_2$ couplings on a square lattice for even $q>4$, revealing a rich phase diagram with emergent discrete spins. Using corner transfer matrix renormalization group and classical Monte Carlo simulations, the authors identify five phases for $J_2/J_1>1/2$, including a novel emergent $bZ_q^-$ phase, an XY-like QLRO region with emergent $U(1)$ symmetry, and Landau-incompatible deconfined transitions that host an emergent $O(2)$ symmetry and central charge $c=1$. A comprehensive effective field theory with dual fields explains the emergence of the $bZ_q^-$ spins via a relevant cosine operator and accounts for the DQC-like continuous transition between Landau-incompatible orders. The results advance understanding of nonstandard emergence and complex phase transitions in frustrated lattice systems and suggest experimental platforms such as Josephson-junction chains to observe these phenomena.

Abstract

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 $\mathbb{Z}_q$-broken ferromagnet, an intermediate XY-like critical quasi-long-range-ordered (QLRO) 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 $\mathbb{Z}_q\times\mathbb{Z}_2$ symmetry, two $\mathbb{Z}_2$-broken nematic phases (one with and one without QLRO), and an exotic stripe phase with emergent discrete $\mathbb{Z}_q$ spin degrees of freedom prohibited in the microscopic Hamiltonian. Remarkably, this seemingly forbidden $\mathbb{Z}_q$ order emerges via a relevant operator in the infrared long-wavelength limit, rather than from an irrelevant perturbation, highlighting a non-standard 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.

Figures (6)

• Figure 1: The phases of the system. (a)-(b) The phase diagram for $q=6$ through the lens of EE of the boundary MPS. For $J_2/J_1 < 1/2$, the system follows the standard 6-state clock model: low-$T$ FM ($\mathbb{Z}_6^+$ FM), intermediate QLRO, and high-$T$ PM phases. For $J_2/J_1 > 1/2$, five distinct phases emerge, including $\mathbb{Z}_6^+$ AFM $\oplus$$\mathbb{Z}_2$ NM at low $T$ and critical QLRO $\oplus$$\mathbb{Z}_2$ NM at intermediate $T$. Furthermore, a narrow non-critical $\mathbb{Z}_2$ NM phase appears at smaller $J_2/J_1 > 1/2$ (see (b)). Most notably, a third non-critical phase ($\mathbb{Z}_6^-$ AFM $\oplus$$\mathbb{Z}_2$ NM) emerges (the lobe in (a)), where effective $\mathbb{Z}_6$ angles shift from $2k\pi/6$ to $(2k+1)\pi/6$. Solid, dashed, and dotted lines indicate BKT, Ising, and first-order transitions, respectively. The solid black lines with red '$+$' symbols indicate the Landau-incompatible continuous transition, see text, with the black squares indicating its extent. The black circle in (a) marks the point beyond which the Ising transition and the $\mathbb{Z}_2$ NM phase disappear within our numerical accuracy, while the black circle in (b) denotes the location where the first-order transition splits into two BKT transitions. The EEs are computed via isotropic CTMRG on the dual lattice with $\chi = 128$. (c) Histograms of the effective clock angles obtained with CMC for a lattice of linear size $L=64$, at $\mathbb{Z}_6^+$ AFM $\oplus$$\mathbb{Z}_2$ NM ($T=0.14$) and $\mathbb{Z}_6^-$ AFM $\oplus$$\mathbb{Z}_2$ NM ($T=0.17$) phases with $J_2/J_1 = 0.52$. (d) Histograms of the local clock angles from a single CMC snapshot, corresponding to the same parameter points as in (c). (e) Observables for $J_2/J_1 = 0.52$ as functions of $T/J_1$, including EEs (in both original and dual lattices), and the order parameters $\braket{e^{i\theta}}$ and $\kappa$. Here, anisotropic CTMRG is used for both the models with $\chi = 128$. The EEs are also computed with $\chi = 96$ and $80$ that shows strong $\chi$ dependence in the critical QLRO $\oplus$$\mathbb{Z}_2$ NM phase. (f)-(h) Frequency distribution of the complex order parameter $\tilde{m}$ in the complex plane, obtained from $2.5 \times 10^6$ CMC snapshots for system sizes $L = 64$ and $384$. Results are shown for three representative points: (f) the $\mathbb{Z}_q^+$ AFM $\oplus$$\mathbb{Z}_2$ NM phase ($T/J_1 = 0.14$), (g) the Landau-incompatible continuous transition ($T/J_1 = 0.154295$), and (h) the $\mathbb{Z}_q^-$ AFM $\oplus$$\mathbb{Z}_2$ NM phase ($T/J_1 = 0.17$), with $J_2/J_1 = 0.52$ fixed. At the transition, the distribution becomes rotationally invariant, reflecting the emergent $O(2)$ symmetry. The apparent hexagonal pattern in (g) is a finite-size artifact that diminishes with increasing system size.
• Figure 2: Characterizing critical transitions. (a) Correlation length $\xi$ in the dual system for MPS bond dimension $\chi=640$ (red solid line) at $J_2/J_1=0.52$, as $T/J_1$ is varied across $\mathbb{Z}_q^-$ AFM $\oplus$$\mathbb{Z}_2$ NM, QLRO $\oplus$$\mathbb{Z}_2$ NM, $\mathbb{Z}_2$ NM, and PM phases. Dashed blue lines show $\xi$ for $\chi = 192, 256, 384, 512$. Expected critical scaling relations for BKT and Ising transitions are verified by fits near the transitions. (b) EE across the Landau-incompatible continuous transition between the $\mathbb{Z}_q^{\pm}$ AFM $\oplus$$\mathbb{Z}_2$ NM phases at $J_2/J_1=0.52$ from anisotropic CTMRG with $\chi = 48, 64, 80, 112$. (c) Finite-entanglement scaling of the EE peak for different bond dimensions $\chi \in [40:112]$ at the Landau-incompatible transition yields central charge $c=1$.
• Figure 3: Phase diagram of the effective field theory. The theory in Eq. \ref{['eq:S']} with $\gamma' > 0$ predicts a phase diagram comprising a disordered phase, a critical phase, and two Landau-incompatible spontaneously symmetry-broken (SSB$^\pm$) phases, each possessing $q$ distinct vacua. Horizontal solid lines indicate BKT transitions, while vertical solid and dashed lines correspond to second-order transitions (with continuously varying critical exponents) and first-order transitions, respectively.
• Figure 4: Phases of the system for $q > 6$. (a) The schematic phase diagram of the system for $q=8$ in the $(J_2/J_1, T/J_1)$-plane. All other details remain the same as in Fig. \ref{['fig:phase']}(a). The schematic diagram is constructed based on the EE pattern, panel (b), obtained from isotropic CTMRG for the dual system with MPS bond dimension $\chi=128$. (c) Histograms of the effective clock angles for $q = 8$ computed via CMC, at the $\mathbb{Z}_8^+$ AFM $\oplus$$\mathbb{Z}_2$ NM ($T=0.08$) and the $\mathbb{Z}_8^-$ AFM $\oplus$$\mathbb{Z}_2$ NM ($T=0.11$) phases $(J_2/J_1 = 0.52)$, for a system of size $L=64$. (d) Histograms of the local clock angles from a single CMC snapshot, corresponding to the same parameter points as in (c). (e) Real-space snapshots of clock states $k \in \{0,1,\dots,q-1\}$ for $q=8$ obtained from CMC for a system of linear size $L=64$. Snapshots are shown for the phases: $\mathbb{Z}_q^+$ AFM $\oplus$$\mathbb{Z}_2$ NM ($T/J_1 = 0.08$), $\mathbb{Z}_q^-$ AFM $\oplus$$\mathbb{Z}_2$ NM ($T/J_1 = 0.11$), QLRO $\oplus$$\mathbb{Z}_2$ NM ($T/J_1 = 0.20$), and PM ($T/J_1 = 0.39$). $J_2/J_1$ is fixed to $0.52$. For clarity of visualization, the AFM order in the two $\sqrt{2} \times \sqrt{2}$ sublattices has been removed by rotating the spins on odd sublattice sites by $\pi$ (see text). (f) The pattern of EE for $q=10$ obtained from isotropic CTMRG for the dual system with MPS bond dimension $\chi=128$. (g) Same as in panel (f), but for $q=12$. The arrows in panels (f) and (g) indicate the phase with emergent $\mathbb{Z}_q^-$-spins.
• Figure 5: Real-space snapshots of clock states for $q=6$. Snapshots are shown for the phases: $\mathbb{Z}_q^+$ AFM $\oplus$$\mathbb{Z}_2$ NM ($T/J_1 = 0.14$), $\mathbb{Z}_q^-$ AFM $\oplus$$\mathbb{Z}_2$ NM ($T/J_1 = 0.17$), QLRO $\oplus$$\mathbb{Z}_2$ NM ($T/J_1 = 0.23$), and PM ($T/J_1 = 0.3$). $J_2/J_1$ is fixed to $0.52$. All other details are same as in Fig. \ref{['fig:phase_8']}(e).