Six-state Clock Model Research Articles

Valley-polarized nematic order in twisted moiré systems: In-plane orbital magnetism and crossover from non-Fermi liquid to Fermi liquid

The interplay between strong correlations and non-trivial topology in twisted moir\'e systems can give rise to a rich landscape of ordered states that intertwine the spin, valley, and charge degrees of freedom. In this paper, we investigate the properties of a system that displays long-range valley-polarized nematic order. Besides breaking the threefold rotational symmetry of the triangular moir\'e superlattice, this type of order also breaks twofold rotational and time-reversal symmetries, which leads to interesting properties. First, we develop a phenomenological model to describe the onset of this ordered state in twisted moir\'e systems and to explore its signatures in their thermodynamic and electronic properties. Its main manifestation is that it triggers the emergence of in-plane orbital magnetic moments oriented along high-symmetry lattice directions. We also investigate the properties of the valley-polarized nematic state at zero temperature. Due to the existence of a dangerously irrelevant coupling $\lambda$ in the six-state clock model that describes the putative valley-polarized nematic quantum critical point, the ordered state displays a pseudo-Goldstone mode. Using a two-patch model, we compute the fermionic self-energy to show that down to very low energies, the Yukawa-like coupling between the pseudo-Goldstone mode and the electronic degrees of freedom promotes the emergence of non-Fermi liquid behavior. Below a crossover energy scale $\Omega^{*}\sim\lambda^{3/2}$, however, Fermi liquid behavior is recovered. Finally, we discuss the applicability of these results to other non-trivial nematic states, such as the spin-polarized nematic phase.

Molecular dipoles in designer honeycomb lattices

Recent advances in ultracold atoms in optical lattices and developments in surface science have allowed for the creation of artificial lattices as well as the control of many-body interactions. Such systems provide new settings to investigate interaction-driven instabilities and non-trivial topology. In this paper, we explore the interplay between molecular electric dipoles on a two-dimensional triangular lattice with fermions hopping on the dual decorated honeycomb lattice which hosts Dirac and flat band states. We show that short-range dipole-dipole interaction can lead to ordering into various stripe and vortex crystal ground states. We study these ordered states and their thermal transitions as a function of the interaction range using simulated annealing and Monte Carlo methods. For the special case of zero wave vector ferrodipolar order, incorporating dipole-electron interactions and integrating out the electrons leads to a six-state clock model for the dipole ordering. Finally, we discuss the impact of the various dipole orders on the electronic band structure and the local tunneling density of states. Our work may be relevant to studies of "molecular graphene" -- CO molecules arranged on the Cu(111) surface -- which have been explored using scanning tunneling spectroscopy, as well as ultracold molecule-fermion mixtures in optical lattices.

Monte Carlo study on domain wall dynamics of J1–J2 triangular spin system

Monte Carlo simulations are employed to study the model of a triangular Ising antiferromagnet with the next-nearest neighbor ferromagnetic interaction (TIAFF), which can be well mapped to a six-state clock model, and provide a new model to explore the excitation of a topological six-state domain vortex. The TIAFF model retains the symmetry of the lattice and provides more details of vortex excitations and micro dynamics. The crossover of the domain walls in three sub-lattices creates the topological six-state domain vortices. The attractive force between the vortex and the anti-vortex is proportional to the number of domain walls that connect them. The movement of the vortex in the lattice is along the domain walls to avoid creating new defects. Our simulation works reveal that vortices obeying Arrhenius law are difficult to excite at a low temperature. Therefore, the surviving vortices are generated through the Kibble–Zurek mechanism (KZM), which explains the formation of topological solitons in a nonequilibrium system. The existing vortices in the system go through a slow random walk and annihilate in the first-order kinetic process.

Finite-m scaling analysis of Berezinskii-Kosterlitz-Thouless phase transitions and entanglement spectrum for the six-state clock model.

We investigate the Berezinskii-Kosterlitz-Thouless transitions for the square-lattice six-state clock model with the corner-transfer matrix renormalization group (CTMRG). Scaling analyses for effective correlation length, magnetization, and entanglement entropy with respect to the cutoff dimension m at the fixed point of the CTMRG provide transition temperatures consistent with a variety of recent numerical studies. We also reveal that the fixed-point spectrum of the corner-transfer matrix in the critical intermediate phase of the six-state clock model is characterized by the scaling dimension consistent with the c=1 boundary conformal field theory associated with the effective Z_{6} dual sine-Gordon model.

Entanglement-Entropy Study of Phase Transitions in Six-State Clock Model

The Berezinskii-Kosterlitz-Thouless (BKT) transitions of the six-state clock model on the square lattice are investigated by means of the corner-transfer matrix renormalization group method. A classical analog of the entanglement entropy $S( L, T )$ is calculated for $L \times L$ square system up to $L = 129$, as a function of temperature $T$. The entropy exhibits a peak at $T = T^*_{~}( L )$, where the temperature depends on both $L$ and the boundary conditions. Applying the finite-size scaling to $T^*_{~}( L )$ and assuming presence of the BKT transitions, the two distinct phase-transition temperatures are estimated to be $T_1^{~} = 0.70$ and $T_2^{~} = 0.88$. The results are in agreement with earlier studies. It should be noted that no thermodynamic functions have been used in this study.

Phase diagram of six-state clock model on rewired square lattices

The six-state clock model (SSCM) on rewired square lattice is studied using Monte Carlo simulation with Wang-Landau algorithm. This is a discrete counterpart of the well-known XY model, the native host of a unique topological phase transition called Kosterlitz-Tholess (KT) transition. The model has two KT transitions, i.e., at temperature T1 and T2, where T1 < T2. The first transition separates the lower temperature magnetic order and the quasi-long range order (QLRO) also known as KT phase; while the second transition separates the QLRO and the higher temperature paramagnetic phase. It has been established that the presence of KT phase is affected by the presence of randomness in the form of site and bond dilution. This intermediate phase is totally ruled out if bonds or sites of the lattice are no longer percolated. Here different type of randomness is probed, namely the rewired lattices, obtained by randomly adding one extra bond to each lattice site, and connect the site to one of its next-nearest neighbors. As a results, the average number of neighbors C increases. The increase of C affects the existence of KT phase. For each value of C, the KT temperatures, T1 and T2, were estimated from the plot of specific heats. Variation of KT temperatures for different values of C is observed, which is plotted with respect to each corresponding C to obtain the system phase diagram.

Universality class of a displacive structural phase transition in two dimensions

The displacive structural phase transition in a two-dimensional model solid due to Benassi and co-workers [PRL 106, 256102 (2011)] is analyzed using Monte Carlo simulations and finite-size scaling. The model is shown to be a member of the two-dimensional six-state clock model universality class. Consequently, the model features two phase transitions, implying the existence of three thermodynamically distinct phases, namely, a low-temperature phase with long-ranged order, an intermediate critical phase with power-law decay of correlations, and a high-temperature phase with short-ranged order.

Phase Transition of the q-State Clock Model: Duality and Tensor Renormalization**Supported by the National Natural Science Foundation of China under Grant No 11474331.

We investigate the critical behavior and the duality property of the ferromagnetic q-state clock model on the square lattice based on the tensor-network formalism. From the entanglement spectra of local tensors defined in the original and dual lattices, we obtain the exact self-dual points for the model with q ≤ 5 and approximate self-dual points for q ≥ 6. We calculate accurately the lower and upper critical temperatures for the six-state clock model from the fixed-point tensors determined using the higher-order tensor renormalization group method and compare with other numerical results.

Anomalous finite-size effect due to quasidegenerate phases in triangular antiferromagnets with long-range interactions and mapping to the generalized six-state clock model

The effect of long-range (LR) interactions on frustrated-spin models is an interesting problem, which provides rich ordering processes. We study the effect of LR interactions on triangular Ising antiferromagnets with the next-nearest-neighbor ferromagnetic interaction (TIAFF). In the thermodynamic limit, the LRTIAFF model should reproduce the corresponding mean-field results, in which successive phase transitions occur among various phases, i.e., the disordered paramagnetic phase, so-called partially disordered phase, three-sublattice ferrimagnetic phase, and two-sublattice ferrimagnetic phase. In the present paper we focus on the magnetic susceptibility at the transition point between the two-sublattice ferrimagnetic and the disordered paramagnetic phases at relatively large ferromagnetic interactions. In the mean-field analysis, the magnetic susceptibility shows no divergence at the transition point. In contrast, a divergencelike enhancement of the susceptibility is observed in Monte Carlo simulations in finite-size systems. We investigate the origin of this difference and find that it is attributed to a virtual degeneracy of the free energies of the partially disordered and 2-FR phases. We also exploit a generalized six-state clock model with an LR interaction, which is a more general system with ${Z}_{6}$ symmetry. We discuss the phase diagram of this model and find that it exhibits richer transition patterns and contains the physics of the LRTIAFF model.

Buckling Transitions and Clock Order of Two-Dimensional Coulomb Crystals

Crystals of repulsively interacting ions in planar traps form hexagonal lattices, which undergo a buckling instability towards a multi-layer structure as the transverse trap frequency is reduced. Numerical and experimental results indicate that the new structure is composed of three planes, whose separation increases continuously from zero. We study the effects of thermal and quantum fluctuations by mapping this structural instability to the six-state clock model. A prominent implication of this mapping is that at finite temperature, fluctuations split the buckling instability into two thermal transitions, accompanied by the appearance of an intermediate critical phase. This phase is characterized by quasi-long-range order in the spatial tripartite pattern. It is manifested by broadened Bragg peaks at new wave vectors, whose line-shape provides a direct measurement of the temperature dependent exponent $\eta(T)$ characteristic of the power-law correlations in the critical phase. A quantum phase transition is found at the largest value of the critical transverse frequency: here the critical intermediate phase shrinks to zero. Moreover, within the ordered phase, we predict a crossover from classical to quantum behavior, signifying the emergence of an additional characteristic scale for clock order. We discuss experimental realizations with trapped ions and polarized dipolar gases, and propose that within accessible technology, such experiments can provide a direct probe of the rich phase diagram of the quantum clock model, not easily observable in condensed matter analogues. Therefore, this works highlights the potential for ionic and dipolar systems to serve as simulators for complex models in statistical mechanics and condensed matter physics.

Emergent criticality and Friedan scaling in a two-dimensional frustrated Heisenberg antiferromagnet

We study a two-dimensional frustrated Heisenberg antiferromagnet on the windmill lattice consisting of triangular and dual honeycomb lattice sites. In the classical ground state the spins on different sublattices are decoupled, but quantum and thermal fluctuations drive the system into a coplanar state via an "order from disorder" mechanism. We obtain the finite temperature phase diagram using renormalization group approaches. In the coplanar regime, the relative U$(1)$ phase between the spins on the two sublattices decouples from the remaining degrees of freedom, and is described by a six-state clock model with an emergent critical phase. At lower temperatures the system enters a $\mathbb{Z}_6$ broken phase with long-range phase correlations. We derive these results by two distinct renormalization group approaches to two-dimensional magnetism: by Wilson-Polyakov scaling and by Friedan's geometric approach to nonlinear sigma models where the scaling of the spin-stiffnesses is governed by the Ricci flow of a 4D metric tensor.

Residual discrete symmetry of the five-state clock model

It is well known that the q-state clock model can exhibit a Kosterlitz-Thouless (KT) transition if q is equal to or greater than a certain threshold, which has been believed to be five. However, recent numerical studies indicate that helicity modulus does not vanish in the high-temperature phase of the five-state clock model as predicted by the KT scenario. By performing Monte Carlo calculations under the fluctuating twist boundary condition, we show that it is because the five-state clock model does not have the fully continuous U(1) symmetry even in the high-temperature phase while the six-state clock model does. We suggest that the upper transition of the five-state clock model is actually a weaker cousin of the KT transition so that it is q≥6 that exhibits the genuine KT behavior.

Ferroelectric annular domains in hexagonal manganites

Experimental observations and theoretical simulations for the ferroelectric (FE) domains in multiferroic RMnO3 (R=Ho-Lu, Y, and Sc) demonstrate that annular domains can appear in the topological Kosterlitz-Thouless (KT) phase in coexistence with the vortices. Moreover, the annular domains often adopt the multilayered structures as indexed by the annular number. The double-layered FE domains, an energetically favored structure, are commonly found in RMnO3. Thermal annealing experiments directly show exchanges between large annular domains and cloverleaf vortices above the trimerization-structural transition temperature T-c. The temperature evolution of the FE domains has been analyzed by using the six-state clock model through Monte Carlo simulations. Based on careful in situ SEM observations, the dynamic properties of annular domains and interplay among domain walls have been briefly discussed. DOI: 10.1103/PhysRevB.87.094106

Phase transitions in a triangular Blume-Capel antiferromagnet

We study the critical behavior of a frustrated Blume-Capel (BC) antiferromagnet on a triangular lattice by Monte Carlo simulations. For a reduced single-ion anisotropy strength $\ensuremath{-}1.47\ensuremath{\lesssim}\ensuremath{\Delta}&lt;0$, we find two phase transitions. The low-temperature phase is characterized by the antiferromagnetic long-range ordering (LRO) on two sublattices with the third one remaining in a nonmagnetic state. At higher temperatures, there is a critical region of the Berezinskii-Kosterlitz-Thouless (BKT) type with a power-law decaying spin-correlation function. For $\ensuremath{-}1.5\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\lesssim}\ensuremath{-}1.47$, there is only one phase transition from the LRO to the paramagnetic region and the transition is of first order. The presence of the BKT phase in the current frustrated BC model is a new feature not observed in its nonfrustrated counterparts. The values of the decay exponent $\ensuremath{\eta}$ of the BKT phase corresponding to upper and lower temperatures appear to be consistent with the theoretical predictions for the six-state clock model.

Emergent magnetic monopoles in frustrated magnetic systems

Emergent magnetic monopoles in frustrated magnetic systems

Magnetic charge and ordering in kagome spin ice

We present a numerical study of magnetic ordering in spin ice on kagome, a two-dimensional lattice of corner-sharing triangles. The magnet has six ground states and the ordering occurs in two stages, as one might expect for a six-state clock model. In spin ice with short-range interactions up to second neighbours, there is an intermediate critical phase separated from the paramagnetic and ordered phases by Kosterlitz-Thouless (KT) transitions. In dipolar spin ice, the intermediate phase has long-range order of staggered magnetic charges. The high- and low-temperature phase transitions are of the Ising and 3-state Potts universality classes, respectively. Freeze-out of defects in the charge order produce a very large spin correlation length in the intermediate phase. As a result of that, the lower-temperature transition appears to be of the KT type.

Emergent Critical Phase and Ricci Flow in a 2D Frustrated Heisenberg Model

We introduce a two-dimensional frustrated Heisenberg antiferromagnet on interpenetrating honeycomb and triangular lattices. Classically the two sublattices decouple, and "order from disorder" drives them into a coplanar state. Applying Friedan's geometric approach to nonlinear sigma models, we obtain the scaling of the spin stiffnesses governed by the Ricci flow of a four-dimensional metric tensor. At low temperatures, the relative phase between the spins on the two sublattices is described by a six-state clock model with an emergent critical phase.

Critical behavior of two-dimensional spin systems under the random-bond six-state clock model

The critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the six-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean μ=1 and different values of variance, from σ2=0.1 to σ2=3.0. An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. The critical temperature Tc for both pure and disordered systems is determined from the critical exponent η(Tc)=1/4. The results showed that a small amount of disorder (small σ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type.

Direct Observation of the Proliferation of Ferroelectric Loop Domains and Vortex-Antivortex Pairs

We discovered stripe patterns of trimerization-ferroelectric domains in hexagonal REMnO(3) (RE=Ho,···,Lu) crystals (grown below ferroelectric transition temperatures (T(c)), reaching up to 1435 °C), in contrast with the vortex patterns in YMnO(3). These stripe patterns roughen with the appearance of numerous loop domains through thermal annealing just below T(c), but the stripe domain patterns turn to vortex-antivortex domain patterns through a freezing process when crystals cross T(c) even though the phase transition appears to not be Kosterlitz-Thouless-type. The experimental systematics are compared with the results of our six-state clock model simulation and also the Kibble-Zurek mechanism for trapped topological defects.

Kosterlitz-Thouless transition of magnetic dipoles on the two-dimensional plane

The universality class of a phase transition is often determined by factors like dimensionality and inherent symmetry. We study the magnetic dipole system in which the ground-state symmetry and the underlying lattice structure are coupled to each other in an intricate way. A two-dimensional (2D) square-lattice system of magnetic dipoles undergoes an order-disorder phase transition belonging to the 2D Ising universality class. According to Prakash and Henley [Phys. Rev. B 42, 6572 (1990)], this can be related to the fourfold-symmetric ground states, which suggests a similarity to the four-state clock model. Provided that this type of symmetry connection holds true, the magnetic dipoles on a honeycomb lattice, which possess sixfold-symmetric ground states, should exhibit a Kosterlitz-Thouless transition in accordance with the six-state clock model. This is verified through numerical simulations in the present investigation. However, it is pointed out that this symmetry argument does not always apply, which suggests that factors other than symmetry can be decisive for the universality class of the magnetic dipole system.