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arXiv:2206.15094 · Physical review. B./Physical review. B

Two-stage melting of an intercomponent Potts long-range order in two dimensions

Feng-Feng Song, Guang-Ming Zhang

Published Apr 14, 2023• 1 citation
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Abstract

The interplay of topology and competing interactions can induce enriched phases and phase transitions at finite temperatures. We consider a weakly coupled two-dimensional hexatic-nematic XY model with a relative ${Z}_{3}$ Potts degrees of freedom, and apply the matrix product state method to solve this model rigorously. Since the partition function is expressed as a product of two-legged one-dimensional transfer matrix operator, an entanglement entropy of the eigenstate corresponding to the maximal eigenvalue of this transfer operator can be used as a stringent criterion to determine various phase transitions precisely. At low temperatures, the intercomponent ${Z}_{3}$ Potts long-range order (LRO) exists, indicating that the hexatic and nematic fields are locked together and their respective vortices exhibit quasi-LRO. In the hexatic regime, below the BKT transition of the hexatic vortices, the intercomponent ${Z}_{3}$ Potts LRO appears, accompanying the binding of nematic vortices. In the nematic regime, however, the intercomponent ${Z}_{3}$ Potts LRO undergoes a two-stage melting process. An intermediate Potts liquid phase emerges between the Potts ordered and disordered phases, characterized by an algebraic correlation with formation of charge-neutral pairs of both hexatic and nematic vortices. These two-stage phase transitions are associated with the proliferation of the domain walls and vortices of the relative ${Z}_{3}$ Potts variable, respectively. Our results thus provide a prototype example of two-stage melting of a two-dimensional LRO driven by multiple topological defects.

DOI
10.1103/physrevb.107.165129

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The interplay of topology and competing interactions can induce enriched phases and phase transitions at finite temperatures. We consider a weakly coupled two-dimensional hexatic-nematic XY model with a relative Z3{Z}_{3}Z3​ Potts degrees of freedom, and apply the matrix product state method to solve this model rigorously. Since the partition function is expressed as a product of two-legged one-dimensional transfer matrix operator, an entanglement entropy of the eigenstate corresponding to the maximal eigenvalue of this transfer operator can be used as a stringent criterion to determine various phase transitions precisely. At low temperatures, the intercomponent Z3{Z}_{3}Z3​ Potts long-range order (LRO) exists, indicating that the hexatic and nematic fields are locked together and their respective vortices exhibit quasi-LRO. In the hexatic regime, below the BKT transition of the hexatic vortices, the intercomponent Z3{Z}_{3}Z3​ Potts LRO appears, accompanying the binding of nematic vortices. In the nematic regime, however, the intercomponent Z3{Z}_{3}Z3​ Potts LRO undergoes a two-stage melting process. An intermediate Potts liquid phase emerges between the Potts ordered and disordered phases, characterized by an algebraic correlation with formation of charge-neutral pairs of both hexatic and nematic vortices. These two-stage phase transitions are associated with the proliferation of the domain walls and vortices of the relative Z3{Z}_{3}Z3​ Potts variable, respectively. Our results thus provide a prototype example of two-stage melting of a two-dimensional LRO driven by multiple topological defects.
Critical properties of the two-dimensional XY model involving solely nematic-like terms of the second and third orders are investigated by spin-wave analysis and Monte Carlo simulation. It is found that, even though neither of the nematic-like terms alone can induce magnetic ordering, their coexistence and competition leads to an extended phase of the magnetic quasi-long-range-order phase, wedged between the two nematic-like phases induced by the respective couplings. Thus, except for the multicritical point, at which all the phases meet, for any finite value of the coupling parameters ratio there are two phase transition: one from the paramagnetic phase to one of the two nematic-like phases followed by another one at lower temperatures to the magnetic phase. The finite-size scaling analysis indicates that the phase transitions between the magnetic and nematic-like phases belong to the Ising and three-state Potts universality classes. Inside the competition-induced algebraic magnetic phase, the spin-pair correlation function is found to decay even much more slowly than in the standard XY model with purely magnetic interactions. Such a magnetic phase is characterized by an extremely low vortex-antivortex pair density attaining a minimum close to the point at which the two couplings are of about equal strength.
It is known that the loss of phase coherence of Cooper pairs in two-dimensional superconductivity corresponds to the unbinding of vortex-antivortex pairs with the quasi-long-range order in the order-parameter phase field, described by the Berezinskii-Kosterlizt-Thouless (BKT) transition of a 2D XY model. Here we show that the second-order Josephson coupling can induce an exotic superconducting phase in a bilayer system. By using tensor-network methods, the partition function of the 2D classical model is expressed as a product of 1D quantum transfer operator, whose eigenequation can be solved by an algorithm of matrix product states rigorously. From the singularity shown by the entanglement entropy of the 1D quantum analog, various phase transitions can be accurately determined. Below the BKT phase transition, an interlayer Ising long-range order is established at T_{Ising}, and the phase coherence of both intralayers and interlayers is locked together. For two identical layers, the Ising transition coincides with the BKT transition at a multicritical point. For two inequivalent layers, however, there emerges an intermediate quasi-long-range order phase (T_{Ising}<T<T_{BKT}), where the vortex-antivortex bindings occur in the layer with the larger intralayer coupling, but only half-vortex pairs with topological strings exist in the other layer, corresponding to the phase coherence of pairs of Cooper pairs. So our study provides a promising way to realize the charge-4e superconductivity in a bilayer system.
We study a bilayer system of coupled two-dimensional XY models by using tensor-network methods, showing that the inter-layer coupling can significantly alter the uncoupled topological Berezinskii-Kosterlizt-Thouless (BKT) transition. In the tensor-network representation, the partition function is mapped into a product of one-dimensional quantum transfer operator, whose eigen-equation can be accurately solved by an algorithm of matrix product states. The entanglement entropy of this one-dimensional quantum correspondence exhibits singularity, which can be used to determine various phase transitions. In the low temperature phase, an inter-layer Ising long-range order is developed,accompanying with vortex-antivortex bindings in both intra-layers and inter-layers. For two identical coupled layers, the Ising transition coincides with the BKT transition at a multi-critical point. For two inequivalent coupled layers, however, there emerges an intermediate quasi-long-range ordered phase, where vortex-antivortex bindings occur only in the layer with the larger intra-layer coupling but half-vortex pairs associated with topological strings emerge in the other layer. These results are supported by the correlation functions of the XY spins and nematic spins, and can be used to understand the novel properties in bilayer systems of superfluids, superconductors and quantum Hall systems.
The Potts model plays an essential role in classical statistical mechanics, illustrating many fundamental phenomena. One example is the existence of partially long-range-ordered states, in which some degrees of freedom remain disordered. This situation may arise from frustration of the interactions, but also from an irregular but unfrustrated lattice structure. We study partial long-range order in a range of antiferromagnetic qqq-state Potts models on different two-dimensional lattices and for all relevant values of qqq. We exploit the power of tensor-based numerical methods to evaluate the partition function of these models and hence to extract the key thermodynamic properties (entropy, specific heat, magnetization, and susceptibility) giving deep insight into the phase transitions and ordered states of each system. Our calculations reveal a range of phenomena related to partial ordering, including different types of entropy-driven phase transition, the role of lattice irregularity, very large values of the critical qc{q}_{c}qc​, and double phase transitions.
We study the quantum melting of stripe phases in models with competing short-range and long-range interactions decaying with distance as $1/{r}^{\ensuremath{\sigma}}$ in two space dimensions. At zero temperature we find a two step disordering of the stripe phases with the growth of quantum fluctuations. A quantum critical point separating a phase with long-range positional order from a phase with long-range orientational order is found when $\ensuremath{\sigma}\ensuremath{\le}4/3$, which includes the Coulomb interaction case $\ensuremath{\sigma}=1$. For $\ensuremath{\sigma}>4/3$ the transition is first order, which includes the dipolar case $\ensuremath{\sigma}=3$. Another quantum critical point separates the orientationally ordered (nematic) phase from a quantum disordered phase for any value of $\ensuremath{\sigma}$. Critical exponents as a function of $\ensuremath{\sigma}$ are computed at one loop order in an $\ensuremath{\epsilon}$ expansion and, whenever available, compared with known results. For finite temperatures it is found that for $\ensuremath{\sigma}\ensuremath{\ge}2$ orientational order decays algebraically with distance until a critical Kosterlitz-Thouless line. Nevertheless, for $\ensuremath{\sigma}<2$ it is found that long-range orientational order can exist at finite temperatures until a critical line which terminates at the quantum critical point at T=0T=0T=0. The temperature dependence of the critical line near the quantum critical point is determined as a function of $\ensuremath{\sigma}$.
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