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arXiv:2304.11944 · Physical review. B./Physical review. B

Tensor network approach to the fully frustrated XY model on a kagome lattice with a fractional vortex-antivortex pairing transition

Feng-Feng Song, Guang-Ming Zhang

Published Jul 20, 2023• 12 citations
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Abstract

We have developed a tensor network approach to the two-dimensional fully frustrated classical XY spin model on the kagome lattice, and clarified the nature of the possible phase transitions of various topological excitations. We find that the standard tensor network representation for the partition function does not work due to the strong frustrations in the low-temperature limit. To avoid the direct truncation of the Boltzmann weight, based on the duality transformation, we introduce a different representation to build the tensor network with local tensors lying on the centers of the elementary triangles of the kagome lattice. Then the partition function is expressed as a product of one-dimensional transfer matrix operators, whose eigenequation can be solved by the variational uniform matrix product state algorithm accurately. The singularity of the entanglement entropy for the one-dimensional quantum operator provides a stringent criterion for the possible phase transitions. Through a systematic numerical analysis of thermodynamic properties and correlation functions in the thermodynamic limit, we prove that the model exhibits a single Berezinskii-Kosterlitz-Thouless phase transition only, which is driven by the unbinding of $\frac{1}{3}$ fractional vortex-antivortex pairs determined at ${T}_{c}\ensuremath{\simeq}0.075{J}_{1}$ accurately. The absence of long-range order of chirality or quasi-long-range order of integer vortices has been verified in the whole finite-temperature range. Thus, the long-standing controversy about the phase transitions in this fully frustrated XY model on the kagome lattice is solved rigorously, which provides a plausible way to understand the charge-$6e$ superconducting phase observed experimentally in the two-dimensional kagome superconductors.

DOI
10.1103/physrevb.108.014424

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We have developed a tensor network approach to the two-dimensional fully frustrated classical XY spin model on the kagome lattice, and clarified the nature of the possible phase transitions of various topological excitations. We find that the standard tensor network representation for the partition function does not work due to the strong frustrations in the low-temperature limit. To avoid the direct truncation of the Boltzmann weight, based on the duality transformation, we introduce a different representation to build the tensor network with local tensors lying on the centers of the elementary triangles of the kagome lattice. Then the partition function is expressed as a product of one-dimensional transfer matrix operators, whose eigenequation can be solved by the variational uniform matrix product state algorithm accurately. The singularity of the entanglement entropy for the one-dimensional quantum operator provides a stringent criterion for the possible phase transitions. Through a systematic numerical analysis of thermodynamic properties and correlation functions in the thermodynamic limit, we prove that the model exhibits a single Berezinskii-Kosterlitz-Thouless phase transition only, which is driven by the unbinding of 13\frac{1}{3}31​ fractional vortex-antivortex pairs determined at Tc\ensuremath≃0.075J1{T}_{c}\ensuremath{\simeq}0.075{J}_{1}Tc​\ensuremath≃0.075J1​ accurately. The absence of long-range order of chirality or quasi-long-range order of integer vortices has been verified in the whole finite-temperature range. Thus, the long-standing controversy about the phase transitions in this fully frustrated XY model on the kagome lattice is solved rigorously, which provides a plausible way to understand the charge-6e6e6e superconducting phase observed experimentally in the two-dimensional kagome superconductors.
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.
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.
In tensor network representation, the partition function of a generalized two-dimensional XY spin model with topological integer and half-integer vortex excitations is mapped to a tensor product of one-dimensional quantum transfer operator, whose eigenequation can be solved by an algorithm of variational uniform matrix product states. Using the singularities of the entanglement entropy, we accurately determine the complete phase diagram of this model. Both the integer vortex-antivortex binding and half-integer vortex-antivortex binding phases are separated from the disordered phase by the usual Berezinskii-Kosterlitz-Thouless (BKT) transitions, while a continuous topological phase transition exists between two different vortex binding phases, exhibiting a logarithmic divergence of the specific heat and exponential divergence of the spin correlation length. A hybrid BKT and Ising universality class of topological phase transition is thus established. We further prove that three phase transition lines meet at a multicritical point from which a deconfinement crossover line extends into the disordered phase.
Frustration is a ubiquitous phenomenon in many-body physics that influences the nature of the system in a profound way with exotic emergent behavior. Despite its long research history, the analytical or numerical investigations on frustrated spin models remain a formidable challenge due to their extensive ground-state degeneracy. In this paper, we propose a unified tensor network theory to numerically solve the frustrated classical spin models on various two-dimensional (2D) lattice geometry with high efficiency. We show that the appropriate encoding of emergent degrees of freedom in each local tensor is of crucial importance in the construction of the infinite tensor network representation of the partition function. The frustrations are thus relieved through the effective interactions between emergent local degrees of freedom. Then the partition function is written as a product of a one-dimensional (1D) transfer operator, whose eigenequation can be solved by the standard algorithm of matrix product states rigorously, and various phase transitions can be accurately determined from the singularities of the entanglement entropy of the 1D quantum correspondence. We demonstrated the power of our general theory by numerically solving 2D fully frustrated XY spin models on the kagome, square, and triangular lattices, giving rise to a variety of thermal phase transitions from infinite-order Brezinskii-Kosterlitz-Thouless transitions, second-order transitions, to first-order phase transitions. Our approach holds the potential application to other types of frustrated classical systems like Heisenberg spin antiferromagnets.
The fundamental concepts of phases of matter and thermal phase transitions constitute the cornerstone of our understanding of the physical universe. The historical development of the phase transition theory from Landau’s spontaneous symmetry breaking paradigm to modern topological phase transition theories represents a major milestone in the evolution of numerous scientific disciplines. From the perspective of emergent philosophy, the interplay of topological excitations leads to enriched physical phenomena. One prominent prototype is the Berezinskii-Kosterlitz-Thouless (BKT) phase transition, where unbinding of integer vortices occurs in the absence of spontaneous breaking of continuous U(1) symmetry. Using the state-of-the-art tensor network methods, we express the partition function of the two-dimensional XY-related system in terms of a product of one-dimensional transfer operators. From the singularities of the entanglement entropy of the one-dimensional transfer operator, we accurately determine the complete phase diagram of the partition function. This method provides new insights into the emergent phenomenon driven by topological excitations, and sheds new light on future studies of 2D systems with continuous symmetries.
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)U(1)U(1) and Z2{Z}_{2}Z2​ symmetries. Our study reveals an emergent intermediate phase around ${J}_{2}/{J}_{1}\ensuremath{\sim}0.505$, which is characterized by a Z2{Z}_{2}Z2​ 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<J2/J1<0.5010.5<{J}_{2}/{J}_{1}<0.5010.5<J2​/J1​<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.