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

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 new 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 eigen-equation 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 $1/3$ fractional vortex-antivortex pairs determined at $T_{c}\simeq 0.075J_{1}$ 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.Comment: 14 pages, 13 figures, submitted version for publicatio

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

Interplay of topology and competing interactions can induce new 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 inter-component $Z_3$ 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 inter-component $Z_3$ Potts LRO appears, accompanying with the binding of nematic vortices. In the nematic regime, however, the inter-component $Z_3$ 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 long-range order, driven by multiple topological defects.Comment: 18 pages, 13 figures. The title is slightly modified, and the supplementary materials are include

Poly[tetra­aquadi-μ4-oxalato-lutetium(III)potassium]

In the title compound, [KLu(C2O4)2(H2O)4]n, the LuIII ion lies on a site of symmetry in a dodeca­hedron defined by eight O atoms from four oxalate ligands. The K atom lies on another site of the same symmetry and is coordinated by four oxalate O atoms and four O water atoms. The mid-point of the C—C bond of the oxalate group lies on an inversion center. In the packing structure, each oxalate ligand links two Lu(III) and two K atoms, forming a three-dimensional open framework with channels running along [001]. Inter­molecular O—H⋯O hydrogen bonds occur