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arXiv:2009.08379 · Physical review. E

Lifshitz phase transitions in a one-dimensional Gamma model

Zi-An Liu, Tian-Cheng Yi, Jin-Hua Sun, Yuli Dong, Wen‐Long You

Published Sep 17, 2020• 29 citations
arXivHTMLPDF

Abstract

In this paper, we study quantum phase transitions and magnetic properties of a one-dimensional spin-1/2 Gamma model, which describes the off-diagonal exchange interactions between edge-shared octahedra with strong spin-orbit couplings along the sawtooth chain. The competing exchange interactions between the nearest neighbors and the second neighbors stabilize the semimetallic ground state in terms of spinless fermions, and give rise to a rich phase diagram, which consists of three gapless phases. We find distinct phases are characterized by the number of Weyl nodes in the momentum space, and such changes in the topology of the Fermi surface without symmetry breaking produce a variety of Lifshitz transitions, in which the Weyl nodes situating at k=π change from type I to type II. A coexistence of type-I and type-II Weyl nodes is found in phase II. The information measures including concurrence, entanglement entropy, and relative entropy can effectively signal the second-order transitions. The results indicate that the Gamma model can act as an exactly solvable model to describe Lifshitz phase transitions in correlated electron systems.

DOI
10.1103/physreve.102.032127

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In this paper, we study quantum phase transitions and magnetic properties of a one-dimensional spin-1/2 Gamma model, which describes the off-diagonal exchange interactions between edge-shared octahedra with strong spin-orbit couplings along the sawtooth chain. The competing exchange interactions between the nearest neighbors and the second neighbors stabilize the semimetallic ground state in terms of spinless fermions, and give rise to a rich phase diagram, which consists of three gapless phases. We find distinct phases are characterized by the number of Weyl nodes in the momentum space, and such changes in the topology of the Fermi surface without symmetry breaking produce a variety of Lifshitz transitions, in which the Weyl nodes situating at k=π change from type I to type II. A coexistence of type-I and type-II Weyl nodes is found in phase II. The information measures including concurrence, entanglement entropy, and relative entropy can effectively signal the second-order transitions. The results indicate that the Gamma model can act as an exactly solvable model to describe Lifshitz phase transitions in correlated electron systems.
In Part I, the relationship between the topology of the Fermi surface and the entanglement entropy S is examined. Spinless fermionic systems on one and two dimensional lattices at fixed chemical potential are considered. The lattice is partitioned into sub-system of length L and environment, and the entanglement of the subsystem with the environment is calculated via the correlation matrix. S is plotted as a function of the next-nearest or next-next nearest neighbor hopping parameter, t. In 1 dimension, the entanglement entropy jumps at lifshitz transitions where the number of Fermi points changes. In 2 dimensions, a neck-collapsing transition is accompanied by a cusp in S, while the formation of electron or hole-like pockets coincides with a kink in the S as a function of the hopping parameter. The entanglement entropy as a function of subsystem length L is also examined. The leading order coefficient of the LlnL term in 2 dimensions was seen to agree well with the Widom conjecture. Of interest is the difference this coefficient and the coefficient of the term linear in L near the neck-collapsing point. The leading order term changes like |t-tc|1/2 whereas the first sub-leading term varies like |t-tc|1/3, where tc is the critical value of the hopping parameter at the transition. In Part II, we study the statistics of fractionalized excitations in a bosonic model which describes strongly interacting excitons in a N-band insulator. The elementary excitations of this system are strings, in a large N limit. A string is made of a series of bosons whose flavors are correlated such that the end points of a string carries a fractionalized flavor quantum number. When the tension of a string vanishes, the end points are deconfined. We determine the statistics of the fractionalized particles described by the end points of strings. We show that either bosons or Fermions can arise depending on the microscopic coupling constants. In the presence of the cubic interaction in the Hamiltonian as the only higher order interaction term, it was shown that bosons are emergent. In the presence of the quartic interaction with a positive coupling constant, it was revealed that the elementary excitations of the system possess Fermion statistics.
Abstract In this study, a spin-1/2 extended anisotropic XY chain has been introduced in which both time reversal and SU(2) symmetries are broken but Z 2 symmetry is preserved. Magnetic and topological phase diagrams in the parameter space have been drawn in the presence of transverse magnetic field. Entanglement measures like mutual information and quantum discord are also evaluated and it indicates that these transitions are second order in nature. Quantum phase transition is noted at zero magnetic field, as well as magnetic long range order is found to withstand magnetic field of any strength. Exact analytic results for spin-spin correlation functions have been obtained in terms of Jordan-Wigner fermionization. Existence of long range magnetic order has been investigated numerically by finding correlation functions as well as the Binder cumulant in the ground state. Dispersion relation, ground state energy, and energy gap are obtained analytically. In order to find the topologically nontrivial phase, sign of Pfaffian invariant and value of winding number have been evaluated. Both magnetic and topological phases are robust against the magnetic field and found to move coercively in the parameter space with the variation of its strength. Long range orders along two orthogonal directions and two different topological phases are found and their one-to-one correspondence has been found. Finally casting the spinless fermions onto Majorana fermions, properties of zero energy edge states are studied. Three different kinds of Majorana pairings are noted. In the trivial phase, next-nearest-neighbor Majorana pairing is found, whereas two different types of nearest-neighbor Majorana pairings are identified in the topological superconducting phase.
Several exactly solvable quantum spin models with incommensurate magnetic structures, which manifest quantum phase transitions between incommensurate and commensurate phases, are proposed. The ground-state and thermodynamic characteristics of these quantum spin antiferromagnetic models with the Dzyaloshinsky-Moriya antisymmetric exchange interaction and spin-frustrating next-nearest-neighbor, three-spin, and four-spin (ring exchange) interactions are exactly studied. We investigated the ground state phases of the considered models, as function of the magnetic anisotropy, Dzyaloshinsky-Moriya interaction, next-nearest-neighbor coupling constants, and external magnetic field. Several quantum phase transitions between magnetically incommensurate phases and between incommensurate and ferromagnetic phases are predicted, and the behavior of thermodynamic characteristics of those models at phase transitions is studied. We have shown that for models with chiral three-spin interactions the Dzyaloshinsky-Moriya coupling yields the onset of the weak ferromagnetism, similar to three-dimensional ordered antiferromagnets, but not present in other quantum spin chains with the antisymmetric exchange. Considered models appeared to be important for several quasi-one-dimensional spin systems, studied in recent experiments.
We study the Kitaev-Heisenberg-$\mathrm{\ensuremath{\Gamma}}$ model with antiferromagnetic Kitaev exchanges in the strong anisotropic (toric code) limit to understand the phases and the intervening phase transitions between the gapped Z2{Z}_{2}Z2​ quantum spin liquid and the spin-ordered (in the Heisenberg limit) as well as paramagnetic phases (in the pseudodipolar, $\mathrm{\ensuremath{\Gamma}}$, limit). We find that the paramagnetic phase obtained in the large $\mathrm{\ensuremath{\Gamma}}$ limit has no topological entanglement entropy and is proximate to a gapless critical point of a system described by equal superposition of differently oriented stacked one-dimensional ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry protected topological phases. Using a combination of exact diagonalization calculations and field theoretic analysis we map out the phases and phase transitions to reveal the complete phase diagram as a function of the Heisenberg, the Kitaev, and the pseudodipolar interactions. Our work shows a rich plethora of unconventional phases and phase transitions and provides a comprehensive understanding of the physics of anisotropic Kitaev-Heisenberg-$\mathrm{\ensuremath{\Gamma}}$ systems along with our recent paper [Phys. Rev. B 102, 235124 (2020)] where the ferromagnetic Kitaev exchange was studied.
We consider a symmetric spin-1/2 Ising-XXZ double sawtooth spin ladder obtained from distorting a spin chain, with the XXZ interaction between the interstitial Heisenberg dimers (which are connected to the spins based on the legs via an Ising-type interaction), the Ising coupling between nearest-neighbor spins of the legs and rungs spins, respectively, and additional cyclic four-spin exchange (ring exchange) in the square plaquette of each block. The presented analysis supplemented by results of the exact solution of the model with infinite periodic boundary implies a rich ground state phase diagram. As well as the quantum phase transitions, the characteristics of some of the thermodynamic parameters such as heat capacity, magnetization and magnetic susceptibility are investigated. We prove here that among the considered thermodynamic and thermal parameters, solely heat capacity is sensitive versus the changes of the cyclic four-spin exchange interaction. By using the heat capacity function, we obtain a singularity relation between the cyclic four-spin exchange interaction and the exchange coupling between pair spins on each rung of the spin ladder. All thermal and thermodynamic quantities under consideration should be investigated by regarding those points which satisfy the singularity relation. The thermal entanglement within the Heisenberg spin dimers is investigated by using the concurrence, which is calculated from a relevant reduced density operator in the thermodynamic limit.
Abstract We perform a comprehensive analytical study of the exotic quantum phases and phase transitions emerging from the cluster-Ising model with off-diagonal Gamma interactions. Specifically, we map out the ground-state phase diagram by analyzing both local and nonlocal order parameters, together with the energy spectra. The results reveal one pair of gapped phases—namely antiferromagnetic (AFM) long-range ordered phase and symmetry-protected topological (SPT) phase—as well as two distinct gapless spiral phases induced by the off-diagonal interactions. Remarkably, the phase diagram contains four distinct transition lines. Two of them, separating different gapped or gapless phases, are governed by the Ising and three-copy Ising conformal field theories, respectively. In contrast, the remaining two transition lines—between the gapless spiral and gapped phases—belong to a non-conformal Lifshitz criticality with dynamical critical exponent z=2z = 2z=2 z = 2 . More importantly, the intersection of these four transition lines gives rise to a new Lifshitz multicritical point exhibiting emergent conformal symmetry, marking a fundamental departure from all previously known non-conformal Lifshitz points. This work provides a valuable reference for future investigations of exotic gapless phases and their transitions in exactly solvable many-body systems.