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arXiv:1706.05057 · Physical review. B./Physical review. B

Finite-temperature phase transition to a Kitaev spin liquid phase on a hyperoctagon lattice: A large-scale quantum Monte Carlo study

Petr A. Mishchenko, Yasuyuki Kato, Yukitoshi Motome

Published Sep 15, 2017• 22 citations
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Abstract

The quantum spin liquid is an enigmatic quantum state in insulating magnets, in which conventional long-range order is suppressed by strong quantum fluctuations. Recently, an unconventional phase transition was reported between the low-temperature quantum spin liquid and the high-temperature paramagnet in the Kitaev model on a three-dimensional hyperhoneycomb lattice. Here, we show that a similar ``liquid-gas'' transition takes place in another three-dimensional lattice, the hyperoctagon lattice. We investigate the critical phenomena by adopting the Green-function based Monte Carlo technique with the kernel polynomial method, which enables systematic analysis of up to 2048 sites. The critical temperature is lower than that in the hyperhoneycomb case, reflecting the smaller flux gap. We also discuss the transition on the basis of an effective model in the anisotropic limit.

DOI
10.1103/physrevb.96.125124

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The quantum spin liquid is an enigmatic quantum state in insulating magnets, in which conventional long-range order is suppressed by strong quantum fluctuations. Recently, an unconventional phase transition was reported between the low-temperature quantum spin liquid and the high-temperature paramagnet in the Kitaev model on a three-dimensional hyperhoneycomb lattice. Here, we show that a similar ``liquid-gas'' transition takes place in another three-dimensional lattice, the hyperoctagon lattice. We investigate the critical phenomena by adopting the Green-function based Monte Carlo technique with the kernel polynomial method, which enables systematic analysis of up to 2048 sites. The critical temperature is lower than that in the hyperhoneycomb case, reflecting the smaller flux gap. We also discuss the transition on the basis of an effective model in the anisotropic limit.
The quantum spin liquid is an enigmatic entity that is often hard to characterize within the conventional framework of condensed matter physics. We here present theoretical and numerical evidence for the characterization of a quantum spin liquid phase extending from the exact ground state to a finite critical temperature. We investigate a three-dimensional variant of the Kitaev model on a hyperhoneycomb lattice in the limit of strong anisotropy; the model is mapped onto an effective Ising-type model, where elementary excitations consist of closed loops of flipped Ising-type variables on a diamond lattice. Analyzing this effective model by Monte Carlo simulation, we find a phase transition from the quantum spin liquid to a paramagnet at a finite critical temperature Tc{T}_{c}Tc​, accompanied by a divergent singularity of the specific heat. We also compute the magnetic properties in terms of the original quantum spins. We find that the magnetic susceptibility exhibits a broad hump above Tc{T}_{c}Tc​, while it obeys the Curie law at high temperature and approaches a nonzero Van Vleck--type constant at low temperature. Although the susceptibility changes continuously at Tc{T}_{c}Tc​, its temperature derivative shows a critical divergence at Tc{T}_{c}Tc​. We also clarify that the dynamical spin correlation function is momentum independent but shows quantized peaks corresponding to discretized excitations. Although the phase transition accompanies no apparent symmetry breaking in terms of the Ising-type variables or of the original quantum spins, we characterize it from a topological viewpoint. We find that, by defining the flux density for loops of the Ising-type variables, the transition can be interpreted as one occurring from the zero-flux quantum spin liquid to the nonzero-flux paramagnet; the latter has a Coulombic nature due to the local constraints. The role of global constraints on the Ising-type variables is examined in comparison with the results in the two-dimensional loop model. The correspondence of our model to the Ising model on a diamond lattice is also discussed. A possible relevance of our results to the recently discovered hyperhoneycomb compound $\ensuremath{\beta}$-Li2{\text{Li}}_{2}Li2​IrO3{\text{IrO}}_{3}IrO3​ is mentioned.
We study the 3D Kitaev and Kitaev-Heisenberg models, respectively, on the hyperhoneycomb and hyperoctagon lattices, both at zero and finite-temperature, in the thermodynamic limit. Our analysis relies on advanced tensor network (TN) simulations based on graph projected entangled-pair states (gPEPS). We map out the TN phase diagrams of the models and characterize their underlying gapped and gapless phases both at zero and finite temperature. In particular, we demonstrate how cooling down the hyperhoneycomb system from high temperature leads to fractionalization of spins to Majorana fermions and gauge fields that occurs in two separate temperature regimes, leaving their fingerprint on specific heat as a double-peak feature as well as on other quantities such as the thermal entropy, spin-spin correlations, and bond entropy. Using the Majorana representation of the Kitaev model, we further show that the low-temperature thermal transition to the Kitaev quantum spin liquid (QSL) phase is associated with the nontrivial Majorana band topology and the presence of Weyl nodes, which manifests itself via nonvanishing Chern number and finite thermal Hall conductivity. Beyond the pure Kitaev limit, we study the 3D Kitaev-Heisenberg (KH) model on the hyperoctagon lattice and extract the full phase diagram for different Heisenberg couplings. We further explore the thermodynamic properties of the magnetically-ordered regions in the KH model and show that, in contrast to the QSL phase, here the thermal phase transition follows the standard Landau symmetry-breaking theory.
The long-sought quantum spin liquid is a quantum-entangled magnetic state leading to the fractionalization of spin degrees of freedom. Quasiparticles emergent from the fractionalization affect not only the ground-state properties but also thermodynamic behavior in a peculiar manner. Here we investigate how the spin dynamics evolves from the high-temperature paramagnet to the quantum spin-liquid ground state, for the Kitaev spin model describing the fractionalization into itinerant matter fermions and localized Z2{Z}_{2}Z2​ fluxes. Beyond the previous studies [J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett. 117, 157203 (2016); J. Yoshitake, J. Nasu, Y. Kato, and Y. Motome, Phys. Rev. B 96, 024438 (2017)], in which the mean-field nature of the cluster dynamical mean-field theory prevented us from studying low-temperature properties, we develop a numerical technique by applying the continuous-time quantum Monte Carlo (CTQMC) method to statistical samples generated by the quantum Monte Carlo (QMC) method in a Majorana fermion representation. This QMC+CTQMC method is fully unbiased and enables us to investigate the low-temperature spin dynamics dominated by thermally excited Z2{Z}_{2}Z2​ fluxes, including the unconventional phase transition caused by Z2{Z}_{2}Z2​ flux loops in three dimensions, which was unreachable by the previous methods. We apply this technique to the Kitaev model in both two and three dimensions. Our results clearly distinguish two cases: while the dynamics changes smoothly through the crossover in the two-dimensional honeycomb case, it exhibits singular behaviors at the phase transition in the three-dimensional hyperhoneycomb case. We show that the low-temperature spin dynamics is a sensitive probe for thermally fluctuating Z2{Z}_{2}Z2​ fluxes that behave very differently between two and three dimensions.
A three-dimensional Kitaev model on a hyperhoneycomb lattice is investigated numerically at finite temperature. The Kitaev model is one of the solvable quantum spin models, where the ground state is given by gapped and gapless spin liquids, depending on the anisotropy of the interactions. This model can be rewritten as a free Majorana fermion system coupled with Z 2 variables. The density of states of Majorana fermions shows an excitation gap in the gapped region, while it is semimetallic in the gapless region reflecting the Dirac node. Performing the Monte Carlo simulation, we calculate the temperature dependence of the Majorana spectra. We find that the semimetallic dip is filled as temperature increases in the gapless region, but surprisingly, the spectrum develops an excitation gap in the region near the gapless-gapped boundary. Such changes of the low-energy spectrum appear sharply at the transition temperature from the spin liquid to the paramagnetic state. The results indicate that thermal fluctuations of the Z 2 fields significantly influence the low-energy state of Majorana fermions, especially in the spin liquid formation.
The quantum spin liquid is an exotic quantum state of matter in magnets. This state is a spin analog of liquid helium that does not solidify down to the lowest temperature due to strong quantum fluctuations. In conventional fluids, the liquid and gas possess the same symmetry and adiabatically connect to each other by bypassing the critical end point. We find that the situation is qualitatively different in quantum spin liquids realized in a three-dimensional Kitaev model; both gapless and gapped quantum spin liquid phases at low temperatures are always distinguished from the high-temperature paramagnet (spin gas) by a phase transition. The results challenge the common belief that the absence of thermodynamic singularity down to the lowest temperature is a symptom of a quantum spin liquid.
Chiral spin liquids (CSLs) in three dimensions and thermal phase transitions to paramagnet are studied by unbiased Monte Carlo simulations. For an extension of the Kitaev model to a three-dimensional tricoordinate network dubbed the hypernonagon lattice, we derive low-energy effective models in two different anisotropic limits. We show that the effective interactions between the emergent Z2{Z}_{2}Z2​ degrees of freedom called fluxes are unfrustrated in one limit, while highly frustrated in the other. In both cases, we find a first-order phase transition to the CSL, where both time-reversal and parity symmetries are spontaneously broken. In the frustrated case, however, the CSL state is highly exotic---the flux configuration is subextensively degenerate while showing a directional order with broken C3{C}_{3}C3​ rotational symmetry. Our results provide two contrasting archetypes of CSLs in three dimensions, both of which allow approximation-free simulation for investigating the thermodynamics.