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Register a new userFinite temperature geometric properties of the Kitaev honeycomb model
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We study finite temperature topological phase transitions of the Kitaevs spin honeycomb model in the vortex-free sector with the use of the recently introduced mean Uhlmann curvature. We employ an appropriate Fermionisation procedure to study the system as a two-band p-wave superconductor described by a BdG Hamiltonian. This allows to study relevant quantities such as Berry and mean Uhlmann curvatures in a simple setting. More specifically, we consider the spin honeycomb in the presence of an external magnetic field breaking time reversal symmetry. The introduction of such an external perturbation opens a gap in the phase of the system characterised by non-Abelian statistics, and makes the model to belong to a symmetry protected class, so that the Uhmann number can be analysed. We first consider the Berry curvature on a particular evolution line over the phase diagram. The mean Uhlmann curvature and the Uhlmann number are then analysed considering the system to be in a Gibbs state at finite temperature. Then, we show that the mean Uhlmann curvature describes a cross-over effect of the phases at high temperature. We also find an interesting nonmonotonic behaviour of the Uhlmann number as a function of the temperature in the trivial phase, which is due to the partial filling of the conduction band around Dirac points.
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We analyse the Kitaev honeycomb model, by means of the Berry curvature with respect to Hamiltonian parameters. We concentrate on the ground-state vortex-free sector, which allows us to exploit an appropriate Fermionisation technique. The parameter space includes a time-reversal breaking term which provides an analytical headway to study the curvature in phases in which it would otherwise vanish. The curvature is then analysed in the limit in which the time-reversal-symmetry-breaking perturbation vanishes. This provides remarkable information about the topological phase transitions of the model. A non-critical behaviour is found in the Berry curvature itself, which shows a distinctive behaviour in the different phases. The analysis of the first derivative shows a critical behaviour around the transition point.
It is widely accepted that topological superconductors can only have an effective interpretation in terms of curved geometry rather than gauge fields due to their charge neutrality. This approach is commonly employed in order to investigate their properties, such as the behaviour of their energy currents. Nevertheless, it is not known how accurately curved geometry can describe actual microscopic models. Here, we demonstrate that the low-energy properties of the Kitaev honeycomb lattice model, a topological superconductor that supports localised Majorana zero modes at its vortex excitations, are faithfully described in terms of Riemann-Cartan geometry. In particular, we show analytically that the continuum limit of the model is given in terms of the Majorana version of the Dirac Hamiltonian coupled to both curvature and torsion. We numerically establish the accuracy of the geometric description for a wide variety of couplings of the microscopic model. Our work opens up the opportunity to accurately predict dynamical properties of the Kitaev model from its effective geometric description.
We investigate the quantum spin liquid (QSL) ground state of anisotropic Kitaev model with antiferromagnetic (AFM) coupling under the $[001]$ magnetic field with the finite-temperature Lanczos method (FTLM). In this anisotropic AFM Kitaev model with $K_{X}=K_{Y}$, $K_{X}+K_{Y}+K_{Z}=-3K$, and $K_{Z}<-K$, with magnetic field increasing, the gapped QSL experiences a transition to a gapless QSL at $h_{c1}=gmu_{B}H_{z1}/K$, to another gapless QSL with $C_{6}$ rotational symmetry at $h_{c2}$, and to a new $U(1)$ gapless QSL between $h_{c3}$ and $h_{c4}$, respectively. These indicate that magnetic field could first turn the anisotropic gapped or gapless QSL back into the isotropic $C_{6}$ gapless one and then make it to undergo the similar evolution as the isotropic case. Moreover, the critical magnetic fields $h_{c1}$, $h_{c2}$, $h_{c3}$, and $h_{c4}$ come up monotonically with the increasing Kitaev coupling; this suggests that the magnetic field can be applied to the modulation of the anisotropic Kitaev materials.
In two dimensions, topological phases of free Majorana fermions coupled to a $mathbb{Z}_2$ gauge field are known to be classified according to the Chern number $ u in mathbb{Z}$. Its value mod 16 specifies the type of anyonic excitations. In this paper, we investigate triangular vortex configurations (and their dual) in the Kitaev honeycomb model and show that fourteen of these sixteen phases can be obtained by adding a time-reversal symmetry-breaking term. Missing phases are $ u=pm 7$. More generally, we prove that any periodic vortex configuration with an odd number of vortices per geometric unit cell can only host even Chern numbers whereas odd Chern numbers can be found in other cases.
Two indicators of finite-temperature topological properties based on the Uhlmann connection, one generalizing the Wilson loop to the Uhlmann-Wilson loop and the other generalizing the Berry phase to the Uhlmann phase, are constructed explicitly for a time-reversal invariant topological insulators with a $Z_2$ index. While the phases of the eigenvalues of the Wilson loop reflect the $Z_2$ index of the model at zero temperature, it is found that the signature from the Uhlmann-Wilson loop gradually fades away as temperature increases. On the other hand, the Berry phase exhibits quantization due to the underlying holonomy group. The Uhlmann phase retains the quantization at finite temperatures and serves as an indicator of topological properties. A phase diagram showing where jumps of the Uhlmann phase can be found is presented. By modifying the model to allow higher winding numbers, finite-temperature topological regimes sandwiched between trivial regimes at high and low temperatures may emerge.
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