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APS

Quantum fidelity in Tomonaga-Luttinger liquid and Kitaev model in the thermodynamic limit

Victor Mukherjee, Amit Dutta, Diptiman Sen

Published Aug 12, 2011• 0 citations
OpenAlex

Abstract

We study the scaling behavior of the fidelity (F ) in the thermodynamic limit using the examples of a Tomonaga-Luttinger liquid in one dimension and the Kitaev model on a honeycomb lattice. We show that the thermodynamic fidelity inside the gapless as well as gapped phases follow powerlaw scalings, with the power given by some of the critical exponents of the system. The generic scaling forms of F for an anisotropic quantum critical point for both thermodynamic and nonthermodynamic limits have been derived and verified for the Kitaev model. The behavior of F inside the gapless phase of the Kitaev model is found to depend strongly on its distance from the anisotropic quantum critical point. Finally, we consider a rotation of each spin in the Kitaev model around the z axis and calculate F through the overlap between the ground states for angle of rotation η and η+ dη, respectively. We thereby show that the associated geometric phase vanishes. We have supplemented our analytical calculations with numerical simulations wherever necessary.

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•Mar 11, 2008•0 citations
We study the scaling behavior of the fidelity (F ) in the thermodynamic limit using the examples of a Tomonaga-Luttinger liquid in one dimension and the Kitaev model on a honeycomb lattice. We show that the thermodynamic fidelity inside the gapless as well as gapped phases follow powerlaw scalings, with the power given by some of the critical exponents of the system. The generic scaling forms of F for an anisotropic quantum critical point for both thermodynamic and nonthermodynamic limits have been derived and verified for the Kitaev model. The behavior of F inside the gapless phase of the Kitaev model is found to depend strongly on its distance from the anisotropic quantum critical point. Finally, we consider a rotation of each spin in the Kitaev model around the z axis and calculate F through the overlap between the ground states for angle of rotation η and η+ dη, respectively. We thereby show that the associated geometric phase vanishes. We have supplemented our analytical calculations with numerical simulations wherever necessary.
We study the scaling behavior of the fidelity (FFF) in the thermodynamic limit using the examples of a system of Dirac fermions in one dimension and the Kitaev model on a honeycomb lattice. We show that the thermodynamic fidelity inside the gapless as well as gapped phases follow power-law scalings, with the power given by some of the critical exponents of the system. The generic scaling forms of FFF for an anisotropic quantum critical point for both the thermodynamic and nonthermodynamic limits have been derived and verified for the Kitaev model. The interesting scaling behavior of FFF inside the gapless phase of the Kitaev model is also discussed. Finally, we consider a rotation of each spin in the Kitaev model around the zzz axis and calculate FFF through the overlap between the ground states for the angle of rotation $\ensuremath{\eta}$ and $\ensuremath{\eta}+d\ensuremath{\eta}$, respectively. We thereby show that the associated geometric phase vanishes. We have supplemented our analytical calculations with numerical simulations wherever necessary.
The ground-state fidelity per lattice site is shown to be able to detect quantum phase transitions for the Kitaev model on the honeycomb lattice, a prototypical example of quantum lattice systems with topological order. It is found that, in the thermodynamic limit, the ground-state fidelity per lattice site is nonanalytic at the phase boundaries; the second-order derivative of its logarithmic function with respect to a control parameter describing the interaction between neighboring spins is divergent. A finite-size scaling analysis is performed, which allows us to extract the correlation length critical exponent from the scaling behaviors of a part of the ground-state fidelity per lattice site.
We study the behavior of the defect and heat densities under sudden quenching near the quantum critical points in the two-dimensional Kitaev honeycomb model both in the thermodynamic and nonthermodynamic limits. We consider quenches starting from a quantum critical point into the gapped as well as the gapless phases. We choose points on the lines of anisotropic quantum critical points as well as different points of intersection of these lines as the initial points from where the quenching starts. We find that the defect and heat densities display the expected power-law scalings along with logarithmic corrections to scaling (or cusp singularities) in certain cases. In the vicinity of some of the intersection points, the scaling behaviors change, indicating an effective dimensional reduction; the scaling behavior near these points depends on the number of critical lines crossed in the process of quenching. All the analytical predictions are also verified by numerical integration.
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.
The characterization of quantum spin liquid phases in Kitaev materials has been a subject of intensive studies over the recent years, both theoretically and experimentally. Most theoretical studies have focused on an isotropically interacting model with its coupling strength being equivalent on each bond in an attempt to simplify the problem. In this Letter, we study an extended spin-1/2 Kitaev- Γ model on a honeycomb lattice with an additional tuning parameter that controls the coupling strength on one of the bonds: We connect the limit of isolated Kitaev- Γ chains, which is known to exhibit an emergent S U ( 2 ) 1 Tomonaga-Luttinger liquid phase [Yang , ], to the two-dimensional model. We report on an instance, in which properties of the Tomonaga-Luttinger liquid persist for finite interchain coupling forming a phase we call the proximate Tomonaga-Luttinger liquid (pTLL). This pTLL is strongly anisotropic in character analogous to , and features spinon-like excitations similar to those of the antiferromagnetic Heisenberg chain. We use numerical exact diagonalization and density matrix renormalization group on various cluster geometries in a complementary way to overcome finite-size limitations. Published by the American Physical Society 2024
We study exactly both the ground-state fidelity susceptibility and bond-bond correlation function in the Kitaev honeycomb model. Our results show that the fidelity susceptibility can be used to identify the topological phase transition from a gapped A phase with Abelian anyon excitations to a gapless B phase with non-Abelian anyon excitations. We also find that the bond-bond correlation function decays exponentially in the gapped phase, but algebraically in the gapless phase. For the former case, the correlation length is found to be 1/ξ=2sinh⁡−1[2Jz−1/(1−Jz)]1/\xi=2\sinh^{-1}[\sqrt{2J_z -1}/(1-J_z)]1/ξ=2sinh−1[2Jz​−1​/(1−Jz​)], which diverges around the critical point Jz=(1/2)+J_z=(1/2)^+Jz​=(1/2)+.