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Detection of quantum phase boundary at finite temperatures in two dimensional Kitaev model.

Protyush Nandi, Sirshendu Bhattacharyya, Subinay Dasgupta

Published Nov 22, 2021• 0 citations
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Abstract

Quantum phase transitions occur when quantum fluctuation destroys order at zero temperature. With an increase in temperature, normally the thermal fluctuation wipes out any signs of this transition. For the two-dimensional Kitaev model, we have identified a physical quantity that shows algebraic divergence at a {\em finite} temperature, when an interaction parameter crosses the phase boundary for quantum phase transition. The quantity under consideration is the long time limit of the rate function related to a form of quantum fidelity under the quench of an interaction parameter. The robustness of the divergence makes the phase boundary detectable at any temperature. Our treatment is analytic and restricted to the vortex-free sector.

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Quantum phase transitions occur when quantum fluctuation destroys order at zero temperature. With an increase in temperature, normally the thermal fluctuation wipes out any signs of this transition. For the two-dimensional Kitaev model, we have identified a physical quantity that shows algebraic divergence at a {\em finite} temperature, when an interaction parameter crosses the phase boundary for quantum phase transition. The quantity under consideration is the long time limit of the rate function related to a form of quantum fidelity under the quench of an interaction parameter. The robustness of the divergence makes the phase boundary detectable at any temperature. Our treatment is analytic and restricted to the vortex-free sector.
Quantum phase transitions occur when quantum fluctuation destroys order at zero temperature. With an increase in temperature, normally the thermal fluctuation wipes out any signs of this transition. Here, we identify a physical quantity that shows nonanalytic behavior at finite temperatures, when an interaction parameter is quenched across the line of quantum phase transition. This quantity under consideration is the long time limit of a form of quantum fidelity. Our treatment is analytic for XY chain and 2D Kitaev model and is numerical for a 3D Hamiltonian applicable to Weyl semimetals.
The signature of quantum phase transition is generally wiped out at finite temperature. A few quantities that have been observed to carry this signature through a nonanalytic behavior are also limited to low temperatures only. With an aim to identify a suitable dynamical quantity at a high temperature, we have recently constructed a function from quantum fidelity, which has the potential to bear a nonanalytic signature at the quantum critical point beyond low-temperature regime. In this paper, we elaborate our earlier paper and demonstrate the behavior of the corresponding rate function and the robustness of the nonanalyticity for a number of many-body Hamiltonians in different dimensions. We have also shown that our rate function reduces to that used in the demonstration of the dynamical quantum phase transition (DQPT) at zero temperature. It has been further observed that, unlike DQPT, the long-time limit of the rate function can faithfully detect the equilibrium quantum phase transition as well.
We revisit the two-dimensional quantum Ising model by computing renormalization group flows close to its quantum critical point. The low but finite temperature regime in the vicinity of the quantum critical point is squashed between two distinct non-Gaussian fixed points: the classical fixed point dominated by thermal fluctuations and the quantum critical fixed point dominated by zero-point quantum fluctuations. Truncating an exact flow equation for the effective action, we derive a set of renormalization group equations and analyze how the interplay of quantum and thermal fluctuations, both non-Gaussian in nature, influences the shape of the phase boundary and the region in the phase diagram where critical fluctuations occur. The solution of the flow equations makes this interplay transparent: we detect finite temperature crossovers by computing critical exponents and we confirm that the power law describing the finite temperature phase boundary as a function of control parameter is given by the correlation length exponent at zero temperature as predicted in an ϵϵϵ-expansion with ϵ=1ϵ=1ϵ=1 by Sachdev [Phys. Rev. B 55, 142 (1997)].
Quantum phase transitions are sudden changes in the ground-state wavefunction of a many-body system that can occur as a control parameter such as a concentration or a field strength is varied. They are driven purely by the competition between quantum fluctuations and mutual interactions among constituents of the system, not by thermal fluctuations; hence they can occur even at zero temperature. Examples of quantum phase transitions in many-body physics may be found in systems ranging from high-temperature superconductors to topological insulators. A quantum phase transition usually can be characterized by nonanalyticity/discontinuity in certain order parameters or divergence of the ground state energy eigenvalue and/or its derivatives with respect to certain physical quantities. Here in a circular one-dimensional spin model with Heisenberg XY interaction and no magnetic field, we observe critical phenomena for the n0=1/N→0n_0=1/N\rightarrow0n0​=1/N→0 Mott insulator caused by a qualitative change of the boundary condition. We demonstrate in the vicinity of the transition point a sudden change in ground-state properties accompanied by an avoided level-crossing between the ground and the first excited states. Notably, our result links conventional quantum phase transitions to microscopic boundary conditions, with significant implications for quantum information, quantum control, and quantum computing.
We systematically explore and show the existence of finite-temperature continuous quantum phase transition (CTQPT) at a critical point, namely, during solidification or melting such that the first-order thermal phase transition is a special case within CTQPT. Infact, CTQPT is related to chemical reaction where quantum fluctuation (due to wavefunction transformation) is caused by thermal energy and it can occur maximally for temperatures much higher than zero Kelvin. To extract the quantity related to CTQPT, we use the ionization energy theory and the energy-level spacing renormalization group method to derive the energy-level spacing entropy, renormalized Bose-Einstein distribution and the time-dependent specific heat capacity. This work unambiguously shows that the quantum phase transition applies for any finite temperatures.
The notion of a dynamical quantum phase transition (DQPT) was recently introduced [Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)] as the nonanalytic behavior of the Loschmidt echo at critical times in the thermodynamic limit. In this work the quench dynamics in the ground state sector of the two-dimensional Kitaev honeycomb model is studied regarding the occurrence of DQPTs. For general two-dimensional systems of BCS type it is demonstrated how the zeros of the Loschmidt echo coalesce to areas in the thermodynamic limit, implying that DQPTs occur as discontinuities in the second derivative. In the Kitaev honeycomb model DQPTs appear after quenches across a phase boundary or within the massless phase. In the 1d limit of the Kitaev honeycomb model it becomes clear that the discontinuity in the higher derivative is intimately related to the higher dimensionality of the nondegenerate model. Moreover, there is a strong connection between the stationary value of the rate function of the Loschmidt echo after long times and the occurrence of DQPTs in this model.