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New Journal of Physics

Hidden continuous quantum phase transition without gap closing in non-Hermitian transverse Ising model

Fei Yang, Heng Wang, Meng-Lei Yang, Cui-Xian Guo, Xiaoran Wang, Gaoyong Sun, Su-Peng Kou

Published Apr 1, 2022• 13 citations
Publisher

Abstract

Abstract Continuous phase transition in quantum matters is a significant issue in condensed matter physics. In general, the continuous quantum phase transitions in many-body systems occur with gap closing. On the other hand, non-Hermitian systems could display quite different properties as their Hermitian counterparts. In this paper, we show that a hidden, continuous quantum phase transition occurs without gap closing in non-Hermitian transverse Ising model. By using a projected Jordan–Wigner transformation, the one-dimensional (1D) non-Hermitian transverse Ising model with ferromagnetic order is mapped on to 1D non-Hermitian Kitaev model with topological superconducting order and becomes exactly solvable. A hidden, continuous quantum phase transition is really normal–abnormal transition for fermionic correlation in the 1D non-Hermitian Kitaev model. In addition, similar hidden, continuous quantum phase transition is discovered in two-dimensional non-Hermitian transverse Ising model and thus becomes a universal feature in certain non-Hermitian many-body systems.

DOI
10.1088/1367-2630/ac652f

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Abstract Continuous phase transition in quantum matters is a significant issue in condensed matter physics. In general, the continuous quantum phase transitions in many-body systems occur with gap closing. On the other hand, non-Hermitian systems could display quite different properties as their Hermitian counterparts. In this paper, we show that a hidden, continuous quantum phase transition occurs without gap closing in non-Hermitian transverse Ising model. By using a projected Jordan–Wigner transformation, the one-dimensional (1D) non-Hermitian transverse Ising model with ferromagnetic order is mapped on to 1D non-Hermitian Kitaev model with topological superconducting order and becomes exactly solvable. A hidden, continuous quantum phase transition is really normal–abnormal transition for fermionic correlation in the 1D non-Hermitian Kitaev model. In addition, similar hidden, continuous quantum phase transition is discovered in two-dimensional non-Hermitian transverse Ising model and thus becomes a universal feature in certain non-Hermitian many-body systems.
Contrary to the conventional wisdom in Hermitian systems, a continuous quantum phase transition between gapped phases is shown to occur without closing the energy gap $\mathrm{\ensuremath{\Delta}}$ in non-Hermitian quantum many-body systems. Here, the relevant length scale $\ensuremath{\xi}\ensuremath{\simeq}{v}_{\mathrm{LR}}/\mathrm{\ensuremath{\Delta}}$ diverges because of the breakdown of the Lieb-Robinson bound on the velocity (i.e., unboundedness of vLR{v}_{\mathrm{LR}}vLR​) rather than vanishing of the energy gap $\mathrm{\ensuremath{\Delta}}$. The susceptibility to a change in the system parameter exhibits a singularity due to nonorthogonality of eigenstates. As an illustrative example, we present an exactly solvable model by generalizing Kitaev's toric-code model to a non-Hermitian regime.
Abstract As a very simple model, the Ising model plays an important role in statistical physics. In the paper, with the help of quantum Liouvillian statistical theory, we study the one-dimensional non-Hermitian Ising model at finite temperature and give its analytical solutions. We find that the non-Hermitian Ising model shows quite different properties from those of its Hermitian counterpart. For example, the ‘pseudo-phase transition’ is explored between the ‘topological’ phase and the ‘non-topological’ phase, at which the Liouvillian energy gap is closed rather than the usual energy gap. In particular, we point out that the one-dimensional non-Hermitian Ising model at finite temperature can be equivalent to an effective anisotropic XY model in the transverse field. This work will help people understand quantum statistical properties of non-Hermitian systems at finite temperatures.
A non-Hermitian deformation of the one-dimensional transverse Ising model is shown to have the property of quasi-hermiticity. The transverse Ising chain is obtained from the starting non-Hermitian Hamiltonian through a similarity transformation. Consequently, both the models have identical eigen spectra, although the eigenfunctions are different. The metric in the Hilbert space, which makes the non-Hermitian model unitary and ensures the completeness of states, has been constructed explicitly. Although the longitudinal correlation functions are identical for both the non-Hermitian and the Hermitian Ising models, the difference shows up in the transverse correlation functions, which have been calculated explicitly and are not always real. A proper set of Hermitian spin operators in the Hilbert space of the non-Hermitian Hamiltonian has been identified, in terms of which all the correlation functions of the non-Hermitian Hamiltonian become real and identical to that of the standard transverse Ising model. Comments on the quantum phase transitions in the non-Hermitian model have been made.
Abstract In this work, we investigate the quantum phase transition in a non-Hermitian XY spin chain. The phase diagram shows that the critical points of Ising phase transition expand into a critical transition zone after introducing a non-Hermitian effect. By analyzing the non-Hermitian gap and long-range correlation function, one can distinguish different phases by means of different gap features and decay properties of correlation function, a tricky problem in traditional XY model. Furthermore, the results reveal the relationship among different regions of the phase diagram, non-Hermitian energy gap and long-range correlation function.
Quantum phase transitions in the one-dimensional extended quantum compass model in a transverse field are studied by using the Jordan-Wigner transformation. This model is always gapful, except at the critical surfaces where the energy gap disappears. We obtain the analytic expressions of all critical fields which drive quantum phase transitions. This model shows a rich phase diagram which includes spin-flop, strip antiferromagnetic, and saturated ferromagnetic phases in addition to the phase with antiparallel ordering of the yyy spin component on odd bonds. We also study also the universality and scaling properties of the transverse susceptibility and nearest-neighbor correlation function derivatives in different regions to confirm the results obtained using the energy gap analysis.
Quantum phase transitions occur at zero temperature upon variation of some nonthermal control parameters. The Ising chain in a transverse field is probably the most-studied model undergoing such a transition, from ferromagnetic to paramagnetic state1,2. This model can be exactly solved by using a Jordan-Wigner transformation, which transforms the spins into noninteracting spinless fermions1. At the quantum critical point, the magnetic excitations can carry arbitrarily low energy and dominate the low temperature properties. Here we report the unveiling of such quantum critical point in quasi-one-dimensional Ising ferromagnet CoNb2O6 by ultra-low-temperature thermal conductivity measurements. We find that in the paramagnetic state, phonons are scattered by the magnetic excitations above certain temperature Ts, which corresponds to a gap. As predicted by the theoretical model1, this gap linearly goes to zero with decreasing the transverse field, thus determining the quantum critical point of the Ising chain.