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arXiv:cond-mat/0606348 · Journal of the Physical Society of Japan

A Non-Hermitian Critical Point and the Correlation Length of Strongly Correlated Quantum Systems

Yuichi Nakamura, Naomichi Hatano

Published Sep 28, 2006• 9 citations
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Abstract

We study a non-Hermitian generalization of quantum systems in which an imaginary vector potential is added to the momentum operator. In the tight-binding approximation, we make the hopping energy asymmetric in the Hermitian Hamiltonian. In a previous article, we conjectured that the non-Hermitian critical point where the energy gap vanishes is equal to the inverse correlation length of the Hermitian system and we confirmed the conjecture for two exactly solvable systems. In this article, we present more evidence for the conjecture. We also argue the basis of our conjecture by noting the dispersion relation of the elementary excitation.

DOI
10.1143/jpsj.75.104001

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We study a non-Hermitian generalization of quantum systems in which an imaginary vector potential is added to the momentum operator. In the tight-binding approximation, we make the hopping energy asymmetric in the Hermitian Hamiltonian. In a previous article, we conjectured that the non-Hermitian critical point where the energy gap vanishes is equal to the inverse correlation length of the Hermitian system and we confirmed the conjecture for two exactly solvable systems. In this article, we present more evidence for the conjecture. We also argue the basis of our conjecture by noting the dispersion relation of the elementary excitation.
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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.
We consider a PT-symmetric Fermi gas with an exceptional point, representing the critical point between PT-symmetric and symmetry broken phases. The low energy spectrum remains linear in momentum and is identical to that of a Hermitian Fermi gas. The fermionic Green's function decays in a power law fashion for large distances, as expected from gapless excitations, although the exponent is reduced from -1 due to the quantum Zeno effect. In spite of the gapless nature of the excitations, the ground state entanglement entropy saturates to a finite value, independent of the subsystem size due to the non-Hermitian correlation length intrinsic to the system. Attractive or repulsive interaction drives the system into the PT-symmetry broken regime or opens up a gap and protects PT symmetry, respectively. Our results challenge the concept of universality in non-Hermitian systems, where quantum criticality can be masked due to non-Hermiticity.
We introduce a formalism for time-dependent correlation functions for systems whose evolutions are governed by non-Hermitian Hamiltonians of general type. It turns out that one can define two different types of time correlation functions. Both these definitions seem to be physically consistent while becoming equivalent only in certain cases. Moreover, when autocorrelation functions are considered, one can introduce another function defined as the relative difference between the two definitions. We conjecture that such a function can be used to assess the positive semidefiniteness of the density operator without computing its eigenvalues. We illustrate these points by studying analytically a number of models with two energy levels.
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We analyze the scattering dynamics and spectrum of a quantum particle on a tight-binding lattice subject to a non-Hermitian (purely imaginary) local potential. The reflection, transmission, and absorption coefficients are studied as a function of the strength of this absorbing potential. The system is found to have an exceptional point at a certain strength of the potential. Unusually, all (or nearly all) of the spectrum pairs up into mutually coalescing eigenstate pairs at this exceptional point. At large potential strengths, the absorption coefficient decreases and the effect of the imaginary potential is similar to that of a real potential. We quantify this similarity by utilizing properties of a localized eigenstate.