-symmetry and its spontaneous breakdown explained by anti-linearity

Abstract

The impact of an anti-unitary symmetry on the spectrum of non-hermitean operators is studied. Wigner's normal form of an anti-unitary operator is shown to account for the spectral properties of non-hermitean, PT -symmetric Hamiltonians. Both the occurrence of single real or complex conjugate pairs of eigenvalues follows from this theory. The corresponding energy eigenstates span either one-or two-dimensional irreducible representations of the symmetry PT . In this framework, the concept of a spontaneously broken PT -symmetry is not needed.Deep in their hearts, many quantum physicists will renounce hermiticity of operators only reluctantly. However, non-hermitean Hamiltonians are applied successfully in nuclear physics, biology and condensed matter, often modelling the interaction of a quantum system with its environment in a phenomenological way. Since 1998, nonhermitean Hamiltonians continue to attract interest from a conceptual point of view [1]: surprisingly, the eigenvalues of a one-dimensional harmonic oscillator Hamiltonian remain real when the complex potentialV = ix 3 is added to it. Numerical, semiclassical, and analytic evidence [2] has been accumulated confirming that bound states with real eigenvalues exist for the vast class of complex potentials satisfying V † (x) = V (−x). In addition, pairs of complex conjugate eigenvalues occur systematically.PT -symmetry has been put forward to explain the observed energy spectra. The Hamiltonian operatorsĤ under scrutiny are invariant under the combined action of parity P and time reversal T , [Ĥ, PT ] = 0 .

A spin chain model with non-Hermitian interaction: the Ising quantum spin chain in an imaginary field

“…He provided an easy explanation for this feature: Acting adjointly on the Hamiltonian with a spin rotation operator R = e Its eigenvalues and those of H(λ, κ) are therefore either all real or occur in complex conjugate pairs. This is precisely the well known behaviour one finds when H(λ, κ) is symmetric with respect to an anti-linear operator [12,13,14,15,16], which as mentioned above has recently attracted a lot of attention. In quantum mechanical or field theoretical models the anti-linear operator is commonly taken to be the PT -operator, which carries out a simultaneous parity transformation P : x → −x and time reversal T : t → −t.…”

Irreversibility of the renormalization group flow in non-unitary quantum field theory

“…In the context of non-hermitian quantum mechanics this is termed a biorthogonal basis [54,55,56]. In particular, we have 1 = n |R n L n |.…”

PT symmetry on the lattice: the quantum group invariantXXZspin chain

“…If a system displays P T symmetry in the weak but not the strong sense, then it is said to display spontaneous breaking of P T symmetry [10].…”