Katherine Jones-Smith

In 1956 Dyson analyzed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of non-Hermitian quantum mechanics formalism at our disposal when considering Dyson’s work, both technically and contextually. Here we recast Dyson’s work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self-adjoint, if not manifestly “Hermitian”. Then we extend his scheme to doped antiferromagnets described by the t-J model, in hopes of shedding light on the physics of high-temperature superconductivity.

http://arxiv.org/abs/1304.5689

Quantum Physics (quant-ph)

Katherine Jones-Smith, Rudolph Kalveks

We present two examples of non-Hermitian Hamiltonians which consist of an unperturbed part plus a perturbation that behaves like a vector, in the framework of PT quantum mechanics. The first example is a generalization of the recent work by Bender and Kalveks, wherein the E2 algebra was examined; here we consider the E3 algebra representing a particle on a sphere, and identify the critical value of coupling constant which marks the transition from real to imaginary eigenvalues. Next we analyze a model with SO(3) symmetry, and in the process extend the application of the Wigner-Eckart theorem to a non-Hermitian setting.

http://arxiv.org/abs/1304.5692

Quantum Physics (quant-ph)

Carl M. Bender, Mariagiovanna Gianfreda

The C operator in PT-symmetric quantum mechanics satisfies a system of three simultaneous algebraic operator equations, \(C^2=1\), \([C,PT]=0\), and \([C,H]=0\). These equations are difficult to solve exactly, so perturbative methods have been used in the past to calculate C. The usual approach has been to express the Hamiltonian as \(H=H_0+\epsilon H_1\), and to seek a solution for C in the form \(C=e^Q P\), where \(Q=Q(q,p)\) is odd in the momentum p, even in the coordinate q, and has a perturbation expansion of the form \(Q=\epsilon Q_1+\epsilon^3 Q_3+\epsilon^5 Q_5+\ldots\). [In previous work it has always been assumed that the coefficients of even powers of $\epsilon$ in this expansion would be absent because their presence would violate the condition that \(Q(p,q)\) is odd in p.] In an earlier paper it was argued that the C operator is not unique because the perturbation coefficient \(Q_1\) is nonunique. Here, the nonuniqueness of C is demonstrated at a more fundamental level: It is shown that the perturbation expansion for Q actually has the more general form \(Q=Q_0+\epsilon Q_1+\epsilon^2 Q_2+\ldots\) in which {\it all} powers and not just odd powers of \(\epsilon\) appear. For the case in which \(H_0\) is the harmonic-oscillator Hamiltonian, \(Q_0\) is calculated exactly and in closed form and it is shown explicitly to be nonunique. The results are verified by using powerful summation procedures based on analytic continuation. It is also shown how to calculate the higher coefficients in the perturbation series for Q.

http://arxiv.org/abs/1302.7047

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Carl M. Bender, V. Branchina, Emanuele Messina

It was shown recently that a PT-symmetric \(i\phi^3\) quantum field theory in \(6-\epsilon\) dimensions possesses a nontrivial fixed point. The critical behavior of this theory around the fixed point is examined and it is shown that the corresponding phase transition is related to the existence of a nontrivial solution of the gap equation. The theory is studied first in the mean-field approximation and the critical exponents are calculated. Then, it is examined beyond the mean-field approximation by using renormalization-group techniques, and the critical exponents for \(6-\epsilon\) dimensions are calculated to order \(\epsilon\). It is shown that because of its stability the PT-symmetric \(i\phi^3\) theory has a higher predictive power than the conventional \(\phi^3\) theory. A comparison of the \(i\phi^3\) model with the Lee-Yang model is given.

http://arxiv.org/abs/1301.6207

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Carl M. Bender, Stefan Boettcher

The paper by Bowen, Mancini, Fessatidis, and Murawski (2012 Phys. Scr. {\bf 85}, 065005) demonstrates in a dramatic fashion the serious difficulties that can arise when one rushes to perform numerical studies before understanding the physics and mathematics of the problem at hand and without understanding the limitations of the numerical methods used. Based on their flawed numerical work, the authors conclude that the work of Bender and Boettcher is wrong even though it has been verified at a completely rigorous level. Unfortunately, the numerical procedures performed and described in the paper by Bowen et al are incorrectly applied and wrongly interpreted.

http://arxiv.org/abs/1210.0426

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Peter N. Meisinger, Michael C. Ogilvie

PT-symmetric Hamiltonians and transfer matrices arise naturally in statistical mechanics. These classical and quantum models often require the use of complex or negative weights and thus fall outside of the conventional equilibrium statistical mechanics of Hermitian systems. PT-symmetric models form a natural class where the partition function is necessarily real, but not necessarily positive. The correlation functions of these models display a much richer set of behaviors than Hermitian systems, displaying sinusoidally-modulated exponential decay, as in a dense fluid, or even sinusoidal modulation without decay. Classical spin models with PT symmetry include Z(N) models with a complex magnetic field, the chiral Potts model and the anisotropic next-nearest-neighbor Ising (ANNNI) model. Quantum many-body problems with a non-zero chemical potential have a natural PT-symmetric representation related to the sign problem. Two-dimensional QCD with heavy quarks at non-zero chemical potential can be solved by diagonalizing an appropriate PT-symmetric Hamiltonian.

http://arxiv.org/abs/1208.5077

Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics – Lattice (hep-lat)

A.S. Rodrigues, K. Li, V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, Carl M. Bender

In this work we analyze PT-symmetric double-well potentials based on a two-mode picture. We reduce the problem into a PT-symmetric dimer and illustrate that the latter has effectively two fundamental bifurcations, a pitchfork (symmetry-breaking bifurcation) and a saddle-center one, which is the nonlinear analog of the PT-phase-transition. It is shown that the symmetry breaking leads to ghost states (amounting to growth or decay); although these states are not true solutions of the original continuum problem, the system’s dynamics closely follows them, at least in its metastable evolution. Past the second bifurcation, there are no longer states of the original continuum system. Nevertheless, the solutions can be analytically continued to yield a new pair of branches, which is also identified and dynamically examined. Our explicit analytical results for the dimer are directly compared to the full continuum problem, yielding a good agreement.

http://arxiv.org/abs/1207.1066

Pattern Formation and Solitons (nlin.PS); Quantum Gases (cond-mat.quant-gas); Mathematical Physics (math-ph)

Carl M. Bender, David J. Weir

Non-Hermitian PT-symmetric quantum-mechanical Hamiltonians generally exhibit a phase transition that separates two parametric regions, (i) a region of unbroken PT symmetry in which the eigenvalues are all real, and (ii) a region of broken PT symmetry in which some of the eigenvalues are complex. This transition has recently been observed experimentally in a variety of physical systems. Until now, theoretical studies of the PT phase transition have generally been limited to one-dimensional models. Here, four nontrivial coupled PT-symmetric Hamiltonians, \(H=p^2/2+x^2/2+q^2/2+y^2/2+igx^2y\), \(H=p^2/2+x^2/2+q^2/2+y^2+igx^2y\), \(H=p^2/2+x^2/2+q^2/2+y^2/2+r^2/2+z^2/2+igxyz\), and \(H=p^2/2+x^2/2+q^2/2+y^2+r^2/2+3z^2/2+igxyz\) are examined. Based on extensive numerical studies, this paper conjectures that all four models exhibit a phase transition. The transitions are found to occur at \(g\approx 0.1\), \(g\approx 0.04\), \(g\approx 0.1\), and \(g\approx 0.05\). These results suggest that the PT phase transition is a robust phenomenon not limited to systems having one degree of freedom.

http://arxiv.org/abs/1206.5100

Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

Carl M. Bender, Bjorn K. Berntson, David Parker, E. Samuel

If a Hamiltonian is PT symmetric, there are two possibilities: Either the eigenvalues are entirely real, in which case the Hamiltonian is said to be in an unbroken-PT-symmetric phase, or else the eigenvalues are partly real and partly complex, in which case the Hamiltonian is said to be in a broken-PT-symmetric phase. As one varies the parameters of the Hamiltonian, one can pass through the phase transition that separates the unbroken and broken phases. This transition has recently been observed in a variety of laboratory experiments. This paper explains the phase transition in a simple and intuitive fashion and then describes an extremely elementary experiment in which the phase transition is easily observed.

http://arxiv.org/abs/1206.4972

Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

Carl M. Bender, Moshe Moshe, Sarben Sarkar

The conventional double-scaling limit of a quartic quantum field theory is inconsistent because the critical coupling constant is negative. Thus, at the critical coupling the Lagrangian appears to define a quantum theory whose energy is complex. Worse yet, the functional integral for the partition function of the theory does not exist. It is shown that one can avoid these difficulties if one approaches this correlated limit in a PT-symmetric fashion. The partition function is calculated explicitly in the double-scaling limit of an zero-dimensional O(N)-symmetric quartic model.

http://arxiv.org/abs/1206.4943

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)