Symmetric Hamiltonians Research Articles

Modal analysis and coupling in metal-insulator-metal waveguides

This paper shows how to analyze plasmonic metal-insulator-metal waveguides using the full modal structure of these guides. The analysis applies to all frequencies, particularly including the near infrared and visible spectrum, and to a wide range of sizes, including nanometallic structures. We use the approach here specifically to analyze waveguide junctions. We show that the full modal structure of the metal-insulator-metal MIM waveguides—which consists of real and complex discrete eigenvalue spectra, as well as the continuous spectrum—forms a complete basis set. We provide the derivation of these modes using the techniques developed for Sturm-Liouville and generalized eigenvalue equations. We demonstrate the need to include all parts of the spectrum to have a complete set of basis vectors to describe scattering within MIM waveguides with the mode-matching technique. We numerically compare the mode-matching formulation with finite-difference frequency-domain analysis and find very good agreement between the two for modal scattering at symmetric MIM waveguide junctions. We touch upon the similarities between the underlying mathematical structure of the MIM waveguide and the PT symmetric quantum-mechanical pseudo-Hermitian Hamiltonians. The rich set of modes that the MIM waveguide supports forms a canonical example against which other more complicated geometries can be compared. Our work here encompasses the microwave results but extends also to waveguides with real metals even at infrared and optical frequencies.

Exact time dependence of solutions to the time-dependent Schrödinger equation

Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.

Class of exactly solvableSO(n)symmetric spin chains with matrix product ground states

We introduce a class of exactly solvable $SO(n)$ symmetric Hamiltonians with matrix product ground states. For an odd $n\ensuremath{\ge}3$ case, the ground state is a translational invariant Haldane gap spin liquid state; while for an even $n\ensuremath{\ge}4$ case, the ground state is a spontaneously dimerized state with twofold degeneracy. In the matrix product ground states for both cases, we identify a hidden antiferromagnetic order, which is characterized by nonlocal string order parameters. The ground-state phase diagram of a generalized $SO(n)$ symmetric bilinear-biquadratic model is discussed.

Quantum toboggans: models exhibiting a multisheeted PT—symmetry

A generalization of the concept of PT—symmetric Hamiltonians H = p2 + V (x) is presented. For the usual analytic potentials V (x) (with singularities) and for the recently widely accepted ‘PT—symmetric’ asymptotic boundary conditions for wave functions ψ(x) (selected inside a pair of complex wedges generalizing the usual x → ±∞ asymptotics), non-equivalent quantum toboggans are defined as integrated along topologically different paths C of coordinates x ε Cl.

Beyond conventional factorization: non-Hermitian Hamiltonians with radial oscillator spectrum

The eigenvalue problem of the spherically symmetric oscillator Hamiltonian is revisited in the context of canonical raising and lowering operators. The Hamiltonian is then factorized in terms of two not mutually adjoint factorizing operators which, in turn, give rise to a non-Hermitian radial Hamiltonian. The set of eigenvalues of this new Hamiltonian is exactly the same as the energy spectrum of the radial oscillator and the new square-integrable eigenfunctions are complex Darboux-deformations of the associated Laguerre polynomials.

Computational power of symmetric Hamiltonians

The presence of symmetries, be they discrete or continuous, in a physical system typically leads to a reduction in the problem to be solved. Here we report that neither translational invariance nor rotational invariance reduce the computational complexity of simulating Hamiltonian dynamics; the problem is still bounded error, quantum polynomial time complete, and is believed to be hard on a classical computer. This is achieved by designing a system to implement a universal quantum interface, a device which enables control of an arbitrary computation through the control of a fixed number of spins, and using it as a building block to entirely remove the need for control, except in the system initialization. Finally, it is shown that cooling such Hamiltonians to their ground states in the presence of random magnetic fields solves a Quantum-Merlin-Arthur-complete problem.

Spherical-separability of Non-Hermitian Hamiltonians and Pseudo- $\mathcal{PT}$ -symmetry

Non-Hermitian but \(\mathcal{P}_{\varphi }\mathcal{T}_{\varphi }\) -symmetrized spherically-separable Dirac and Schrödinger Hamiltonians are considered. It is observed that the descendant Hamiltonians H r , H θ , and H φ play essential roles and offer some “user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a \(\mathcal{P}_{\varphi }\mathcal{T}_{\varphi }\) -symmetrized H φ , we have shown that the conventional Dirac (relativistic) and Schrödinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction V(θ)≠0 in the descendant Hamiltonian H θ would manifest a change in the angular θ-dependent part of the general solution too. Whilst some \(\mathcal{P}_{\varphi }\mathcal{T}_{\varphi }\) -symmetrized H φ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the \(\mathcal{PT}\) -symmetric ones (here the non-Hermitian \(\mathcal{P}_{\varphi }\mathcal{T}_{\varphi }\) -symmetric Hamiltonians) are nicknamed as pseudo- \(\mathcal{PT}\) -symmetric.

Interactions of Hermitian and non-Hermitian Hamiltonians

The coupling of non-Hermitian -symmetric Hamiltonians to standard Hermitian Hamiltonians, each of which individually has a real energy spectrum, is explored by means of a number of soluble models. It is found that in all cases the energy remains real for small values of the coupling constant, but becomes complex if the coupling becomes stronger than some critical value. For a quadratic non-Hermitian -symmetric Hamiltonian coupled to an arbitrary real Hermitian -symmetric Hamiltonian, the reality of the ground-state energy for small enough coupling constant is established up to second order in perturbation theory.

Supersymmetric quasi-Hermitian Hamiltonians with point interactions on a loop

We explore some aspects of -symmetric Hamiltonians with two point interactions. We determine classes of point interactions for which the Hamiltonians are supersymmetric. We prove that these Hamiltonians are quasi-Hermitian and find a very simple formula for the metric operator Θ and its square root ϱ as well. Further, we present the quasi-Hermitian Hamiltonian (with one-point interaction) with a continuous spectrum.

Horizons of stability

Although the quantum bound-state energies may be generated by the so-called -symmetric Hamiltonians where is, typically, parity, the spectrum only remains real and observable (i.e., in the language of physics, the -symmetry remains unbroken) inside a domain of couplings. We show that the boundary (i.e., certain stability and observability horizon formed by Kato's exceptional points) remains algebraic (i.e., we determine it by closed formulae) for a certain toy-model family of N-dimensional anharmonic-oscillator-related matrix Hamiltonians H(N) with N = 2, 3, …, 11.

Calculation of the metric in the Hilbert space of a -symmetric model via the spectral theorem

In a previous paper [1] we introduced a very simple -symmetric non-Hermitian Hamiltonian with a real spectrum and derived a closed formula for the metric operator relating the problem to a Hermitian one. In this paper we propose an alternative formula for the metric operator, which we believe is more elegant and whose construction—based on a backward use of the spectral theorem for self-adjoint operators—provides new insights into the nature of the model.

Metrics and isospectral partners for the most generic cubic -symmetric non-Hermitian Hamiltonian

We investigate properties of the most general -symmetric non-Hermitian Hamiltonian of cubic order in the annihilation and creation operators as a ten-parameter family. For various choices of the parameters we systematically construct an exact expression for a metric operator and an isospectral Hermitian counterpart in the same similarity class by exploiting the isomorphism between operator and Moyal products. We elaborate on the subtleties of this approach. For special choices of the ten parameters the Hamiltonian reduces to various models previously studied, such as to the complex cubic potential, the so-called Swanson Hamiltonian or the transformed version of the from below unbounded quartic −x4-potential. In addition, it also reduces to various models not considered in the present context, namely the single-site lattice Reggeon model and a transformed version of the massive sextic ±x6-potential, which plays an important role as a toy model to identify theories with vanishing cosmological constant.

Exact isospectral pairs of symmetric Hamiltonians

A technique for constructing an infinite tower of pairs of symmetric Hamiltonians, and (n = 2, 3, 4, …), that have exactly the same eigenvalues is described and illustrated by means of three examples (n = 2, 3, 4). The eigenvalue problem for the first Hamiltonian of the pair must be posed in the complex domain, so its eigenfunctions satisfy a complex differential equation and fulfill homogeneous boundary conditions in Stokes' wedges in the complex plane. The eigenfunctions of the second Hamiltonian of the pair obey a real differential equation and satisfy boundary conditions on the real axis. This equivalence constitutes a proof that the eigenvalues of both Hamiltonians are real. Although the eigenvalue differential equation associated with is real, the Hamiltonian exhibits quantum anomalies (terms proportional to powers of ℏ). These anomalies are remnants of the complex nature of the equivalent Hamiltonian . For the cases n = 2, 3, 4 in the classical limit in which the anomaly terms in are discarded, the pair of Hamiltonians Hn,classical and Kn,classical have closed classical orbits whose periods are identical.

No-Ghost Theorem for the Fourth-Order Derivative Pais-Uhlenbeck Oscillator Model

A new realization of the fourth-order derivative Pais-Uhlenbeck oscillator is constructed. This realization possesses no states of negative norm and has a real energy spectrum that is bounded below. The key to this construction is the recognition that in this realization the Hamiltonian is not Dirac Hermitian. However, the Hamiltonian is symmetric under combined space reflection P and time reversal T. The Hilbert space that is appropriate for this PT-symmetric Hamiltonian is identified and it is found to have a positive-definite inner product. Furthermore, the time-evolution operator is unitary.

Phase-space equilibrium distributions and their applications to spin systems with nonaxially symmetric Hamiltonians

The Fourier series representation of the equilibrium quasiprobability density function ${W}_{S}(\ensuremath{\vartheta},\ensuremath{\varphi})$ or Wigner function of spin ``orientations'' for arbitrary spin Hamiltonians in a representation (phase) space of the polar angles $(\ensuremath{\vartheta},\ensuremath{\varphi})$ (analogous to the Wigner function for translational motion) arising from the generalized coherent state representation of the density operator is evaluated explicitly for some nonaxially symmetric problems including a uniaxial paramagnet in a transverse external field, a biaxial, and a cubic system. It is shown by generalizing transition state theory to spins [i.e., calculating the escape rate using the equilibrium density function ${W}_{S}(\ensuremath{\vartheta},\ensuremath{\varphi})$ only] that one may evaluate the reversal time of the magnetization. The quantum corrections to the transition state theory escape rate equation for classical magnetic dipoles appear both in the prefactor and in the exponential part of the escape rate and exhibit a marked dependence on the spin number. Furthermore, the phase-space representation allows us to estimate the switching field curves and/or surfaces for spin systems because quantum effects in these fields can be estimated via Thiaville's geometrical method [Phys. Rev. B 61, 12221 (2000)] for the study of the magnetization reversal of single-domain ferromagnetic particles. The calculation is accomplished (just as the determination of the equilibrium quasiprobability distributions in the phase space of the polar angles) by calculating switching field curves and/or surfaces using the Weyl symbol ($c$-number representation) of the Hamiltonian operator for given magnetocrystalline-Zeeman energy terms. Examples of such calculations for various spin systems are presented. Moreover, the reversal time of the magnetization allows us to estimate thermal effects on the switching fields for spin systems.

Thermodynamic Casimir effects involving interacting field theories with zero modes

Systems with an $O(n)$ symmetrical Hamiltonian are considered in a $d$-dimensional slab geometry of macroscopic lateral extension and finite thickness $L$ that undergo a continuous bulk phase transition in the limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$. The effective forces induced by thermal fluctuations at and above the bulk critical temperature ${T}_{c,\ensuremath{\infty}}$ (thermodynamic Casimir effect) are investigated below the upper critical dimension ${d}^{*}=4$ by means of field-theoretic renormalization-group methods for the case of periodic and special-special boundary conditions, where the latter correspond to the critical enhancement of the surface interactions on both boundary planes. As shown previously [Europhys. Lett. 75, 241 (2006)], the zero modes that are present in Landau theory at ${T}_{c,\ensuremath{\infty}}$ make conventional renormalization-group-improved perturbation theory in $4\ensuremath{-}ϵ$ dimensions ill-defined. The revised expansion introduced there is utilized to compute the scaling functions of the excess free energy and the Casimir force for temperatures $T\ensuremath{\geqslant}{T}_{c,\ensuremath{\infty}}$ as functions of $\mathsf{L}\ensuremath{\equiv}L∕{\ensuremath{\xi}}_{\ensuremath{\infty}}$, where ${\ensuremath{\xi}}_{\ensuremath{\infty}}$ is the bulk correlation length. Scaling functions of the $L$-dependent residual free energy per area are obtained, whose $\mathsf{L}\ensuremath{\rightarrow}0$ limits are in conformity with previous results for the Casimir amplitudes ${\ensuremath{\Delta}}_{C}$ to $O({ϵ}^{3∕2})$ and display a more reasonable small-$\mathsf{L}$ behavior inasmuch as they approach the critical value ${\ensuremath{\Delta}}_{C}$ monotonically as $\mathsf{L}\ensuremath{\rightarrow}0$. Extrapolations to $d=3$ for the Ising case $n=1$ with periodic boundary conditions are in fair agreement with Monte Carlo results. In the case of special-special boundary conditions, extrapolations to $d=3$ are hampered by the fact that the one-loop result for the inverse finite-size susceptibility becomes negative for some values of $\mathsf{L}$ when $ϵ\ensuremath{\gtrsim}0.83$.

-symmetry and weak pseudo-hermiticity

For an invertible (bounded) linear operator acting in a Hilbert space , we consider the consequences of the -symmetry of a non-hermitian Hamiltonian where is the time-reversal operator. If H is symmetric in the sense that , then -symmetry is equivalent to -weak-pseudo-hermiticity. But in general this equivalence does not hold. We show this using some specific examples. Among these is a large class of non--symmetric Hamiltonians that share the spectral properties of -symmetric Hamiltonians.

η-Weak-Pseudo-Hermiticity Generators and Radially Symmetric Hamiltonians

A class η-weak-pseudo-Hermiticity generators for spherically symmetric non-Hermitian Hamiltonians are presented. An operators-based procedure is introduced so that the results for the 1D Schrodinger Hamiltonian may very well be reproduced. A generalization beyond the nodeless states is proposed. Our illustrative examples include η- weak-pseudo-Hermiticity generators for the non-Hermitian weakly perturbed 1D and radial oscillators, and the non-Hermitian perturbed radial Coulomb.

Faster than Hermitian Time Evolution

For any pair of quantum states, an initial state |Ii and a final quantum state |Fi, in a Hilbert space, there are many Hamiltonians H under which |Ii evolves into |Fi. Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, Emax and Emin, is held fixed. We can then determine the Hamiltonian H that satisfies this constraint and achieves the transformation from the initial state to the final state in the least possible time . For Hermitian Hamiltonians, has a nonzero lower bound. However, among non-Hermitian PT -symmetric Hamiltonians satisfying the same energy constraint, can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of can be made arbitrarily small because for PT -symmetric Hamiltonians the path from the vector |Ii to the vector |Fi, as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.

Tridiagonal -symmetric N-by-N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime

A generic -symmetric Hamiltonian is assumed tridiagonalized and truncated to N < ∞ dimensions, H → H(chain model), and all its up–down symmetrized special cases with J = [N/2] real couplings are considered, H(chain model) → H(N). Using symbolic manipulation and extrapolation techniques we find out that in the strongly non-Hermitian regime the secular equation gets partially factorized at all N. This enables us to reveal a fine-tuned alignment of the dominant couplings implying an asymptotically sharply spiked shape of the boundary of the J-dimensional quasi-Hermiticity domain in which all the spectrum of energies E(N)n remains real and observable.