Symmetric Hamiltonians Research Articles

Analytic approximation for eigenvalues of a class of PT -symmetric Hamiltonians

An analytical approximation for the eigenvalues of $\mathcal{PT}$ symmetric Hamiltonian $\mathsf{H} = -d^{2}/dx^{2} - (\mathrm{i}x)^{\epsilon+2}$, $\epsilon > -1$ is developed via simple basis sets of harmonic-oscillator wave functions with variable frequencies and equilibrium positions. We demonstrate that our approximation provides high accuracy for any given energy level for all values of $\epsilon > -1$.

Irreversibility of the renormalization group flow in non-unitary quantum field theory**Dedicated to John Cardy on the occasion of his 70th birthday.

We show irreversibility of the renormalization group flow in non-unitary but -invariant quantum field theory in two space-time dimensions. In addition to unbroken -symmetry and a positive energy spectrum, we assume standard properties of quantum field theory including a local energy-momentum tensor and relativistic invariance. This generalizes Zamolodchikov’s c-theorem to -symmetric Hamiltonians. Our proof follows closely Zamolodchikov’s arguments. We show that a function of the renormalization group parameter s exists which is non-negative and monotonically decreasing along renormalization group flows. Its value at a critical point is the ‘effective central charge’ entering the specific free energy. At least in rational models, this equals , where c is the central charge and Δ is the lowest primary field dimension in the conformal field theory which describes the critical point.

Boltzmann–Gibbs states in topological quantum walks and associated many-body systems: fidelity and Uhlmann parallel transport analysis of phase transitions

We perform the fidelity analysis for Boltzmann–Gibbs-like states in order to investigate whether the topological order of 1D fermionic systems at zero temperature is maintained at finite temperatures. We use quantum walk protocols that are known to simulate topological phases and the respective quantum phase transitions for chiral symmetric Hamiltonians. Using the standard approaches of the fidelity analysis and the study of edge states, we conclude that no thermal-like phase transitions occur as temperature increases, i.e. the topological behaviour is washed out gradually. We also show that the behaviour of the Uhlmann geometric factor associated to the considered fidelity exhibits the same behaviour as the latter, thus confirming the results obtained using the previously established approaches.

An approach for obtaining integrable Hamiltonians from Poisson-commuting polynomial families

We discuss a general approach permitting the identification of a broad class of sets of Poisson-commuting Hamiltonians, which are integrable in the sense of Liouville. It is shown that all such Hamiltonians can be solved explicitly by a separation of variables ansatz. The method leads in particular to a proof that the so-called “goldfish” Hamiltonian is maximally superintegrable and leads to an elementary identification of a full set of integrals of motion. The Hamiltonians in involution with the “goldfish” Hamiltonian are also explicitly integrated. New integrable Hamiltonians are identified, among which some have the property of being isochronous, that is, all their orbits have the same period. Finally, a peculiar structure is identified in the Poisson brackets between the elementary symmetric functions and the set of Hamiltonians commuting with the “goldfish” Hamiltonian: these can be expressed as products between elementary symmetric functions and Hamiltonians. The structure displays an invariance property with respect to one element and has both a symmetry and a closure property. The meaning of this structure is not altogether clear to the author, but it turns out to be a powerful tool.

Translation-Invariant Parent Hamiltonians of Valence Bond Crystals.

We present a general method to construct translation-invariant and SU(2) symmetric antiferromagnetic parent Hamiltonians of valence bond crystals (VBCs). The method is based on a canonical mapping transforming S=1/2 spin operators into a bilinear form of a new set of dimer fermion operators. We construct parent Hamiltonians of the columnar and the staggered VBCs on the square lattice, for which the VBC is an eigenstate in all regimes and the exact ground state in some region of the phase diagram. We study the departure from the exact VBC regime upon tuning the anisotropy by means of the hierarchical mean field theory and exact diagonalization on finite clusters. In both Hamiltonians, the VBC phase extends over the exact regime and transits to a columnar antiferromagnet (CAFM) through a window of intermediate phases, revealing an intriguing competition of correlation lengths at the VBC-CAFM transition. The method can be readily applied to construct other VBC parent Hamiltonians in different lattices and dimensions.

Confluent Crum-Darboux transformations in Dirac Hamiltonians withPT-symmetric Bragg gratings

We consider optical systems where propagation of light can be described by a Dirac-like equation with $PT$-symmetric Hamiltonian. In order to construct exactly solvable configurations, we extend the confluent Crum-Darboux transformation for the one-dimensional Dirac equation. The properties of the associated intertwining operators are discussed and the explicit form for higher-order transformations is presented. We utilize the results to derive a multi-parametric class of exactly solvable systems where the balanced gain and loss represented by the $PT$-symmetric refractive index can imply localization of the electric field in the material.

Localization, quantum resonances, and ratchet acceleration in a periodically kickedPT-symmetric quantum rotator

We consider wave transport phenomena in a $\mathcal{PT}$-symmetric extension of the periodically-kicked quantum rotator model and reveal that dynamical localization assists the unbroken $\mathcal{PT}$ phase. In the delocalized (quantum resonance) regime, $\mathcal{PT}$ symmetry is always in the broken phase and ratchet acceleration arises as a signature of unidirectional non-Hermitian transport. An optical implementation of the periodically-kicked $\mathcal{PT}$-symmetric Hamiltonian, based on transverse beam propagation in a passive optical resonator with combined phase and loss gratings, is suggested to visualize acceleration modes in fractional Talbot cavities.

On the quantum mechanics of bicomplex Hamiltonian system

We investigate the Schrödinger equation in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. We propose an analytical method to solve bicomplex-version of Schrödinger equation corresponding to the systems of Hamiltonians of both hermitian (self-adjoint) and non-hermitian PT symmetric type. In our approach we extend the existing mathematical formulation of quantum system searching for the exact or quasi-exact solution for ground state energy eigenvalues and associated wave functions acting in bicomplex Hilbert space. The model concerning hermitian Hamiltonians is then applied to the problems of two bicomplex valued polynomial oscillators one involving x2 and another of isotonic type. The ground states and associated energy values for both the oscillators are found to be hyperbolic in nature. The model in connection to the unbroken PT symmetric Hamiltonians is then applied to illustrate the problems of complex and bicomplex valued shifted oscillators.

The stabilizer group of honeycomb lattices and its application to deformed monolayers

Isospectral transformations of exactly solvable models constitute a fruitful method for obtaining new structures with prescribed properties. In this paper we study the stability group of the Dirac algebra in honeycomb lattices representing graphene or boron nitride. New crystalline arrays with conical (Dirac) points are obtained; in particular, a model for dichalcogenide monolayers is proposed and analysed. In our studies we encounter unitary and non-unitary transformations. We show that the latter give rise to -symmetric Hamiltonians, in compliance with known results in the context of boosted Dirac equations. The results of the unitary part are applied to the description of invariant bandgaps and dispersion relations in materials such as MoS2. A careful construction based on atomic orbitals is proposed and the resulting dispersion relation is compared with previous results obtained through DFT.

Non-Hermitian heat engine with all-quantum-adiabatic-process cycle

As a quantum device, a quantum heat engine (QHE) is described by a Hermitian Hamiltonian. However, since it is an open system, reservoirs must be imposed phenomenologically without any description in the context of quantum mechanics. A non-Hermitian system is expected to describe an open system that exchanges energy and particles with external reservoirs. Correspondingly, such an exchange can be adiabatic in the context of quantum mechanics. We first propose a non-Hermitian QHE by a concrete simple two-level system, which is an spin in a complex external magnetic field. The non-Hermitian -symmetric Hamiltonian, as a self-contained one, describes both the working medium and reservoirs. A heat engine cycle is composed of completely quantum adiabatic processes. Surprisingly, the heat efficiency is obtained to be the same as that of the Hermitian quantum Otto cycle. A classical analog of this scheme is also presented. Our finding paves the way for revealing the role of a non-Hermitian Hamiltonian in physics.

PT-symmetry breaking in waveguides with competing loss-gain pairs

We consider a periodic waveguide array whose unit cell consists of a $\mathcal{PT}$-symmetric quadrimer with two competing loss/gain parameter pairs which lead to qualitatively different symmetry-broken phases. It is shown that the transitions between the phases are described by a symmetry-adapted nonlocal current which maps the spectral properties to the spatially resolved field, for the lattice as well as for the isolated quadrimer. Its site-average acts like a natural order parameter for the general class of one-dimensional $\mathcal{PT}$-symmetric Hamiltonians, vanishing in the unbroken phase and being nonzero in the broken phase. We investigate how the beam dynamics in the array is affected by the presence of competing loss/gain rates in the unit cell, showing that the enriched band structure yields the possibility to control the propagation length before divergence when the system resides in the broken $\mathcal{PT}$ phase.

Inverse dispersion method for calculation of complex photonic band diagram andPTsymmetry

We suggest an inverse dispersion method for calculating photonic band diagram for materials with arbitrary frequency-dependent dielectric functions. The method is able to calculate the complex wave vector for a given frequency by solving the eigenvalue problem with a non-Hermitian operator. The analogy with $\cal{PT}$-symmetric Hamiltonians reveals that the operator corresponds to the momentum as a physical quantity and the singularities at the band edges are related to the branch points and responses for the features on the band edges. The method is realized using plane wave expansion technique for two-dimensional periodical structure in the case of TE- and TM-polarization. We illustrate the applicability of the method by calculation of the photonic band diagrams of an infinite two-dimension square lattice composed of dielectric cylinders using the measured frequency dependent dielectric functions of different materials (amorphous hydrogenated carbon, silicon, and chalcogenide glass). We show that the method allows to distinguish unambiguously between Bragg and Mie gaps in the spectra.

Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation

We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit.

Coqblin-Schrieffer model for an ultracold gas of ytterbium atoms with metastable state

Motivated by the impressive recent advance in manipulating cold ytterbium atoms we explore and substantiate the feasibility of realizing the Coqblin-Schrieffer model in a gas of cold fermionic $^{173}$Yb atoms. Making use of different AC polarizabillity of the electronic ground state (electronic configuration $^1S_0$) and the long lived metastable state (electronic configuration $^3P_0$), it is substantiated that the latter can be localized and serve as a magnetic impurity while the former remains itinerant. The exchange mechanism between the itinerant $^1S_0$ and the localized $^3P_0$ atoms is analyzed and shown to be antiferromagnetic. The ensuing SU(6) symmetric Coqblin-Schrieffer Hamiltonian is constructed, and, using the calculated exchange constant $J$, perturbative RG analysis yield the Kondo temperature $T_K$ that is experimentally accessible. A number of thermodynamic measurable observables are calculated in the weak coupling regime $T>T_K$ (using perturbative RG analysis) and in the strong coupling regime $T<T_K$ (employing known Bethe ansatz techniques).

Non-Hermitian neutrino oscillations in matter with PT symmetric Hamiltonians

We introduce and develop a novel approach to extend the ordinary two-flavor neutrino oscillation formalism in matter using a non-Hermitian PT symmetric effective Hamiltonian. The condition of PT symmetry is weaker and less mathematical than that of hermicity, but more physical, and such an extension of the formalism can give rise to sub-leading effects in neutrino flavor transitions similar to the effects by so-called non-standard neutrino interactions. We derive the necessary conditions for the spectrum of the effective Hamiltonian to be real as well as the mappings between the fundamental and effective parameters. We find that the real spectrum of the effective Hamiltonian will depend on all new fundamental parameters introduced in the non-Hermitian PT symmetric extension of the usual neutrino oscillation formalism and that either i) the spectrum is exact and the effective leptonic mixing must always be maximal or ii) the spectrum is approximate and all new fundamental parameters must be small.

Quantum eigenvalue estimation for irreducible non-negative matrices

Quantum phase estimation algorithm (PEA) has been successfully adapted as a sub frame of many other algorithms applied to a wide variety of applications in different fields. However, the requirement of a good approximate eigenvector given as an input to the algorithm hinders the application of the algorithm to the problems where we do not have any prior knowledge about the eigenvector. In this paper, we show that the principal eigenvalue of an irreducible non-negative operator can be determined by using an equal superposition initial state in the PEA. This removes the necessity of the existence of an initial good approximate eigenvector. Moreover, we show that the success probability of the algorithm is related to the closeness of the operator to a stochastic matrix. Therefore, we draw an estimate for the success probability by using the variance of the column sums of the operator. This provides a priori information which can be used to know the success probability of the algorithm beforehand for the non-negative matrices and apply the algorithm only in cases when the estimated probability is reasonably high. Finally, we discuss the possible applications and show the results for random symmetric matrices and 3-local Hamiltonians with non-negative off-diagonal elements.

PassivePT-symmetric couplers without complex optical potentials

In addition to the implementation of parity-time ($\mathcal{PT}$)-symmetric optical systems by carefully and actively controlling the gain and loss, we show that a $2\times 2$ $\mathcal{PT}$-symmetric Hamiltonian has a unitarily equivalent representation without complex optical potentials in the resulting optical coupler. Through the Naimark dilation in operator algebra, passive $\mathcal{PT}$-symmetric couplers can thus be implemented with a refractive index of real values and asymmetric coupling coefficients. This opens up the possibility to implement general $\mathcal{PT}$-symmetric systems with state-of-the-art asymmetric slab waveguides, dissimilar optical fibers, or cavities with chiral mirrors.

Exactly solvable -symmetric models in two dimensions

Non-Hermitian, -symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, potentials for a non-relativistic particle confined in a circular geometry. We show that the -symmetry threshold can be tuned by introducing a second gain-loss potential or its Hermitian counterpart. Our results explicitly demonstrate that breaking in two dimensions has a rich phase diagram, with multiple re-entrant -symmetric phases.

Painlevé transcendents and -symmetric Hamiltonians

Unstable separatrix solutions for the first and second Painlevé transcendents are studied both numerically and analytically. For a fixed initial condition, say there is a discrete set of initial slopes that give rise to separatrix solutions. Similarly, for a fixed initial slope, say y’(0) = 0, there is a discrete set of initial values that give rise to separatrix solutions. For Painlevé I the large-n asymptotic behavior of bn is () and that of cn is (), and for Painlevé II the large-n asymptotic behavior of bn is () and that of cn is (). The constants and which are the coefficients in these asymptotic behaviors, are first determined numerically. Then, by using asymptotic methods, they are found analytically by reducing the nonlinear equations to the linear eigenvalue problems associated with the cubic and quartic -symmetric Hamiltonians and

Coherent optical control of polarization with a critical metasurface

We describe the mechanism by which a metamaterial surface can act as an ideal phase-controlled rotatable linear polarizer. With equal-power linearly polarized beams incident on each side of the surface, varying the relative phase rotates the polarization angles of the output beams, while keeping the polarization exactly linear. The explanation is based on coupled-mode theory and the idea of coherent perfect absorption into auxiliary polarization channels. The polarization-rotating behavior occurs at a critical point of the coupled-mode theory, which can be associated with the exceptional point of a parity-time (PT) symmetric effective Hamiltonian.