Symmetric Hamiltonians Research Articles

The critical Casimir effect in films for generic non-symmetry-breaking boundary conditions

Systems described by an O(n) symmetrical ϕ4 Hamiltonian are considered in a d-dimensional film geometry at their bulk critical points. A detailed renormalization group (RG) study of the critical Casimir forces induced between the film's boundary planes , by thermal fluctuations is presented for the case where the O(n) symmetry remains unbroken by the surfaces. The boundary planes are assumed to cause short-range disturbances of the interactions that can be modelled by standard surface contributions ∝ϕ2 corresponding to subcritical or critical enhancement of the surface interactions. This translates into mesoscopic boundary conditions of the generic symmetry-preserving Robin type . RG-improved perturbation theory and Abel–Plana techniques are used to compute the L-dependent part fres of the reduced excess free energy per film area A → ∞ to two-loop order. When d < 4, it takes the scaling form fres ≈ D(c1LΦ/ν,c2LΦ/ν)/Ld−1 as L → ∞, where ci are scaling fields associated with the surface-enhancement variables c˚i, while Φ is a standard surface crossover exponent. The scaling function D(c1,c2) and its analogue for the Casimir force are determined via expansion in ϵ = 4 − d and extrapolated to d = 3 dimensions. In the special case c1 = c2 = 0, the expansion becomes fractional. Consistency with the known fractional expansions of D(0,0) and to order ϵ3/2 is achieved by appropriate reorganization of RG-improved perturbation theory. For appropriate choices of c1 and c2, the Casimir forces can have either sign. Furthermore, crossovers from attraction to repulsion and vice versa may occur as L increases.

Energy level structure and quantum phase transitions of spin systems with nonaxially symmetric Hamiltonians

A general spin system with a nonaxially symmetric Hamiltonian containing Jx, Jz-linear and Jz-quadratic terms, widely used in many-body fermionic and bosonic systems and in molecular magnetism, is considered for the variations of general parameters describing intensity interaction changes of each of its terms. For this model Hamiltonian, a semiclassical energy surface (ES) is obtained by means of the coherent-state formalism. An analysis of this ES function, based on catastrophe theory, determines the separatrix in the control parameter space of the system Hamiltonian: the loci of singularities representing semiclassical phase transitions. Here we show that distinct regions of qualitatively different spectrum structures, as well as a singular behavior of quantum states, are ruled by this separatrix: here we show that the separatrix not only describes ground-state singularities, which have been associated with quantum phase transitions, but also reveals the structure of the excited spectrum, distinguishing different quantum phases within the parameter space. Finally, we consider magnetic susceptibility and heat capacity of the system at finite temperature, in order to study thermal properties and thermodynamical phase transitions in the perspective of the separatrix of this Hamiltonian system.

Absence of topological insulator phases in non-HermitianPT-symmetric Hamiltonians

In this work, we consider a generalization of the symmetry classification of topological insulators to non-Hermitian Hamiltonians, which satisfy a combined $PT$ symmetry (parity and time reversal). We show via examples and explicit proofs from separate bulk and gapless boundary-state perspectives that the typical paradigm of forming topological insulator states from Dirac Hamiltonians is not compatible with the construction of non-Hermitian $PT$-symmetric Hamiltonians. The topological insulator states are $PT$-breaking phases and have energy spectra that are complex (not real) and, thus, are not consistent quantum theories.

$\mathcal{PT}$ -Symmetric Periodic Optical Potentials

In quantum theory, any Hamiltonian describing a physical system is mathematically represented by a self-adjoint linear operator to ensure the reality of the associated observables. In an attempt to extend quantum mechanics into the complex domain, it was realized few years ago that certain non-Hermitian parity-time (\(\mathcal{PT}\)) symmetric Hamiltonians can exhibit an entirely real spectrum. Much of the reported progress has been remained theoretical, and therefore hasn’t led to a viable experimental proposal for which non Hermitian quantum effects could be observed in laboratory experiments. Quite recently however, it was suggested that the concept of \(\mathcal{PT}\)-symmetry could be physically realized within the framework of classical optics. This proposal has, in turn, stimulated extensive investigations and research studies related to \(\mathcal{PT}\)-symmetric Optics and paved the way for the first experimental observation of \(\mathcal{PT}\)-symmetry breaking in any physical system. In this paper, we present recent results regarding \(\mathcal{PT}\)-symmetric Optics.

Exchange Perturbation Theory for Multiatomic Electron System and Its Application to Spin Arrangement in Manganite Chains

A new methodology of binding energy calculation with respect to different spin arrangements for a multiatomic electron system is developed from the first principle in the frame of the exchange perturbation theory (EPT). We developed EPT formalism in the general form of the Rayleigh‐Srchödinger expansion with a symmetric Hamiltonian, taking into account an exchange and nonadditive contributions of a superexchange interaction. The expressions of all corrections to the energy and wave function were reduced to the nonsymmetric Hamiltonian form. The EPT method is extended for the case of degeneracy in the total spin of a system. As an example of the application of the developed EPT formalism for the degeneracy case, spin arrangements were considered for the key 〈Mn〉–O–〈Mn〉 (〈Mn〉: Mn3+ or Mn4+) fragments in manganites. In 〈Mn〉–O–〈Mn〉 for La1/3Ca2/3MnO3 are in good agreement the obtained estimations of Heisenberg parameter and binding energy with the available experimental data.

$\mathcal{PT}$ Symmetry in Statistical Mechanics and the Sign Problem

Generalized \(\mathcal{PT}\) symmetry provides crucial insight into the sign problem for two classes of models. In the case of quantum statistical models at non-zero chemical potential, the free energy density is directly related to the ground state energy of a non-Hermitian, but generalized \(\mathcal{PT}\)-symmetric Hamiltonian. There is a corresponding class of \(\mathcal{PT}\)-symmetric classical statistical mechanics models with non-Hermitian transfer matrices. We discuss a class of Z(N) spin models with explicit \(\mathcal{PT}\) symmetry and also the ANNNI model, which has a hidden \(\mathcal{PT}\) symmetry. For both quantum and classical models, the class of models with generalized \(\mathcal{PT}\) symmetry is precisely the class where the complex weight problem can be reduced to real weights, i.e., a sign problem. The spatial two-point functions of such models can exhibit three different behaviors: exponential decay, oscillatory decay, and periodic behavior. The latter two regions are associated with \(\mathcal{PT}\) symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with \(\mathcal{PT}\) symmetry breaking of the ground state, and is generically a first-order transition. In the region where \(\mathcal{PT}\) symmetry is unbroken, the sign problem can always be solved in principle using the equivalence to a Hermitian theory in this region. The ANNNI model provides an example of a model with \(\mathcal{PT}\) symmetry which can be simulated for all parameter values, including cases where \(\mathcal{PT}\) symmetry is broken.

A uniform object-oriented solution to the eigenvalue problem for real symmetric and Hermitian matrices

We present a system of classes, SHMatrix, to deal in a unified way with the computation of eigenvalues and eigenvectors in real symmetric and Hermitian matrices. Thus, two descendant classes, one for the real symmetric and other for the Hermitian cases, override the abstract methods defined in a base class. The use of the inheritance relationship and polymorphism allows handling objects of any descendant class using a single reference of the base class. The system of classes is intended to be the core element of more sophisticated methods to deal with large eigenvalue problems, as those arising in the variational treatment of realistic quantum mechanical problems. The present system of classes allows computing a subset of all the possible eigenvalues and, optionally, the corresponding eigenvectors. Comparison with well established solutions for analogous eigenvalue problems, as those included in LAPACK, shows that the present solution is competitive against them. Program summary Program title: SHMatrix Catalogue identifier: AEHZ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHZ_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2616 No. of bytes in distributed program, including test data, etc.: 127 312 Distribution format: tar.gz Programming language: Standard ANSI C++. Computer: PCs and workstations. Operating system: Linux, Windows. Classification: 4.8. Nature of problem: The treatment of problems involving eigensystems is a central topic in the quantum mechanical field. Here, the use of the variational approach leads to the computation of eigenvalues and eigenvectors of real symmetric and Hermitian Hamiltonian matrices. Realistic models with several degrees of freedom leads to large (sometimes very large) matrices. Different techniques, such as divide and conquer, can be used to factorize the matrices in order to apply a parallel computing approach. However, it is still interesting to have a core procedure able to tackle the computation of eigenvalues and eigenvectors once the matrix has been factorized to pieces of enough small size. Several available software packages, such as LAPACK, tackled this problem under the traditional imperative programming paradigm. In order to ease the modelling of complex quantum mechanical models it could be interesting to apply an object-oriented approach to the treatment of the eigenproblem. This approach offers the advantage of a single, uniform treatment for the real symmetric and Hermitian cases. Solution method: To reach the above goals, we have developed a system of classes: SHMatrix. SHMatrix is composed by an abstract base class and two descendant classes, one for real symmetric matrices and the other for the Hermitian case. The object-oriented characteristics of inheritance and polymorphism allows handling both cases using a single reference of the base class. The basic computing strategy applied in SHMatrix allows computing subsets of eigenvalues and (optionally) eigenvectors. The tests performed show that SHMatrix is competitive, and more efficient for large matrices, than the equivalent routines of the LAPACK package. Running time: The examples included in the distribution take only a couple of seconds to run.

Extending $\mathcal{P}\mathcal{T}$ Symmetry from Heisenberg Algebra to E2 Algebra

The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=−iu, [u,v]=0. We can construct the Hamiltonian H=J 2+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the $\mathcal{P}\mathcal{T}$ -symmetric and non-Hermitian Hamiltonian H=J 2+igu, where again g is real. As in the case of $\mathcal{P}\mathcal{T}$ -symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this $\mathcal{P}\mathcal{T}$ -symmetric Hamiltonian, a region of unbroken $\mathcal{P}\mathcal{T}$ symmetry in which all the eigenvalues are real and a region of broken $\mathcal{P}\mathcal{T}$ symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.

Fock space expansion of σ meson in leading- Nc

We examine the leading-Nc behavior of the masses and transition matrix elements of some low-lying, few-particle configurations in QCD. A truncation of the Fock space produces an effective, symmetric Hamiltonian that we diagonalize. The lowest eigenvalue is identified as the sigma meson if the Hamiltonian is chosen to represent the scalar sector. As an application, the coefficients of the Nc powers are then fit to two-loop Unitarized SU(2) Chiral Perturbation Theory results for the sigma mass and width as a function of the number of colors, and we show that those results can be accommodated using the QCD Nc dependence previously derived for matrix elements, without the need for unnatural parameters or fine tunings. Finally, we show a very preliminary good quality fit, estimating the proportion of tetraquark/molecule-like (dominant), q-anti-q-like (subdominant) and exotic-like (marginal) configurations in the sigma.

ChemInform Abstract: The Microwave Spectrum of Cyclobutadiene Iron Tricarbonyl.

Abstract The rotational spectrum of cyclobutadiene iron tricarbonyl was measured using a Flygare-Balle-type microwave spectrometer. Twenty a -dipole transitions were measured in the 5–16 GHz range for this prolate symmetric-top molecule. A least-squares fit of the data to a distortable symmetric top Hamiltonian yielded B = 961.9856(8) MHz, D J = 0.184(8) kHz, D JK = 1.20(3) kHz. These results indicate that the vibrationally averaged structure of the cyclobutadiene ring is square and perpendicular to the a -molecular axis.

Order-dependent mappings: Strong-coupling behavior from weak-coupling expansions in non-Hermitian theories

A long time ago, it has been conjectured that a Hamiltonian with a potential of the form x2+ivx3, v real, has a real spectrum. This conjecture has been generalized to a class of the so-called PT symmetric Hamiltonians and some proofs have been given. Here, we show by numerical investigation that the divergent perturbation series can be summed efficiently by an order-dependent mapping (ODM) in the whole complex plane of the coupling parameter v2, and that some information about the location of level-crossing singularities can be obtained in this way. Furthermore, we discuss to which accuracy the strong-coupling limit can be obtained from the initially weak-coupling perturbative expansion, by the ODM summation method. The basic idea of the ODM summation method is the notion of order-dependent “local” disk of convergence and analytic continuation by an ODM of the domain of analyticity augmented by the local disk of convergence onto a circle. In the limit of vanishing local radius of convergence, which is the limit of high transformation order, convergence is demonstrated both by numerical evidence as well as by analytic estimates.

Symmetry as a generalization of Hermiticity

The Hilbert space in -symmetric quantum mechanics is formulated as a linear vector space with a dynamic inner product. The most general -symmetric matrix Hamiltonians are constructed for the 2 × 2 and 3 × 3 cases. In the former case, the -symmetric Hamiltonian represents the most general matrix Hamiltonian with a real spectrum. In both cases, Hermitian matrices are shown to be special cases of -symmetric matrices. This finding confirms and strengthens the early belief that the -symmetric quantum mechanics is a generalization of the conventional Hermitian quantum mechanics.

Quantum Noise and Self-Sustained Radiation ofPT-Symmetric Systems

The observation that PT-symmetric Hamiltonians can have real-valued energy levels even if they are non-Hermitian has triggered intense activities, with experiments, in particular, focusing on optical systems, where Hermiticity can be broken by absorption and amplification. For classical waves, absorption and amplification are related by time-reversal symmetry. This work shows that microreversibility-breaking quantum noise turns PT-symmetric systems into self-sustained sources of radiation, which distinguishes them from ordinary, Hermitian quantum systems.

Pairing in 4-component fermion systems: The bulk limit of SU(4)-symmetric Hamiltonians

Fermion systems with more than two components can exhibit pairing condensates of a much more complex structure than the well-known single BCS condensate of spin-up and spin-down fermions. In the framework of the exactly solvable SO(8) Richardson–Gaudin (RG) model with SU(4)-symmetric Hamiltonians, we show that the BCS approximation remains valid in the thermodynamic limit of large systems for describing the ground-state energy and the canonical and quasiparticle excitation gaps. Correlations beyond BCS pairing give rise to a spectrum of collective excitations, but these do not affect the bulk energy and quasiparticle gaps.

Formula omitted] symmetry and necessary and sufficient conditions for the reality of energy eigenvalues

The mathematical requirement that a Hamiltonian H be Hermitian is sufficient to guarantee the reality of its eigenvalues, but it is not necessary. By establishing three theorems, this Letter gives conditions that are both necessary and sufficient. First, it is shown that if the secular equation is real, the Hamiltonian is necessarily PT symmetric. Second, the set of all operators C that nontrivially obey the two equations [ C , H ] = 0 and C 2 = 1 is introduced. For a diagonalizable PT -symmetric Hamiltonian it is shown that the set is nonempty and that the energy eigenvalues of H are all real only if every such C commutes with PT . There is at least one complex-conjugate pair of eigenvalues whenever at least one such C does not commute with PT . Third, the eigenvalues of any nondiagonalizable Jordan-block matrix that possesses just one eigenvector are all real if the block is PT symmetric, and are all complex if it is not. These theorems hold for matrix Hamiltonians of any finite dimensionality.

PT Symmetric Schrödinger Operators: Reality of the Perturbed Eigenvalues

We prove the reality of the perturbed eigenvalues of some PT symmetric Hamiltonians of physical interest by means of stability methods. In particular we study 2-dimensional generalized harmonic oscillators with polynomial perturbation and the one-dimensional $x^2(ix)^{\epsilon}$ for $-1<\epsilon<0$.

Symmetry breaking and exceptional points for a class of inhomogeneous complex potentials

We study a three-parameter family of -symmetric Hamiltonians, related via the ODE/IM correspondence to the Perk–Schultz models. We show that real eigenvalues merge and become complex at quadratic and cubic exceptional points, and explore the corresponding Jordan block structures by exploiting the quasi-exact solvability of a subset of the models. The mapping of the phase diagram is completed using a combination of numerical, analytical and perturbative approaches. Among other things this reveals some novel properties of the Bender–Dunne polynomials, and gives new insight into a phase transition to infinitely many complex eigenvalues that was first observed by Bender and Boettcher. A new exactly solvable limit, the inhomogeneous complex square well, is also identified.

Small-ϵ behavior of the non-Hermitian -symmetric Hamiltonian H = p2 + x2(ix)ϵ

The energy eigenvalues of the class of non-Hermitian -symmetric Hamiltonians H = p2 + x2(ix)ϵ (ϵ ⩾ 0) are real, positive and discrete. The behavior of these eigenvalues has been studied perturbatively for small ϵ. However, until now no other features of H have been examined perturbatively. In this paper the small-ϵ expansion of the operator and the equivalent isospectral Dirac–Hermitian Hamiltonian h are derived.

PT Symmetry and QCD: Finite Temperature and Density

The relevance of PT symmetry to quantum chromodynamics (QCD), the gauge theory of the strong interactions, is explored in the context of finite temperature and density. Two significant problems in QCD are studied: the sign problem of finite-density QCD, and the problem of confinement. It is proven that the effective action for heavy quarks at finite density is PT -symmetric. For the case of 1+1 dimensions, the PT -symmetric Hamiltonian, although not Hermitian, has real eigenvalues for a range of values of the chemical potential µ, solving the sign problem for this model. The effective action for heavy quarks is part of a potentially large class of generalized sine-Gordon models which are non-Hermitian but are PT -symmetric. Generalized sine-Gordon models also occur naturally in gauge theories in which magnetic monopoles lead to confinement. We explore gauge theories where monopoles cause confinement at arbitrarily high temperatures. Several different classes of monopole gases exist, with each class leading to different string tension scaling laws. For one class of monopole gas models, the PT -symmetric affine Toda field theory emerges naturally as the effective theory. This in turn leads to sine-law scaling for string tensions, a behavior consistent with lattice simulations.

Adding a complex shift to HVPT

A complex coordinate shift is incorporated in renormalized hypervirial perturbation theory. Test calculations on harmonic oscillators with cubic perturbations show that the resulting method produces accurate numerical results for the real energies of PT symmetric Hamiltonians with complex potentials and for the complex resonance energies of Hamiltonians with real potentials.