Spectrum-generating Algebra Research Articles

η -pairing in Hubbard models: From spectrum generating algebras to quantum many-body scars

We revisit the $\eta$-pairing states in Hubbard models and explore their connections to quantum many-body scars to discover a universal scars mechanism. $\eta$-pairing occurs due to an algebraic structure known as a Spectrum Generating Algebra (SGA), giving rise to equally spaced towers of eigenstates in the spectrum. We generalize the original $\eta$-pairing construction and show that several Hubbard-like models on arbitrary graphs exhibit SGAs, including ones with disorder and spin-orbit coupling. We further define a Restricted Spectrum Generating Algebra (RSGA) and give examples of perturbations to the Hubbard-like models that preserve an equally spaced tower of the original model as eigenstates. The states of the surviving tower exhibit a sub-thermal entanglement entropy, and we analytically obtain parameter regimes for which they lie in the bulk of the spectrum, showing that they are exact quantum many-body scars. The RSGA framework also explains the equally spaced towers of eigenstates in several well-known models of quantum scars, including the AKLT model.

From worldline to quantum superconformal mechanics with and without oscillatorial terms: D(2,1;α) and sl(2|1) models

In this paper, we quantize superconformal $\ensuremath{\sigma}$ models defined by worldline supermultiplets. Two types of superconformal mechanics, with and without a De Alfaro Fubini Furlan (DFF) term, are considered. Without a DFF term (Calogero potential only), the supersymmetry is unbroken. The models with a DFF term correspond to deformed (if the Calogero potential is present) or undeformed oscillators. For these (un)deformed oscillators, the classical invariant superconformal algebra acts as a spectrum-generating algebra of the quantum theory. Besides the $osp(1|2)$ examples, we explicitly quantize the superconformally invariant worldline $\ensuremath{\sigma}$ models defined by the $\mathcal{N}=4$ (1, 4, 3) supermultiplet [with $D(2,1;\ensuremath{\alpha})$ invariance, for $\ensuremath{\alpha}\ensuremath{\ne}0,\ensuremath{-}1$] and by the $\mathcal{N}=2$ (2, 2, 0) supermultiplet [with two-dimensional target and $sl(2|1)$ invariance]. The parameter $\ensuremath{\alpha}$ is the scaling dimension of the (1, 4, 3) supermultiplet and, in the DFF case, has a direct interpretation as a vacuum energy. In the DFF case, for the $sl(2|1)$ models, the scaling dimension $\ensuremath{\lambda}$ is quantized (either $\ensuremath{\lambda}=\frac{1}{2}+\mathbb{Z}$ or $\ensuremath{\lambda}=\mathbb{Z}$). The ordinary two-dimensional oscillator is recovered, after imposing a superselection restriction, from the $\ensuremath{\lambda}=\ensuremath{-}\frac{1}{2}$ model. In particular, a single bosonic vacuum is selected. The spectrum of the unrestricted two-dimensional theory is decomposed into an infinite set of lowest-weight representations of $sl(2|1)$. Extra fermionic raising operators, not belonging to the original $sl(2|1)$ superalgebra, allow (for $\ensuremath{\lambda}=\frac{1}{2}+\mathbb{Z}$) to construct the whole spectrum from the two degenerate (one bosonic and one fermionic) vacua.

A comparative study of the vibrational spectra of HCN in exact vibron model and in mean field approximation

The aim of this work is to study the highly excited vibrational states of hydrogen cyanide HCN in the exact vibron model and with mean field approximation in the vibron model. Considering the U(4) ⊗ U(4) spectrum-generating algebra for linear triatomic molecules, the standard Hamiltonian is constructed using the linear and quadratic combination of Casimir operators. For higher order corrections, the quadratic contributions of Casimir operators are used to construct the Hamiltonian. Using this Hamiltonian the higher excited vibrational levels of HCN are calculated in the local mode approximation. The energy levels are observed as a function of vibron number N. The best fit is obtained for N = 184 (N1 = 139, N2 = 45) with root mean square (r.m.s.) deviation 5.598 cm−1. The intermodal coupling within the same polyad is studied and addressed properly by introducing Majorana operator. The r.m.s. deviation is then reduced to 4.755 cm−1. The modification is negligible, which indicates the local nature of HCN. In this work, 35 experimental levels are taken for fit, out of which only two sets of levels are accidentally degenerate. The Fermi resonances of the accidentally degenerate levels are studied using the Fermi operator and r.m.s. deviation becomes 4.835. The coefficient of Majorana and Fermi coupling for different levels are obtained by diagonalzing the Majorana and Fermi matrices for each polyad. The Majorana and Fermi matrices for each polyad are diagonalized with the MATRIX CALCULATOR program. The algebraic parameters are evaluated by a least square fit against the experimental data using MATLAB R2015a. Using this model, a set of energy levels is predicted up to 30 000 cm−1, with very good accuracy. HCN is chosen for this study, because, its vibrational states can be fairly described without any modification due to Fermi resonance. The fundamental vibrational levels of HCN are again calculated, using mean field approximation and compared to those obtained using the vibron model. A good agreement is observed.

AdS perturbations, isometries, selection rules and the Higgs oscillator

Dynamics of small-amplitude perturbations in the global anti-de Sitter (AdS) spacetime is restricted by selection rules that forbid effective energy transfer between certain sets of normal modes. The selection rules arise algebraically because some integrals of products of AdS mode functions vanish. Here, we reveal the relation of these selection rules to AdS isometries. The formulation we discover through this systematic approach is both simpler and stronger than what has been reported previously. In addition to the selection rule considerations, we develop a number of useful representations for the global AdS mode functions, with connections to algebraic structures of the Higgs oscillator, a superintegrable system describing a particle on a sphere in an inverse cosine-squared potential, where the AdS isometries play the role of a spectrum-generating algebra.

Classical Analog of Extended Phase Space SUSY and Its Breaking

We derive the classical analog of the extended phase space quantum mechanics of the particle with odd degrees of freedom which gives rise to ()-realization of supersymmetry (SUSY) algebra. By means of an iterative procedure, we find the approximate ground state solutions to the extended Schrödinger-like equation and use these solutions further to calculate the parameters which measure the breaking of extended SUSY such as the ground state energy. Consequently, we calculate a more practical measure for the SUSY breaking which is the expectation value of an auxiliary field. We analyze nonperturbative mechanism for extended phase space SUSY breaking in the instanton picture and show that this has resulted from tunneling between the classical vacua of the theory. Particular attention is given to the algebraic properties of shape invariance and spectrum generating algebra.

Spectrum generating algebra for the continuous spectrum of a free particle in Lobachevski space

In this paper, we construct a Spectrum Generating Algebra (SGA) for a quantum system with purely continuous spectrum: the quantum free particle in a Lobachevski space with constant negative curvature. The SGA contains the geometrical symmetry algebra of the system plus a subalgebra of operators that give the spectrum of the system and connects the eigenfunctions of the Hamiltonian among themselves. In our case, the geometrical symmetry algebra is $\mathfrak {so}(3,1)$so(3,1) and the SGA is $\mathfrak {so}(4,2)$so(4,2). We start with a representation of $\mathfrak {so}(4,2)$so(4,2) by functions on a realization of the Lobachevski space given by a two-sheeted hyperboloid, where the Lie algebra commutators are the usual Poisson-Dirac brackets. Then, we introduce a quantized version of the representation in which functions are replaced by operators on a Hilbert space and Poisson-Dirac brackets by commutators. Eigenfunctions of the Hamiltonian are given and “naive” ladder operators are identified. The previously defined “naive” ladder operators shift the eigenvalues by a complex number so that an alternative approach is necessary. This is obtained by a non-self-adjoint function of a linear combination of the ladder operators, which gives the correct relation among the eigenfunctions of the Hamiltonian. We give an eigenfunction expansion of functions over the upper sheet of a two-sheeted hyperboloid in terms of the eigenfunctions of the Hamiltonian.

From Coalgebra to Bialgebra for the Six-Vertex Model: The Star-Triangle Relation as a Necessary Condition for Commuting Transfer Matrices

Using the most elementary methods and considerations, the solution of the star-triangle condition (a2+b2-c2)/2ab = ((a’)^2+(b’)^2-(c’))^2/2a’b’ is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representations of the bialgebra can be constructed. The star-triangle relation is proved to be necessary for the commutativity of the transfer matrices T(a, b, c) and T(a’, b’, c’).

Exact solutions of the Schrödinger equation for the pseudoharmonic potential: an application to some diatomic molecules

For arbitrary values n and l quantum numbers, we present the solutions of the 3-dimensional Schrodinger wave equation with the pseudoharmonic potential via the SU(1, 1) Spectrum Generating Algebra (SGA) approach. The explicit bound state energies and eigenfunctions are obtained. The matrix elements r 2 and $${r\frac{d}{dr}}$$ are obtained (in a closed form) directly from the creation and annihilation operators. In addition, by applying the Hellmann–Feynman theorem, the expectation values of r 2 and p 2 are obtained. The energy states, the expectation values of r 2 and p 2 and the Heisenberg uncertainty products (HUP) for set of diatomic molecules (CO, NO, O2, N2, CH, H2, ScH) for arbitrary values of n and l quantum numbers are obtained. The results obtained are in excellent agreement with the available results in the literature. It is also shown that the HUP is obeyed for all diatomic molecules considered.

Finite-size corrections in the bosonic algebraic approach to two-dimensional systems

Analytical results beyond the mean-field-limit approximation for several observables of the two-dimensional limit of the vibron model are presented. Finite-size scaling exponent values are also analytically derived. The computed corrections and scaling exponents are compared to numerical calculations for this algebraic model, based on a bosonic u(3) spectrum-generating algebra, obtaining good agreement.

Spectrum generating algebras for the free motion in S3

We construct the spectrum generating algebra (SGA) for a free particle in the three-dimensional sphere S3 for both classical and quantum descriptions. In the classical approach, the SGA supplies time-dependent constants of motion that allow to solve algebraically the motion. In the quantum case, the SGA includes the ladder operators that give the eigenstates of the free Hamiltonian. We study this quantum case from two equivalent points of view.

Closed superstrings in a uniform magnetic field and regularization criterion

We summarize exact solutions of closed superstrings in a constant magnetic field, from a view point of the regularization criterion. Some models will be excluded according to this criterion. The spectrum-generating algebra is also constructed in these interacting models.

Algorithm for simulation of quantum many-body dynamics using dynamical coarse-graining

An algorithm for simulation of quantum many-body dynamics having su(2) spectrum-generating algebra is developed. The algorithm is based on the idea of dynamical coarse-graining. The original unitary dynamics of the target observables--the elements of the spectrum-generating algebra--is simulated by a surrogate open-system dynamics, which can be interpreted as weak measurement of the target observables, performed on the evolving system. The open-system state can be represented by a mixture of pure states, localized in the phase space. The localization reduces the scaling of the computational resources with the Hilbert-space dimension n by factor n{sup 3/2}(ln n){sup -1} compared to conventional sparse-matrix methods. The guidelines for the choice of parameters for the simulation are presented and the scaling of the computational resources with the Hilbert-space dimension of the system is estimated. The algorithm is applied to the simulation of the dynamics of systems of 2x10{sup 4} and 2x10{sup 6} cold atoms in a double-well trap, described by the two-site Bose-Hubbard model.

The globally stable solution of a stochastic nonlinear Schrödinger equation

Weak measurement of a subset of noncommuting observables of a quantum system can be modeled by the open-system evolution, governed by the master equation in the Lindblad form. The open-system density operator can be represented as a statistical mixture over non-unitarily evolving pure states, driven by the stochastic nonlinear Schrödinger equation (sNLSE). The globally stable solution of the sNLSE is obtained in the case where the measured subset of observables comprises the spectrum-generating algebra of the system. This solution is a generalized coherent state (GCS), associated with the algebra. The result is based on proving that the GCS minimizes the trace-norm of the covariance matrix, associated with the spectrum-generating algebra.

Efficient simulation of quantum evolution using dynamical coarse graining

A scheme to simulate the evolution of a restricted set of observables of a quantum system is proposed. The set comprises the spectrum-generating algebra of the Hamiltonian. Focusing on the simulation of the restricted set allows to drastically reduce the cost of the simulation. This reduction is the result of replacing the original unitary dynamics by a special open-system evolution. This open-system evolution can be interpreted as a process of weak measurement of the distinguished observables performed on the evolving system of interest. Under the condition that the observables are ``classical'' and the Hamiltonian is moderately nonlinear, the open-system dynamics displays a large time-scale separation between the relaxation of the observables and the decoherence of a generic state. This time-scale separation allows the unitary dynamics of the observables to be efficiently simulated by the open-system dynamics on the intermediate time scale. The simulation employs unraveling of the corresponding master equations into pure-state evolutions, governed by the stochastic nonlinear Schr\odinger equation. The stochastic pure-state evolution can be simulated efficiently using a representation of the state in the time-dependent basis of the generalized coherent states, associated with the spectrum-generating algebra.

Discrete representations for the deformed su(1, 1) algebra via the magnetic monopole harmonics

It is well known that the magnetic quantum number m of monopole harmonics describes quantization corresponding to the z-component of the angular momentum operator in the framework of su(2) symmetry algebra. Here, it is shown that the azimuthal quantum number l allocates itself a ladder symmetry by the operators which are described in terms of l. Furthermore, quantization of both quantum numbers l and m leads to extracting positive and negative irreducible discrete representations of the deformed su(1, 1) algebra for l − m and l + m, respectively. It will also be shown in detail how they in turn lead to the spectrum-generating algebra for all monopole harmonics.

Equations of motion for a spectrum-generating algebra: Lipkin–Meshkov–Glick model

For a spectrum-generating Lie algebra, a generalized equations-of-motion scheme determines numerical values of excitation energies and algebra matrix elements. In the approach to the infinite particle number limit or, more generally, whenever the dimension of the quantum state space is very large, the equations-of-motion method may achieve results that are impractical to obtain by diagonalization of the Hamiltonian matrix. To test the method's effectiveness, we apply it to the well-known Lipkin–Meshkov–Glick (LMG) model to find its low-energy spectrum and associated generator matrix elements in the eigenenergy basis. When the dimension of the LMG representation space is 106, computation time on a notebook computer is a few minutes. For a large particle number in the LMG model, the low-energy spectrum makes a quantum phase transition from a nondegenerate harmonic vibrator to a twofold degenerate harmonic oscillator. The equations-of-motion method computes critical exponents at the transition point.

Spectrum-Generating Algebra for Charged String

Spectrum-Generating Algebra for Charged String

N = 4 Supersymmetric Morse Oscillator andIts Spectrum-Generating Algebra**The projectsupported by National Natural Science Foundation of Chinaunder Grant No. 10275038, and partly by the State Key BasicResearch Development Program under Grant No. G2000077400,and Natural Science Foundation of Tsinghua University

In this paper N = 4 supersymmetry of generalized Morse oscillators inone dimension is studied. Both bound states and scattering states of its foursuperpartner Hamiltonians are analyzed by using unitary irreducible representationsof the noncompact Lie algebra su(1,1).The spectrum-generating algebra governing the Hamiltonian of the N = 4supersymmetric Morse oscillator is shown to be connected with the realization of Liesuperalgebra osp(1,2) or B(0,1) in terms of the variables ofa supersymmetric two-dimensional harmonic oscillator.

Analysis of the spectrum generating algebra method for obtaining energy spectra

We analyze and clarify how the SGA (spectrum generating algebra) method has been applied to different potentials. We emphasize that each energy level Eν obtained originally by Morse belongs to a different so(2,1) multiplet. The corresponding wave functions Ψν are eigenfuntions of the compact generators J0ν with the same eigenvalue k0, but with different eigenvalues qν of the Casimir operators Q. We derive a general expression for all effective potentials which have Ψλν,ν+m(r)∝(J+ν)mΨλν,ν(r) as eigenfunctions, without using supersymmetry formalism. The different actions of SGA is further illustrated by two diagrams.

Generalized Coherent States Associated with the Cλ-Extended Oscillator

Two new types of coherent states associated with the Cλ-extended oscillator, where Cλ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut–Girardello and the Perelomov su(1,1) coherent states for λ=2, as well as the annihilation-operator coherent states of the Cλ-extended oscillator spectrum-generating algebra for higher λ values. The second ones, which are eigenstates of the Cλ-extended oscillator annihilation operator, extend to higher λ values the paraboson coherent states, to which they reduce for λ=2. All these states satisfy a unity resolution relation in the Cλ-extended oscillator Fock space (or in some subspace thereof). They give rise to Bargmann representations of the latter, wherein the generators of the Cλ-extended oscillator algebra are realized as differential-operator-valued matrices (or differential operators). The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited.