Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless fermion (or boson) systems, with say m$m$ fermions (or bosons) in N$N$ single particle states and interacting via k$k$-body interactions, we have EGUE(k$k$) [embedded GUE of k$k$-body interactions] with GUE embedding and the embedding algebra is U(N)$U\left(N\right)$. A finite quantum system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different), particle addition to or removal from a given system and so on. Towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and final energies), we have derived formulas for the lower order bivariate moments of the strength densities generated by a variety of transition operators. Firstly, for a spinless fermion system, using EGUE(k$k$) representation for a Hamiltonian that is k$k$-body and an independent EGUE(t$t$) representation for a transition operator that is t$t$-body and employing the embedding U(N)$U\left(N\right)$ algebra, finite-N$N$ formulas for moments up to order four are derived, for the first time, for the transition strength densities. Secondly, formulas for the moments up to order four are also derived for systems with two types of spinless fermions and a transition operator similar to beta decay and neutrinoless beta decay operators. In addition, moments formulas are also derived for a transition operator that removes k₀$k_0$ number of particles from a system of m$m$ spinless fermions. In the dilute limit, these formulas are shown to reduce to those for the EGOE version derived using the asymptotic limit theory of Mon and French (1975). Numerical results obtained using the exact formulas for two-body (k=2$k=2$) Hamiltonians (in some examples for k=3$k=3$ and 4$4$) and the asymptotic formulas clearly establish that in general the smoothed (with respect to energy) form of the bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extensions of these results to bosonic systems and EGUE ensembles with further symmetries are discussed.     

Thermalization in one- plus two-body ensembles for dense interacting boson systems

Employing one- plus two-body random matrix ensembles for bosons, temperature and entropy are calculated, using different definitions, as a function of the two-body interaction strength λ   for a system with 10 bosons (m=10$m=10$) in five single-particle levels (N=5$N=5$). It is found that in a region λ∼_λt$\lambda \sim {\lambda }_t$, different definitions give essentially the same values for temperature and entropy, thus defining a thermalization region. Also, (m,N)$(m,N)$ dependence of _λt${\lambda }_t$ has been derived. It is seen that _λt${\lambda }_t$ is much larger than the λ values where level fluctuations change from Poisson to GOE and strength functions change from Breit–Wigner to Gaussian.     

On the Riemannian geometry of Seiberg–Witten moduli spaces

We construct a natural L²$L^2$-metric on the perturbed Seiberg–Witten moduli spaces _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ of a compact 4-manifold M$M$, and we study the resulting Riemannian geometry of _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$. We derive a formula which expresses the sectional curvature of _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ in terms of the Green operators of the deformation complex of the Seiberg–Witten equations. In case M$M$ is simply connected, we construct a Riemannian metric on the Seiberg–Witten principal U(1)$U\left(1\right)$ bundle P→_Mμ+$\mathfrak{P}\to {\mathfrak{M}}_{{\mu }^{+}}$ such that the bundle projection becomes a Riemannian submersion. On a Kähler surface M$M$, the L²$L^2$-metric on _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ coincides with the natural Kähler metric on moduli spaces of vortices.     

Entanglement thresholds for displaying the quantum nature of teleportation

A protocol for transferring an unknown single qubit state evidences quantum features when the average fidelity of the outcomes is, in principle, greater than 2/3$2/3$. We propose to use the probabilistic and unambiguous state extraction scheme   as a mechanism to redistribute the fidelity in the outcome of the standard teleportation when the process is performed with an X$X$-state as a noisy quantum channel. We show that the entanglement of the channel is necessary but not sufficient in order for the average fidelity f_X$f_X$ to display quantum features, i.e., we find a threshold C_X$C_X$ for the concurrence of the channel. On the other hand, if the mechanism for redistributing fidelity is successful then we find a filterable outcome with average fidelity f_X,0$f_{X,0}$ that can be greater than f_X$f_X$. In addition, we find the threshold concurrence of the channel C_X,0$C_{X,0}$ in order for the average fidelity f_X,0$f_{X,0}$ to display quantum features and surprisingly, the threshold concurrence C_X,0$C_{X,0}$ can be less than C_X$C_X$. Even more, we find some special cases for which the threshold values become zero.     

The H2+ molecular ion: Low-lying states

Matching for a wavefunction the WKB expansion at large distances and Taylor expansion at small distances leads to a compact, few-parametric uniform approximation found in Turbiner and Olivares-Pilon (2011). The ten low-lying eigenstates of H₂⁺ of the quantum numbers (n,m,Λ,±)$(n,m,\Lambda ,\pm )$  with n=m=0$n=m=0$ at Λ=0,1,2$\Lambda =0,1,2$, with n=1$n=1$, m=0$m=0$ and n=0$n=0$, m=1$m=1$ at Λ=0$\Lambda =0$ of both parities are explored for all interproton distances R$R$. For all these states this approximation provides the relative accuracy ?10⁻⁵$?10^{-5}$ (not less than 5 s.d.) locally, for any real coordinate x$x$ in eigenfunctions, when for total energy E(R)$E\left(R\right)$ it gives 10-11 s.d. for R∈[0,50]$R\in [0,50]$  a.u. Corrections to the approximation are evaluated in the specially-designed, convergent perturbation theory. Separation constants are found with not less than 8 s.d. The oscillator strength for the electric dipole transitions E1$E1$ is calculated with not less than 6 s.d. A dramatic dip in the E1$E1$ oscillator strength f_{1sσg−3pσu}$f_{1s{\sigma }_g-3p{\sigma }_u}$ at R∼R_eq$R\sim R_{eq}$ is observed. The magnetic dipole and electric quadrupole transitions are calculated for the first time with not less than 6 s.d. in oscillator strength. For two lowest states (0,0,0,±)$(0,0,0,\pm )$ (or, equivalently, 1sσ_g$1s{\sigma }_g$ and 2pσ_u$2p{\sigma }_u$ states) the potential curves are checked and confirmed in the Lagrange mesh method within 12 s.d. Based on them the Energy Gap between 1sσ_g$1s{\sigma }_g$ and 2pσ_u$2p{\sigma }_u$ potential curves is approximated with modified Pade Re^−R[Pade(8/7)](R)$R{e}^{-R}\left[Pade(8/7)\right]\left(R\right)$ with not less than 4-5 figures at R∈[0,40]$R\in [0,40]$ a.u. Sum of potential curves E_1sσg+E_2pσu$E_{1s{\sigma }_g}+E_{2p{\sigma }_u}$ is approximated by Pade 1/R[Pade(5/8)](R)$1/R\left[Pade(5/8)\right]\left(R\right)$ in R∈[0,40]$R\in [0,40]$ a.u. with not less than 3-4 figures.     

The Bethe lattice treatment of sound attenuation for a spin- 3/2 Ising model

The sound attenuation phenomena is investigated for a spin- 3/2 Ising model on the Bethe lattice in terms of the recursion relations by using the Onsager theory of irreversible thermodynamics. The dependencies of sound attenuation on the temperature (T$T$), frequency (w$w$), Onsager coefficient (γ$\gamma$) and external magnetic field (H$H$) near the second-order (_Tc)$\left(T_c\right)$ and first-order (_Tt)$\left(T_t\right)$ phase transition temperatures are examined for given coordination numbers q$q$ on the Bethe lattice. It is assumed that the sound wave couples to the order-parameter fluctuations which decay mainly via the order-parameter relaxation process, thus two relaxation times are obtained and which are used to obtain an expression for the sound attenuation coefficient (α)$\left(\alpha \right)$. Our investigations revealed that only one peak is obtained near _Tt$T_t$ and three peaks are found near _Tc$T_c$ when the Onsager coefficient is varied at a given constant frequency for q=3$q=3$. Fixing the Onsager coefficient and varying the frequency always leads to two peaks for q=3,4$q=3,4$ and 6 near _Tc$T_c$. The sound attenuation peaks are observed near _Tt$T_t$ at lower values of external magnetic field, but as it increases the sound attenuation peaks decrease and eventually disappear.     

A hidden analytic structure of the Rabi model

The Rabi model describes the simplest interaction between a cavity mode with a frequency ω_c${\omega }_c$ and a two-level system with a resonance frequency ω₀${\omega }_0$. It is shown here that the spectrum of the Rabi model coincides with the support of the discrete Stieltjes integral measure in the orthogonality relations of recently introduced orthogonal polynomials. The exactly solvable limit of the Rabi model corresponding to Δ=ω₀/(2ω_c)=0$\Delta ={\omega }_0/\left(2{\omega }_c\right)=0$, which describes a displaced harmonic oscillator, is characterized by the discrete Charlier polynomials in normalized energy ?$?$, which are orthogonal on an equidistant lattice. A non-zero value of Δ$\Delta$ leads to non-classical discrete orthogonal polynomials ?_k(?)${?}_k\left(?\right)$ and induces a deformation of the underlying equidistant lattice. The results provide a basis for a novel analytic method of solving the Rabi model. The number of ca. 1350 calculable energy levels per parity subspace obtained in double precision (cca 16 digits) by an elementary stepping algorithm is up to two orders of magnitude higher than is possible to obtain by Braak’s solution. Any first n$n$ eigenvalues of the Rabi model arranged in increasing order can be determined as zeros of ?_N(?)${?}_N\left(?\right)$ of at least the degree N=n+n_t$N=n+n_t$. The value of n_t>0$n_t>0$, which is slowly increasing with n$n$, depends on the required precision. For instance, n_t?26$n_t?26$ for n=1000$n=1000$ and dimensionless interaction constant κ=0.2$\kappa =0.2$, if double precision is required. Given that the sequence of the l$l$th zeros x_nl$x_{nl}$’s of ?_n(?)${?}_n\left(?\right)$’s defines a monotonically decreasing discrete flow with increasing n$n$, the Rabi model is indistinguishable from an algebraically solvable model in any finite precision. Although we can rigorously prove our results only for dimensionless interaction constant κ<1$\kappa <1$, numerics and exactly solvable example suggest that the main conclusions remain to be valid also for κ≥1$\kappa \ge 1$.     

Schizophrenic active neutrinos and exotic sterile neutrinos

We implement a schizophrenic scenario for the active neutrinos in a model in which there are also exotic right-handed neutrinos making a model with a local U_(1)B−L$U{(1)}_{B-L}$ anomaly free. Two of right-handed neutrinos carry B−L=−4$B-L=-4$ while the third one carries B−L=5$B-L=5$. Unlike the non-exotic version of the model, in which all right-handed neutrinos carry the same B−L=−1$B-L=-1$ charge, in this case the neutrinos have their own scalar sector and no hierarchy in the Yukawa coupling in the Dirac mass term is necessary.     

Tenth Peregrine breather solution to the NLS equation

In this paper we construct a particularly important solution to the focusing NLS equation, namely a Peregrine breather of the rank 10 which we call, P₁₀$P_{10}$ breather. The related explicit formula is given by the ratio of two polynomials of degree 110 with integer coefficients times trivial exponential factor. This formula drastically simplifies for the “initial values” namely for t=0$t=0$ or x=0$x=0$. This formula confirms a general conjecture saying that between all quasi-rational solutions of the rank N$N$ fixed by the condition that its absolute value tends to 1 at infinity and its highest maximum is located at the point (x=0,t=0)$(x=0,t=0)$, the P_N$P_N$ breather is distinguished by the fact that P_N(0,0)=2N+1$P_N(0,0)=2N+1$ and, in the aforementioned class of quasi-rational solutions, it is an absolute maximum. At the end we also make a few remarks concerning the rational deformations of P₁₀$P_{10}$ breather involving 2N−2$2N-2$ free real parameters chosen in a way that P_N$P_N$ breather itself corresponds to the zero values of these parameters although we have no intention to discuss the properties of these deformations here.     

On leafwise conformal diffeomorphisms

For every diffeomorphism φ:M→N$\phi :M\to N$ between 3-dimensional Riemannian manifolds M$M$ and N$N$, there are locally two 2-dimensional distributions _D±$D_{\pm }$ such that φ$\phi$ is conformal on both of them. We state necessary and sufficient conditions for a distribution to be one of _D±$D_{\pm }$. These are algebraic conditions expressed in terms of the self-adjoint and positive definite operator induced from _φ∗${\phi }_{\ast }$. We investigate the integrability condition of _D+$D_{+}$ and _D−$D_{-}$. We also show that it is possible to choose coordinate systems in which leafwise conformal diffeomorphism is holomorphic on leaves.     

Calculating statistical distributions from operator relations: The statistical distributions of various intermediate statistics

In this paper, we give a general discussion on the calculation of the statistical distribution from a given operator relation of creation, annihilation, and number operators. Our result shows that as long as the relation between the number operator and the creation and annihilation operators can be expressed as a^†b=Λ(N)$a^{†}b=\Lambda \left(N\right)$ or N=Λ⁻¹(a^†b)$N={\Lambda }^{-1}\left(a^{†}b\right)$, where N$N$, a^†$a^{†}$, and b$b$ denote the number, creation, and annihilation operators, i.e., N$N$ is a function of quadratic product of the creation and annihilation operators, the corresponding statistical distribution is the Gentile distribution, a statistical distribution in which the maximum occupation number is an arbitrary integer. As examples, we discuss the statistical distributions corresponding to various operator relations. In particular, besides the Bose–Einstein and Fermi–Dirac cases, we discuss the statistical distributions for various schemes of intermediate statistics, especially various q$q$-deformation schemes. Our result shows that the statistical distributions corresponding to various q$q$-deformation schemes are various Gentile distributions with different maximum occupation numbers which are determined by the deformation parameter q$q$. This result shows that the results given in much literature on the q$q$-deformation distribution are inaccurate or incomplete.     

Trigonometrical sums connected with the chiral Potts model,Verlinde dimension formula,two-dimensional resistor network,and number theory

We have recently developed methods for obtaining exact two-point resistance of the complete graph minus N$N$ edges. We use these methods to obtain closed formulas of certain trigonometrical sums that arise in connection with one-dimensional lattice, in proving Scott’s conjecture on permanent of Cauchy matrix, and in the perturbative chiral Potts model. The generalized trigonometrical sums of the chiral Potts model are shown to satisfy recursion formulas that are transparent and direct, and differ from those of Gervois and Mehta. By making a change of variables in these recursion formulas, the dimension of the space of conformal blocks of SU(2)$SU\left(2\right)$ and SO(3)$SO\left(3\right)$ WZW models may be computed recursively. Our methods are then extended to compute the corner-to-corner resistance, and the Kirchhoff index of the first non-trivial two-dimensional resistor network, 2×N$2\times N$. Finally, we obtain new closed formulas for variant of trigonometrical sums, some of which appear in connection with number theory.     

Commuting conformal and dual conformal symmetries in the Regge limit

In this paper we continue our study of the dual SL(2,C)$\mathit{SL}(2,C)$ symmetry of the BFKL equation, analogous to the dual conformal symmetry of N=4$N=4$ super-Yang–Mills. We find that the ordinary and dual SL(2,C)$\mathit{SL}(2,C)$ symmetries do not generate a Yangian, in contrast to the ordinary and dual conformal symmetries in the four-dimensional gauge theory. The algebraic structure is still reminiscent of that of N=4$N=4$ SYM, however, and one can extract a generator from the dual SL(2,C)$\mathit{SL}(2,C)$ close to the bi-local form associated with Yangian algebras. We also discuss the issue of whether the dual SL(2,C)$\mathit{SL}(2,C)$ symmetry, which in its original form is broken by IR effects, is broken in a controlled way, similar to the way the dual conformal symmetry of N=4$N=4$ satisfies an anomalous Ward identity. At least for the lowest orders it seems possible to recover the dual SL(2,C)$\mathit{SL}(2,C)$ by deforming its representation, keeping open the possibility that it is an exact symmetry of BFKL. Independently of a possible relation to N=4$N=4$ scattering amplitudes, this opens an avenue for explaining the integrability of BFKL in terms of two finite-dimensional subalgebras.     

Critical dynamics in systems controlled by fractional kinetic equations

The field theory renormalization group is used for analyzing the fractional Langevin equation with the order of the temporal derivative 0<α<1$0<\alpha <1$, fractional Laplacian of the order σ$\sigma$, and Gaussian noise correlator. The case of non-linearity φ^m${\phi }^m$ with odd m≥3$m\ge 3$ is considered. It is proved that the model is multiplicatively renormalizable. Propagators were found in the momentum and coordinate representation, expressed in terms of Fox’s H functions.     

Is space-time symmetry a suitable generalization of parity-time symmetry?

We discuss space-time symmetric Hamiltonian operators of the form H=H₀+igH^′$H=H_0+ig{H}^{\prime }$, where H₀$H_0$ is Hermitian and g$g$ real. H₀$H_0$ is invariant under the unitary operations of a point group G$G$ while H^′$H^{\prime }$ is invariant under transformation by elements of a subgroup G^′$G^{\prime }$ of G$G$. If G$G$ exhibits irreducible representations of dimension greater than unity, then it is possible that H$H$ has complex eigenvalues for sufficiently small nonzero values of g$g$. In the particular case that H$H$ is parity-time symmetric then it appears to exhibit real eigenvalues for all 0<g<g_c$0<g<g_c$, where g_c$g_c$ is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether H$H$ may exhibit real or complex eigenvalues for g>0$g>0$. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.