Three PT-symmetric Hamiltonians with completely different spectra

We discuss three Hamiltonians, each with a central-field part H₀$H_0$ and a PT-symmetric perturbation igz$igz$. When H₀$H_0$ is the isotropic Harmonic oscillator the spectrum is real for all g$g$ because H$H$ is isospectral to H₀+g²/2$H_0+g^2/2$. When H₀$H_0$ is the Hydrogen atom then infinitely many eigenvalues are complex for all g$g$. If the potential in H₀$H_0$ is linear in the radial variable r$r$ then the spectrum of H$H$ exhibits real eigenvalues for 0<g<g_c$0<g<g_c$ and a PT phase transition at g_c$g_c$.     

Quantum phase diagram of the half filled Hubbard model with bond-charge interaction

Using quantum field theory and bosonization, we determine the quantum phase diagram of the one-dimensional Hubbard model with bond-charge interaction X in addition to the usual Coulomb repulsion U at half-filling, for small values of the interactions. We show that it is essential to take into account formally irrelevant terms of order X  . They generate relevant terms proportional to X²$X^2$ in the flow of the renormalization group (RG). These terms are calculated using operator product expansions. The model shows three phases separated by a charge transition at U=Uc$U=U_c$ and a spin transition at U=Us>Uc$U=U_s>U_c$. For U<Uc$U<U_c$ singlet superconducting correlations dominate, while for U>Us$U>U_s$, the system is in the spin-density wave phase as in the usual Hubbard model. For intermediate values Uc<U<Us$U_c<U<U_s$, the system is in a spontaneously dimerized bond-ordered wave phase, which is absent in the ordinary Hubbard model with X=0$X=0$. We obtain that the charge transition remains at Uc=0$U_c=0$ for X≠0$X\ne 0$. Solving the RG equations for the spin sector, we provide an analytical expression for Us(X)$U_s(X)$. The results, with only one adjustable parameter, are in excellent agreement with numerical ones for X<t/2$X<t/2$ where t is the hopping.     

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.     

Dynamical SUSY breaking in heterotic M-theory

It is shown that four-dimensional N=1$N=1$ supersymmetric QCD with massive flavors in the fundamental representation of the gauge group can be realized in the hidden sector of E₈×E₈$E_8\times E_8$ heterotic string vacua. The number of flavors can be chosen to lie in the range of validity of the free-magnetic dual, using which one can demonstrate the existence of long-lived meta-stable non-supersymmetric vacua. This is shown explicitly for the gauge group Spin(10)$\mathit{Spin}(10)$, but the methods are applicable to Spin(Nc)$\mathit{Spin}(N_c)$, SU(Nc)$\mathit{SU}(N_c)$ and Sp(Nc)$\mathit{Sp}(N_c)$ for a wide range of color index Nc$N_c$. Hidden sectors of this type can potentially be used as a mechanism to break supersymmetry within the context of heterotic M-theory.     

Quantum spherical model with competing interactions

We analyse the phase diagram of a quantum mean spherical model in terms of the temperature T$T$, a quantum parameter g$g$, and the ratio p=−_J2/_J1$p=-J_2/J_1$, where _J1>0$J_1>0$ refers to ferromagnetic interactions between first-neighbour sites along the d$d$ directions of a hypercubic lattice, and _J2<0$J_2<0$ is associated with competing antiferromagnetic interactions between second neighbours along m≤d$m\le d$ directions. We regain a number of known results for the classical version of this model, including the topology of the critical line in the g=0$g=0$ space, with a Lifshitz point at p=1/4$p=1/4$, for d>2$d>2$, and closed-form expressions for the decay of the pair correlations in one dimension. In the T=0$T=0$ phase diagram, there is a critical border, _gc=_gc(p)$g_c=g_c\left(p\right)$ for d≥2$d\ge 2$, with a singularity at the Lifshitz point if d<(m+4)/2$d<(m+4)/2$. We also establish upper and lower critical dimensions, and analyse the quantum critical behavior in the neighborhood of p=1/4$p=1/4$.     

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.     

Domain wall propagation in Permalloy nanowires with a thickness gradient

The magnetization reversal behavior of Permalloy nanowires has been investigated using a magneto-optic Kerr effect setup. Nanowires with various widths, w=250$w=250$ nm to 3 μm and a thickness of t=10$t=10$ nm were fabricated by electron-beam lithography and subsequent lift-off. Furthermore, similar nanowires but with a thickness gradient along the nanowire axis have been prepared to investigate the influence of the gradient on the magnetic domain wall propagation. Magnetization hysteresis loops recorded on individual nanowires without a gradient are compared to corresponding wires with a thickness gradient. The dependence of the coercive field, _Hc$H_c$ vs. t/w$t/w$ shows a linear behavior for wires without a gradient. However, wires with a gradient display a more complex crossover behavior. We find a plateau in the _Hc$H_c$ vs. t/w$t/w$ curve at values of w$w$, where a transformation from transverse to vortex domain wall type is expected.     

Right unitarity triangles,stable CP-violating phases and approximate quark–lepton complementarity

Current experimental data indicate that two unitarity triangles of the CKM quark mixing matrix V   are almost the right triangles with α≈90°$\alpha \approx 90°$. We highlight a very suggestive parametrization of V and show that its CP-violating phase ? is nearly equal to α   (i.e., ?−α≈1.1°$?-\alpha \approx 1.1°$). Both ? and α   are stable against the renormalizaton-group evolution from the electroweak scale MZ$M_Z$ to a superhigh energy scale MX$M_X$ or vice versa, and thus it is impossible to obtain α=90°$\alpha =90°$ at MZ$M_Z$ from ?=90°$?=90°$ at MX$M_X$. We conjecture that there might also exist a maximal CP-violating phase φ≈90°$\phi \approx 90°$ in the MNS lepton mixing matrix U. The approximate quark–lepton complementarity relations, which hold in the standard parametrizations of V and U, can also hold in our particular parametrizations of V and U   simply due to the smallness of |V_ub|$|V_{ub}|$ and |V_e3|$|V_{e3}|$.     

Zero finite-temperature charge stiffness within the half-filled 1D Hubbard model

Even though the one-dimensional (1D) Hubbard model is solvable by the Bethe ansatz, at half-filling its finite-temperature T>0$T>0$ transport properties remain poorly understood. In this paper we combine that solution with symmetry to show that within that prominent T=0$T=0$ 1D insulator the charge stiffness D(T)$D\left(T\right)$ vanishes for T>0$T>0$ and finite values of the on-site repulsion U$U$ in the thermodynamic limit. This result is exact and clarifies a long-standing open problem. It rules out that at half-filling the model is an ideal conductor in the thermodynamic limit. Whether at finite T$T$ and U>0$U>0$ it is an ideal insulator or a normal resistor remains an open question. That at half-filling the charge stiffness is finite at U=0$U=0$ and vanishes for U>0$U>0$ is found to result from a general transition from a conductor to an insulator or resistor occurring at U=U_c=0$U=U_c=0$ for all finite temperatures T>0$T>0$. (At T=0$T=0$ such a transition is the quantum metal to Mott-Hubbard-insulator transition.) The interplay of the η$\eta$-spin SU(2)$SU\left(2\right)$ symmetry with the hidden U(1)$U\left(1\right)$ symmetry beyond SO(4)$SO\left(4\right)$ is found to play a central role in the unusual finite-temperature charge transport properties of the 1D half-filled Hubbard model.     

Comparative study between a two-dimensional anisotropic Heisenberg antiferromagnet with easy-axis single-ion anisotropy and one with easy-axis exchange anisotropy

Generally, in literature, easy-axis single ion anisotropy and easy-axis exchange anisotropy was treated in indistinct way. In this work we propose to perform a comparative study of the effects of these two easy-axis anisotropies on the behavior of the magnetization and the critical temperature (_Tc)$(T_{\mathrm{c}})$ in the 2D classical Heisenberg antiferromagnetic model. In order to study the low-temperature thermodynamics of this model, we should consider the contribution of anisotropic spin waves, using a self-consistent harmonic approximation (SCHA) theory. We compare the predictions of SCHA with numerical simulations on L×L$L\times L$ square lattices using Monte Carlo (MC) simulations, which include effects due to all thermodynamically allowed excitations. Our SCHA results are in good agreement with our MC simulations results and have shown that the strong K$K$ limit gives two different Ising-like behavior. In the exchange anisotropic case, the dependence of _Tc$T_{\mathrm{c}}$ on anisotropic parameter K$K$ becomes linear and in the single-ion anisotropic case, _Tc$T_{\mathrm{c}}$ becomes independent of K$K$. Also, using MC simulations and finite size scaling, we show that the critical exponents in the two anisotropic case are compatible with the 2D Ising values α=0.125$\alpha =0.125$ and γ=1.75$\gamma =1.75$.     

Non-Abelian fusion rules from an Abelian system

We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a n$n$ dimensional vector space which we call H_n$H_n$. The Z_p${\mathbb{Z}}_p$ gauge particles act on the vertex particles and thus H_n$H_n$ can be thought of as a C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ module. An exactly solvable model is built with operators acting in this Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of n$n$ and p$p$, though we believe this feature holds for all n>p$n>p$. We will see that non-Abelian anyons of the quantum double of C(S₃)$\mathbb{C}\left(S_3\right)$ are obtained as part of the vertex excitations of the model with n=6$n=6$ and p=3$p=3$. Ising anyons are obtained in the model with n=4$n=4$ and p=2$p=2$. The n=3$n=3$ and p=2$p=2$ case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than Z_p${\mathbb{Z}}_p$. This makes them possible candidates for realizing quantum computation.     

Elliptic flow arising from ridges due to semi-hard scattering

Azimuthal anisotropy in heavy-ion collisions is studied by taking into account the ridges generated by semi-hard scattering of intermediate-momentum partons, which can be sensitive to the initial spatial configuration of the medium in non-central collisions. In a simple treatment of the problem where the recombination of only thermal partons is considered, analytical formulas can be derived that yield results in accord with the data on v₂$v_2$ for pT<1–2 GeV/c$p_T<1\text{–}2\text{GeV}/c$. Centrality dependence is described by a geometrical factor. Ridge phenomenology is used to determine the initial slopes of v₂$v_2$ at low pT$p_T$ for both pion and proton. For higher pT$p_T$, shower partons from high-pT$p_T$ jets must be included, but they are not considered here.     

Pseudo-topological transitions in 2D gravity models coupled to massless scalar fields

We study the geometries generated by two-dimensional causal dynamical triangulations (CDT) coupled to d   massless scalar fields. Using methods similar to those used to study four-dimensional CDT we show that there exists a c=1$c=1$ “barrier”, analogous to the c=1$c=1$ barrier encountered in non-critical string theory, only the CDT transition is easier to be detected numerically. For d?1$d?1$ we observe time-translation invariance and geometries entirely governed by quantum fluctuations around the uniform toroidal topology put in by hand. For d>1$d>1$ the effective average geometry is no longer toroidal but “semiclassical” and spherical with Hausdorff dimension _dH=3$d_H=3$. In the d>1$d>1$ sector we study the time dependence of the semiclassical spatial volume distribution and show that the observed behavior is described by an effective mini-superspace action analogous to the actions found in the de Sitter phase of three- and four-dimensional pure CDT simulations and in the three-dimensional CDT-like Ho?ava–Lifshitz models.     

On analogues of black brane solutions in the model with multicomponent anisotropic fluid

A family of spherically symmetric solutions with horizon in the model with m  -component anisotropic fluid is presented. The metrics are defined on a manifold that contains a product of n−1$n-1$ Ricci-flat “internal” spaces. The equation of state for any s  -th component is defined by a vector Us$U^s$ belonging to Rn+1${\mathbb{R}}^{n+1}$. The solutions are governed by moduli functions Hs$H_s$ obeying non-linear differential equations with certain boundary conditions imposed. A simulation of black brane solutions in the model with antisymmetric forms is considered. An example of solution imitating M₂–M₅$M_2\text{–}{M}_5$ configuration (in D=11$D=11$ supergravity) corresponding to Lie algebra A₂$A_2$ is presented.     

Thermodynamics properties of an anisotropic Ising antiferromagnet in both external longitudinal and transverse fields

We have studied the anisotropic two-dimensional nearest-neighbor Ising model with competitive interactions in both uniform longitudinal field H$H$ and transverse magnetic field Ω$\Omega$. Using the effective-field theory (EFT) with correlation in cluster with N=1$N=1$ spin we calculate the thermodynamic properties as a function of temperature with values H$H$ and Ω$\Omega$ fixed. The model consists of ferromagnetic interaction _Jx$J_x$ in the x$x$ direction and antiferromagnetic interaction _Jy$J_y$ in the y$y$ direction, and it is found that for H/_Jy∈[0,2]$H/J_y\in [0,2]$ the system exhibits a second-order phase transition. The thermodynamic properties are obtained for the particular case of λ=_Jx/_Jy=1$\lambda =J_x/J_y=1$ (isotropic square lattice).     

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$\kappa \ge 1$.     

Inflationary predictions at small γ

This Letter explores single field inflation models with a constant, but arbitrary speed of sound cs$c_s$, obtained by deforming the kinetic energy terms to a Dirac–Born–Infeld form. Allowing cs<1$c_s<1$ provides a simple parametrization of non-Gaussianity. The dependence of inflationary observables on the parameter cs$c_s$ is considered in the leading order slow roll approximation. The results show that in most cases the dependence is actually rather weak for the range of cs$c_s$ allowed by existing bounds on non-Gaussianity.     

Critical behavior of an anisotropic Ising antiferromagnet in both external longitudinal and transverse fields

In this paper we study the critical behavior of a two-sublattice Ising model on an anisotropic square lattice in both uniform longitudinal (H  ) and transverse (Ω$\Omega$) fields by using the effective-field theory. The model consists of ferromagnetic interaction J_x in the x direction and antiferromagnetic interaction J_y in the y direction in the presence of the H   and Ω$\Omega$ fields. We obtain the phase diagrams in the H–T$H–T$ and Ω–T$\Omega –T$ planes changing values of the Ω$\Omega$ and H   parameters, respectively for fixed value at λ=_Jx/_Jy=1$\lambda =J_x/J_y=1$. At null temperature, the ground state phase diagram in the Ω–H$\Omega –H$ plane for several values of λ$\lambda$ parameter is analyzed. In the particular case of λ=1$\lambda =1$ we compare our results with mean-field theory (MFT) and was not observed reentrant behavior around of the critical field _Hc/_Jy=2.0$H_c/J_y=2.0$ for Ω=0$\Omega =0$ by using EFT.     

A study on the relations between the topological parameter and entanglement

In this Letter, some relations between the topological parameter d   and concurrences of the projective entangled states have been presented. It is shown that for the case with d=n$d=n$, all the projective entangled states of two n  -dimensional quantum systems are the maximally entangled states (i.e. C=1$C=1$). And for another case with d≠n$d\ne n$, C   both approach 0 when d→+∞$d\to +\infty$ for n=2$n=2$ and 3. Then we study the thermal entanglement and the entanglement sudden death (ESD) for a kind of Yang–Baxter Hamiltonian. It is found that the parameter d   influences not only the critical temperature _Tc$T_c$ but also the maximum entanglement value that the system can arrive at. And we also find that the parameter d has a great influence on the ESD.