Thermodynamics of antiferromagnetic alternating spin chains

We consider integrable quantum spin chains with alternating spins (S₁,S₂)$(S_1,S_2)$. We derive a finite set of non-linear integral equations for the thermodynamics of these models by use of the quantum transfer matrix approach. Numerical solutions of the integral equations are provided for quantities like specific heat, magnetic susceptibility and in the case S₁=S₂$S_1=S_2$ for the thermal Drude weight. At low temperatures one class of models shows finite magnetization and the other class presents antiferromagnetic behaviour. The thermal Drude weight behaves linearly on T at low temperatures and is proportional to the central charge c   of the system. Quite generally, we observe residual entropy for S₁≠S₂$S_1\ne S_2$.     

Analytical approximation schemes for solving exact renormalization group equations. II Conformal mappings

We present a new efficient analytical approximation scheme to two-point boundary value problems of ordinary differential equations (ODEs) adapted to the study of the derivative expansion of the exact renormalization group equations. It is based on a compactification of the complex plane of the independent variable using a mapping of an angular sector onto a unit disc. We explicitly treat, for the scalar field, the local potential approximations of the Wegner–Houghton equation in the dimension d=3$d=3$ and of the Wilson–Polchinski equation for some values of d∈]2,3]$d\in ]2,3]$. We then consider, for d=3$d=3$, the coupled ODEs obtained by Morris at the second order of the derivative expansion. In both cases the fixed points and the eigenvalues attached to them are estimated. Comparisons of the results obtained are made with the shooting method and with the other analytical methods available. The best accuracy is reached with our new method which presents also the advantage of being very fast. Thus, it is well adapted to the study of more complicated systems of equations.     

Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey–Wilson polynomials

Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey–Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey–Wilson polynomials in terms of a degree ?   (?=1,2,…$?=1,2,\dots$) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree ??1$??1$ and thus not constrained by any generalisation of Bochner's theorem.     

The QCD spin chain S matrix

Beisert et al. have identified an integrable SU(2,2)$\mathit{SU}(2,2)$ quantum spin chain which gives the one-loop anomalous dimensions of certain operators in large Nc$N_c$ QCD. We derive a set of nonlinear integral equations (NLIEs) for this model, and compute the scattering matrix of the various (in particular, magnon) excitations.     

Rational r-matrices,higher rank Lie algebras and integrable proton–neutron BCS models

We study integrable cases of pairing BCS hamiltonians containing several types of fermions. We prove that there exist three classes of such integrable models associated with classical rational r  -matrices and Lie algebras gl(2m)$\mathit{gl}(2m)$, sp(2m)$\mathit{sp}(2m)$ and so(2m)$\mathit{so}(2m)$ correspondingly. We diagonalize the constructed hamiltonians by means of the algebraic Bethe ansatz. In the partial case of two types of fermions (m=2$m=2$) the obtained models may be interpreted as N=Z$N=Z$ proton–neutron integrable models. In particular, in the case of sp(4)$\mathit{sp}(4)$ we recover the famous integrable proton–neutron model of Richardson.     

On the renormalization of the bosonized multi-flavor Schwinger model

The phase structure of the bosonized multi-flavor Schwinger model is investigated by means of the differential renormalization group (RG) method. In the limit of small fermion mass the linearized RG flow is sufficient to determine the low-energy behavior of the N  -flavor model, if it has been rotated by a suitable rotation in the internal space. For large fermion mass, the exact RG flow has been solved numerically. The low-energy behavior of the multi-flavor model is rather different depending on whether N=1$N=1$ or N>1$N>1$, where N   is the number of flavors. For N>1$N>1$ the reflection symmetry always suffers breakdown in both the weak and strong coupling regimes, in contrary to the N=1$N=1$ case, where it remains unbroken in the strong coupling phase.     

Differential operator realizations of superalgebras and free field representations of corresponding current algebras

Based on the particular orderings introduced for the positive roots of finite-dimensional basic Lie superalgebras, we construct the explicit differential operator representations of the osp(2r|2n)$\mathit{osp}(2r|2n)$ and osp(2r+1|2n)$\mathit{osp}(2r+1|2n)$ superalgebras and the explicit free field realizations of the corresponding current superalgebras ospk_(2r|2n)$\mathit{osp}{(2r|2n)}_k$ and ospk_(2r+1|2n)$\mathit{osp}{(2r+1|2n)}_k$ at an arbitrary level k. The free field representations of the corresponding energy–momentum tensors and screening currents of the first kind are also presented.     

Particle diagrams and statistics of many-body random potentials

We present a method using Feynman-like diagrams to calculate the statistical properties of random many-body potentials. This method provides a promising alternative to existing techniques typically applied to this class of problems, such as the method of supersymmetry and the eigenvector expansion technique pioneered in Benet et al. (2001). We use it here to calculate the fourth, sixth and eighth moments of the average level density for systems with m$m$ bosons or fermions that interact through a random k$k$-body Hermitian potential (k≤m$k\le m$); the ensemble of such potentials with a Gaussian weight is known as the embedded Gaussian Unitary Ensemble   (eGUE) (Mon and French, 1975). Our results apply in the limit where the number l$l$ of available single-particle states is taken to infinity. A key advantage of the method is that it provides an efficient way to identify only those expressions which will stay relevant in this limit. It also provides a general argument for why these terms have to be the same for bosons and fermions. The moments are obtained as sums over ratios of binomial expressions, with a transition from moments associated to a semi-circular level density for m<2k$m<2k$ to Gaussian moments in the dilute limit k?m?l$k?m?l$. Regarding the form of this transition, we see that as m$m$ is increased, more and more diagrams become relevant, with new contributions starting from each of the points m=2k,3k,…,nk$m=2k,3k,\dots ,nk$ for the 2n$2n$th moment.     

On the space of quantum fields in massive two-dimensional theories

For a large class of integrable quantum field theories we show that the S-matrix determines a space of fields which decomposes into subspaces labeled, besides the charge and spin indices, by an integer k. For scalar fields k   is non-negative and is naturally identified as an off-critical extension of the conformal level. To each particle we associate an operator acting in the space of fields whose eigenvectors are primary (k=0$k=0$) fields of the massive theory. We discuss how the existing results for models as different as Zn$Z_n$, sine-Gordon or Ising with magnetic field fit into this classification.     

Bogolyubov approximation for diagonal model of an interacting Bose gas

We study, using the Bogolyubov approximation, the thermodynamic behavior of a superstable Bose system whose energy operator in the second-quantized form contains a nonlinear expression in the occupation numbers operators. We prove that for all values of the chemical potential satisfying μ>λ(0)$\mu >\lambda (0)$, where λ(0)?0$\lambda (0)?0$ is the lowest energy value, the system undergoes Bose–Einstein condensation.     

Avoiding the big-rip jeopardy in a quintom dark energy model with higher derivatives

In the framework of a single scalar field quintom model with higher derivative, we construct in this Letter a dark energy model of which the equation of state (EOS) w   crosses over the cosmological constant boundary. Interestingly during the evolution of the universe w<−1$w<-1$ happens just for a period of time with a distinguished feature that w   starts with a value above −1, transits into w<−1$w<-1$, then comes back to w>−1$w>-1$. This avoids the big-rip jeopardy induced by w<−1$w<-1$.     

Spin chains in magnetic field,non-skew-symmetric classical r-matrices and BCS-type integrable systems

We construct generalized Gaudin systems in an external magnetic field corresponding to arbitrary so(3)$\mathit{so}(3)$-valued non-skew-symmetric r-matrices with spectral parameters and non-homogeneous external magnetic fields. In the case of r  -matrices diagonal in the sl(2)$\mathit{sl}(2)$ basis we calculate the spectrum and the eigen-values of the corresponding generalized Gaudin hamiltonians using the algebraic Bethe ansatz. We explicitly consider several one-parametric families of non-skew-symmetric classical r-matrices and the corresponding generalized Gaudin systems in a magnetic field. We apply these results to fermionic systems and obtain a wide class of new integrable fermionic BCS-type hamiltonians.     

A generalized Beraha conjecture for non-planar graphs

We study the partition function _ZG(nk,k)(Q,v)$Z_{G(nk,k)}(Q,v)$ of the Q  -state Potts model on the family of (non-planar) generalized Petersen graphs G(nk,k)$G(nk,k)$. We study its zeros in the plane (Q,v)$(Q,v)$ for 1?k?7$1?k?7$. We also consider two specializations of _ZG(nk,k)$Z_{G(nk,k)}$, namely the chromatic polynomial _PG(nk,k)(Q)$P_{G(nk,k)}(Q)$ (corresponding to v=−1$v=-1$), and the flow polynomial _ΦG(nk,k)(Q)${\Phi }_{G(nk,k)}(Q)$ (corresponding to v=−Q$v=-Q$). In these two cases, we study their zeros in the complex Q  -plane for 1?k?7$1?k?7$. We pay special attention to the accumulation loci of the corresponding zeros when n→∞$n\to \infty$. We observe that the Berker–Kadanoff phase that is present in two-dimensional Potts models, also exists for non-planar recursive graphs. Their qualitative features are the same; but the main difference is that the role played by the Beraha numbers for planar graphs is now played by the non-negative integers for non-planar graphs. At these integer values of Q, there are massive eigenvalue cancellations, in the same way as the eigenvalue cancellations that happen at the Beraha numbers for planar graphs.