Spin Calogero Model Research Articles

FZZT branes and non-singlets of matrix quantum mechanics

We explore the non-singlet sector of matrix quantum mechanics dual to c = 1 Liouville theory. The non-singlets are obtained by adding Nf× N bi-fundamental fields in the gauged matrix quantum mechanics model as well as a one dimensional Chern-Simons term. The present model is associated with a spin-Calogero model in the presence of an external magnetic field. In chiral variables, the low energy excitations-currents satisfy an SU {left(2{N}_fright)}_{tilde{k}} Kăc-Moody algebra at large N. We analyse the canonical partition function as well as two and four point correlation functions, discuss a Gross-Witten-Wadia phase transition at large N, Nf and study different limits of the parameters that allow us to recover the matrix model of Kazakov-Kostov-Kutasov conjectured to describe a two dimensional black hole. The grand canonical partition function is a τ- function obeying discrete soliton equations. We finally conjecture a possible dynamical picture for the formation of a black hole in terms of condensation of long-strings in the strongly coupled region of the Liouville direction.

Supersymmetric Calogero and Calogero-Sutherland models from gauging

We describe how the new kinds of N = 2 and N = 4 supersymmetric extensions of the rational and hyperbolic Calogero models can be derived by gauging U(n) symmetry of the appropriate superfield matrix models. These systems feature non-standard numbers N n2 of physical fermionic variables as compared with N n in the standard case. An essential ingredient of N = 4 models is the necessary presence of semi-dynamical spin variables described by d = 1 Wess-Zumino terms. The bosonic cores of N = 4 models are U(2) spin Calogero and Calogero-Sutherland models. In the hyperbolic case two non-equivalent N = 4 extensions exist, with and without the interacting center-of-mass coordinate in the bosonic sector. The talk is based on joint works with Sergey Fedoruk and Olaf Lechtenfeld.

Supersymmetric many-body Euler–Calogero–Moser model

We explicitly construct a supersymmetric so(n) spin-Calogero model with an arbitrary even number N of supersymmetries. It features 12Nn(n+1) rather than Nn fermionic coordinates and a very simple structure of the supercharges and the Hamiltonian. The latter, together with additional conserved currents, form an osp(N|2) superalgebra. We provide a superspace description for the simplest case, namely N=2 supersymmetry. The reduction to an N-extended supersymmetric goldfish model is also discussed.

Formula omitted]-extended supersymmetric Calogero models

We propose a new N-extended supersymmetric su(n) spin-Calogero model. Employing a generalized Hamiltonian reduction adopted to the supersymmetric case, we explicitly construct a novel rational n-particle Calogero model with an arbitrary even number of supersymmetries. It features Nn2 rather than Nn fermionic coordinates and increasingly high fermionic powers in the supercharges and the Hamiltonian.

Supersymmetric analogue of BCN type rational integrable models with polarized spin reversal operators

We derive the exact spectra as well as partition functions for a class of BCN type of spin Calogero models, whose Hamiltonians are constructed by using supersymmetric analogues of polarized spin reversal operators (SAPSRO). The strong coupling limit of these spin Calogero models yields BCN type of Polychronakos–Frahm (PF) spin chains with SAPSRO. By applying the freezing trick, we obtain an exact expression for the partition functions of such PF spin chains. We also derive a formula which expresses the partition function of any BCN type of PF spin chain with SAPSRO in terms of partition functions of several AK types of supersymmetric PF spin chains, where K⩽N−1. Subsequently we show that an extended boson–fermion duality relation is obeyed by the partition functions of the BCN type of PF chains with SAPSRO. Some spectral properties of these spin chains, like level density distribution and nearest neighbor spacing distribution, are also studied.

Rational quantum integrable systems of [formula omitted] type with polarized spin reversal operators

We study the spin Calogero model of DN type with polarized spin reversal operators, as well as its associated spin chain of Haldane–Shastry type, both in the antiferromagnetic and ferromagnetic cases. We compute the spectrum and the partition function of the former model in closed form, from which we derive an exact formula for the chain's partition function in terms of products of partition functions of Polychronakos–Frahm spin chains of type A. Using a recursion relation for the latter partition functions that we derive in the paper, we are able to numerically evaluate the partition function, and thus the spectrum, of the DN-type spin chain for relatively high values of the number of spins N. We analyze several global properties of the chain's spectrum, such as the asymptotic level density, the distribution of consecutive spacings of the unfolded spectrum, and the average degeneracy. In particular, our results suggest that this chain is invariant under a suitable Yangian group, and that its spectrum coincides with that of a Yangian-invariant vertex model with linear energy function and dispersion relation.

Generalized Calogero system for simple Lie algebra series

We derive equations of motions and solutions for the generalized spin-Calogero models associated for the classical Lie algebra series Bn, Cn BCn, and Dn using the pole expansion.

Partition functions of Polychronakos like spin chains associated with polarized spin reversal operators

We construct polarized spin reversal operator (PSRO) which yields a class of representations for the BCN type of Weyl algebra, and subsequently use this PSRO to find out novel exactly solvable variants of the BCN type of spin Calogero model. The strong coupling limit of such spin Calogero models generates the BCN type of Polychronakos spin chains with PSRO. We derive the exact spectra of the BCN type of spin Calogero models with PSRO and compute the partition functions of the related spin chains by using the freezing trick. We also find out an interesting relation between the partition functions of the BCN type and AN−1 type of Polychronakos spin chains. Finally, we study spectral properties like level density and distribution of spacing between consecutive energy levels for BCN type of Polychronakos spin chains with PSRO.

Cold Fermi gas with inverse square interaction in a harmonic trap

We study equilibrium density and spin density profiles for a model of cold one-dimensional spin 1/2 fermions interacting via inverse square interaction and exchange in an external harmonic trap. This model is the well-known spin-Calogero model (sCM) and its fully nonlinear collective field theory description is known. We extend the field theory description to the presence of an external harmonic trap and obtain analytic results for statics and dynamics of the system. For instance, we find how the equilibrium density profile changes upon tuning the interaction strength. The results we obtain for equilibrium configurations are very similar to the ones obtained recently by Ma and Yang (2010) [1] for a model of fermions with short ranged interactions. Our main approximation is the neglect of the terms of higher order in spatial derivatives in equations of motion – gradientless approximation (Kulkarni et al., 2009) [2]. Within this approximation the hydrodynamic equations of motion can be written as a set of decoupled forced Riemann–Hopf equations for the dressed Fermi momenta of the model. This enables us to write analytical solutions for the dynamics of spin and charge. We describe the time evolution of the charge density when an initial non-equilibrium profile is created by cooling the gas with an additional potential in place and then suddenly removing the potential. We present our results as a simple “single-particle” evolution in the phase space reminiscing a similar description of the dynamics of noninteracting one-dimensional fermions.

Publisher's Note: Nonlinear dynamics of spin and charge in spin-Calogero model [Phys. Rev. B80, 165105 (2009)

Publisher's Note: Nonlinear dynamics of spin and charge in spin-Calogero model [Phys. Rev. B<b>80</b>, 165105 (2009)

Nonlinear dynamics of spin and charge in spin-Calogero model

The fully nonlinear dynamics of spin and charge in spin-Calogero model is studied. The latter is an integrable one-dimensional model of quantum spin-1/2 particles interacting through inverse-square interaction and exchange. Classical hydrodynamic equations of motion are written for this model in the regime where gradient corrections to the exact hydrodynamic formulation of the theory may be neglected. In this approximation variables separate in terms of dressed Fermi momenta of the model. Hydrodynamic equations reduce to a set of decoupled Riemann-Hopf (or inviscid Burgers') equations for the dressed Fermi momenta. We study the dynamics of some nonequilibrium spin-charge configurations for times smaller than the time scale of the gradient catastrophe. We find an interesting interplay between spin and charge degrees of freedom. In the limit of large coupling constant the hydrodynamics reduces to the spin hydrodynamics of the Haldane-Shastry model.

Emptiness and depletion formation probability in spin models with inverse square interaction

We calculate the Emptiness Formation Probability (EFP) in the spin-Calogero Model (sCM) and Haldane–Shastry Model (HSM) using their hydrodynamic description. The EFP is the probability that a region of space is completely void of particles in the ground state of a quantum many body system. We calculate this probability in an instanton approach, by considering the more general problem of an arbitrary depletion of particles (DFP). In the limit of large size of depletion region the probability is dominated by a classical configuration in imaginary time that satisfies a set of boundary conditions and the action calculated on such solution gives the EFP/DFP with exponential accuracy. We show that the calculation for sCM can be elegantly performed by representing the gradientless hydrodynamics of spin particles as a sum of two spin-less Calogero collective field theories in auxiliary variables. Interestingly, the result we find for the EFP can be casted in a form reminiscing of spin-charge separation, which should be violated for a non-linear effect such as this. We also highlight the connections between sCM, HSM and λ = 2 spin-less Calogero model from a EFP/DFP perspective.

Notes on collective field theory of matrix and spin Calogero models

Matrix models and related Spin-Calogero-Sutherland models are of major relevance in a variety of subjects, ranging from condensed matter physics to QCD and low dimensional string theory. They are characterized by integrability and exact solvability. Their continuum, field theoretic representations are likewise of definite interest. In this paper we describe various continuum, field theoretic representations of these models based on bosonization and collective field theory techniques. We compare various known representations and describe some nontrivial applications.

Spin Calogero models associated with Riemannian symmetric spaces of negative curvature

The Hamiltonian symmetry reduction of the geodesics system on a symmetric space of negative curvature by the maximal compact subgroup of the isometry group is investigated at an arbitrary value of the momentum map. Restricting to regular elements in the configuration space, the reduction generically yields a spin Calogero model with hyperbolic interaction potentials defined by the root system of the symmetric space. These models come equipped with Lax pairs and many constants of motion, and can be integrated by the projection method. The special values of the momentum map leading to spinless Calogero models are classified under some conditions, explaining why the B C n models with two independent coupling constants are associated with SU ( n + 1 , n ) / S ( U ( n + 1 ) × U ( n ) ) as found by Olshanetsky and Perelomov. In the zero curvature limit our models reproduce rational spin Calogero models studied previously and similar models correspond to other (affine) symmetric spaces, too. The construction works at the quantized level as well.

Spin Calogero models obtained from dynamical r-matrices and geodesic motion

We study classical integrable systems based on the Alekseev–Meinrenken dynamical r-matrices corresponding to automorphisms of self-dual Lie algebras, G . We prove that these r-matrices are uniquely characterized by a non-degeneracy property and apply a construction due to Li and Xu to associate spin Calogero type models with them. The equation of motion of any model of this type is found to be a projection of the natural geodesic equation on a Lie group G with Lie algebra G , and its phase space is interpreted as a Hamiltonian reduction of an open submanifold of the cotangent bundle T ∗ G , using the symmetry arising from the adjoint action of G twisted by the underlying automorphism. This shows the integrability of the resulting systems and gives an algorithm to solve them. As illustrative examples we present new models built on the involutive diagram automorphisms of the real split and compact simple Lie algebras, and also explain that many further examples fit in the dynamical r-matrix framework.

Exact Spectrum and Partition Function of SU(m|n) Supersymmetric Polychronakos Model

By using the fact that Polychronakos-like models can be obtained through the `freezing limit' of related spin Calogero models, we calculate the exact spectrum as well as partition function of SU(m|n) supersymmetric Polychronakos (SP) model. It turns out that, similar to the non-supersymmetric case, the spectrum of SU(m|n) SP model is also equally spaced. However, the degeneracy factors of corresponding energy levels crucially depend on the values of bosonic degrees of freedom (m) and fermionic degrees of freedom (n). As a result, the partition functions of SP models are expressed through some novel q-polynomials. Finally, by interchanging the bosonic and fermionic degrees of freedom, we obtain a duality relation among the partition functions of SP models.

The symplectic structure of the spin Calogero model

We compute the symplectic structure of the spin Calogero model in terms of algebro-geometric data on the associated spectral curve.

The Yangian symmetry in the spin Calogero model and its applications

By using the non-symmetric Hermite polynomials and a technique based on the Yangian Gelfand-Zetlin bases, we decompose the space of states of the Calogero model with spin into irreducible Yangian modules, construct an orthogonal basis of eigenvectors and derive product-type formulas for norms of these eigenvectors.