arXiv subjects

B. Basu-Mallick

Publications and source records attributed to B. Basu-Mallick.

At least 37 records · Page 2Linked to original sources

arXiv2008

Exactly solvable D_N-type quantum spin models with long-range interaction

We derive the spectra of the D_N-type Calogero (rational) su(m) spin model, including the degeneracy factors of all energy levels. By taking the strong coupling limit of this model, in which its spin and dynamical degrees of freedom decouple, we compute the exact partition function of the su(m) Polychronakos-Frahm spin chain of D_N type. With the help of this partition function we study several statistical properties of the chain's spectrum, such as the density of energy levels and the distribution of spacings between consecutive levels.

cond-mat.str-el↗

arXiv2008

Inequivalent quantization of the rational Calogero model with a Coulomb type interaction

We consider the inequivalent quantizations of a $N$-body rational Calogero model with a Coulomb type interaction. It is shown that for certain range of the coupling constants, this system admits a one-parameter family of self-adjoint extensions. We analyze both the bound and scattering state sectors and find novel solutions of this model. We also find the ladder operators for this system, with which the previously known solutions can be constructed.

hep-th↗

arXiv2007

Low energy properties of the SU(m|n) supersymmetric Haldane-Shastry spin chain

The ground state and low energy excitations of the SU(m|n) supersymmetric Haldane-Shastry spin chain are analyzed. In the thermodynamic limit, it is found that the ground state degeneracy is finite only for the SU(m|0) and SU(m|1) spin chains, while the dispersion relation for the low energy and low momentum excitations is linear for all values of m and n. We show that the low energy excitations of the SU(m|1) spin chain are described by a conformal field theory of m non-interacting Dirac fermions which have only positive energies; the central charge of this theory is m/2. Finally, for n \ge 1, the partition functions of the SU(m|n) Haldane-Shastry spin chain and the SU(m|n) Polychronakos spin chain are shown to be related in a simple way in the thermodynamic limit at low temperatures.

hep-th↗

arXiv2007

Boson-fermion duality in SU(m|n) supersymmetric Haldane-Shastry spin chain

By using the Y(gl(m|n)) super Yangian symmetry of the SU(m|n) supersymmetric Haldane-Shastry spin chain, we show that the partition function of this model satisfies a duality relation under the exchange of bosonic and fermionic spin degrees of freedom. As a byproduct of this study of the duality relation, we find a novel combinatorial formula for the super Schur polynomials associated with some irreducible representations of the Y(gl(m|n)) Yangian algebra. Finally, we reveal an intimate connection between the global SU(m|n) symmetry of a spin chain and the boson-fermion duality relation.

hep-th↗

arXiv2006

Exact partition function of SU(m|n) supersymmetric Haldane-Shastry spin chain

By taking the freezing limit of a spin Calogero-Sutherland model containing `anyon like' representation of the permutation algebra, we derive the exact partition function of SU(m|n) supersymmetric Haldane-Shastry (HS) spin chain. This partition function allows us to study global properties of the spectrum like level density distribution and nearest neighbour spacing distribution. It is found that, for supersymmetric HS spin chains with large number of lattice sites, continuous part of the energy level density obeys Gaussian distribution with a high degree of accuracy. The mean value and standard deviation of such Gaussian distribution can be calculated exactly. We also conjecture that the partition function of supersymmetric HS spin chain satisfies a duality relation under the exchange of bosonic and fermionic spin degrees of freedom.

hep-th↗

arXiv2006

Quantization and Conformal Properties of a Generalized Calogero Model

We analyze a generalization of the quantum Calogero model with the underlying conformal symmetry, paying special attention to the two-body model deformation. Owing to the underlying $ SU(1,1) $ symmetry, we find that the analytic solutions of this model can be described within the scope of the Bargmann representation analysis and we investigate its dynamical structure by constructing the corresponding Fock space realization. The analysis from the standpoint of supersymmetric quantum mechanics (SUSYQM), when applied to this problem, reveals that the model is also shape invariant. For a certain range of the system parameters, the two-body generalization of the Calogero model is shown to admit a one-parameter family of self-adjoint extensions, leading to inequivalent quantizations of the system.

hep-th↗

arXiv2005

Multi-band structure of the quantum bound states for a generalized nonlinear Schrodinger model

By using the method of coordinate Bethe ansatz, we study N-body bound states of a generalized nonlinear Schrodinger model having two real coupling constants c and η. It is found that such bound states exist for all possible values of c and within several nonoverlapping ranges (called bands) of η. The ranges of ηwithin each band can be determined completely using Farey sequences in number theory. We observe that N-body bound states appearing within each band can have both positive and negative values of the momentum and binding energy.

hep-th↗

arXiv2004

Quantum bound states for a derivative nonlinear Schrodinger model and number theory

A derivative nonlinear Schrodinger model is shown to support localized N-body bound states for several ranges (called bands) of the coupling constant eta. The ranges of eta within each band can be completely determined using number theoretic concepts such as Farey sequences and continued fractions. For N > 2, the N-body bound states can have both positive and negative momentum. For eta > 0, bound states with positive momentum have positive binding energy, while states with negative momentum have negative binding energy.

hep-th↗

arXiv2004

Phase shift analysis of PT-symmetric nonhermitian extension of A_{N-1} Calogero model without confining interaction

We discuss a many-particle quantum system, which is obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the $A_{N-1}$ rational Calogero model without confining potential. This model gives rise to scattering states with continuous real spectrum. The scattering phase shift is determined through the exchange statistics parameter. We find that, unlike the case of usual Calogero model, the exclusion and exchange statistics parameter differ from each other in the presence of PT invariant interaction.

nlin.SI↗

arXiv2003

Construction of some special subsequences within a Farey sequence

Recently it has been found that some special subsequences within a Farey sequence play a crucial role in determining the ranges of coupling constant for which quantum soliton states can exist for an integrable derivative nonlinear Schrodinger model. In this article, we find a novel mapping which connects two such subsequences belonging to Farey sequences of different orders. By using this mapping, we construct an algorithm to generate all of these special subsequences within a Farey sequence. We also derive the continued fraction expansions for all the elements belonging to a subsequence and observe a close connection amongst the corresponding expansion coefficients.

math-ph↗

arXiv2003

Bound and scattering states of extended Calogero model with an additional PT invariant interaction

Here we discuss two many-particle quantum systems, which are obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the Calogero model with and without confining potential. It is shown that the energy eigenvalues are real for both of these quantum systems. For the case of extended Calogero model with confining potential, we obtain discrete bound states satisfying generalised exclusion statistics. On the other hand, the extended Calogero model without confining term gives rise to scattering states with continuous spectrum. The scattering phase shift for this case is determined through the exchange statistics parameter. We find that, unlike the case of usual Calogero model, the exclusion and exchange statistics parameter differ from each other in the presence of PT invariant interaction.

hep-th↗

arXiv2003

Exact solvability and quantum integrability of a derivative nonlinear Schrodinger model

By using a variant of quantum inverse scattering method (QISM) which is directly applicable to field theoretical systems, we derive all possible commutation relations among the operator valued elements of the monodromy matrix associated with an integrable derivative nonlinear Schrodinger (DNLS) model. From these commutation relations we obtain the exact Bethe eigenstates for the quantum conserved quantities of DNLS model. We also explicitly construct the first few quantum conserved quantities including the Hamiltonian in terms of the basic field operators of this model. It turns out that this quantum Hamiltonian has a new kind of coupling constant which is quite different from the classical one. This fact allows us to apply QISM to generate the spectrum of quantum DNLS Hamiltonian for the full range of its coupling constant.

hep-th↗

arXiv2003

Novel multi-band quantum soliton states for a derivative nonlinear Schrodinger model

We show that localized N-body soliton states exist for a quantum integrable derivative nonlinear Schrodinger model for several non-overlapping ranges (called bands) of the coupling constant η. The number of such distinct bands is given by Euler's ϕ-function which appears in the context of number theory. The ranges of ηwithin each band can also be determined completely using concepts from number theory such as Farey sequences and continued fractions. We observe that N-body soliton states appearing within each band can have both positive and negative momentum. Moreover, for all bands lying in the region η> 0, soliton states with positive momentum have positive binding energy (called bound states), while the states with negative momentum have negative binding energy (anti-bound states).

hep-th↗

arXiv2003

Jost solutions and quantum conserved quantities of an integrable derivative nonlinear Schrodinger model

We study differential and integral relations for the quantum Jost solutions associated with an integrable derivative nonlinear Schrodinger (DNLS) model. By using commutation relations between such Jost solutions and the basic field operators of DNLS model, we explicitly construct first few quantum conserved quantities of this system including its Hamiltonian. It turns out that this quantum Hamiltonian has a new kind of coupling constant which is quite different from the classical one. This modified coupling constant plays a crucial role in our comparison between the results of algebraic and coordinate Bethe ansatz for the case of DNLS model. We also find out the range of modified coupling constant for which the quantum $N$-soliton state of DNLS model has a positive binding energy.

hep-th↗

arXiv2003

Bound and anti-bound soliton states for a quantum integrable derivative nonlinear Schrodinger model

We find that localized quantum N-body soliton states exist for a derivative nonlinear Schrodinger (DNLS) model within an extended range of coupling constant (ξ_q) given by 0 < | ξ_q | < 1/\hbar \tan [π/(N-1)]. We also observe that soliton states with both positive and negative momentum can appear for a fixed value of ξ_q. Thus the chirality property of classical DNLS solitons is not preserved at the quantum level. Furthermore, it is found that the solitons with positive (negative) chirality have positive (negative) binding energy.

hep-th↗

arXiv2003

Inequivalent Quantizations of the Rational Calogero Model

We show that the rational Calogero model with suitable boundary condition admits quantum states with non-equispaced energy levels. Such a spectrum generically consists of infinitely many positive energy states and a single negative energy state. The new states appear for arbitrary number of particles and for specific ranges of the coupling constant. These states owe their existence to the self-adjoint extensions of the corresponding Hamiltonian, which are labelled by a real parameter z. Each value of z corresponds to a particular spectrum, leading to inequivalent quantizations of the model.

hep-th↗

arXiv2003

Novel Quantum States of the Rational Calogero Models Without the Confining Interaction

We show that the N-particle A_{N-1} and B_N rational Calogero models without the harmonic interaction admit a new class of bound and scattering states. These states owe their existence to the self-adjoint extensions of the corresponding Hamiltonians, labelled by a real parameter z. It is shown that the new states appear for all values of N and for specific ranges of the coupling constants. Moreover, they are shown to exist even in the excited sectors of the Calogero models. The self-adjoint extension generically breaks the classical scaling symmetry, leading to quantum mechanical scaling anomaly. The scaling symmetry can however be restored for certain values of the parameter z. We also generalize these results for many particle systems with classically scale invariant long range interactions in arbitrary dimensions.

hep-th↗

arXiv2002

Algebraic Bethe ansatz for a quantum integrable derivative nonlinear Schrodinger model

We find that the quantum monodromy matrix associated with a derivative nonlinear Schrodinger (DNLS) model exhibits U(2) or U(1,1) symmetry depending on the sign of the related coupling constant. By using a variant of quantum inverse scattering method which is directly applicable to field theoretical models, we derive all possible commutation relations among the operator valued elements of such monodromy matrix. Thus, we obtain the commutation relation between creation and annihilation operators of quasi-particles associated with DNLS model and find out the $S$-matrix for two-body scattering. We also observe that, for some special values of the coupling constant, there exists an upper bound on the number of quasi-particles which can form a soliton state for the quantum DNLS model.

hep-th↗