Highest Weight States Research Articles

Seven years of the proxy-SU(3) shell model symmetry

The proxy-SU(3) symmetry was first presented in HINPw4 in Ioannina in May2017, justified within the Nilsson model and applied to parameter-free predictions of the collective variables β and γ in medium-mass and heavy nuclei. Major steps forward, including the connection of the proxy-SU(3) symmetry to the shell model, the justification of the dominance of highest weight states in terms of the short range nature of the nucleon-nucleon interaction, as well as the first proposal of appearance of islands of shape coexistence on the nuclear chart, have been presented in HINPw6 in Athens in May 2021. The recently hot topic of the prevalence of triaxial shapes in heavy nuclei will also be briefly outlined in the proxy-SU(3) framework.

Constant primary operators and where to find them: the strange case of BPS defects in ABJ(M) theory

We investigate the one-dimensional defect SCFT defined on the 1/2 BPS Wilson line/loop in ABJ(M) theory. We show that the supermatrix structure of the defect imposes a covariant supermatrix representation of the supercharges. Exploiting this covariant formulation, we prove the existence of a long multiplet whose highest weight state is a constant supermatrix operator. At weak coupling, we study this operator in perturbation theory and confirm that it acquires a non-trivial anomalous dimension. At strong coupling, we conjecture that this operator is dual to the lowest bound state of fluctuations of the fundamental open string in AdS4 × ℂℙ3 around the classical 1/2 BPS solution. Quite unexpectedly, this operator also arises in the cohomological equivalence between bosonic and fermionic Wilson loops. We also discuss some regularization subtleties arising in perturbative calculations on the infinite Wilson line.

Celestial Klein spaces

We consider the analytic continuation of ($p+q$)-dimensional Minkowski space (with $p$ and $q$ even) to $(p,q)$ signature, and study the conformal boundary of the resulting ``Klein space.'' Unlike the familiar $(\ensuremath{-}+++\ensuremath{\cdots})$ signature, now the null infinity $\mathcal{I}$ has only one connected component. The spatial and timelike infinities (${i}^{0}$ and ${i}^{\ensuremath{'}}$) are quotients of generalizations of AdS spaces to nonstandard signature. Together, $\mathcal{I}$, ${i}^{0}$, and ${i}^{\ensuremath{'}}$ combine to produce the topological boundary ${S}^{p+q\ensuremath{-}1}$ as an ${S}^{p\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{S}^{q\ensuremath{-}1}$ fibration over a null segment. The highest weight states (the $L$-primaries) and descendants of $SO(p,q)$ with integral weights give rise to natural scattering states. One can also define $H$-primaries which are highest weight with respect to a signature-mixing version of the Cartan-Weyl generators that leave a point on the celestial ${S}^{p\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{S}^{q\ensuremath{-}1}$ fixed. These correspond to massless particles that emerge at that point and are Mellin transforms of plane wave states.

How Coordinate Bethe Ansatz Works for Inozemtsev Model

Three decades ago, Inozemtsev found an isotropic long-range spin chain with elliptic pair potential that interpolates between the Heisenberg and Haldane-Shastry (HS) spin chains while admitting an exact solution throughout, based on a connection with the elliptic quantum Calogero-Sutherland model. Though Inozemtsev's spin chain is widely believed to be quantum integrable, the underlying algebraic reason for its exact solvability is not yet well understood. As a step in this direction we refine Inozemtsev's `extended coordinate Bethe ansatz' and clarify various aspects of the model's exact spectrum and its limits. We identify quasimomenta in terms of which the $M$-particle energy is close to being (functionally) additive, as one would expect from the limiting models; our expression is additive iff the energy of the elliptic Calogero-Sutherland system is so. This enables us to rewrite the energy and Bethe-ansatz equations on the elliptic curve, turning the spectral problem into a rational problem as might be expected for an isotropic spin chain. We treat the $M=2$ particle sector and its limits in detail. We identify an $S$-matrix that is independent of positions. We show that the Bethe-ansatz equations reduce to those of Heisenberg in one limit and give rise to the `motifs' of HS in the other limit. We show that, as the interpolation parameter changes, the `scattering states' from Heisenberg become Yangian highest-weight states for HS, while bound states become ($\mathfrak{sl}_2$-highest weight versions of) affine descendants of the magnons from $M=1$. For bound states we find a generalisation of the known equation for the `critical length' for the Heisenberg spin chain. We discuss completeness for $M=2$ by passing to the elliptic curve. Our review of the two-particle sectors of the Heisenberg and HS spin chains may be of independent interest.

Revisiting the Ramond sector of the N=1 superconformal minimal models

Key to the exact solubility of the unitary minimal models in two-dimensional conformal field theory is the organization of their Hilbert space into Verma modules, whereby all eigenstates of the Hamiltonian are obtained by the repeated action of Virasoro lowering operators onto a finite set of highest-weight states. The usual representation-theoretic approach to removing from all modules zero-norm descendant states generated in such a way is based on the assumption that those states form a nested sequence of Verma submodules built upon singular vectors, i.e., descendant highest-weight states. We show that this fundamental assumption breaks down for the Ramond-sector Verma module with highest weight $c/24$ in the even series of $\mathcal{N}\!=\!1$ superconformal minimal models with central charge $c$. To resolve this impasse, we conjecture, and prove at low orders, the existence of a nested sequence of linear-dependence relations that enables us to compute the character of the irreducible $c/24$ module. Based on this character formula, we argue that imposing modular invariance of the torus partition function requires the introduction of a non-null odd-parity Ramond-sector ground state. This symmetrization of the ground-state manifold allows us to uncover a set of conformally invariant boundary conditions not previously discussed and absent in the odd series of superconformal minimal models, and to derive for the first time a complete set of fusion rules for the even series of those models.

Logarithmic correlation functions for critical dense polymers on the cylinder

We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter nn. In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity \alpha \in (0,\infty)α∈(0,∞). These correlators are defined as ratios Z(x)/Z_0Z(x)/Z0 of partition functions, where Z_0Z0 is a reference partition function wherein only simple half-arcs are attached to the boundary of the cylinder. For Z(x)Z(x), the boundary of the cylinder is also decorated with simple half-arcs, but it also has two special positions 11 and xx where the boundary condition is different. We investigate two such kinds of boundary conditions: (i) there is a single node at each of these points where a long arc is attached, and (ii) there are pairs of adjacent nodes at these points where two long arcs are attached. We find explicit expressions for these correlators for finite nn using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as n\to \inftyn→∞ are expressed as simple integrals that depend on the scaling parameter \tau = \frac {x-1} n \in (0,1)τ=x−1n∈(0,1). For small \tauτ, the leading behaviours are proportional to \tau^{1/4}τ1/4, \tau^{1/4}\log \tauτ1/4logτ, \log \taulogτ and \log^2 \taulog2τ. We interpret the lattice results in terms of ratios of conformal correlation functions. We assume that the corresponding boundary changing fields are highest weight states in irreducible, Kac or staggered Virasoro modules, with central charge c=-2c=−2 and conformal dimensions \Delta = -\frac18Δ=−18 or \Delta = 0Δ=0. With these assumptions, we obtain differential equations of order two and three satisfied by the conformal correlation functions, solve these equations in terms of hypergeometric functions, and find a perfect agreement with the lattice results. We use the lattice results to compute structure constants and ratios thereof which appear in the operator product expansions of the boundary condition changing fields. The fusion of these fields is found to be non-abelian.

Partition functions with spin in AdS2 via quasinormal mode methods

We extend the results of arXiv:1401.7016, computing one loop partition functions for massive fields with spin half in AdS_2 using the quasinormal mode method proposed by Denef, Hartnoll, and Sachdev in arXiv:0908.2657. We find the finite representations of SO(2,1) for spin zero and spin half, consisting of a highest weight state |h\rangle and descendants with non-unitary values of h. These finite representations capture the poles and zeroes of the one loop determinants. Together with the asymptotic behavior of the partition functions (which can be easily computed using a large mass heat kernel expansion), these are sufficient to determine the full answer for the one loop determinants. We also discuss extensions to higher dimensional AdS_{2n} and higher spins.

Universal Bethe ansatz solution for the Temperley–Lieb spin chain

We consider the Temperley–Lieb (TL) open quantum spin chain with “free” boundary conditions associated with the spin-s representation of quantum-deformed sl(2). We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.

On skew tau-functions in higher spin theory

Recent studies of higher spin theory in three dimensions concentrate on Wilson loops in Chern-Simons theory, which in the classical limit reduce to peculiar corner matrix elements between the highest and lowest weight states in a given representation of SL(N). Despite these "skew" tau-functions can seem very different from conventional ones, which are the matrix elements between the two highest weight states, they also satisfy the Toda recursion between different fundamental representations. Moreover, in the most popular examples they possess simple representations in terms of matrix models and Schur functions. We provide a brief introduction to this new interesting field, which, after quantization, can serve as an additional bridge between knot and integrability theories.

Proof of the Wehrl-type Entropy Conjecture for Symmetric $${SU(N)}$$ Coherent States

The Wehrl entropy conjecture for coherent (highest weight) states in representations of the Heisenberg group, which was proved in 1978 and recently extended by us to the group $${SU(2)}$$ , is further extended here to symmetric representations of the groups $${SU(N)}$$ for all N. This result gives further evidence for our conjecture that highest weight states minimize group integrals of certain concave functions for a large class of Lie groups and their representations.

Nontrivial magnetoresistive behavior of a single-wall carbon nanotube with an attached molecular magnet

The spin-resolved transport properties of a single-wall carbon nanotube quantum dot, with an attached single molecular magnet, are studied theoretically. With the aid of the real-time diagrammatic technique in the lowest-order perturbation expansion with respect to the tunnel coupling, the current, differential conductance, and the tunnel magnetoresistance (TMR) are determined in both the linear and nonlinear response regimes. It is shown that transport properties depend greatly on both the shell filling sequence of the carbon nanotube and the type of exchange interaction between the molecular magnet and nanotube. This results in highly nontrivial behavior of the TMR, which is especially visible in the low bias voltage regime. Depending on the gate voltage and parameters of the system, we find transport regimes where either a greatly enhanced or negative TMR develops. The mechanism leading to such behavior is associated with nonequilibrium spin accumulation, which builds up in the antiparallel magnetic configuration of the device. We show that it is crucial whether the spin accumulation occurs in the highest-weight spin states or in states with lower spin values. While in the former case it leads to enhanced TMR, in the latter case it may result in negative tunnel magnetoresistance. In addition, we analyze how the above effects depend on the magnitude of the molecular magnet's spin, and show that this dependence is generally nonmonotonic.

Macroscopic quantum spin tunneling with two interacting spins

We study the simple Hamiltonian, $H=\ensuremath{-}K({S}_{1z}^{2}+{S}_{2z}^{2})+\ensuremath{\lambda}{\stackrel{P\vec}{S}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{P\vec}{S}}_{2}$, of two large, coupled spins which are taken equal, each of total spin $s$ with $\ensuremath{\lambda}$ the exchange coupling constant. The exact ground state of this simple Hamiltonian is not known for an antiferromagnetic coupling corresponding to the $\ensuremath{\lambda}>0$. In the absence of the exchange interaction, the ground state is fourfold degenerate, corresponding to the states where the individual spins are in their highest weight or lowest weight states, $|\ensuremath{\uparrow},\ensuremath{\uparrow}\ensuremath{\rangle},|\ensuremath{\downarrow},\ensuremath{\downarrow}\ensuremath{\rangle},|\ensuremath{\uparrow},\ensuremath{\downarrow}\ensuremath{\rangle},|\ensuremath{\downarrow},\ensuremath{\uparrow}\ensuremath{\rangle}$, in obvious notation. The first two remain exact eigenstates of the full Hamiltonian. However, we show that the two states $|\ensuremath{\uparrow},\ensuremath{\downarrow}\ensuremath{\rangle},|\ensuremath{\downarrow},\ensuremath{\uparrow}\ensuremath{\rangle}$ organize themselves into the combinations $|\ifmmode\pm\else\textpm\fi{}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|\ensuremath{\uparrow},\ensuremath{\downarrow}\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\rangle})$, up to perturbative corrections. For the antiferromagnetic case, we show that the ground state is nondegenerate, and we find the interesting result that for integer spins the ground state is $|+\ensuremath{\rangle}$ and the first excited state is the antisymmetric combination $|\ensuremath{-}\ensuremath{\rangle}$ while for half odd integer spin, these roles are exactly reversed. The energy splitting, however, is proportional to ${\ensuremath{\lambda}}^{2s}$, as expected by perturbation theory to the $2s$th order. We obtain these results through the spin coherent state path integral.

BCS model with asymmetric pair scattering: a non-Hermitian, exactly solvable Hamiltonian exhibiting generalized exclusion statistics

We demonstrate the occurrence of free quasi-particle excitations obeying generalized exclusion statistics in a BCS model with asymmetric pair scattering. The results are derived from an exact solution of the Hamiltonian, which was obtained via the algebraic Bethe ansatz utilizing the representation theory of an underlying Yangian algebra. The free quasi-particle excitations are associated with highest weight states of the Yangian algebra, corresponding to a class of analytic solutions of the Bethe ansatz equations.

Computer implementation of the Kerov–Kirillov–Reshetikhin algorithm

The Kerov–Kirillov–Reshetikhin (KKR) algorithm establishes a bijection between semistandard Weyl tableaux of the shape λ and weight μ and rigged string configurations of type (λ,μ). This algorithm can be applied to Heisenberg magnetic chains and their generalisations by use of Schur–Weyl duality, and gives a way to classify all eigenstates of the chain in a methodological manner. Program summaryProgram title: KKR algorithmCatalogue identifier: AELQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AELQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 974No. of bytes in distributed program, including test data, etc.: 11 118Distribution format: tar.gzProgramming language: MapleComputer: PCOperating system: Windows, LinuxRAM: 1 GBClassification: 2.7, 4.2Nature of problem: The method of classification of all eigenstates and eigenvalues is given for the generalised model of the one-dimensional Heisenberg magnet consisting of N nodes.Solution method: We use a prolongation of the RSK bijection [1] to the set of all eigenstates of an integrable model, in terms of rigged string configurations. More specifically, one selects from the irreducible basis birr of the Schur–Weyl duality all highest weight states |λthwy〉 with respect to the unitary group U(n) and generates a rigged string configurations |ν→,{J}〉=KKR(|λthwy〉) for each y∈SYT(λ).Running time: The running time is about 1 ms.Reference:[1]S. Veigneau, Algebraic Combinatorics Environment for the computer algebra system, Userʼs Reference Manual, Version 3.0 98-11, Université de Marne-la-Vallé, Institut Gaspard-Monge, CNRS, 1998.

Solutions of free higher spins in AdS

We consider free massive and massless higher integer spins in AdS backgrounds in general D dimensions. We obtain the solutions corresponding to the highest-weight state of the spin-ℓ representations of the SO(2,D−1) isometry groups. The solution for the spin-ℓ field is expressed recursively in terms of that for the spin-(ℓ−1). Thus starting from the explicit spin-0, all the higher-spin solutions can be obtained. These solutions allow us to derive the generalized Breitenlohner–Freedman bound, and analyze the asymptotic falloffs. In particular, solutions with negative mass square in general have falloffs slower than those of the Schwarzschild AdS black holes in the AdS boundaries.

Highest weight Macdonald and Jack polynomials

Fractional quantum Hall states of particles in the lowest Landau levels are described by multivariate polynomials. The incompressible liquid states when described on a sphere are fully invariant under the rotation group. Excited quasiparticle/quasihole states are members of multiplets under the rotation group and generically there is a nontrivial highest weight member of the multiplet from which all states can be constructed. Some of the trial states proposed in the literature belong to classical families of symmetric polynomials. In this paper we study Macdonald and Jack polynomials that are highest weight states. For Macdonald polynomials, it is a (q, t)-deformation of the raising angular momentum operator that defines the highest weight condition. By specialization of the parameters we obtain a classification of the highest weight Jack polynomials. Our results are valid in the case of staircase and rectangular partition indexing the polynomials.

Particle-hole gel'fand states and quasidegenerate perturbation theory

The unitary group formulation of many-body theory is based on particle-hole Gel'fand states, states labeled only by the excitation process: h(hole orbitals)-p(particle orbitals). The particle-hole Gel'fand states are constructed from generator states, states obtained by applying weight-lowering products of generators to the highest weight state and then evaluating the matrix elements Lie algebraically. We make application to spin-projected, quasidegenerate perturbation theory.

The unitary-group formulation of coupled-cluster many-electron theory

Coupled-cluster many-electron theory (CCMET) has been developed in the second-quantized formulation with and without spin projection. In this article CCMET is developed in the unitary-group formulation where the group is U(p) and p is the number of orbitals in the basis set. The zero-order ground state is the highest-weight state of an irreducibly invariant subspace of U(p) and the excitation operators are infinitesimal generators. These irreducible spaces are uniquely labeled by Young diagrams which the Pauli principle limits to no more than one column for fermion orbitals and no more than two columns for freeon (spin-free) orbitals. In the latter case the spin is one-half the difference in the lengths of the two columns. We employ both the Gel'fand and the generator bases for the irreducible invariant vector spaces. Matrix elements for the former are evaluated by techniques due to Gel'fand, Biedenharn, Louck, Paldus, and Shavitt and for the latter directly by Lie algebraic and diagrammatic techniques as a simple function of the weight components of the highest-weight state. For freeon or fermion orbitals the results are equivalent to those obtained by the second-quantized formulation with or without spin projection.

Operator mixing and the AdS/CFT correspondence

We provide a direct prescription for computing the mixing among gauge invariant operators in = 4 SYM. Our approach is based on the action of the superalgebra on the states of the theory and thus it can be also applied to resolve the mixing in the dual string description. As an example, we focus on the supermultiplet containing the BMN operators with two impurities. On the field theory side, we derive the leading planar quantum corrections to the naive expression of the highest weight state. Then we use the same prescription in the BMN limit of the AdS5 × S5 string theory and derive the form of the 2-impurity highest weight state. The string expression matches nicely the SYM result and provides a prediction for the mixing due to higher order quantum corrections in field theory.

Modular differential equations for torus one-point functions

It is shown that in a rational conformal field theory every torus one-point function of a given highest weight state satisfies a modular differential equation. We derive and solve these differential equations explicitly for some Virasoro minimal models. In general, however, the resulting amplitudes do not seem to be expressible in terms of standard transcendental functions.