Haldane-Shastry Model Research Articles

Infinite matrix product states for long-range [formula omitted] spin models

We construct 1D and 2D long-range SU(N) spin models as parent Hamiltonians associated with infinite matrix product states. The latter are constructed from correlators of primary fields in the SU(N)1 WZW model. Since the resulting groundstates are of Gutzwiller–Jastrow type, our models can be regarded as lattice discretizations of fractional quantum Hall systems. We then focus on two specific types of 1D spin chains with spins located on the unit circle, a uniform and an alternating arrangement. For an equidistant distribution of identical spins we establish an explicit connection to the SU(N) Haldane–Shastry model, thereby proving that the model is critical and described by a SU(N)1 WZW model. In contrast, while turning out to be critical as well, the alternating model can only be treated numerically. Our numerical results rely on a reformulation of the original problem in terms of loop models.

Quantum spin models for the [formula omitted] Wess–Zumino–Witten model

We propose 1D and 2D lattice wave functions constructed from the SU(n)1 Wess–Zumino–Witten (WZW) model and derive their parent Hamiltonians. When all spins in the lattice transform under SU(n) fundamental representations, we obtain a two-body Hamiltonian in 1D, including the SU(n) Haldane–Shastry model as a special case. In 2D, we show that the wave function converges to a class of Halperin's multilayer fractional quantum Hall states and belongs to chiral spin liquids. Our result reveals a hidden SU(n) symmetry for this class of Halperin states. When the spins sit on bipartite lattices with alternating fundamental and conjugate representations, we provide numerical evidence that the state in 1D exhibits quantum criticality deviating from the expected behaviors of the SU(n)1 WZW model, while in 2D they are chiral spin liquids being consistent with the prediction of the SU(n)1 WZW model.

Laughlin Spin-Liquid States on Lattices Obtained from Conformal Field Theory

We propose a set of spin system wave functions that is very similar to lattice versions of the Laughlin states. The wave functions are conformal blocks of conformal field theories and for a filling factor of ν = 1/2 we provide a parent Hamiltonian, which is valid for any even number of spins and is at the same time a 2D generalization of the Haldane-Shastry model. We also demonstrate that the Kalmeyer-Laughlin state is reproduced as a particular case of this model. Finally, we discuss various properties of the spin states and point out several analogies to known results for the Laughlin states.

Family of spin-Schain representations of SU(2)kWess-Zumino-Witten models

We investigate a family of spin-$S$ chain Hamiltonians recently introduced by one of us [Greiter, Mapping of Parent Hamiltonians, Springer Tracts in Modern Physics, Vol. 244 (Springer, Berlin, 2011)]. For $S=1/2$, it corresponds to the Haldane-Shastry model. For general spin $S$, we find indication that the low-energy theory of these spin chains is described by the SU(2)${}_{k}$ Wess-Zumino-Witten model with coupling $k=2S$. In particular, we investigate the $S=1$ model whose ground state is given by a Pfaffian for even number of sites $N$. We reconcile aspects of the spectrum of the Hamiltonian for arbitrary $N$ with trial states obtained by Schwinger projection of two Haldane-Shastry chains.

Quantum spin Hamiltonians for theSU(2)k WZW model

We propose to use null vectors in conformal field theories to derive modelHamiltonians of quantum spin chains and the corresponding ground statewavefunction(s). The approach is quite general, and we illustrate it by constructinga family of Hamiltonians whose ground states are the chiral correlators of theSU(2)k WZW model for integer values of the levelk. The simplest examplecorresponds to k = 1 and is essentially a nonuniform generalization of theHaldane–Shastry model with long-range exchange couplings. At levelk = 2, we analysethe model for N spin 1 fields. We find that the Renyi entropy and the two-point spin correlator show,respectively, logarithmic growth and algebraic decay. Furthermore, we use the null vectors toderive a set of algebraic, linear equations relating spin correlators within each model. At levelk = 1, these equations allow us to compute the two-point spin correlators analytically for thefinite chain uniform Haldane–Shastry model and to obtain numerical results for thenonuniform case and for higher-point spin correlators in a very simple way and withoutresorting to Monte Carlo techniques.

Conservation laws, integrability, and transport in one-dimensional quantum systems

In integrable one-dimensional quantum systems an infinite set of local conserved quantities exists which can prevent a current from decaying completely. For cases like the spin current in the $\mathit{XXZ}$ model at zero magnetic field or the charge current in the attractive Hubbard model at half filling, however, the current operator does not have overlap with any of the local conserved quantities. We show that in these situations transport at finite temperatures is dominated by a diffusive contribution with the Drude weight being either small or even zero. For the $\mathit{XXZ}$ model we discuss in detail the relation between our results, the phenomenological theory of spin diffusion, and measurements of the spin-lattice relaxation rate in spin chain compounds. Furthermore, we study the Haldane-Shastry model where a conserved spin current exists.

Infinite matrix product states, conformal field theory, and the Haldane-Shastry model

We generalize the matrix product states method using the chiral vertex operators of conformal field theory and apply it to study the ground states of the XXZ spin chain, the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model and random Heisenberg models. We compute the overlap with the exact wave functions, spin-spin correlators, and the Renyi entropy, showing that critical systems can be described by this method. For rotational invariant ansatzs we construct an inhomogenous extension of the Haldane-Shastry model with long-range exchange interactions.

Entanglement in a spin system with inverse square statistical interaction

We investigate the entanglement content of the ground state of a system characterized by effective elementary degrees of freedom with fractional statistics. To this end, we explicitly construct the ground state for a chain of N spins with inverse square interaction (the Haldane–Shastry model) in the presence of an external uniform magnetic field. For such a system at zero temperature, we evaluate the entanglement in the ground state both at finite size and in the thermodynamic limit. We relate the behavior of the quantum correlations with the spinon condensation phenomenon occurring at the saturation field.

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.

Derivation of Green's function of a spin Calogero–Sutherland model by Uglov's method

The hole propagator of a spin 1/2 Calogero–Sutherland model is derived using Uglov's method, which maps the exact eigenfunctions of the model, called the Yangian Gelfand-Zetlin basis, to a limit of Macdonald polynomials (gl2-Jack polynomials). To apply this mapping method to the calculation of 1-particle Green's function, we confirm that the sum of the field annihilation operator ψ↑ + ψ↓ on a Yangian Gelfand-Zetlin basis is transformed to the field annihilation operator ψ on gl2-Jack polynomials by the mapping. The resultant expression for the hole propagator for a finite-size system is written in terms of renormalized momenta and spin of quasi-holes, and the expression in the thermodynamic limit coincides with the earlier result derived by another method. We also discuss the singularity of the spectral function for a specific coupling parameter where the hole propagator of the spin Calogero–Sutherland model becomes equivalent to the dynamical colour correlation function of an SU(3) Haldane–Shastry model.

Applying matrix product operators to model systems with long-range interactions

An algorithm is presented which computes a translationally invariant matrix product state approximation of the ground state of an infinite one-dimensional (1D) system. It does this by embedding sites into an approximation of the infinite ``environment'' of the chain, allowing the sites to relax and then merging them with the environment in order to refine the approximation. By making use of matrix product operators, our approach is able to directly model any long-range interaction that can be systematically approximated by a series of decaying exponentials. We apply these techniques to compute the ground state of the Haldane-Shastry model [Phys. Rev. Lett. 60, 635 (1988) and Phys. Rev. Lett. 60, 639 (1988)] and present the results.

Interaction and thermodynamics of spinons in the XX chain

The mapping between the fermion and spinon compositions of eigenstates in the one-dimensional spin-1/2 XX model on a lattice with N sites is used to describe the spinon interaction from two different perspectives: (i) for finite N the energy of all eigenstates is expressed as a function of spinon momenta and spinon spins, which, in turn, are solutions of a set of Bethe ansatz equations. The latter are the basis of an exact thermodynamic analysis in the spinon representation of the XX model. (ii) For N → ∞ the energy per site of spinon configurations involving any number of spinon orbitals is expressed as a function of reduced variables representing momentum, filling and magnetization of each orbital. The spins of spinons in a single orbital are found to be coupled in a manner well described by an Ising-like equivalent-neighbor interaction, switching from ferromagnetic to antiferromagnetic as the filling exceeds a critical level. Comparisons are made with results for the Haldane–Shastry model.

Two-spin-subsystem entanglement in spin-1/2 rings with long-range interactions

We consider the two-spin-subsystem entanglement for eigenstates of the Hamiltonian $H={\ensuremath{\sum}}_{1\ensuremath{\le}j<k\ensuremath{\le}N}{(\frac{1}{{r}_{j,k}})}^{\ensuremath{\alpha}}{\ensuremath{\sigma}}_{j}\ensuremath{\cdot}{\ensuremath{\sigma}}_{k}$ for a ring of $N$ spin-1/2 particles with associated spin vector operator $(\ensuremath{\hbar}/2){\ensuremath{\sigma}}_{j}$ for the $j$th spin. Here ${r}_{j,k}$ is the chord distance between sites $j$ and $k$. The case $\ensuremath{\alpha}=2$ corresponds to the solvable Haldane-Shastry model whose spectrum has very high degeneracies not present for $\ensuremath{\alpha}\ensuremath{\ne}2$. Two-spin-subsystem entanglement shows high sensitivity and distinguishes $\ensuremath{\alpha}=2$ from $\ensuremath{\alpha}\ensuremath{\ne}2$. There is no entanglement beyond nearest neighbors for all eigenstates when $\ensuremath{\alpha}=2$. Whereas for $\ensuremath{\alpha}\ensuremath{\ne}2$ one has selective entanglement at any distance for eigenstates of sufficiently high energy in a certain interval of $\ensuremath{\alpha}$ which depends on the energy. The ground state (which is a singlet only for even $N$) does not have entanglement beyond nearest neighbors, and the nearest-neighbor entanglement is virtually independent of the range of the interaction controlled by $\ensuremath{\alpha}$.

Many-spinon states and representations of Yangians in the SU(n) Haldane–Shastry model

We study the relation of Yangians and spinons in the SU(n) Haldane–Shastry model. The representation theory of the Yangian is shown to be intimately related to the fractional statistics of the spinons. We construct the spinon Hilbert space from tensor products of the fundamental representations of the Yangian.

Many-Spinon States and the Secret Significance of Young Tableaux

We establish a one-to-one correspondence between the Young tableaux classifying the total spin representations of N spins and the exact eigenstates of the Haldane-Shastry model for a chain with N sites classified by the total spins and the fractionally spaced single-particle momenta of the spinons.

Cooper pairs and exclusion statistics from coupled free-fermion chains

We show how to couple two free-fermion chains so that the excitations consist of Cooperpairs with zero energy, and free particles obeying (mutual) exclusion statistics. Thisbehaviour is reminiscent of anyonic superconductivity, and of a ferromagnetic version of theHaldane–Shastry spin chain, although here the interactions are local. We solve this modelusing the nested Bethe ansatz, and find all the eigenstates; the Cooper pairscorrespond to exact-string or ‘0/0’ solutions of the Bethe equations. We showhow the model possesses an infinite-dimensional symmetry algebra, which is asupersymmetric version of the Yangian symmetry algebra for the Haldane–Shastry model.

Coloron excitations of the SU(3) Haldane-Shastry model

In a recent publication, we proposed two possible wave functions for the elementary excitations of the SU(3) Haldane-Shastry model, but argued on very general grounds that only one or the other can be a valid excitation. Here we provide the explicit details of our calculation proving that the wave function describing a coloron excitation which transforms according to representation $\overline{3}$ under SU(3) rotations if the spins of the original model transform according to representation 3, is exact. We further provide an explicit construction of the exact color-polarized two-coloron eigenstates, and thereby show that colorons are free but that their relative momentum spacings are shifted according to fractional statistics with parameter $g=2∕3$. We evaluate the SU(3) spin currents. Finally, we interpret our results within the framework of the asymptotic Bethe ansatz and generalize some of them to the case of $\mathrm{SU}(n)$.

No attraction between spinons in the Haldane-Shastry model

While the Bethe ansatz solution of the Haldane-Shastry model appears to suggest that the spinons represent a free gas of half-fermions Bernevig, Giuliano, and Laughlin (BGL) [Phys. Rev. Lett. 86, 3392 (2001); Phys. Rev. B 64, 24425 (2001)] have concluded recently that there is an attractive interaction between spinons. We argue that the dressed scattering matrix obtained with the asymptotic Bethe ansatz is to be interpreted as the true and physical scattering matrix of the excitations, and hence, that the result by BGL is inconsistent with an earlier result by Essler [Phys. Rev. B 51, 13357 (1995)]. We critically reexamine the analysis of BGL, and conclude that there is no interaction between spinons or spinons and holons in the Haldane-Shastry model.

Spinons in the Haldane–Shastry model: An ideal gas of half-fermions

The dynamics of spinons in the Haldane–Shastry model, a paradigm for antiferromagnetic spin systems, has been subject to intense recent interest. Bernevig, Giuliano and Laughlin have presented arguments for the existence of a short-ranged, attractive interaction between the spinon excitations in this model. We re-examine their analysis and conclude that their interpretation is not correct; the spinon excitations in the Haldane–Shastry model do not interact. The spinons rather represent an ideal gas of half-fermions.