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1 A Note on Inverse-Square Exchange Models

Frank Göhmann, Miki Wadati

Published Jan 1, 1995• 5 citations
OpenAlex

Abstract

The su(1|1) symmetric version of the Haldane-Shastry spin chain is diagonalized by means of a linear transformation. The same transformation applied to the original su(2) model yields simple expressions for the Hamiltonian and the generators of the Yangian symmetry of the model in terms of spin wave operators. Several years ago Haldane and Shastry independently discovered a soluble spin chain with exchange interaction falling off as the inverse square of the distance between the spins [1, 2]. This model can be thought of as the static limit [3] of the Sutherland model with internal degrees of freedom [4, 5, 6]. The idea of putting internal degrees of freedom into the Calogero-Sutherland models has proved to be very fruitful since then. A rich family of related models has been found. These long range interacting systems are under active investigation now. A common algebraic structure underlying their integrability is still waiting for its discovery. In this letter we discuss a linear transformation, that turns the su(m|n) supersymmetric extension of the original Haldane-Shastry Hamiltonian into a more compact form,

Related papers

A Note on Inverse-Square Exchange Models

Frank Göhmann, Miki Wadati

The su(1$|$1) symmetric version of the Haldane-Shastry spin chain is diagonalized by means of a linear transformation. The same transformation applied to the original su(2) model yields simple expressions for the Hamiltonian and the generators of the Yangian symmetry of the model in terms of spin wave operators.

arXiv:cond-mat/9506077•Journal of the Physical Society of Japan•Oct 15, 1995•9 citations

Spectrum of a spin chain with inverse square exchange

Holger Frahm

The spectrum of a one-dimensional chain of SU(n) spins positioned at the static equilibrium positions of the particles in the corresponding classical Calogero system with an exchange interaction inversely proportional to the square of their distance is studied. As in the translationally invariant Haldane-Shastry model the spectrum is found to exhibit a very simple structure containing highly degenerate 'super-multiplets'. The algebra underlying this structure is identified and several sets of raising and lowering operators are given explicitly. On the basis of this algebra and numerical studies the authors give the complete spectrum and thermodynamics of the SU(2) system.

arXiv:cond-mat/9303050•Journal of Physics A Mathematical and General•May 21, 1993•91 citations

A novel class of translationally invariant spin chains with long-range interactions

B. Basu-Mallick, F. Finkel, A. González-López

Abstract We introduce a new class of open, translationally invariant spin chains with long-range interactions depending on both spin permutation and (polarized) spin reversal operators, which includes the Haldane-Shastry chain as a particular degenerate case. The new class is characterized by the fact that the Hamiltonian is invariant under “twisted” translations, combining an ordinary translation with a spin flip at one end of the chain. It includes a remarkable model with elliptic spin-spin interactions, smoothly interpolating between the XXX Heisenberg model with anti-periodic boundary conditions and a new open chain with sites uniformly spaced on a half-circle and interactions inversely proportional to the square of the distance between the spins. We are able to compute in closed form the partition function of the latter chain, thereby obtaining a complete description of its spectrum in terms of a pair of independent su(1|1) and su(m/2) motifs when the number m of internal degrees of freedom is even. This implies that the even m model is invariant under the direct sum of the Yangians Y (gl(1|1)) and Y (gl(0|m/2)). We also analyze several statistical properties of the new chain’s spectrum. In particular, we show that it is highly degenerate, which strongly suggests the existence of an underlying (twisted) Yangian symmetry also for odd m.

arXiv:2003.07217•Journal of High Energy Physics•Aug 1, 2020•3 citations

From fermionic spin-Calogero–Sutherland models to the Haldane–Shastry chain by freezing

Jules Lamers, Didina Serban

Abstract The Haldane–Shastry spin chain has a myriad of remarkable properties, including Yangian symmetry and, for spin 1 / 2 , explicit highest-weight eigenvectors featuring (the case α = 1 / 2 of) Jack polynomials. This stems from the spin-Calogero–Sutherland model, which reduces to Haldane–Shastry in a special ‘freezing’ limit. In this work we clarify various points that, to the best of our knowledge, were missing in the literature. We have two main results. First, we show that freezing the fermionic spin-1/2 Calogero–Sutherland model naturally accounts for the precise form of the Haldane–Shastry wave functions, including the Vandermonde factor squared. Second, we use the fermionic framework to prove the claim of Bernard–Gaudin–Haldane–Pasquier that the Yangian highest-weight eigenvectors of the SU ( r ) -version of the Haldane–Shastry chain arise by freezing SU ( r − 1 ) spin-Calogero–Sutherland eigenvectors at α = 1 / 2 .

Journal of Physics A Mathematical and Theoretical•May 14, 2024•7 citations

SU(K) and supersymmetric Yangians

Yoshio Kuramoto, Yusuke S. Kato

We now describe the properties of the SU( >K ) chain in more algebraic terms. The relevant symmetry is the SU( K ) Yangian, which is usually referred to as Y ( sl K ). On the other hand, the Yangian symmetry of the Sutherland model with spin 1/2 is given by Y ( gl 2 ). The difference between Y ( sl K ) and Y ( gl K ) is that the charge degrees of freedom U(1) is absent in Y ( sl K ), while the spin degrees of freedom is included in both Y ( sl K ) and Y ( sl K ). In Section 4.8.3, we have removed the U(1) component by means of the freezing trick, i.e., by taking the infinitely large repulsion parameter. In this chapter we take a more algebraic approach to study Y ( sl K ), which makes the relation between Y ( gl K ) and Y ( sl K ) clearer. In Section 9.1, we begin with the construction of the SU( K ) monodromy matrix as the product form. The commuting property of the Cherednik–Dunkl operators plays a crucial role here. The monodromy matrix is the basic building block of the Yangian algbebra, as discussed in Section 8.2. Because the Hamiltonian of the SU( K ) spin chain and the monodromy matrix commute, each eigenstate of the spin chain constitutes a basis for representations of the SU( K ) Yangian. In Section 9.2 and 9.3, we discuss the quantum determinant by comparing it with the ordinary determinant. It is demonstrated in Section 9.4 that the quantum determinant has a scalar nature that leads to a simple evaluation. Because of the scalar nature, however, the quantum determinant cannot distinguish different SU( K ) Yangian states.

Cambridge University Press eBooks•Aug 6, 2009•0 citations

Integrability and Applications of the Exactly-Solvable Haldane-Shastry One-Dimensional Quantum Spin Chain

J. C. Talstra

Recently, the one dimensional model of $N$ spins with $S=\frac{1}{2}$ on a circle, interacting with an exchange that falls off with the inverse square of the separation: $H_{\rm ISE} =\sum_{i\neq j} \frac{1}{[\frac{N}π \sin(\frac{i-j}{N}π)]^2}(\svec_i\cdot \svec_j-\frac{1}{4})$, or ISE-model, has received ample attention. Its special features include: relatively simple eigenfunctions, non-interacting elementary excitations that obey semionic statistics (spinons), and a large ``quantum group'' symmetry algebra called the Yangian. This model is fully integrable, albeit in a slightly different sense than the more traditional nearest neighbor exchange (NNE) Heisenberg chain. This thesis comes in 4 chapters. Chapter 1 introduces the model and presents the construction of a subset of the eigenfunctions. The other eigenfunctions are shown to be generated by the action of the Yangian symmetry algebra of $H_{\rm ISE}$. Chapter 2 presents a method to construct the set of constants of the motion of the ISE-model. The ISE-model is tractable enough to obtain its zero-magnetic-field dynamical structure factors. Chapter 3 attempts to extend this to a non-zero magnetic field, where, due to the presence of spinons in the groundstate, more complicated excitations contribute: small numbers of magnons and spinons. discuss the relation to the more complicated NNE-model. Finally chapter 4 illustrates how a recently conjectured new form of Off Diagonal Long Range Order in antiferromagnetic spin chains can be reinterpreted as a spinon propagator in the ISE-model, and verified numerically. We briefly comment on its relevance to stabilizing superconductivity in the layered cuprates.

arXiv:cond-mat/9509178•Sep 30, 1995•4 citations
The su(1|1) symmetric version of the Haldane-Shastry spin chain is diagonalized by means of a linear transformation. The same transformation applied to the original su(2) model yields simple expressions for the Hamiltonian and the generators of the Yangian symmetry of the model in terms of spin wave operators. Several years ago Haldane and Shastry independently discovered a soluble spin chain with exchange interaction falling off as the inverse square of the distance between the spins [1, 2]. This model can be thought of as the static limit [3] of the Sutherland model with internal degrees of freedom [4, 5, 6]. The idea of putting internal degrees of freedom into the Calogero-Sutherland models has proved to be very fruitful since then. A rich family of related models has been found. These long range interacting systems are under active investigation now. A common algebraic structure underlying their integrability is still waiting for its discovery. In this letter we discuss a linear transformation, that turns the su(m|n) supersymmetric extension of the original Haldane-Shastry Hamiltonian into a more compact form,
The spectrum of a one-dimensional chain of SU(n) spins positioned at the static equilibrium positions of the particles in the corresponding classical Calogero system with an exchange interaction inversely proportional to the square of their distance is studied. As in the translationally invariant Haldane-Shastry model the spectrum is found to exhibit a very simple structure containing highly degenerate 'super-multiplets'. The algebra underlying this structure is identified and several sets of raising and lowering operators are given explicitly. On the basis of this algebra and numerical studies the authors give the complete spectrum and thermodynamics of the SU(2) system.
Abstract We introduce a new class of open, translationally invariant spin chains with long-range interactions depending on both spin permutation and (polarized) spin reversal operators, which includes the Haldane-Shastry chain as a particular degenerate case. The new class is characterized by the fact that the Hamiltonian is invariant under “twisted” translations, combining an ordinary translation with a spin flip at one end of the chain. It includes a remarkable model with elliptic spin-spin interactions, smoothly interpolating between the XXX Heisenberg model with anti-periodic boundary conditions and a new open chain with sites uniformly spaced on a half-circle and interactions inversely proportional to the square of the distance between the spins. We are able to compute in closed form the partition function of the latter chain, thereby obtaining a complete description of its spectrum in terms of a pair of independent su(1|1) and su(m/2) motifs when the number m of internal degrees of freedom is even. This implies that the even m model is invariant under the direct sum of the Yangians Y (gl(1|1)) and Y (gl(0|m/2)). We also analyze several statistical properties of the new chain’s spectrum. In particular, we show that it is highly degenerate, which strongly suggests the existence of an underlying (twisted) Yangian symmetry also for odd m.
Abstract The Haldane–Shastry spin chain has a myriad of remarkable properties, including Yangian symmetry and, for spin 1 / 2 , explicit highest-weight eigenvectors featuring (the case α = 1 / 2 of) Jack polynomials. This stems from the spin-Calogero–Sutherland model, which reduces to Haldane–Shastry in a special ‘freezing’ limit. In this work we clarify various points that, to the best of our knowledge, were missing in the literature. We have two main results. First, we show that freezing the fermionic spin-1/2 Calogero–Sutherland model naturally accounts for the precise form of the Haldane–Shastry wave functions, including the Vandermonde factor squared. Second, we use the fermionic framework to prove the claim of Bernard–Gaudin–Haldane–Pasquier that the Yangian highest-weight eigenvectors of the SU ( r ) -version of the Haldane–Shastry chain arise by freezing SU ( r − 1 ) spin-Calogero–Sutherland eigenvectors at α = 1 / 2 .
We now describe the properties of the SU( >K ) chain in more algebraic terms. The relevant symmetry is the SU( K ) Yangian, which is usually referred to as Y ( sl K ). On the other hand, the Yangian symmetry of the Sutherland model with spin 1/2 is given by Y ( gl 2 ). The difference between Y ( sl K ) and Y ( gl K ) is that the charge degrees of freedom U(1) is absent in Y ( sl K ), while the spin degrees of freedom is included in both Y ( sl K ) and Y ( sl K ). In Section 4.8.3, we have removed the U(1) component by means of the freezing trick, i.e., by taking the infinitely large repulsion parameter. In this chapter we take a more algebraic approach to study Y ( sl K ), which makes the relation between Y ( gl K ) and Y ( sl K ) clearer. In Section 9.1, we begin with the construction of the SU( K ) monodromy matrix as the product form. The commuting property of the Cherednik–Dunkl operators plays a crucial role here. The monodromy matrix is the basic building block of the Yangian algbebra, as discussed in Section 8.2. Because the Hamiltonian of the SU( K ) spin chain and the monodromy matrix commute, each eigenstate of the spin chain constitutes a basis for representations of the SU( K ) Yangian. In Section 9.2 and 9.3, we discuss the quantum determinant by comparing it with the ordinary determinant. It is demonstrated in Section 9.4 that the quantum determinant has a scalar nature that leads to a simple evaluation. Because of the scalar nature, however, the quantum determinant cannot distinguish different SU( K ) Yangian states.
Recently, the one dimensional model of NNN spins with S=12S=\frac{1}{2}S=21​ on a circle, interacting with an exchange that falls off with the inverse square of the separation: $H_{\rm ISE} =\sum_{i\neq j} \frac{1}{[\frac{N}π \sin(\frac{i-j}{N}π)]^2}(\svec_i\cdot \svec_j-\frac{1}{4})$, or ISE-model, has received ample attention. Its special features include: relatively simple eigenfunctions, non-interacting elementary excitations that obey semionic statistics (spinons), and a large ``quantum group'' symmetry algebra called the Yangian. This model is fully integrable, albeit in a slightly different sense than the more traditional nearest neighbor exchange (NNE) Heisenberg chain. This thesis comes in 4 chapters. Chapter 1 introduces the model and presents the construction of a subset of the eigenfunctions. The other eigenfunctions are shown to be generated by the action of the Yangian symmetry algebra of HISEH_{\rm ISE}HISE​. Chapter 2 presents a method to construct the set of constants of the motion of the ISE-model. The ISE-model is tractable enough to obtain its zero-magnetic-field dynamical structure factors. Chapter 3 attempts to extend this to a non-zero magnetic field, where, due to the presence of spinons in the groundstate, more complicated excitations contribute: small numbers of magnons and spinons. discuss the relation to the more complicated NNE-model. Finally chapter 4 illustrates how a recently conjectured new form of Off Diagonal Long Range Order in antiferromagnetic spin chains can be reinterpreted as a spinon propagator in the ISE-model, and verified numerically. We briefly comment on its relevance to stabilizing superconductivity in the layered cuprates.