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Continuum Limit of gl(M/N) Spin Chains

Constantin Candu

Published Jan 1, 2011• 6 citations
OpenAlex

Abstract

We study the spectrum of an integrable antiferromagnetic Hamiltonian of the gl(M|N) spin chain of alternating fundamental and dual representations. After extensive numerical analysis, we identify the vacuum and low lying excitations and with this knowledge perform the continuum limit, while keeping a finite gap. All antiferromagnetic gl(n + N|N)spin chains with n, N > 0 are shown to possess in the continuum limit 2n −2 multiplets of massive particles which scatter with gl(n)Gross-Neveu like S-matrices, meaning that their eigenvalues do not depend on N. We argue that the continuum theory is the gl(M|N) Gross-Neveu model, that is the massive deformation of the $ {{s\hat{l}}}{\left( {M\left| N \right.} \right)_1} $ Wess-Zumino-Witten model. As we can see in the example of gl(2 m|1) spin chains, the full particle spectrum is much richer. Our analysis suggests that for a complete characterization of the latter it is not enough to restrict to large volume calculations, as we do in this work.

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The continuum limit of gl(M|N) spin chains

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We study the spectrum of an integrable antiferromagnetic Hamiltonian of the gl(M|N) spin chain of alternating fundamental and dual representations. After extensive numerical analysis, we identify the vacuum and low lying excitations and with this knowledge perform the continuum limit, while keeping a finite gap. All antiferromagnetic gl(n + N|N)spin chains with n, N > 0 are shown to possess in the continuum limit 2n −2 multiplets of massive particles which scatter with gl(n)Gross-Neveu like S-matrices, meaning that their eigenvalues do not depend on N. We argue that the continuum theory is the gl(M|N) Gross-Neveu model, that is the massive deformation of the $ {{s\hat{l}}}{\left( {M\left| N \right.} \right)_1} $ Wess-Zumino-Witten model. As we can see in the example of gl(2 m|1) spin chains, the full particle spectrum is much richer. Our analysis suggests that for a complete characterization of the latter it is not enough to restrict to large volume calculations, as we do in this work.

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Spin chains and vertex models based on superalgebras

Konstantin Hobuß

The thermodynamic limit of superspin chains can show several intriguing properties, including the emergence of continua of scaling dimensions and the appearance of discrete states when imposing toroidal boundary conditions. Nevertheless, the former’s exhaustive characterization in terms of Conformal Field Theories is still lacking. In order to study the thermodynamic limit of the U_q[sl(2|1)] 3 ⊗ 3̄ superspin chain in the antiferromagnetic regime, we analyze the low lying excitations by means of the model’s exact solution using the Algebraic Bethe Ansatz. In the isotropic limit, this model may be used as a toy model for the description of plateau transitions in Quantum Hall systems. The definition of a quasimomentum operator allows for a characterization of the continua of scaling dimensions, thereby giving rise to a quantum number for the corresponding non-compact component of the spectrum in the thermodynamic limit. The associated degeneracies are lifted on the lattice by logarithmic fine structures. Based on the extrapolation of our finite size data we find that under a variation of the boundary conditions from periodic to antiperiodic for the fermionic degrees of freedom, levels from the continuous part of the spectrum flow into discrete levels and vice versa. Investigating the thermodynamic limit of the q-deformed osp(3|2) superspin chain, corresponding to an intersecting loop model in the rational limit, we seek to uncover its low-lying critical exponents. We present evidences that the latter are built in terms of composites of anomalous dimensions of two Coulomb gases with distinct radii and exponents associated to Z(2) degrees of freedom. This view is supported by the fact that the S = 1 XXZ integrable chain spectrum is present in some of the sectors of the superspin chain at a particular value of the deformation parameter. We find that the fine structure of finite-size effects is very rich for a typical anisotropic spin chain. In fact, we argue on the existence of a family of states with the same conformal dimension whose lattice degeneracies are lifted by logarithmic corrections. On the other hand, we also report on states of the spectrum whose finite-size corrections seem to be governed by a power law behaviour. We finally observe that under toroidal boundary conditions the ground state dependence on the twist angle has two distinct analytical structures.

•Jan 1, 2019•0 citations
We study the spectrum of an integrable antiferromagnetic Hamiltonian of the gl(M|N) spin chain of alternating fundamental and dual representations. After extensive numerical analysis, we identify the vacuum and low lying excitations and with this knowledge perform the continuum limit, while keeping a finite gap. All antiferromagnetic gl(n + N|N)spin chains with n, N > 0 are shown to possess in the continuum limit 2n −2 multiplets of massive particles which scatter with gl(n)Gross-Neveu like S-matrices, meaning that their eigenvalues do not depend on N. We argue that the continuum theory is the gl(M|N) Gross-Neveu model, that is the massive deformation of the sl^(M∣N)1{{s\hat{l}}}{\left( {M\left| N \right.} \right)_1}sl^(M∣N)1​ Wess-Zumino-Witten model. As we can see in the example of gl(2 m|1) spin chains, the full particle spectrum is much richer. Our analysis suggests that for a complete characterization of the latter it is not enough to restrict to large volume calculations, as we do in this work.
We study the spectrum of an integrable antiferromagnetic Hamiltonian of the gl(M|N) spin chain of alternating fundamental and dual representations. After extensive numerical analysis, we identify the vacuum and low lying excitations and with this knowledge perform the continuum limit, while keeping a finite gap. All antiferromagnetic gl(n + N|N)spin chains with n, N > 0 are shown to possess in the continuum limit 2n −2 multiplets of massive particles which scatter with gl(n)Gross-Neveu like S-matrices, meaning that their eigenvalues do not depend on N. We argue that the continuum theory is the gl(M|N) Gross-Neveu model, that is the massive deformation of the $ {{s\hat{l}}}{\left( {M\left| N \right.} \right)_1} $ Wess-Zumino-Witten model. As we can see in the example of gl(2 m|1) spin chains, the full particle spectrum is much richer. Our analysis suggests that for a complete characterization of the latter it is not enough to restrict to large volume calculations, as we do in this work.
A class of SU(n)\mathrm{SU}(n)SU(n) antiferromagnetic quantum "spin" chains is solved in the large-nnn limit. While the low-lying spectrum of the normal spin chain (n=2n=2n=2) consists of massless solitons, in the large-nnn limit the soliton is massive. Furthermore, the models exhibit many features of the CPnC{P}^{n}CPn field theories at nonzero topological charge, which, it was recently argued, they become in the limit of a large representation.
Abstract We construct a Hamiltonian lattice regularisation of the N-flavour Gross-Neveu model that manifestly respects the full O(2N) symmetry, preventing the appearance of any unwanted marginal perturbations to the quantum field theory. In the context of this lattice model, the dynamical mass generation is intimately related to the Coleman-Mermin-Wagner and Lieb-Schultz-Mattis theorems. In particular, the model can be interpreted as lying at the first order phase transition line between a trivial and symmetry-protected topological (SPT) phase, which explains the degeneracy of the elementary kink excitations. We show that our Hamiltonian model can be solved analytically in the large N limit, producing the correct expression for the mass gap. Furthermore, we perform extensive numerical matrix product state simulations for N = 2, thereby recovering the emergent Lorentz symmetry and the proper non-perturbative mass gap scaling in the continuum limit. Finally, our simulations also reveal how the continuum limit manifests itself in the entanglement spectrum. As expected from conformal field theory we find two conformal towers, one tower spanned by the linear representations of O(4), corresponding to the trivial phase, and the other by the projective (i.e. spinor) representations, corresponding to the SPT phase.
We show that while the exactly integrable spin-(3/2 Heisenberg antiferromagnet is a representation of the nonlinear \ensuremath{\sigma} model with nontrivial Wess-Zumino term, the generic massless antiferromagnet has long-wavelength behavior equivalent to spin (1/2. This is demonstrated by comparing the predictions of conformal symmetry and rotational symmetry to the gaps calculated for finite chains. The leading corrections are eliminated leading to stable estimates of the conformal central charge and exponents for the spin-correlation functions.
The thermodynamic limit of superspin chains can show several intriguing properties, including the emergence of continua of scaling dimensions and the appearance of discrete states when imposing toroidal boundary conditions. Nevertheless, the former’s exhaustive characterization in terms of Conformal Field Theories is still lacking. In order to study the thermodynamic limit of the U_q[sl(2|1)] 3 ⊗ 3̄ superspin chain in the antiferromagnetic regime, we analyze the low lying excitations by means of the model’s exact solution using the Algebraic Bethe Ansatz. In the isotropic limit, this model may be used as a toy model for the description of plateau transitions in Quantum Hall systems. The definition of a quasimomentum operator allows for a characterization of the continua of scaling dimensions, thereby giving rise to a quantum number for the corresponding non-compact component of the spectrum in the thermodynamic limit. The associated degeneracies are lifted on the lattice by logarithmic fine structures. Based on the extrapolation of our finite size data we find that under a variation of the boundary conditions from periodic to antiperiodic for the fermionic degrees of freedom, levels from the continuous part of the spectrum flow into discrete levels and vice versa. Investigating the thermodynamic limit of the q-deformed osp(3|2) superspin chain, corresponding to an intersecting loop model in the rational limit, we seek to uncover its low-lying critical exponents. We present evidences that the latter are built in terms of composites of anomalous dimensions of two Coulomb gases with distinct radii and exponents associated to Z(2) degrees of freedom. This view is supported by the fact that the S = 1 XXZ integrable chain spectrum is present in some of the sectors of the superspin chain at a particular value of the deformation parameter. We find that the fine structure of finite-size effects is very rich for a typical anisotropic spin chain. In fact, we argue on the existence of a family of states with the same conformal dimension whose lattice degeneracies are lifted by logarithmic corrections. On the other hand, we also report on states of the spectrum whose finite-size corrections seem to be governed by a power law behaviour. We finally observe that under toroidal boundary conditions the ground state dependence on the twist angle has two distinct analytical structures.