SciPost Physics

SciPost Phys. 6, 071 (2019) · published 21 June 2019

• doi: 10.21468/SciPostPhys.6.6.071
• pdf
• BiBTeX
• RIS
• Submissions/Reports
• 

Abstract

We apply our new approach of quantum Separation of Variables (SoV) to the complete characterization of the transfer matrix spectrum of quantum integrable lattice models associated to gl(n)-invariant R-matrices in the fundamental representations. We consider lattices with N sites and quasi-periodic boundary conditions associated to an arbitrary twist K having simple spectrum (but not necessarily diagonalizable). In our approach the SoV basis is constructed in an universal manner starting from the direct use of the conserved charges of the models, i.e., from the commuting family of transfer matrices. Using the integrable structure of the models, incarnated in the hierarchy of transfer matrices fusion relations, we prove that our SoV basis indeed separates the spectrum of the corresponding transfer matrices. Moreover, the combined use of the fusion rules, of the known analytic properties of the transfer matrices and of the SoV basis allows us to obtain the complete characterization of the transfer matrix spectrum and to prove its simplicity. Any transfer matrix eigenvalue is completely characterized as a solution of a so-called quantum spectral curve equation that we obtain as a difference functional equation of order n. Namely, any eigenvalue satisfies this equation and any solution of this equation having prescribed properties leads to an eigenvalue. We construct the associated eigenvector, unique up to normalization, by computing its decomposition on the SoV basis that is of a factorized form written in terms of the powers of the corresponding eigenvalues. If the twist matrix K is diagonalizable with simple spectrum then the transfer matrix is also diagonalizable with simple spectrum. In that case, we give a construction of the Baxter Q-operator satisfying a T-Q equation of order n, the quantum spectral curve equation, involving the hierarchy of the fused transfer matrices.

• Niccoli et al., On correlation functions for the open XXZ chain with non-longitudinal boundary fields: The case with a constraint
  SciPost Phys. 16, 099 (2024) [Crossref]
• Huang et al., Exact solution of the Heisenberg XX model with twisted boundary conditions
  Physics Letters A 384, 126474 126474 (2020) [Crossref]
• Maillet et al., On quantum separation of variables beyond fundamental representations
  SciPost Phys. 10, 026 (2021) [Crossref]
• Maillet et al., Separation of variables bases for integrable $gl_{\mathcal{M}|\mathcal{N}}$ and Hubbard models
  SciPost Phys. 9, 060 (2020) [Crossref]
• Niccoli et al., Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields
  J. Phys. A: Math. Theor. 55, 405203 (2022) [Crossref]
• Chernyak et al., Completeness of Wronskian Bethe Equations for Rational $${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$$ Spin Chains
  Commun. Math. Phys. 391, 969 (2022) [Crossref]
• Levkovich-Maslyuk, A review of the AdS/CFT Quantum Spectral Curve
  J. Phys. A: Math. Theor. 53, 283004 (2020) [Crossref]
• Ryan et al., Separation of Variables for Rational $$\mathfrak {gl}(\mathsf {n})$$ Spin Chains in Any Compact Representation, via Fusion, Embedding Morphism and Bäcklund Flow
  Commun. Math. Phys. 383, 311 (2021) [Crossref]
• Gromov et al., Determinant form of correlators in high rank integrable spin chains via separation of variables
  J. High Energ. Phys. 2021, 169 (2021) [Crossref]
• Derkachov et al., Mirror channel eigenvectors of the d-dimensional fishnets
  J. High Energ. Phys. 2021, 174 (2021) [Crossref]
• Maillet et al., On scalar products in higher rank quantum separation of variables
  SciPost Phys. 9, 086 (2020) [Crossref]
• Niccoli et al., Correlation functions by separation of variables: the XXX spin chain
  SciPost Phys. 10, 006 (2021) [Crossref]
• Gromov et al., Form-factors and complete basis of observables via separation of variables for higher rank spin chains
  J. High Energ. Phys. 2022, 39 (2022) [Crossref]
• Pei et al., On scalar products and form factors by separation of variables: the antiperiodic XXZ model
  J. Phys. A: Math. Theor. 55, 015205 (2022) [Crossref]
• Cavaglià et al., Separation of variables and scalar products at any rank
  J. High Energ. Phys. 2019, 52 (2019) [Crossref]
• Maillet et al., Complete spectrum of quantum integrable lattice models associated to $\boldsymbol {\mathcal{U}_{q} (\widehat{gl_{n}})}$ by separation of variables
  J. Phys. A: Math. Theor. 52, 315203 (2019) [Crossref]
• Maillet et al., On separation of variables for reflection algebras
  J. Stat. Mech. 2019, 094020 (2019) [Crossref]