We consider the open XXZ spin chain with unparallel boundary fields, in the special case in which the Vertex-IRF transform of the K-matrix labelling the boundary field at site 1 is diagonal and invariant under the value of the gauge parameters, the other K-matrix being arbitrary. We discuss here several properties related to this special gauge invariant boundary condition. Such a model can be solved in the Vertex-IRF framework with arbitrary gauge parameters: we can diagonalise the transfer matrix by means of generalised Sklyanin's Separation of Variable (SoV) approach in a family of SoV bases that depend on two arbitrary gauge parameters $α$ and $β$ , $β$ being the so-called dynamical parameter and $α$ an arbitrary shift of the spectral parameter. We show here that such SoV bases depend actually on $α$ and $β$ through the difference $α-β$ only. Considering the ungauged limit in which $α-β$ tends to infinity, we show how to compute all elementary building blocks for the correlation functions of this model. We finally show how to compute overlaps between eigenstates of this model and eigenstates of a spin chain sharing the same boundary condition at site $N$ and in which the gauge invariant boundary condition at site 1 has been changed to a more general one.We consider the open XXZ spin chain with unparallel boundary fields, in the special case in which the Vertex-IRF transform of the K-matrix labelling the boundary field at site 1 is diagonal and invariant under the value of the gauge parameters, the other K-matrix being arbitrary. We discuss here several properties related to this special gauge invariant boundary condition. Such a model can be solved in the Vertex-IRF framework with arbitrary gauge parameters: we can diagonalise the transfer matrix by means of generalised Sklyanin's Separation of Variable (SoV) approach in a family of SoV bases that depend on two arbitrary gauge parameters $α$ and $β$, $β$ being the so-called dynamical parameter and $α$ an arbitrary shift of the spectral parameter. We show here that such SoV bases depend actually on $α$ and $β$ through the difference $α-β$ only. Considering the ungauged limit in which $α-β$ tends to infinity, we show how to compute all elementary building blocks for the correlation functions of this model. We finally show how to compute overlaps between eigenstates of this model and eigenstates of a spin chain sharing the same boundary condition at site $N$ and in which the gauge invariant boundary condition at site 1 has been changed to a more general one.

arXiv:2608.22365Aug 23, 20260 citations

Application of Advanced Diagonalization Methods to Quantum Spin Systems

Jie Yu Wang, Ralf Meyer

Springer proceedings in mathematics & statisticsJan 1, 20151 citation

Entanglement entropy in quantum spin chains with broken parity number symmetry

Arash Jafarizadeh, M. A. Rajabpour

Consider a generic quantum spin chain that can be mapped to free quadratic fermions via Jordan-Wigner (JW) transformation. In the presence of arbitrary boundary magnetic fields, this Hamiltonian is no longer a quadratic Hamiltonian after JW transformation. Using ancillary sites and enlarging the Hamiltonian we first introduce a bigger quadratic Hamiltonian. Then we diagonalize this enlarged Hamiltonian in its most generic form and show that all the states are degenerate because of the presence of a zero mode. The eigenstates of the original spin chain with boundary magnetic fields can be derived after appropriate projection. We study in depth the properties of the eigenstates of the enlarged Hamiltonian. In particular we find: 1) the eigenstates in configuration bases, 2) calculate all the correlation functions, 3) find the reduced density matrices, 4) calculate the entanglement entropy. We show that the generic eigenstate of the enlarged Hamiltonian (including the eigenstates of the original spin chain) breaks the parity number symmetry and consequently one needs to take care of some technicalities regarding the calculation of the reduced density matrix and entanglement entropy. Interestingly we show that the entanglement structure of these eigenstates is quite universal and independent of the Hamiltonian. We support our results by applying them to a couple of examples.Consider a generic quantum spin chain that can be mapped to free quadratic fermions via Jordan-Wigner (JW) transformation. In the presence of arbitrary boundary magnetic fields, this Hamiltonian is no longer a quadratic Hamiltonian after JW transformation. Using ancillary sites and enlarging the Hamiltonian we first introduce a bigger quadratic Hamiltonian. Then we diagonalize this enlarged Hamiltonian in its most generic form and show that all the states are degenerate because of the presence of a zero mode. The eigenstates of the original spin chain with boundary magnetic fields can be derived after appropriate projection. We study in depth the properties of the eigenstates of the enlarged Hamiltonian. In particular we find: 1) the eigenstates in configuration bases, 2) calculate all the correlation functions, 3) find the reduced density matrices, 4) calculate the entanglement entropy. We show that the generic eigenstate of the enlarged Hamiltonian (including the eigenstates of the original spin chain) breaks the parity number symmetry and consequently one needs to take care of some technicalities regarding the calculation of the reduced density matrix and entanglement entropy. Interestingly we show that the entanglement structure of these eigenstates is quite universal and independent of the Hamiltonian. We support our results by applying them to a couple of examples.

arXiv:2109.06359SciPost PhysicsJun 14, 20228 citations

Generalization to Arbitrary Spin $\boldsymbol{\it{S}}$ Generalization to Arbitrary Spin $\boldsymbol{\it{S}}$

Martin Greiter

Springer tracts in modern physicsJan 1, 20110 citations

α

and

β

β

being the so-called dynamical parameter and

α

an arbitrary shift of the spectral parameter. We show here that such SoV bases depend actually on

α

and

β

through the difference

α-β

only. Considering the ungauged limit in which

α-β

tends to infinity, we show how to compute all elementary building blocks for the correlation functions of this model. We finally show how to compute overlaps between eigenstates of this model and eigenstates of a spin chain sharing the same boundary condition at site

N

and in which the gauge invariant boundary condition at site 1 has been changed to a more general one.

Generalization to Arbitrary Spin

\boldsymbol{\it{S}}