Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Spin operator matrix elements in the XX spin chain with application to the dynamics of the interacting Dicke model

Ning Wu

Published Sep 3, 2017• 0 citations
OpenAlex

Abstract

For the one-dimensional spin-1/2 XX model with either periodic or open boundary conditions, it is shown by using a fermionic approach that the matrix element of the spin operator $S^-_j$ ($S^-_{j}S^+_{j'}$) between two eigenstates with numbers of excitations $n$ and $n+1$ ($n$ and $n$) can be expressed as the determinant of an appropriate $(n+1)\times (n+1)$ matrix whose entries involve the coefficients of the canonical transformations diagonalizing the model. In the special case of a homogeneous periodic XX chain, the matrix element of $S^-_j$ reduces to a variant of the Cauchy determinant that can be evaluated analytically to yield a factorized expression. The obtained compact representations of these matrix elements are then applied to two physical scenarios: (i) Nonlinear optical response of molecular aggregates, for which the determinant representation of the transition dipole matrix elements between eigenstates provides a convenient way to calculate the third-order nonlinear responses for aggregates from small to large sizes compared with the optical wavelength, and (ii) real-time dynamics of an interacting Dicke model consisting of a single bosonic mode coupled to a one-dimensional XX spin bath. In this setup, full quantum calculation up to $N\leq 16$ spins for vanishing intrabath coupling shows that the decay of the reduced bosonic occupation number approaches a finite plateau value (in the long-time limit) that depends on the ratio between the number of excitations and the total number of spins. Our results can find useful applications in various system-bath systems, with the system part inhomogeneously coupled to an interacting XX chain.

Related papers

Determinant representations of spin-operator matrix elements in the XX spin chain and their applications

Ning Wu

For the one-dimensional spin-1/2 XX model with either periodic or open boundary conditions, it is shown by using a fermionic approach that the matrix element of the spin operator ${S}_{j}^{\ensuremath{-}}$ (${S}_{j}^{\ensuremath{-}}{S}_{{j}^{\ensuremath{'}}}^{+}$) between two eigenstates with numbers of excitations $n$ and $n+1$ ($n$ and $n$) can be expressed as the determinant of an appropriate $(n+1)\ifmmode\times\else\texttimes\fi{}(n+1)$ matrix whose entries involve the coefficients of the canonical transformations diagonalizing the model. In the special case of a homogeneous periodic XX chain, the matrix element of ${S}_{j}^{\ensuremath{-}}$ reduces to a variant of the Cauchy determinant that can be evaluated analytically to yield a factorized expression. The obtained compact representations of these matrix elements are then applied to two physical scenarios: (i) Nonlinear optical response of molecular aggregates, for which the determinant representation of the transition dipole matrix elements between eigenstates provides a convenient way to calculate the third-order nonlinear responses for aggregates from small to large sizes compared with the optical wavelength; and (ii) real-time dynamics of an interacting Dicke model consisting of a single bosonic mode coupled to a one-dimensional XX spin bath. In this setup, full quantum calculation up to $N\ensuremath{\le}16$ spins for vanishing intrabath coupling shows that the decay of the reduced bosonic occupation number approaches a finite plateau value (in the long-time limit) that depends on the ratio between the number of excitations and the total number of spins. Our results can find useful applications in various ``system-bath'' systems, with the system part inhomogeneously coupled to an interacting XX chain.

arXiv:1709.00682•Physical review. B./Physical review. B•Jan 3, 2018•11 citations

The open XXZ spin chain in the SoV framework: scalar product of separate states

N Kitanine, Jean Michel Maillet, Giuliano Niccoli +1 more

Abstract In our previous paper (Kitanine et al 2017 J. Phys. A: Math. Theor . 50 224001) we have obtained, for the XXX spin-1/2 Heisenberg open chain, new determinant representations for the scalar products of separate states in the quantum separation of variables (SoV) framework. In this article we perform a similar study in a more complicated case: the XXZ open spin-1/2 chain with the most general integrable boundary terms. We first revisit the SoV solution of this model using a matrix version of the Vertex-IRF gauge transformation so as to reduce one of the boundary K -matrices to a diagonal form. As usual within the SoV approach, the scalar products of separate states are computed in terms of dressed Vandermonde determinants having an intricate dependency on the inhomogeneity parameters. We show that these determinants can be transformed into different ones in which the homogeneous limit can be taken straightforwardly. These representations generalize in a non-trivial manner to the trigonometric case the expressions found previously in the rational case. We also show that generically all scalar products can be expressed in a form which is similar to—although more cumbersome than—the well-known Slavnov determinant representation for the scalar products of the Bethe states of the periodic chain. Considering a special choice of the boundary parameters relevant in the thermodynamic limit to describe the half infinite chain with a general boundary, we particularize these representations to the case of one of the two states being an eigenstate. We obtain simplified formulas that should be of direct use to compute the form factors and correlation functions of this model.

arXiv:1807.05197•Journal of Physics A Mathematical and Theoretical•Oct 10, 2018•24 citations

The open XXX spin chain in the SoV framework: scalar product of separate states

N Kitanine, Jean Michel Maillet, Giuliano Niccoli +1 more

Abstract We consider the XXX open spin-1/2 chain with the most general non-diagonal boundary terms, that we solve by means of the quantum separation of variables (SoV) approach. We compute the scalar products of separate states, a class of states which notably contains all the eigenstates of the model. As usual for models solved by SoV, these scalar products can be expressed as some determinants with a non-trivial dependance in terms of the inhomogeneity parameters that have to be introduced for the method to be applicable. We show that these determinants can be transformed into alternative ones in which the homogeneous limit can easily be taken. These new representations can be considered as generalizations of the well-known determinant representation for the scalar products of the Bethe states of the periodic chain. In the particular case where a constraint is applied on the boundary parameters, such that the transfer matrix spectrum and eigenstates can be characterized in terms of polynomial solutions of a usual T - Q equation, the scalar product that we compute here corresponds to the scalar product between two off-shell Bethe-type states. If in addition one of the states is an eigenstate, the determinant representation can be simplified, hence leading in this boundary case to direct analogues of algebraic Bethe ansatz determinant representations of the scalar products for the periodic chain.

arXiv:1606.06917•Journal of Physics A Mathematical and Theoretical•May 9, 2017•58 citations

Boundary-driven magnetization transport in the spin- 1 2 XXZ chain: Role of the system-bath coupling strength and timescales

Mariel Kempa, Markus Kraft, Sourav Nandy +4 more

Understanding the transport properties of quantum many-body systems is a central challenge in condensed matter and statistical physics. Theoretical studies usually rely on two main approaches: Dynamics of linear-response functions in closed systems and boundary-driven dynamics governed by Markovian master equations for open systems. While the equivalence of their dynamical behavior has been explored in recent studies, a systematic comparison of the transport coefficients obtained from these two classes of approaches remains a largely open question. Here, we address this gap by comparing and contrasting the dc diffusion constant $\mathcal{D}_{\text{dc}}$ according to the two approaches, focusing on the specific example of magnetization transport in the spin-$1/2$ XXZ chain. Using exact numerical simulations for finite system sizes, we find (i) a clear mismatch between the two $\mathcal{D}_{\text{dc}}$ and (ii) a strong dependence of $\mathcal{D}_{\text{dc}}$ on the system-bath coupling strength for the open system, where neither (i) nor (ii) tend to vanish in the thermodynamic limit. These findings suggest limitations of the open-system approach to transport coefficients. To gain insight into the origin of (i) and (ii), we go beyond $\mathcal{D}_{\text{dc}}$ and analyze the full time dependence of the diffusion coefficient $D(t)$ in the open system. In this way, we find that both (i) and (ii) vanish up to a finite time scale. While this time scale gradually increases with system size and tends to be macroscopic in the thermodynamic limit, this increase is still slow compared to the increase of the time to reach the steady state, where (i) and (ii) do not vanish. This observation can be seen as a wrong, yet unavoidable order of limits of long times first and large system sizes afterwards.

arXiv:2507.16528•Physical review. B./Physical review. B•Jan 15, 2026•0 citations

Exact solution for the Lindbladian dynamics for the open XX spin chain with boundary dissipation

Kohei Yamanaka, Tomohiro Sasamoto

We obtain exact formulas for the time-dependence of a few physical observables for the open XX spin chain with Lindbladian dynamics. Our analysis is based on the fact that the Lindblad equation for an arbitrary open quadratic system of N N fermions is explicitly solved in terms of diagonalization of a 4N\times4N 4 N × 4 N matrix called structure matrix by following the scheme of the third quantization. We mainly focus on the time-dependence of magnetization and spin current. As a short-time behavior at a given site, we observe the plateau regime except near the center of the chain. Basic features of this are explained by the light-cone structure created by propagations of boundary effects from the initial time, but we can explain their more detailed properties analytically using our exact formulas. On the other hand, after the plateau regime, the magnetization and spin current exhibit a slow decay to the steady state values described by the Liouvillian gap. We analytically establish its O(N^{-3}) O ( N − 3 ) scaling and also determine its coefficient.

arXiv:2104.11479•SciPost Physics•May 15, 2023•19 citations

Spin-operator form factors of the critical Ising chain and their finite volume scaling limits

Yizhuang Liu

In this work, we provide a self-contained derivation of the spin-operator matrix elements in the fermionic basis, for the critical periodic Ising chain at a generic system length $N\in 2Z_{\ge 2}$. The approach relies on the near-Cauchy property of certain matrices formed by the Toeplitz symbols in the critical model, and leads to a few square-root products for the leg functions. The square root products allow simple integral representations, that further reduce to the Binet's second integral and its generalization by Hermite, in the finite volume scaling limit. This leads to product formulas for the spin operator matrix elements in the scaling limit, providing explicit expressions for the spin-operator form factors of the Ising CFT in the fermionic basis, that were computed iteratively in Yurov:1991my. They are all rational numbers up to $\sqrt{2}$. We also determine the normalization factor of the spin-operator and show explicitly how the coefficient $G(\frac{1}{2})G(\frac{3}{2})$ appear through a ground state overlap. Moreover, by expanding the spin-spin two point correlator in the fermionic basis, we observed a Fredholm determinant identity that allows to show the convergence of the rescaled two-point correlator to the CFT version on a cylinder.

arXiv:2601.00751•Jan 2, 2026•0 citations
For the one-dimensional spin-1/2 XX model with either periodic or open boundary conditions, it is shown by using a fermionic approach that the matrix element of the spin operator Sj−S^-_jSj−​ (Sj−Sj′+S^-_{j}S^+_{j'}Sj−​Sj′+​) between two eigenstates with numbers of excitations nnn and n+1n+1n+1 (nnn and nnn) can be expressed as the determinant of an appropriate (n+1)×(n+1)(n+1)\times (n+1)(n+1)×(n+1) matrix whose entries involve the coefficients of the canonical transformations diagonalizing the model. In the special case of a homogeneous periodic XX chain, the matrix element of Sj−S^-_jSj−​ reduces to a variant of the Cauchy determinant that can be evaluated analytically to yield a factorized expression. The obtained compact representations of these matrix elements are then applied to two physical scenarios: (i) Nonlinear optical response of molecular aggregates, for which the determinant representation of the transition dipole matrix elements between eigenstates provides a convenient way to calculate the third-order nonlinear responses for aggregates from small to large sizes compared with the optical wavelength, and (ii) real-time dynamics of an interacting Dicke model consisting of a single bosonic mode coupled to a one-dimensional XX spin bath. In this setup, full quantum calculation up to N≤16N\leq 16N≤16 spins for vanishing intrabath coupling shows that the decay of the reduced bosonic occupation number approaches a finite plateau value (in the long-time limit) that depends on the ratio between the number of excitations and the total number of spins. Our results can find useful applications in various system-bath systems, with the system part inhomogeneously coupled to an interacting XX chain.
For the one-dimensional spin-1/2 XX model with either periodic or open boundary conditions, it is shown by using a fermionic approach that the matrix element of the spin operator ${S}_{j}^{\ensuremath{-}}$ (${S}_{j}^{\ensuremath{-}}{S}_{{j}^{\ensuremath{'}}}^{+}$) between two eigenstates with numbers of excitations nnn and n+1n+1n+1 (nnn and nnn) can be expressed as the determinant of an appropriate $(n+1)\ifmmode\times\else\texttimes\fi{}(n+1)$ matrix whose entries involve the coefficients of the canonical transformations diagonalizing the model. In the special case of a homogeneous periodic XX chain, the matrix element of ${S}_{j}^{\ensuremath{-}}$ reduces to a variant of the Cauchy determinant that can be evaluated analytically to yield a factorized expression. The obtained compact representations of these matrix elements are then applied to two physical scenarios: (i) Nonlinear optical response of molecular aggregates, for which the determinant representation of the transition dipole matrix elements between eigenstates provides a convenient way to calculate the third-order nonlinear responses for aggregates from small to large sizes compared with the optical wavelength; and (ii) real-time dynamics of an interacting Dicke model consisting of a single bosonic mode coupled to a one-dimensional XX spin bath. In this setup, full quantum calculation up to $N\ensuremath{\le}16$ spins for vanishing intrabath coupling shows that the decay of the reduced bosonic occupation number approaches a finite plateau value (in the long-time limit) that depends on the ratio between the number of excitations and the total number of spins. Our results can find useful applications in various ``system-bath'' systems, with the system part inhomogeneously coupled to an interacting XX chain.
Abstract In our previous paper (Kitanine et al 2017 J. Phys. A: Math. Theor . 50 224001) we have obtained, for the XXX spin-1/2 Heisenberg open chain, new determinant representations for the scalar products of separate states in the quantum separation of variables (SoV) framework. In this article we perform a similar study in a more complicated case: the XXZ open spin-1/2 chain with the most general integrable boundary terms. We first revisit the SoV solution of this model using a matrix version of the Vertex-IRF gauge transformation so as to reduce one of the boundary K -matrices to a diagonal form. As usual within the SoV approach, the scalar products of separate states are computed in terms of dressed Vandermonde determinants having an intricate dependency on the inhomogeneity parameters. We show that these determinants can be transformed into different ones in which the homogeneous limit can be taken straightforwardly. These representations generalize in a non-trivial manner to the trigonometric case the expressions found previously in the rational case. We also show that generically all scalar products can be expressed in a form which is similar to—although more cumbersome than—the well-known Slavnov determinant representation for the scalar products of the Bethe states of the periodic chain. Considering a special choice of the boundary parameters relevant in the thermodynamic limit to describe the half infinite chain with a general boundary, we particularize these representations to the case of one of the two states being an eigenstate. We obtain simplified formulas that should be of direct use to compute the form factors and correlation functions of this model.
Abstract We consider the XXX open spin-1/2 chain with the most general non-diagonal boundary terms, that we solve by means of the quantum separation of variables (SoV) approach. We compute the scalar products of separate states, a class of states which notably contains all the eigenstates of the model. As usual for models solved by SoV, these scalar products can be expressed as some determinants with a non-trivial dependance in terms of the inhomogeneity parameters that have to be introduced for the method to be applicable. We show that these determinants can be transformed into alternative ones in which the homogeneous limit can easily be taken. These new representations can be considered as generalizations of the well-known determinant representation for the scalar products of the Bethe states of the periodic chain. In the particular case where a constraint is applied on the boundary parameters, such that the transfer matrix spectrum and eigenstates can be characterized in terms of polynomial solutions of a usual T - Q equation, the scalar product that we compute here corresponds to the scalar product between two off-shell Bethe-type states. If in addition one of the states is an eigenstate, the determinant representation can be simplified, hence leading in this boundary case to direct analogues of algebraic Bethe ansatz determinant representations of the scalar products for the periodic chain.
Understanding the transport properties of quantum many-body systems is a central challenge in condensed matter and statistical physics. Theoretical studies usually rely on two main approaches: Dynamics of linear-response functions in closed systems and boundary-driven dynamics governed by Markovian master equations for open systems. While the equivalence of their dynamical behavior has been explored in recent studies, a systematic comparison of the transport coefficients obtained from these two classes of approaches remains a largely open question. Here, we address this gap by comparing and contrasting the dc diffusion constant Ddc\mathcal{D}_{\text{dc}}Ddc​ according to the two approaches, focusing on the specific example of magnetization transport in the spin-1/21/21/2 XXZ chain. Using exact numerical simulations for finite system sizes, we find (i) a clear mismatch between the two Ddc\mathcal{D}_{\text{dc}}Ddc​ and (ii) a strong dependence of Ddc\mathcal{D}_{\text{dc}}Ddc​ on the system-bath coupling strength for the open system, where neither (i) nor (ii) tend to vanish in the thermodynamic limit. These findings suggest limitations of the open-system approach to transport coefficients. To gain insight into the origin of (i) and (ii), we go beyond Ddc\mathcal{D}_{\text{dc}}Ddc​ and analyze the full time dependence of the diffusion coefficient D(t)D(t)D(t) in the open system. In this way, we find that both (i) and (ii) vanish up to a finite time scale. While this time scale gradually increases with system size and tends to be macroscopic in the thermodynamic limit, this increase is still slow compared to the increase of the time to reach the steady state, where (i) and (ii) do not vanish. This observation can be seen as a wrong, yet unavoidable order of limits of long times first and large system sizes afterwards.
We obtain exact formulas for the time-dependence of a few physical observables for the open XX spin chain with Lindbladian dynamics. Our analysis is based on the fact that the Lindblad equation for an arbitrary open quadratic system of N N fermions is explicitly solved in terms of diagonalization of a 4N\times4N 4 N × 4 N matrix called structure matrix by following the scheme of the third quantization. We mainly focus on the time-dependence of magnetization and spin current. As a short-time behavior at a given site, we observe the plateau regime except near the center of the chain. Basic features of this are explained by the light-cone structure created by propagations of boundary effects from the initial time, but we can explain their more detailed properties analytically using our exact formulas. On the other hand, after the plateau regime, the magnetization and spin current exhibit a slow decay to the steady state values described by the Liouvillian gap. We analytically establish its O(N^{-3}) O ( N − 3 ) scaling and also determine its coefficient.
In this work, we provide a self-contained derivation of the spin-operator matrix elements in the fermionic basis, for the critical periodic Ising chain at a generic system length N∈2Z≥2N\in 2Z_{\ge 2}N∈2Z≥2​. The approach relies on the near-Cauchy property of certain matrices formed by the Toeplitz symbols in the critical model, and leads to a few square-root products for the leg functions. The square root products allow simple integral representations, that further reduce to the Binet's second integral and its generalization by Hermite, in the finite volume scaling limit. This leads to product formulas for the spin operator matrix elements in the scaling limit, providing explicit expressions for the spin-operator form factors of the Ising CFT in the fermionic basis, that were computed iteratively in Yurov:1991my. They are all rational numbers up to 2\sqrt{2}2​. We also determine the normalization factor of the spin-operator and show explicitly how the coefficient G(12)G(32)G(\frac{1}{2})G(\frac{3}{2})G(21​)G(23​) appear through a ground state overlap. Moreover, by expanding the spin-spin two point correlator in the fermionic basis, we observed a Fredholm determinant identity that allows to show the convergence of the rescaled two-point correlator to the CFT version on a cylinder.