scholarly journals On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model

2021 ◽  
Author(s):  
Hao Pei ◽  
Veronique Terras
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Separation Of Variables ◽  
Form Factors ◽  
Periodic Case ◽  
Matrix Elements ◽  
Heisenberg Chain ◽  
Xxz Model ◽  
Matrix Eigenvalues ◽  
Scalar Products

Abstract We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by Separation of Variables (SoV), and the eigenstates can be constructed in terms of Q-functions, solution of a Baxter TQ-equation, which have double periodicity compared to the periodic case. We compute in this framework the scalar products of a particular class of separate states which notably includes the eigenstates of the transfer matrix. We also compute the form factors of local spin operators, i.e. their matrix elements between two eigenstates of the transfer matrix. We show that these quantities admit determinant representations with rows and columns labelled by the roots of the Q-functions of the corresponding separate states, as in the periodic case, although the form of the determinant are here slightly different. We also propose alternative types of determinant representations written directly in terms of the transfer matrix eigenvalues.

scholarly journals Correlation functions by separation of variables: the XXX spin chain

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Giuliano Niccoli ◽  
Hao Pei ◽  
Véronique Terras
Keyword(s):  
Integral Representation ◽  
Boundary Conditions ◽  
Thermodynamic Limit ◽  
Correlation Functions ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Periodic Boundary Conditions ◽  
Multiple Integral ◽  
Periodic Case ◽  
Zero Temperature

We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.

scholarly journals Separation of variables bases for integrable $gl_{\mathcal{M}|\mathcal{N}}$ and Hubbard models

2020 ◽  
Vol 9 (4) ◽  
Author(s):  
Jean Michel Maillet ◽  
Giuliano Niccoli ◽  
Louis Vignoli
Keyword(s):  
Boundary Conditions ◽  
Hubbard Model ◽  
Spectral Problem ◽  
Transfer Matrix ◽  
Separation Of Variables ◽  
Integrable Models ◽  
Simple Spectrum ◽  
Transfer Matrices ◽  
Matrix Eigenvalue ◽  
Twisted Boundary Conditions

We construct quantum Separation of Variables (SoV) bases for both the fundamental inhomogeneous % gl_{\mathcal{M}|\mathcal{N}}glℳ|𝒩 supersymmetric integrable models and for the inhomogeneous Hubbard model both defined with quasi-periodic twisted boundary conditions given by twist matrices having simple spectrum. The SoV bases are obtained by using the integrable structure of these quantum models, i.e. the associated commuting transfer matrices, following the general scheme introduced in [1]; namely, they are given by set of states generated by the multiple actions of the transfer matrices on a generic co-vector. The existence of such SoV bases implies that the corresponding transfer matrices have non-degenerate spectrum and that they are diagonalizable with simple spectrum if the twist matrices defining the quasi-periodic boundary conditions have that property. Moreover, in these SoV bases the resolution of the transfer matrix eigenvalue problem leads to the resolution of the full spectral problem, i.e. both eigenvalues and eigenvectors. Indeed, to any eigenvalue is associated the unique (up to a trivial overall normalization) eigenvector whose wave-function in the SoV bases is factorized into products of the corresponding transfer matrix eigenvalue computed on the spectrum of the separated variables. As an application, we characterize completely the transfer matrix spectrum in our SoV framework for the fundamental gl_{1|2}gl1|2 supersymmetric integrable model associated to a special class of twist matrices. From these results we also prove the completeness of the Bethe Ansatz for that case. The complete solution of the spectral problem for fundamental inhomogeneous gl_{\mathcal{M}|\mathcal{N}}glℳ|𝒩 supersymmetric integrable models and for the inhomogeneous Hubbard model under the general twisted boundary conditions will be addressed in a future publication.

TRANSFER-MATRIX METHOD FOR SOLVING THE SPIN 1/2 ANTIFERROMAGNETIC HEISENBERG CHAIN

1988 ◽  
Vol 02 (05) ◽  
pp. 1035-1036 ◽  
Author(s):  
M. A. Garcia-Bach ◽  
D. J. Klein ◽  
R. Valenti
Keyword(s):  
Wave Function ◽  
Transfer Matrix ◽  
Ground State ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Valence Bond ◽  
Exact Results ◽  
Matrix Elements ◽  
Heisenberg Chain ◽  
Spin Configuration

Following the discovery of high Tc superconductivity in the copper oxides, there has been a great deal of interest in the RVB wave function proposed by Anderson [1]. As a warm-up exercise we have considered a valence-bond wave function for the one dimensional spin-1/2 Heisenberg chain. The main virtue of our work is to propose a new variational singlet wavefunction which is almost analytically tractable by a transfer-matrix technique. We have obtained the ground state energy for finite as well as infinite chains, in good agreement with exact results. Correlation functions, excited states, and the effects of other interactions (e.g., spin-Peierls) are also accessible within this scheme [2]. Since the ground state of the chain is known to be a singlet (Lieb & Mattis [3]), we write the appropriate wave function as a superposition of valence-bond singlets, [Formula: see text], where | k > is a spin configuration obtained by pairing all spins into singlet pairs, in a way which is common in valence-bond calculations of large molecules. As in that case, each configuration, | k >, can be represented by a Rümer diagram, with directed bonds connecting each pair of spins on the chain. The c k 's are variational co-efficients, the form of which is determined as follows: Each singlet configuration (Rümer diagram) is divided into "zones", a "zone" corresponding to the region between two consecutive sites. Each zone is indexed by its distance from the end of the chain and by the number of bonds crossing it. Our procedure assigns a variational parameter, x ij , to the j th zone, when crossed by i bonds. The resulting wavefunction for an N-site chain is written as [Formula: see text] where m ij(k) equals 1 when zone j is crossed by i bonds and zero otherwise. To make the calculation tractable we reduce the number of variational parameters by disallowing configurations with bonds connecting any two sites separated by more than 2M lattice points. (For simplicity, we have limited ourselves to M=3, but the scheme can be used for any M). With the simple ansatz, matrix elements can be calculated by a transfer-matrix method. To understand the transfer-matrix method note that since only local zone parameters appear in the description of each state | k >, matrix elements and overlaps, [Formula: see text] and < k | k '>, are completely specified by a small number of "local states" associated with each zone. Within a given zone a local state is defined by (i) the number of bonds crossing the zone and (ii), by whether the bonds originate from the initial (| k >) or final (| k '>) state. It is then easy to see that "local states" of consecutive zones are connected by a 15 × 15 transfer matrix (for the case M=3). Furthermore, the overlap matrix element can be written as a product of transfer-matrices associated with each zone of the chain. When calculating matrix elements of the Hamiltonian, an additional matrix, U , must be defined, to represent the particular zone involving the two spins connected by the Heisenberg interaction. The relevant details as well as the comparison with exact results will be given elsewhere. We are planning to ultimately apply this method to the two dimensional case, and hope to include the effects of holes.

scholarly journals On scalar products in higher rank quantum separation of variables

2020 ◽  
Vol 9 (6) ◽  
Author(s):  
Jean Michel Maillet ◽  
Giuliano Niccoli ◽  
Louis Vignoli
Keyword(s):  
Transfer Matrix ◽  
Separation Of Variables ◽  
Lattice Models ◽  
Simple Spectrum ◽  
Conserved Charges ◽  
Integrable Lattice ◽  
Rank One ◽  
Higher Rank ◽  
Left And Right ◽  
Scalar Products

Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models , we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states, that are transfer matrix eigenstates or some simple generalization of them. In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector \langle\underline{\mathbf{h}}\rangle⟨𝐡̲⟩, there could be more than one non-vanishing overlap \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle⟨𝐡̲|𝐤̲⟩ with the vectors |{\underline{\mathbf{k}}}\rangle|𝐤̲⟩ of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the \mathcal{Y}(gl_3)𝒴(gl3) lattice model with NN sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality (by standard we mean \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle \propto \delta_{\underline{\mathbf{h}}, \underline{\mathbf{k}}}⟨𝐡̲|𝐤̲⟩∝δ𝐡̲,𝐤̲). In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the \mathcal{Y}(gl_2)𝒴(gl2) rank one case.

scholarly journals Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II

2018 ◽  
Vol 5 (3) ◽  
Author(s):  
Jean Michel Maillet ◽  
Giuliano Niccoli ◽  
Baptiste Pezelier
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Transfer Matrix ◽  
Separation Of Variables ◽  
General Boundary ◽  
Triangular Form ◽  
Transfer Matrix Spectrum ◽  
Lax Operator ◽  
Matrix Spectrum ◽  
Reflection Algebra

This article is a direct continuation of where we begun the study of the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax operator. There we addressed this problem for the case where one of the KK-matrices describing the boundary conditions is triangular. In the present article we consider the most general integrable boundary conditions, namely the most general boundary KK-matrices satisfying the reflection equation. The spectral analysis is developed by implementing the method of Separation of Variables (SoV). We first design a suitable gauge transformation that enable us to put into correspondence the spectral problem for the most general boundary conditions with another one having one boundary KK-matrix in a triangular form. In these settings the SoV resolution can be obtained along an extension of the method described in . The transfer matrix spectrum is then completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions and equivalently as the set of solutions to an analogue of Baxter’s T-Q functional equation. We further describe scalar product properties of the separate states including eigenstates of the transfer matrix.

scholarly journals Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I

2017 ◽  
Vol 2 (1) ◽  
Author(s):  
Jean Michel Maillet ◽  
Giuliano Niccoli ◽  
Baptiste Pezelier
Keyword(s):  
Spectral Analysis ◽  
Boundary Conditions ◽  
Transfer Matrix ◽  
Separation Of Variables ◽  
Open Boundary ◽  
Open Boundary Conditions ◽  
Transfer Matrix Spectrum ◽  
Lax Operator ◽  
Matrix Spectrum ◽  
Reflection Algebra

We study the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazanov-Stroganov Lax operator. The results apply as well to the spectral analysis of the lattice sine-Gordon model with integrable open boundary conditions. This spectral analysis is developed by implementing the method of separation of variables (SoV). The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions. Moreover, we prove an equivalent characterization as the set of solutions to a Baxter’s like T-Q functional equation and rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form. In order to explain our method in a simple case, the present paper is restricted to representations containing one constraint on the boundary parameters and on the parameters of the Bazanov-Stroganov Lax operator. In a next article, some more technical tools (like Baxter’s gauge transformations) will be introduced to extend our approach to general integrable boundary conditions.

scholarly journals Light-cone sum rules for proton decay

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Ulrich Haisch ◽  
Amando Hala
Keyword(s):  
Lattice Qcd ◽  
Light Cone ◽  
Sum Rules ◽  
State Of The Art ◽  
Form Factors ◽  
Baryon Number ◽  
Proton Decay ◽  
Matrix Elements ◽  
Decay Channels ◽  
Systematic Uncertainties

Abstract We estimate the form factors that parametrise the hadronic matrix elements of proton-to-pion transitions with the help of light-cone sum rules. These form factors are relevant for semi-leptonic proton decay channels induced by baryon-number violating dimension-six operators, as typically studied in the context of grand unified theories. We calculate the form factors in a kinematical regime where the momentum transfer from the proton to the pion is space-like and extrapolate our final results to the regime that is relevant for proton decay. In this way, we obtain estimates for the form factors that show agreement with the state-of-the-art calculations in lattice QCD, if systematic uncertainties are taken into account. Our work is a first step towards calculating more involved proton decay channels where lattice QCD results are not available at present.

scholarly journals Protecting quantum correlations of the XXZ model by topological boundary conditions

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Shi-Ping Zeng ◽  
Hai-Long Shi ◽  
Xu Zhou ◽  
Xiao-Hui Wang ◽  
Si-Yuan Liu ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Quantum Correlations ◽  
Xxz Model ◽  
Topological Boundary

Flexural Vibration Response of Beams With General Boundary Conditions: A Transfer Matrix Approach

1991 ◽  
Author(s):  
T. Önsay
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vibration Response ◽  
General Boundary ◽  
Phase Variable ◽  
General Boundary Conditions ◽  
Transfer Matrices ◽  
Analytical Results

Abstract The wave-mode representation is utilized to obtain a more efficient form to the conventional transfer matrix method for bending vibrations of beams. The proposed improvement is based on a phase-variable canonical state representation of the equation governing the time-harmonic flexural vibrations of a beam. Transfer matrices are obtained for external forces, step-change of beam properties, intermediate supports and for boundaries. The transfer matrices are utilized to obtain the vibration response of a point-excited single-span beam with general boundary conditions. The general characteristic equation and the transfer mobility of a single-span beam are determined. The application of the analytical results are demonstrated on physical structures with different boundary conditions. A hybrid model is developed to incorporate measured impedance of nonideal boundaries into the transfer matrix method. The analytical results are found to be in excellent agreement with experimental measurements.

Bc → J/ψ tensor form factors at large momentum recoil

2021 ◽  
pp. 2150135
Author(s):  
Dandan Shen ◽  
Huimin Ren ◽  
Fan Wu ◽  
Ruilin Zhu
Keyword(s):  
Form Factors ◽  
Distance Matrix ◽  
Relativistic Correction ◽  
Large Momentum ◽  
Matrix Elements ◽  
Leading Order ◽  
Lattice Data ◽  
Tensor Form ◽  
Long Distance ◽  
Current Form

We present a next-to-leading order (NLO) relativistic correction to [Formula: see text] tensor form factors within nonrelativistic QCD (NRQCD). We also consider complete Dirac bilinears [Formula: see text] with [Formula: see text] matrices [Formula: see text] in the [Formula: see text] transition. The relation among different current form factors is given and it shows that symmetries emerge in the heavy bottom quark limit. For a phenomenological extension, we propose to extract the long-distance matrix elements (LDMEs) for [Formula: see text] meson from the recent HPQCD lattice data and the NLO form factors at large momentum recoil.

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