scholarly journals Free fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Marius de Leeuw ◽  
Chiara Paletta ◽  
Anton Pribytok ◽  
Ana L. Retore ◽  
Alessandro Torrielli
Keyword(s):  
Spectral Problem ◽  
Transfer Matrix ◽  
Arbitrary Number ◽  
Free Fermion ◽  
Nearest Neighbour ◽  
Bogoliubov Transformation ◽  
Free Fermions ◽  
New Models ◽  
Lower Dimensional

Abstract In this paper we first demonstrate explicitly that the new models of integrable nearest-neighbour Hamiltonians recently introduced in PRL 125 (2020) 031604 [36] satisfy the so-called free fermion condition. This both implies that all these models are amenable to reformulations as free fermion theories, and establishes the universality of this condition. We explicitly recast the transfer matrix in free fermion form for arbitrary number of sites in the 6-vertex sector, and on two sites in the 8-vertex sector, using a Bogoliubov transformation. We then put this observation to use in lower-dimensional instances of AdS/CFT integrable R-matrices, specifically pure Ramond-Ramond massless and massive AdS3, mixed-flux relativistic AdS3 and massless AdS2. We also attack the class of models akin to AdS5 with our free fermion machinery. In all cases we use the free fermion realisation to greatly simplify and reinterpret a wealth of known results, and to provide a very suggestive reformulation of the spectral problem in all these situations.

Corner transfer matrix of generalized free-fermion vertex systems

1993 ◽  
Vol 26 (23) ◽  
pp. 6807-6823 ◽  
Author(s):  
H -P Eckle ◽  
T T Truong
Keyword(s):  
Transfer Matrix ◽  
Free Fermion

scholarly journals Hubbard Models as Fusion Products of Free Fermions

1998 ◽  
Vol 12 (19) ◽  
pp. 1893-1906 ◽  
Author(s):  
Z. Maassarani
Keyword(s):  
Bethe Ansatz ◽  
Simple Proof ◽  
Complete Integrability ◽  
Baxter Equation ◽  
Free Fermion ◽  
Free Fermions ◽  
Hubbard Models ◽  
Algebraic Bethe Ansatz ◽  
Local Symmetries ◽  
Fusion Products

A class of recently introduced su (n) "free-fermion" models has recently been used to construct generalized Hubbard models. I derive an algebra defining the "free-fermion" models and give new classes of solutions. I then introduce a conjugation matrix and give a new and simple proof of the corresponding decorated Yang–Baxter equation. This provides the algebraic tools required to couple in an integrable way two copies of free-fermion models. Complete integrability of the resulting Hubbard-like models is shown by exhibiting their L and R matrices. Local symmetries of the models are discussed. The diagonalization of the free-fermion models is carried out using the algebraic Bethe Ansatz.

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.

scholarly journals THE TRANSFER MATRIX METHOD FOR MODAL ANALYSIS OF CRACKED MULTISTEP BEAM

2017 ◽  
Vol 55 (5) ◽  
pp. 598
Author(s):  
Nguyen Tien Khiem ◽  
Vu Thi An Ninh ◽  
Tran Thanh Hai
Keyword(s):  
Modal Analysis ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Arbitrary Number ◽  
Matrix Method ◽  
Crack Detection ◽  
Natural Frequencies ◽  
Modal Testing ◽  
Testing Technique

The present study addresses the modal analysis of multistep beam with arbitrary number of cracks by using the transfer matrix method and modal testing technique. First, there is conducted general solution of free vibration problem for uniform beam element with arbitrary number of cracks that allows one to simplify the transfer matrix for cracked multistep beam. The transferring beam state needs to undertake only at the steps of beam but not through crack positions. Such simplified the transfer matrix method makes straightforward to investigate effect of cracks mutually with cross-section step in beam on natural frequencies. It is revealed that step-down and step-up in beam could modify notably sensitivity of natural frequencies to crack so that the analysis provides useful indication for crack detection in multistep beam. The proposed theory was validated by an experimental case study

scholarly journals Characterization of solvable spin models via graph invariants

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 278 ◽  
Author(s):  
Adrian Chapman ◽  
Steven T. Flammia
Keyword(s):  
Complete Characterization ◽  
Free Fermion ◽  
Many Body ◽  
Spin Models ◽  
Free Fermions ◽  
Graph Theoretic ◽  
Theoretic Problem ◽  
Free Fermion Model ◽  
Complete Set

Exactly solvable models are essential in physics. For many-body spin-1/2 systems, an important class of such models consists of those that can be mapped to free fermions hopping on a graph. We provide a complete characterization of models which can be solved this way. Specifically, we reduce the problem of recognizing such spin models to the graph-theoretic problem of recognizing line graphs, which has been solved optimally. A corollary of our result is a complete set of constant-sized commutation structures that constitute the obstructions to a free-fermion solution. We find that symmetries are tightly constrained in these models. Pauli symmetries correspond to either: (i) cycles on the fermion hopping graph, (ii) the fermion parity operator, or (iii) logically encoded qubits. Clifford symmetries within one of these symmetry sectors, with three exceptions, must be symmetries of the free-fermion model itself. We demonstrate how several exact free-fermion solutions from the literature fit into our formalism and give an explicit example of a new model previously unknown to be solvable by free fermions.

scholarly journals Irrelevant deformations of chiral bosons

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Subhroneel Chakrabarti ◽  
Divyanshu Gupta ◽  
Arkajyoti Manna ◽  
Madhusudhan Raman
Keyword(s):  
Closed Form ◽  
Stress Tensor ◽  
Linear Order ◽  
Equations Of Motion ◽  
Local Action ◽  
Free Fermion ◽  
Free Fermions ◽  
Non Local ◽  
Left And Right ◽  
Chiral Bosons

Abstract We study $$ \mathrm{T}\overline{\mathrm{T}} $$ T T ¯ deformations of chiral bosons using the formalism due to Sen. For arbitrary numbers of left- and right-chiral bosons, we find that the $$ \mathrm{T}\overline{\mathrm{T}} $$ T T ¯ -deformed Lagrangian can be computed in closed form, giving rise to a novel non-local action in Sen’s formalism. We establish that at the limit of infinite $$ \mathrm{T}\overline{\mathrm{T}} $$ T T ¯ coupling, the equations of motion of deformed theory exhibits chiral decoupling. We then turn to a discussion of $$ \mathrm{T}\overline{\mathrm{T}} $$ T T ¯ -deformed chiral fermions, and point out that the stress tensor of the $$ \mathrm{T}\overline{\mathrm{T}} $$ T T ¯ -deformed free fermion coincides with the undeformed seed theory. We explain this behaviour of the stress tensor by noting that the deformation term in the action is purely topological in nature and closely resembles the fermionic Wess-Zumino term in the Green-Schwarz formalism. In turn, this observation also explains a puzzle in the literature, viz. why the $$ \mathrm{T}\overline{\mathrm{T}} $$ T T ¯ deformation of multiple free fermions truncate at linear order. We conclude by discussing the possibility of an interplay between $$ \mathrm{T}\overline{\mathrm{T}} $$ T T ¯ deformations and bosonisation.

A NEW SOLVABLE 3D MODEL OF STATISTICAL MECHANICS: THE FREE-FERMION SOLUTION

1996 ◽  
Vol 10 (04) ◽  
pp. 443-453 ◽  
Author(s):  
A.E. BOROVICK ◽  
S.I. KULINICH ◽  
V. Yu. POPKOV ◽  
Yu. M. STRZHEMECHNY
Keyword(s):  
Phase Diagram ◽  
Statistical Mechanics ◽  
Bethe Ansatz ◽  
3D Model ◽  
Nearest Neighbor ◽  
Baxter Equation ◽  
Free Fermion ◽  
Vertex Model ◽  
Free Fermions ◽  
Exactly Solvable

We obtain a new exactly solvable K-plane vertex model. This 3D model is one wih real Boltzmann weights and nearest neighbor interactions. The corresponding Yang-Baxter equation is proved. The Bethe ansatz has also been found, enabling us to completely investigate the phase diagram in the “free fermions” case.

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