Closed Spin Chains Research Articles

A Q-Operator for Open Spin Chains II: Boundary Factorization

One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine sl2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}_2$$\\end{document} and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.

Nested algebraic Bethe ansatz for deformed orthogonal and symplectic spin chains

We construct exact eigenvectors and eigenvalues for Uq(sp2n)- and Uq(so2n)-symmetric closed spin chains by means of a nested algebraic Bethe ansatz method. We use a fusion procedure to construct higher-dimensional Lax operators. Our approach generalises and extends the results obtained by Reshetikhin and De Vega–Karowski. We also present a generalisation of Tarasov–Varchenko trace formula for nested Bethe vectors.

Comparative investigation on correlation properties of XXZ open and closed spin chain of Heisenberg model in one-dimension

Comparative investigation on correlation properties of XXZ open and closed spin chain of Heisenberg model in one-dimension

Reggeon Webs, Spin Chains and the Odderon

At high center-of-mass energies scattering amplitudes enjoy a hidden integrability. An important example in QCD is Odderon exchange, a composite state of three reggeized gluons, with can be understood as a closed spin chain with periodic boundary conditions. In the $$N = 4$$ supersymmetric Yang–Mills theory a similar open spin chain appears for the planar eight-point amplitude. We solve these two examples in transverse momentum and rapidity space, introducing the concept of weighted graph complexity and its associated emerging scaling laws.

Open and closed spin chains as multiprocessor wires: Optimal engineering and reachability

We consider the perfect transfer of a state between arbitrary nodes of one-dimensional spin-1/2 chain with optimally engineered couplings. Motivated by the fact that such a system could be used as a data bus for connecting multiple quantum processors, we derive two necessary and sufficient conditions that have to be met in order to perfectly transfer a state between any two nodes and we employ them to examine both open and closed geometries. Analytical calculations and numerical optimizations are performed for both cases in order to determine the reachability of certain target states and to provide optimal values for the couplings which ensure perfect fidelity. An important finding is that even-sized closed chains allow for perfect transfer between any pair of sites and therefore are a promising platform for the implementation of data bus protocols.

A fusion for the periodic Temperley-Lieb algebra and its continuum limit

The equivalent of fusion in boundary conformal field theory (CFT) can be realized quite simply in the context of lattice models by essentially glueing two open spin chains. This has led to many developments, in particular in the context of chiral logarithmic CFT.We consider in this paper a possible generalization of the idea to the case of bulk conformal field theory. This is of course considerably more difficult, since there is no obvious way of merging two closed spin chains into a big one. In an earlier paper, two of us had proposed a “topological” way of performing this operation in the case of models based on the affine Temperley-Lieb (ATL) algebra, by exploiting the associated braid group representation and skein relations. In the present work, we establish — using, in particular, Frobenius reciprocity — the resulting fusion rules for standard modules of ATL in the generic as well as partially degenerate cases. These fusion rules have a simple interpretation in the continuum limit. However, unlike in the chiral case this interpretation does not match the usual fusion in non-chiral CFTs. Rather, it corresponds to the glueing of the right moving component of one conformal field with the left moving component of the other.

Open spin chains and complexity in the high energy limit

In the high energy limit of scattering amplitudes in Quantum Chromodynamics and supersymmetric theories the dominant Feynman diagrams are characterized by a hidden integrability in the planar limit. A well-known example is that of Odderon exchange, which can be described as a composite state of three reggeized gluons and corresponds to a closed spin chain with periodic boundary conditions. In the N=4 supersymmetric Yang–Mills theory a similar spin chain arises in the multi-Regge asymptotics of the eight-point amplitude in the planar limit. We investigate the associated open spin chain in transverse momentum and rapidity variables solving the corresponding effective Feynman diagrams. We introduce the concept of complexity in the high energy effective field theory and study its emerging scaling laws.

Penrose limits and spin chains in the GJV/CS-SYM duality

We examine Penrose limits of the duality proposed by Guarino, Jafferis and Varela between a type IIA massive background of the type of a warped, squashed AdS4 × S6, and a 2+1 dimensional IR fixed point of mathcal{N}=8 super Yang-Mills deformed by Chern-Simons terms to mathcal{N}=2 supersymmetry. One type of Penrose limit for closed strings corresponds to a large charge closed spin chain, and another, for open strings on giant graviton D-branes, corresponds to an open spin chain on sub-determinant operators. For the first limit, we find that like in the ABJM case, there are functions fa(λ) that interpolate between the perturbative and nonperturbative (string) regions for the magnon energy. For the second, we are unable to match the gravity result with the expected field theory result, making this model more interesting than ones with more supersymmetry.

Remarks towards the spectrum of the Heisenberg spin chain type models

The integrable close and open chain models can be formulated in terms of generators of the Hecke algebras. In this review paper, we describe in detail the Bethe ansatz for the XXX and the XXZ integrable close chain models. We find the Bethe vectors for two--component and inhomogeneous models. We also find the Bethe vectors for the fermionic realization of the integrable XXX and XXZ close chain models by means of the algebraic and coordinate Bethe ansatz. Special modification of the XXZ closed spin chain model ("small polaron model") is consedered. Finally, we discuss some questions relating to the general open Hecke chain models.

Three-point functions and s u 1 | 1 $$ \mathfrak{s}\mathfrak{u}\left(1\Big|1\right) $$ spin chains

We compute three-point functions of general operators in the su(1|1) sector of planar N = 4 SYM in the weak coupling regime, both at tree-level and one-loop. Each operator is represented by a closed spin chain Bethe state characterized by a set of momenta parameterizing the fermionic excitations. At one-loop, we calculate both the two-loop Bethe eigenstates and the relevant Feynman diagrams for the three-point functions within our setup. The final expression for the structure constants is surprisingly simple and hints at a possible form factor based approach yet to be unveiled.

Correlation functions of the Aharony-Bergman-Jafferis-Maldacena model

In the ABJM model, we study the three-point function of two heavy operators and an (ir)relevant one. Following the AdS/CFT correspondence, the structure constant in the large 't Hooft coupling limit can be factorized into two parts. One is the structure constant with a marginal operator, which is fully determined by the physical quantities of heavy operator and gives rise to the consistent result with the RG analysis. The other can be expressed as the universal form depending only on the conformal dimension of an (ir)relevant operator. We also investigate the new size effect of a circular string dual to a certain closed spin chain.

Tailoring three-point functions and integrability

We use Integrability techniques to compute structure constants in N=4 SYM to leading order. Three closed spin chains, which represent the single trace gauge-invariant operators in N=4 SYM, are cut into six open chains which are then sewed back together into some nice pants, the three-point function. The algebraic and coordinate Bethe ansatz tools necessary for this task are reviewed. Finally, we discuss the classical limit of our results, anticipating some predictions for quasi-classical string correlators in terms of algebraic curves.

The nested Bethe ansatz for ‘all’ closed spin chains

We present in a unified and detailed way the nested Bethe ansatz for closed spin chains based on or (super)algebras, with arbitrary representations (i.e. ‘spins’) on each site of the chain. In particular, the case of indecomposable representations of superalgebras is studied. The construction extends and unifies the results already obtained for spin chains based on or and for some particular super-spin chains. We give the Bethe equations and the form of the Bethe vectors. The case of gl(2|1), gl(2|2) and gl(4|4) superalgebras (that are related to AdS/CFT correspondence) is also detailed.

Integrable open spin chains and the doubling trick in Script N = 2 SYM with fundamental matter

We demonstrate that the one-loop anomalous dimension matrix in N = 2 SYM with a single chiral hypermultiplet of fundamental matter, which is dual to AdS_5 X S^5 with a D7-brane filling AdS_5 and wrapped around an $^3 in the S^5, is an integrable open spin chain Hamiltonian. We also use the doubling trick to relate these open spin chains to closed spin chains in pure N = 4 SYM. By using the AdS/CFT correspondence, we find a relation between the corresponding open and closed strings that differs from a simple doubling trick by terms that vanish in the semiclassical limit. We also demonstrate that in some cases the closed string is simpler and easier to study than the corresponding open string, and we speculate on the nature of corrections due to the presence of D-branes that this implies.

Analytical Bethe ansatz for closed and open gl(\U0001d4a9)-spin chains in any representation

We present an ‘algebraic treatment’ of the analytical Bethe ansatz. For this purpose, weintroduce abstract monodromy and transfer matrices which provide an algebraic frameworkfor the analytical Bethe ansatz. It allows us to deal with a generic gl(\U0001d4a9)-spin chain possessing on each site an arbitrary gl(\U0001d4a9)-representation. For open spin chains, we use the classification of the reflection matrices totreat all the diagonal boundary cases.As a result, we obtain the Bethe equations in their full generality for closed and openspin chains. The classifications of finite-dimensional irreducible representationsfor the Yangian (closed spin chains) and for the reflection algebras (open spinchains) are directly linked to the calculation of the transfer matrix eigenvalues.The local Hamiltonian associated with a given integrable spin chain needs to be computedcase by case. A general formula is still lacking at that point. As examples, we recover theusual Hamiltonian and Bethe equations for closed and open spin chains; we treat thealternating spin chains and the closed spin chain with an impurity. We also compute theHamiltonian and Bethe equations for the open spin chain model used in the context oflarge-N QCD.

Separation of variables for the quantum SL(2,Bbb R) spin chain

We construct representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same "pyramid diagram" as appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state.

Alternating antiferromagnetic and ferromagnetic exchange interactions in the S = 1 Heisenberg chain. Theory and magnetic properties

We focus on the magnetic properties of the S = 1 Heisenberg chain with alternating antiferromagnetic and ferromagnetic exchange interactions J 1 and J 2. The magnetic behavior of this system is calculated as a function of the alternation parameter α = J 2/| J 1|, from a general numerical procedure based on closed spin chains of increasing length. These theoretical results are fitted to rational unified expressions, which are subsequently used to describe the magnetic behavior of a nickel (II) complex, [Ni(bipy)(N 3) 2] n , exhibiting an alternating chain structure with a dominant ferromagnetic exchange.