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Register a new userLong-Range GL(n) Integrable Spin Chains and Plane-Wave Matrix Theory
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Quantum spin chains arise naturally from perturbative large-N field theories and matrix models. The Hamiltonian of such a model is a long-range deformation of nearest-neighbor type interactions. Here, we study the most general long-range integrable spin chain with spins transforming in the fundamental representation of gl(n). We derive the Hamiltonian and the corresponding asymptotic Bethe ansatz at the leading four perturbative orders with several free parameters. Furthermore, we propose Bethe equations for all orders and identify the moduli of the integrable system. We finally apply our results to plane-wave matrix theory and show that the Hamiltonian in a closed sector is not of this form and therefore not integrable beyond the first perturbative order. This also implies that the complete model is not integrable.
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We construct the most general perturbatively long-range integrable spin chain with spins transforming in the fundamental representation of gl(N) and open boundary conditions. In addition to the previously determined bulk moduli we find a new set of parameters determining the reflection phase shift. We also consider finite-size contributions and comment on their determination.
For a transverse-field Ising chain with weak long-range interactions we develop a perturbative scheme, based on quantum kinetic equations, around the integrable nearest-neighbour model. We introduce, discuss, and benchmark several truncations of the time evolution equations up to eighth order in the Jordan-Wigner fermionic operators. The resulting set of differential equations can be solved for lattices with $O(10^2)$ sites and facilitates the computation of spin expectation values and correlation functions to high accuracy, at least for moderate timescales. We use this scheme to study the relaxation dynamics of the model, involving prethermalisation and thermalisation. The techniques developed here can be generalised to other spin models with weak integrability-breaking terms.
This review is dedicated to two-dimensional sigma models with flag manifold target spaces, which are generalizations of the familiar $CP^{n-1}$ and Grassmannian models. They naturally arise in the description of continuum limits of spin chains, and their phase structure is sensitive to the values of the topological angles, which are determined by the representations of spins in the chain. Gapless phases can in certain cases be explained by the presence of discrete t Hooft anomalies in the continuum theory. We also discuss integrable flag manifold sigma models, which provide a generalization of the theory of integrable models with symmetric target spaces. These models, as well as their deformations, have an alternative equivalent formulation as bosonic Gross-Neveu models, which proves useful for demonstrating that the deformed geometries are solutions of the renormalization group (Ricci flow) equations, as well as for the analysis of anomalies and for describing potential couplings to fermions.
We study the generalisations of the Craps-Sethi-Verlinde matrix big bang model to curved, in particular plane wave, space-times, beginning with a careful discussion of the DLCQ procedure. Singular homogeneous plane waves are ideal toy-models of realistic space-time singularities since they have been shown to arise universally as their Penrose limits, and we emphasise the role played by the symmetries of these plane waves in implementing the flat space Seiberg-Sen DLCQ prescription for these curved backgrounds. We then analyse various aspects of the resulting matrix string Yang-Mills theories, such as the relation between strong coupling space-time singularities and world-sheet tachyonic mass terms. In order to have concrete examples at hand, in an appendix we determine and analyse the IIA singular homogeneous plane wave - null dilaton backgrounds.
In this paper we define and study a matrix model describing the M-theory plane wave background with a single Horava-Witten domain wall. In the limit of infinite mu, the matrix model action becomes quadratic and we can identify the matrix Hamiltonian with a regularized Hamiltonian for hemispherical membranes that carry fermionic degrees of freedom on their boundaries. The number of fermionic degrees of freedom must be sixteen; this condition arises naturally in the framework of the matrix model. We can also prove the exact E_8 symmetry of the spectrum around the membrane vacua at infinite mu, which arises as a current algebra at level one just as in the heterotic string. We also find the full E_8 gauge multiplet as well as the multiple-gluon states, carried by collections of hemispherical membranes. Finally we discuss the dual description of the hemispherical membranes in terms of spherical fivebranes immersed in the domain wall; we identify the correct vacuum of the matrix model and make some preliminary comparisons with the (1,0) superconformal field theory.
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