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arXiv:2111.12080

$T\bar T$ deformation of Calogero-Sutherland model via dimensional reduction

D. Pavshinkin

Published Nov 23, 2021• 0 citations
arXivHTMLPDF

Abstract

We perform the dynamical change of coordinates to derive a generalization of the trace relation and apply it to the non-linear Schrödinger model. After that, we work out the dimensional reduction of the bilinear $T\bar T$ operator and obtain the new $T\bar T$-like deformation of the quantum mechanics of free non-relativistic fermions and interacting Calogero-Sutherland particles. The deformation modifies the spectrum of the Hamiltonian but does not alter its eigenfunctions. The deformed classical Lagrangian is also obtained. Finally, we study a particular deformation of the two-dimensional Yang-Mills theory that maps the gauge theory onto a system of $T\bar T$-perturbed fermions.

DOI
10.48550/arxiv.2111.12080

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TTˉT\bar TTTˉ deformation of Calogero-Sutherland model via dimensional reduction
We perform the dynamical change of coordinates to derive a generalization of the trace relation and apply it to the non-linear Schrödinger model. After that, we work out the dimensional reduction of the bilinear TTˉT\bar TTTˉ operator and obtain the new TTˉT\bar TTTˉ-like deformation of the quantum mechanics of free non-relativistic fermions and interacting Calogero-Sutherland particles. The deformation modifies the spectrum of the Hamiltonian but does not alter its eigenfunctions. The deformed classical Lagrangian is also obtained. Finally, we study a particular deformation of the two-dimensional Yang-Mills theory that maps the gauge theory onto a system of TTˉT\bar TTTˉ-perturbed fermions.
TT‾T\overline TTT deformation of the Calogero–Sutherland model via dimensional reduction
We derive a general expression for the trace of the energy–momentum tensor TT‾TT \kern0.9pt\overline{\vphantom{T}\kern6.0pt}\kern-6.9pt T TTT -deformed field theories using a dynamical change of coordinates. Then we perform a dimensional reduction of the bilinear TT‾TT \kern0.9pt\overline{\vphantom{T}\kern6.0pt}\kern-6.9pt T TTT operator and obtain a new TT‾TT \kern0.9pt\overline{\vphantom{T}\kern6.0pt}\kern-6.9pt T TTT -like deformation of the quantum mechanics of free nonrelativistic fermions and the interacting Calogero–Sutherland system. The deformation leads to a change in the energy spectrum but does not affect the eigenfunctions. Furthermore, an expression for the deformed classical Lagrangian is obtained. We also study the correspondence between the two-dimensional Yang–Mills theory and the Calogero–Sutherland system in the presence of the deformation.
The 40-year-old Calogero-Sutherland (CS) model remains a source of inspirations for understanding 1d interacting fermions. At β=1,or0β=1, \text{or}0β=1,or0, the CS model describes a free non-relativistic fermion, or boson theory, while for generic βββ, the system can be interpreted either as interacting fermions or bosons, or free anyons depending on the context. However, we shall show in this letter that the fermionic picture is advantageous in diagonalizing the CS Hamiltonian. Comparing to the previously known multi-integral representation or the Dunkl operator formalism for the CS wave functions, our method depends on the (upper or lower) triangular nature of the fermion interaction, which is resolved in perturbation theory of the second quantized form. The eigenstate is constructed from a multiplet of unperturbed states and the perturbation is of finite order. The full construction is a similarity transformation from the free fermion theory, in the same spirit as the Landau Fermi liquid theory and the 1d Luttinger liquid theory. That means quasi-particles or anyons can be represented in terms of free fermion modes (or bosonic modes via bosonization). The method is applicable to other (higher than one space dimension) systems for which the adiabatic theorem applies.
We review recent results on new physical models constructed as PT-symmetrical deformations or extensions of different types of integrable models. We present non-Hermitian versions of quantum spin chains, multi-particle systems of Calogero-Moser-Sutherland type and nonlinear integrable field equations of Korteweg-de Vries type. The quantum spin chain discussed is related to the first example in the series of the non-unitary models of minimal conformal field theories. For the Calogero-Moser-Sutherland models, we provide three alternative deformations: a complex extension for models related to all types of Coxeter/Weyl groups; models describing the evolution of poles in constrained real-valued field equations of nonlinear integrable systems; and genuine deformations based on antilinearly invariant deformed root systems. Deformations of complex nonlinear integrable field equations of Korteweg-de Vries type are studied with regard to different kinds of PT-symmetrical scenarios. A reduction to simple complex quantum mechanical models currently under discussion is presented.
The deformed Calogero-Sutherland (CS) model is a quantum integrable system with arbitrary numbers of two types of particles and reducing to the standard CS model in special cases. We show that a known collective field description of the CS model, which is based on conformal field theory (CFT), is actually a collective field description of the deformed CS model. This provides a natural application of the deformed CS model in Wen’s effective field theory of the fractional quantum Hall effect (FQHE), with the two kinds of particles corresponding to electrons and quasi-hole excitations. In particular, we use known mathematical results about super-Jack polynomials to obtain simple explicit formulas for the orthonormal CFT basis proposed by van Elburg and Schoutens in the context of the FQHE.
Abstract We study solvable deformations of two-dimensional quantum field theories driven by a bilinear operator constructed from a pair of conserved U(1) currents Ja. We propose a quantum formulation of these deformations, based on the gauging of the corresponding symmetries in a path integral. This formalism leads to an exact dressing of the S-matrix of the system, similarly as what happens in the case of a TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ deformation. For conformal theories the deformations under study are expected to be exactly marginal. Still, a peculiar situation might arise when the conserved currents Ja are not well-defined local operators in the original theory. A simple example of this kind of system is provided by rotation currents in a theory of multiple free, massless, non-compact bosons. We verify that, somewhat unexpectedly, such a theory is indeed still conformal after deformation and that it coincides with a TsT transformation of the original system. We then extend our formalism to the case in which the conserved currents are non-Abelian and point out its connection with Deformed T-dual Models and homogeneous Yang-Baxter deformations. In this case as well the deformation is based on a gauging of the symmetries involved and it turns out to be non-trivial only if the symmetry group admits a non-trivial central extension. Finally we apply what we learned by relating the TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ deformation to the central extension of the two-dimensional Poincaré algebra.