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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Conserved charges from self-duality

L. Dolan, Michael Grady

Published Mar 15, 1982• 164 citations
Publisher

Abstract

Given a simple self-dual quantum Hamiltonian $H=KB+\ensuremath{\Gamma}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}$, where $K$ and $\ensuremath{\Gamma}$ are coupling constants, and the condition that $[B,[B,[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}]]]=16[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}]$, then we construct an infinite set of conserved charges ${Q}_{2n}$; $[H,{Q}_{2n}]=0$. In simple models, like the two-dimensional Ising or Baxter eight-vertex, these charges appear in the associated quantum theories and are equivalent to those which result from the transfer-matrix formulation and exact quantum integrability of the system. The power of our result is that it is an operator statement and does not refer to the number of dimensions or the nature of the space-time manifold: lattice, continuum, or loop space. It is suggested how the establishment of this link between duality and integrability could be used to exploit the Kramers-Wannier-type self-duality of the four-dimensional $\mathrm{SU}(N)$ gauge theory to find hidden symmetry.

DOI
10.1103/physrevd.25.1587

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Given a simple self-dual quantum Hamiltonian H=KB+\ensuremathΓB\ifmmode~\else˜\fiH=KB+\ensuremath{\Gamma}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}H=KB+\ensuremathΓB\ifmmode~\else˜\fi​, where KKK and \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ are coupling constants, and the condition that [B,[B,[B,B\ifmmode~\else˜\fi]]]=16[B,B\ifmmode~\else˜\fi][B,[B,[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}]]]=16[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}][B,[B,[B,B\ifmmode~\else˜\fi​]]]=16[B,B\ifmmode~\else˜\fi​], then we construct an infinite set of conserved charges Q2n{Q}_{2n}Q2n​; [H,Q2n]=0[H,{Q}_{2n}]=0[H,Q2n​]=0. In simple models, like the two-dimensional Ising or Baxter eight-vertex, these charges appear in the associated quantum theories and are equivalent to those which result from the transfer-matrix formulation and exact quantum integrability of the system. The power of our result is that it is an operator statement and does not refer to the number of dimensions or the nature of the space-time manifold: lattice, continuum, or loop space. It is suggested how the establishment of this link between duality and integrability could be used to exploit the Kramers-Wannier-type self-duality of the four-dimensional SU(N)\mathrm{SU}(N)SU(N) gauge theory to find hidden symmetry.
The thesis consists of nine chapters and two appendices. In chapter II I discuss the connection given by the transfer matrix formulation between a classical statistical mechanical system and an associated quantum Hamiltonian, using as an example the familiar two- 4 dimensional Ising model. Next, in chapter III, I show how this applies to Baxter's 8-vertex model and the XYZ Hamiltonian, and how an infinite set of conserved charges for the XYZ model arises from properties of the transfer matrix. In Chapter IV I discuss the meaning of self-duality as it applies to quantum Hamiltonians, and develop a general form for a self-dual Hamiltonian. In Chapter V I show how one can find the explicit form for conserved charges in the XY model by a heuristic technique which takes advantage of the models' self-duality. In chapter VI it is shown how the solution to the XYZ model by the quantum inverse scattering method leads to the infinite set of conserved charges associated with it. The thesis culminates in the statement and proof of the theorem mentioned above, which stipulates a condition under which self-dual theories possess an infinite set of conserved charges (Chapter VII), after which I briefly discuss applications of the theorem (VIII) and draw some conclusions (IX).
In this work, we investigate the one-dimensional XXXXXX lattice model and its cousins through the lens of momentum and winding U(1)U(1)U(1) symmetries. We distinguish two closely related Z2\mathbb{Z}_2Z2​ symmetries based on their relation to the U(1)U(1)U(1) symmetries, and establish a web of Z2\mathbb{Z}_2Z2​-gauging relations among these models, rooted in two fundamental seeds: the $ XY \pm YX$ models. These two seeds, each self-dual under gauging of the respective Z2\mathbb{Z}_2Z2​-symmetries, possess manifestly symmetric conserved charges, making transparent the connection between the noninvertible symmetries and the Kramers-Wannier duality. By leveraging the self-dualities of these two seed models, we derive the self-dualities of their cousins, including the XXXXXX model and the Levin-Gu model, through appropriate gauging procedures. Moreover, under these gauging schemes, the lattice T-duality matrices take the form of the identity matrix. These lattice models flow to the c=1c =1c=1 compact boson conformal field theory, with a twist that depends on the lattice size modulo four. Finally, we unify the mapping structures of local conserved charges across these models, providing a comprehensive framework for understanding their symmetries and dualities.
This is a review on infinite non-abelian symmetries in two-dimensional field theories. We show how any integrable QFT enjoys the existence of infinitely many {\bf conserved} charges. These charges {\bf do not commute} between them and satisfy a Yang--Baxter algebra. They are generated by quantum monodromy operators and provide a representation of q−q-q−deformed affine Lie algebras $U_q({\hat\G})$. We review the work by de Vega, Eichenherr and Maillet on the bootstrap construction of the quantum monodromy operators in classically scale invariant theories where the classical monodromy matrix is conserved. Then, the recent generalization to the sine--Gordon (or massive Thirring) model, where such operators do not possess a classical analogue is given (This provides a representation of SU^(2)qS{\hat U}(2)_qSU^(2)q​). It is then reported on the recent work by Destri and de Vega, where both commuting and non-commuting integrals of motion are systematically obtained by Bethe Ansatz in the light-cone approach. The eigenvalues of the six--vertex alternating transfer matrix τ(ł)τ(ł)τ(ł) are explicitly computed on a generic physical state through algebraic Bethe ansatz. In the thermodynamic limit τ(ł)τ(ł)τ(ł) turns out to be a two--valued periodic function. One determination generates the local abelian charges, including energy and momentum, while the other yields the abelian subalgebra of the (non--local) YB algebra. In particular, the bootstrap results coincide with the ratio between the two determinations of the lattice transfer matrix.[Based on a Lecture at the Clausthal Symposium].
Self-dual S3S_3S3​ gauge theory in 2+1d: lattice model and topological phase transitions
Electric-magnetic self-duality of the Z2\mathbb{Z}_2Z2​ gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the S3S_3S3​ quantum double D(S3)\mathcal{D}(S_3)D(S3​) on a tensor product Hilbert space in which the Z2em\mathbb{Z}^{\mathrm{em}}_2Z2em​ anyon-permutation symmetry, exchanging the non-Abelian chargeon CCC and fluxon FFF, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent Z2em\mathbb{Z}_2^{\mathrm{em}}Z2em​-preserving bosonic perturbations, driving D(S3)\mathcal{D}(S_3)D(S3​) into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles D(D2n)\mathcal{D}(D_{2n})D(D2n​). Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.
In classical mechanics, two functions over phase space are said to be in involution if their Poisson bracket vanishes. Since Liouville's time, a dynamical system whose phase space is 2 N dimensional is called completely integrable if there are N functions, or “hamiltonians”, or “charges” in involution. A field theoretic system is called integrable if it possesses an infinity of mutually commuting conserved observables. All these mutually commuting conserved charges or hamiltonians allow us to solve the system exactly, without resorting to approximation schemes. Integrability is an unusual wealth of symmetry we might not think of requiring on realistic physical models. Rather, we should expect the complexity of nature not to be exactly solvable. However, integrability is of epistemological importance: exact solutions allow more perfect understanding. Toy models, of which physicists have been very fond since antiquity, often exist just so that exact and complete solutions can be found, in order to grasp the nature of the phenomenon being modeled. Furthermore, and quite surprisingly, physical systems with an infinite symmetry do exist: any non-linear system with soliton solutions is integrable. We shall be interested in discovering under which circumstances certain kinds of physical systems admit complete integrability, what types of systems these are, and in pointing out the physical roots of such a wonderful property. In the process, we shall have the occasion to use some of the most powerful tools elaborated by workers in mathematics. Given our present understanding of two-dimensional models, integrability appears as a consequence of very simple dynamics, characterized by factorized scattering matrices.