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On the Fermionic Quasi-particle Interpretation in Minimal Models of Conformal Field Theory

Alexander A. Belavin, A. (Freie Univ. Berlin (Germany). Inst. fuer Theoretische Physik) Fring, Technische Univ. Berlin (Germany). Sonderforschungsbereich 288 -Differentialgeometrie und Quantenphysik

Published Jan 1, 1996• 2 citations
OpenAlex

Abstract

The conjecture that the states of the ferminonic quasi-particles in minimal conformal field theories are eigenstates of the integrals of motion to certain eigenvalues is checked and shown to be correct only for the Ising model.

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Scaling Limit of Anyonic Chains and Quantum Simulation of Conformal Field Theory

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Author(s): Shokrian Zini, Modjtaba | Advisor(s): Wang, Zhenghan | Abstract: We provide a mathematical definition of a low energy scaling limit of a sequence of general non-relativistic quantum theories in any dimension, and apply our formalism to anyonic chains. We formulate Conjecture 2.5.3 on conditions when a unitary rational (1+1)-d conformal field theory would arise as such a limit and verify the conjecture for the Ising minimal model M(4,3) using su(2)_2 anyonic chains. Our work is based on the existence of a Fourier transform relation between Temperley-Lieb generators {e_i} and some finite stage operators of the Virasoro generators {L_m+L_{-m}} and {i(L_m-L_{-m})} for unitary minimal models M(k+2,k+1) (proven for k=2). An earlier attempt [1], called the Koo-Saleur formula, has slower convergence with characteristics that hinder the convergence of algebras of observables, an important contribution of this work. Assuming Conjecture 2.5.3, most of our main results for M(4,3) hold for higher (k g= 3) unitary minimal models M(k+2,k+1) as well. Our approach is supported by extensive numerical simulation and physical proofs in the physics literature. It is also inspired by an eventual application to an efficient simulation of conformal field theories by quantum computers. We approach the definition of the unitary evolution and correlator simulation problems in the same spirit of topological quantum field theory simulation as established by M. Freedman et. al. [2]. Under certain conditions, we present complexity theoretic hardness results on the simulation problems by using the framework of fermionic quantum computation by Bravyi and Kitaev [3].

•Jan 1, 2020•0 citations
We generalize the results of [Comm.Math.Phys.299 (2010), 825-866] (hidden Grassmann structure IV) to the case of excited states of the transfer matrix of the sixvertex model acting in the so-called Matsubara direction.We establish an equivalence between a scaling limit of the partition function of the six-vertex model on a cylinder with quasi-local operators inserted and special boundary conditions, corresponding to particlehole excitations, on the one hand, and certain three-point correlation functions of conformal field theory (CFT) on the other hand.As in hidden Grassmann structure IV, the fermionic basis developed in previous papers and its conformal limit are used for a description of the quasi-local operators.In paper IV we claimed that in the conformal limit the fermionic creation operators generate a basis equivalent to the basis of the descendant states in the conformal field theory modulo integrals of motion suggested by A. Zamolodchikov (1987).Here we argue that, in order to completely determine the transformation between the above fermionic basis and the basis of descendants in the CFT, we need to involve excitations.On the side of the lattice model we use the excited-state TBA approach.We consider in detail the case of the descendant at level 8.
The dilute A3{\mathrm{A}}_{3}A3​ model is a solvable interaction round a face model with three local states and adjacency conditions encoded by the Dynkin diagram of the Lie algebra A3{\mathrm{A}}_{3}A3​. It can be regarded as a solvable spin-1 Ising model at the critical temperature in a magnetic field. One therefore expects the scaling limit to be governed by Zamolodchikov's integrable perturbation of the c=1/2 conformal field theory. Indeed, a recent thermodynamic Bethe ansatz approach succeeded in unveiling the corresponding E8{\mathrm{E}}_{8}E8​ structure under certain assumptions on the nature of the Bethe ansatz solutions. In order to check these conjectures, we perform a detailed numerical investigation of the solutions of the Bethe ansatz equations for the critical and off-critical models. Scaling functions for the ground-state corrections and for the lowest spectral gaps are obtained, which give very precise numerical results for the lowest mass ratios in the massive scaling limit. While these agree perfectly with the E8{\mathrm{E}}_{8}E8​ mass ratios, we observe one state that seems to violate the assumptions underlying the thermodynamic Bethe ansatz calculation. We also analyze the critical spectrum of the dilute A3{\mathrm{A}}_{3}A3​ model, which exhibits excitations with a finite gap on top of the massless spectrum of the Ising conformal field theory.
Author(s): Shokrian Zini, Modjtaba | Advisor(s): Wang, Zhenghan | Abstract: We provide a mathematical definition of a low energy scaling limit of a sequence of general non-relativistic quantum theories in any dimension, and apply our formalism to anyonic chains. We formulate Conjecture 2.5.3 on conditions when a unitary rational (1+1)-d conformal field theory would arise as such a limit and verify the conjecture for the Ising minimal model M(4,3) using su(2)_2 anyonic chains. Our work is based on the existence of a Fourier transform relation between Temperley-Lieb generators {e_i} and some finite stage operators of the Virasoro generators {L_m+L_{-m}} and {i(L_m-L_{-m})} for unitary minimal models M(k+2,k+1) (proven for k=2). An earlier attempt [1], called the Koo-Saleur formula, has slower convergence with characteristics that hinder the convergence of algebras of observables, an important contribution of this work. Assuming Conjecture 2.5.3, most of our main results for M(4,3) hold for higher (k g= 3) unitary minimal models M(k+2,k+1) as well. Our approach is supported by extensive numerical simulation and physical proofs in the physics literature. It is also inspired by an eventual application to an efficient simulation of conformal field theories by quantum computers. We approach the definition of the unitary evolution and correlator simulation problems in the same spirit of topological quantum field theory simulation as established by M. Freedman et. al. [2]. Under certain conditions, we present complexity theoretic hardness results on the simulation problems by using the framework of fermionic quantum computation by Bravyi and Kitaev [3].