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

Modjtaba Shokrian Zini

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

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].

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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].
We study the emergence of 2D conformal symmetry in critical quantum spin chains on the finite circle. Our goal is to characterize the conformal field theory (CFT) describing the universality class of the corresponding quantum phase transition. Using only the lattice Hamiltonian H=∑jhjH = \sum_j h_jH=∑j​hj​ as an input, we construct operators HnH_nHn​ (Fourier modes of the local Hamiltonian terms hjh_jhj​) that transform the low-energy eigenstates of HHH in the same way as certain combinations of the Virasoro generators do in the CFT. These operators were introduced for certain integrable lattice models by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper we propose and demonstrate the use of operators HnH_nHn​ to extract conformal data in nonintegrable quantum spin chains. In particular, we explain how to estimate the central charge from a simple ground-state expectation value and to systematically identify the low-energy states of HHH corresponding to Virasoro primary operators, as must be done in order to identify the CFT.
We study the emergence of two-dimensional conformal symmetry in critical quantum spin chains on the finite circle. Our goal is to characterize the conformal field theory (CFT) describing the universality class of the corresponding quantum phase transition. As a means to this end, we propose and demonstrate automated procedures which, using only the lattice Hamiltonian $H={\ensuremath{\sum}}_{j}{h}_{j}$ as an input, systematically identify the low-energy eigenstates corresponding to Virasoro primary and quasiprimary operators, and assign the remaining low-energy eigenstates to conformal towers. The energies and momenta of the primary operator states are needed to determine the primary operator scaling dimensions and conformal spins, an essential part of the conformal data that specifies the CFT. Our techniques use the action, on the low-energy eigenstates of HHH, of the Fourier modes Hn{H}_{n}Hn​ of the Hamiltonian density hj{h}_{j}hj​. The Hn{H}_{n}Hn​ were introduced as lattice representations of the Virasoro generators by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper, we demonstrate that these operators can be used to extract conformal data in a nonintegrable quantum spin chain.
Anyonic Chains – α\alpha α-Induction – CFT – Defects – Subfactors
Abstract Given a unitary fusion category, one can define the Hilbert space of a so-called “anyonic spin-chain” and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such systems can lead to 1+11+11+1 1+1 -dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, the spin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. One of our main results is the construction of a novel algebra of MPOs acting on a bipartite anyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of 1+11+11+1 1+1 CFTs as constructed by Fröhlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called “double triangle algebra” arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by Böckenhauer et al. based on the braided structure of the categories and on α\alpha α α -induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.
Abstract We provide a rigorous lattice approximation of conformal field theories given in terms of lattice fermions in 1+1-dimensions, focussing on free fermion models and Wess–Zumino–Witten models. To this end, we utilize a recently introduced operator-algebraic framework for Wilson–Kadanoff renormalization. In this setting, we prove the convergence of the approximation of the Virasoro generators by the Koo–Saleur formula. From this, we deduce the convergence of lattice approximations of conformal correlation functions to their continuum limit. In addition, we show how these results lead to explicit error estimates pertaining to the quantum simulation of conformal field theories.
Abstract We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules Wj,1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights (hr,s, hr,−s) and (hr,−s, hr,s) that give rise to dual structures: Verma or co-Verma modules. The limit of W0,q±2 {W}_{0,{\mathfrak{q}}^{\pm 2}} W0,q±2​ W 0 , q ± 2 contains diagonal fields (hr,1, hr,1) and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in q±2 {\mathfrak{q}}^{\pm 2} q±2 q ± 2 . In order to obtain matrix elements of Koo-Saleur generators at large system size N we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.
Abstract We study two families of quantum models which have been used previously to investigate the effect of topological symmetries in one-dimensional correlated matter. Various striking similarities are observed between certain Z n quantum clock models, spin chains generalizing the Ising model, and chains of non-Abelian anyons constructed from the so ( n ) 2 fusion category for odd n , both subject to periodic boundary conditions. In spite of the differences between these two types of quantum chains, e.g. their Hilbert spaces being spanned by tensor products of local spin states or fusion paths of anyons, the symmetries of the lattice models are shown to be closely related. Furthermore, under a suitable mapping between the parameters describing the interaction between spins and anyons the respective Hamiltonians share part of their energy spectrum (although their degeneracies may differ). This spin-anyon correspondence can be extended by fine-tuning of the coupling constants leading to exactly solvable models. We show that the algebraic structures underlying the integrability of the clock models and the anyon chain are the same. For n = 3,5,7 we perform an extensive finite size study—both numerical and based on the exact solution—of these models to map out their ground state phase diagram and to identify the effective field theories describing their low energy behaviour. We observe that the continuum limit at the integrable points can be described by rational conformal field theories with extended symmetry algebras which can be related to the discrete ones of the lattice models.