Quantum Computation and the Localization of Modular Functors

Freedman, Michael

doi:10.1007/s102080010006

Cited by 48 publications

(45 citation statements)

References 18 publications

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“…There are a number of obvious generalizations and open questions which we hope to address later, such as the instabilities of these critical theories, especially at the magic values d = 2 cos(π/(k + 2)); the addition of Chern-Simons terms to the action; correlation functions of (9) and the precise relationship between g and d; other gauge groups; complex d and θ-terms in the action; the effects of instantons in (9); and the implications of our results for topological quantum computation [1,11].…”

Section: Mapping Of the Ground-state To A Statistical-mechanicsmentioning

confidence: 86%

Line of Critical Points in2+1Dimensions: Quantum Critical Loop Gases and Non-Abelian Gauge Theory

2005

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We (1) construct a one-parameter family of lattice models of interacting spins; (2) obtain their exact ground states; (3) derive a statistical-mechanical analogy which relates their ground states to O(n) loop gases; (4) show that the models are critical for d ≤ √ 2, where d parametrizes the models; (5) note that for the special values d = 2 cos(π/(k + 2)), they are related to doubled level-k SU(2) Chern-Simons theory; (6) conjecture that they are in the universality class of a non-relativistic SU(2) gauge theory; and (7) show that its one-loop β-function vanishes for all values of the coupling constant, implying that it is also on a critical line.Family of Microscopic Lattice Models. Consider the following models of s = 1/2 spins on the links of a honeycomb lattice, inspired by a model of Kitaev [1]:where N (v) is the set of 3 links neighboring vertex v, andif 1, 2, . . . , 6 label the six edges of plaquette p. This Hamiltonian is a sum of projection operators with positive coefficients and, therefore, is positive-definite. The eigenvalues of the first term in (1) are 2, 0, corresponding to whether there is an even or odd number of σ z = −1 spins neighboring this vertex. The zero eigenvalue corresponds to the latter case. When eigenvalue zero is obtained at every vertex, the σ z = 1 links form connected loops. Hence, the zero-energy subspace of the first term is spanned by all configurations of multi-loops. Loops on the honeycomb lattice cannot cross. The term on the second line is a projection operator which annihilates a state |Ψ if the amplitude for all of the spins on a given plaquette to be up is a factor of d times the amplitude for them to all be down, i.e. if the amplitude for a configuration with a small loop encircling a single plaquette is a factor of d times the amplitude for an otherwise identical configuration without the small loop. The other three lines of the Hamiltonian vanish on a state |Ψ if it accords the same value to a configuration if a loop is deformed to enclose an additional plaquette. It is useful to think of these multi-loops as continuous curves, in which case, the ground state is invariant under smooth deformation of a curve and it loses a factor of d if a small, contractible curve is erased. Together, these conditions have been dubbed 'd-isotopy' [2].Exact Ground State Wavefunctions. The terms on the final four lines of (1) do not commute with each other, but they are compatible in the sense that they all annihilate the ground state, which is a superposition of all configurations α of multiloops weighted by a factor of d to the number of loops n α in each configuration:On an L × L torus, the ground state degeneracy is ∼ L 2 because the Hamiltonian does not mix states |α with different winding numbers. The different ground states are given by (3) but with the sum over α restricted to a single topological class. More generally, the ground state on any genus g ≥ 1 surface is infinitely degenerate in the thermodynamic limit. At d = 1, there is an additional condition called the JonesW...

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Section: Mapping Of the Ground-state To A Statistical-mechanicsmentioning

confidence: 86%

Line of Critical Points in2+1Dimensions: Quantum Critical Loop Gases and Non-Abelian Gauge Theory

2005

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“…[7][8][9][10][11] For example, the bracket model 12,13,15,19 for the Jones polynomial can be realized by a generalization of the Penrose SU (2) spin nets to the quantum group SU (2) q .…”

Section: Discussionmentioning

confidence: 99%

Quantum knots

Kauffman

Lomonaco

2004

SPIE Proceedings

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This paper proposes the definition of a quantum knot as a linear superposition of classical knots in three dimensional space. The definition is constructed and applications are discussed. Then the paper details extensions and also limitations of the Aravind Hypothesis for comparing quantum measurement with classical topological measurement. We propose a separate, network model for quantum evolution and measurement, where the background space is replaced by an evolving network. In this model there is an analog of the Aravind Hypothesis that promises to directly illuminate relationships between physics, topology and quantum knots.

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“…[11,12] Although our model contains topological sectors, distinguished by their total arrow polarizations in x and y-directions, the lowest energy belongs to the trivial topological sector of DDW, with zero total polarization (on an even-even lattice), see Fig. 1.…”

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confidence: 99%

Resonating Plaquette Phase of a Quantum Six-Vertex Model

Syljuåsen

Chakravarty

2006

Phys. Rev. Lett.

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The simplest quantum generalization of the six-vertex model describes fluctuations of the order parameter of the d-density wave (DDW), believed to compete with superconductivity in the highTc superconductors. The ground state of this model undergoes a first order transition from the DDW phase to a resonating plaquette phase as the quantum fluctuations are increased, which is explored with the help of quantum Monte Carlo simulations and analytic considerations involving the n-vector (n = 2) model with cubic anisotropy. In addition to finding a new quantum state, we show that the DDW is robust against a class of quantum fluctuations of its order parameter. The inferred finite temperature phase diagram contains unsuspected multicritical points.PACS numbers: 71.10.Hf, 74.10.+v The vertex models have unusual building blocks: arrows joined at a site forming the vertex. Nonetheless the statistical mechanics of these models are well defined, elegant, and have unexpected connections to more intuitively familiar models. For example, the transfer matrix of the classical six-vertex model is the XXZHamiltonian [1] of Heisenberg spins and the corresponding transfer matrix of the eight vertex model is the XYZHamiltonian.[2] Both of these models are solved by powerful mathematical methods with far reaching consequences. These vertex models are not simply products of mathematical imagination but were originally proposed to describe phases of ferro and antiferroelectric materials.[3] Thus they are effective descriptions of complex organization of matter. These vertex models are classical because they are made out of commuting variables.The interest in the quantum six-vertex model, to be defined below, is rooted in the unusual phenomenology of the cuprate high temperature superconductors.[4] It has been argued that the origin of the pseudogap is an unconventional broken symmetry in which a particle and a hole is bound in an angular momentum l = 2 state, resulting in an order parameter, the d-density wave (DDW), which is effectively hidden.[5] The statistical mechanical description of DDW, which includes both thermal and quantum fluctuations of the directions of the bond currents, is the quantum six-vertex model. [4] In particular, the description of DDW in terms of the six-vertex model resolves the puzzle as to why there are no associated specific heat anomalies, as the phase transition corresponds to an essential singularity in the free energy. In this Letter we consider commensurate DDW as possible incommensuration wave vectors tend to be small in extended Hubbard models. [6] Among the observed broken symmetries in the particleparticle channel are s, p, and d-wave superconductors, where the Cooper pairs bind in the angular momentum channels l = 0, 1, and 2 respectively. It is remarkable however, that in the particle-hole channel, the observed broken symmetries are mainly confined to s-wave symmetry: s-wave singlet and triplet density waves, known as the charge and spin density waves respectively. [7] Here we investigate, for...

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Quantum Computation and the Localization of Modular Functors

Cited by 48 publications

References 18 publications

Line of Critical Points in2+1Dimensions: Quantum Critical Loop Gases and Non-Abelian Gauge Theory

Line of Critical Points in2+1Dimensions: Quantum Critical Loop Gases and Non-Abelian Gauge Theory

Quantum knots

Resonating Plaquette Phase of a Quantum Six-Vertex Model

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