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27
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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Cited by 35 (2 self)
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
Spectral gap and exponential decay of correlations
 Comm. Math. Phys
"... We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with shortrange interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, ..."
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Cited by 19 (1 self)
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We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with shortrange interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain selfsimilarity in (fractal) dimensions D < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this selfsimilarity condition is automatically satisfied. We also treat systems with longrange, powerlaw decaying interactions. Spectral Gap and Exponential Decay of Correlations 2 1
Adiabatic Charge Transport And The Kubo Formula For 2D Hall Conductance
 Comm. Pure Appl. Math
, 2004
"... We study adiabatic charge transport in a two dimensional lattice model of electron gas at zero temperature. It is proved that if the Fermi level falls in the localization regime then, for a slowly varied weak electric eld, in the adiabatic limit the accumulated excess Hall transport is correctly ..."
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Cited by 11 (1 self)
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We study adiabatic charge transport in a two dimensional lattice model of electron gas at zero temperature. It is proved that if the Fermi level falls in the localization regime then, for a slowly varied weak electric eld, in the adiabatic limit the accumulated excess Hall transport is correctly described by the linear response Kubo Streda formula. Corrections to the leading term are given in an asymptotic series for the Hall current in powers of the adiabatic parameter. The analysis is based on an extension of an expansion of Nenciu, with the spectral gap condition replaced by localization bounds.
LiebRobinson Bounds for Harmonic and Anharmonic Lattice Systems
"... We prove LiebRobinson bounds for systems defined on infinite dimensional Hilbert spaces and described by unbounded Hamiltonians. In particular, we consider harmonic and certain anharmonic lattice systems. 1 ..."
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Cited by 11 (9 self)
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We prove LiebRobinson bounds for systems defined on infinite dimensional Hilbert spaces and described by unbounded Hamiltonians. In particular, we consider harmonic and certain anharmonic lattice systems. 1
Quantum dynamics with mean field interactions: a new approach
, 2008
"... We propose a new approach for the study of the time evolution of a factorized Nparticle bosonic wave function with respect to a meanfield dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads t ..."
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Cited by 9 (0 self)
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We propose a new approach for the study of the time evolution of a factorized Nparticle bosonic wave function with respect to a meanfield dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the manyparticle Schrödinger dynamics and the oneparticle nonlinear Hartree dynamics. In particular the oneparticle density matrix associated with the solution to the Nparticle Schrödinger equation is shown to converge to the projection onto the onedimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order 1/N for all fixed times.
Equality of the bulk and edge Hall conductances in a mobility gap
 Commun. Math. Phys
, 2005
"... Abstract: We consider the edge and bulk conductances for 2D quantum Hall systems in which the Fermi energy falls in a band where bulk states are localized. We show that the resulting quantities are equal, when appropriately defined. An appropriate definition of the edge conductance may be obtained t ..."
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Cited by 9 (0 self)
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Abstract: We consider the edge and bulk conductances for 2D quantum Hall systems in which the Fermi energy falls in a band where bulk states are localized. We show that the resulting quantities are equal, when appropriately defined. An appropriate definition of the edge conductance may be obtained through a suitable time averaging procedure or by including a contribution from states in the localized band. In a further result on the Harper Hamiltonian, we show that this contribution is essential. In an appendix we establish quantized plateaus for the conductance of systems which need not be translation ergodic. 1.
Quantum Stochastic Dynamics I: Spin Systems on a Lattice
, 1995
"... : In the context of noncommutative IL p spaces we discuss the conditions for existence and ergodicity of translation invariant stochastic spin flip and diffusion dynamics for quantum spin systems with finite range interactions on a lattice. Key words: Noncommutative IL p spaces, stochastic spin ..."
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Cited by 9 (0 self)
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: In the context of noncommutative IL p spaces we discuss the conditions for existence and ergodicity of translation invariant stochastic spin flip and diffusion dynamics for quantum spin systems with finite range interactions on a lattice. Key words: Noncommutative IL p spaces, stochastic spin flip and diffusion dynamics, quantum spins, systems on a lattice, finite range interactions. 1.Introduction The analysis in the interpolating family of IL p spaces associated to a probability measure plays an essential role in the study of the classical Markov semigroups. In general it is important for their construction as well as for the investigation of the ergodicity properties. It is especially useful if the underlying configuration space is infinite dimensional. In this paper we introduce some basic ideas concerning the application of interpolating IL p spaces to study Markov semigroups in the noncommutative context of quantum spin systems on a lattice. In Section 2 we show that usi...
On the existence of the dynamics for anharmonic quantum oscillator systems
 Rev. Math. Phys
, 2010
"... Abstract. We construct a W ∗dynamical system describing the dynamics of a class of anharmonic quantum oscillator lattice systems in the thermodynamic limit. Our approach is based on recently proved LiebRobinson bounds for such systems on finite lattices [19]. 1. ..."
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Cited by 7 (5 self)
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Abstract. We construct a W ∗dynamical system describing the dynamics of a class of anharmonic quantum oscillator lattice systems in the thermodynamic limit. Our approach is based on recently proved LiebRobinson bounds for such systems on finite lattices [19]. 1.
Quantum Boolean Functions
, 2009
"... In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f² = I. We describe several generalisations of wellknown results in the theory of boolean functions, including quantum property testing; a quantum version of the Goldreich ..."
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Cited by 6 (4 self)
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In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f² = I. We describe several generalisations of wellknown results in the theory of boolean functions, including quantum property testing; a quantum version of the GoldreichLevin algorithm for finding the large Fourier coefficients of boolean functions; and two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions. In order to obtain one of these generalisations, we prove a quantum extension of the hypercontractive inequality of
A MULTIDIMENSIONAL LIEBSCHULTZMATTIS THEOREM
, 2007
"... Abstract. For a large class of finiterange quantum spin models with halfinteger spins, we prove that uniqueness of the ground state implies the existence of a lowlying excited state. For systems of linear size L, with arbitrary finite dimension, we obtain an upper bound on the excitation energy ( ..."
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Cited by 5 (1 self)
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Abstract. For a large class of finiterange quantum spin models with halfinteger spins, we prove that uniqueness of the ground state implies the existence of a lowlying excited state. For systems of linear size L, with arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form (C log L)/L. This result can be regarded as a multidimensional LiebSchultzMattis theorem [14] and provides a rigorous proof of the main result in [8]. 1. Introduction and