Results 1  10
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15
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 30 (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.
LiebRobinson Bounds and the Exponential Clustering Theorem
"... We give a LiebRobinson bound for the group velocity of a large class of discrete quantum systems and use it to prove that a nonvanishing spectral gap implies exponential clustering in the ground state of such systems. 1 ..."
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Cited by 20 (13 self)
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We give a LiebRobinson bound for the group velocity of a large class of discrete quantum systems and use it to prove that a nonvanishing spectral gap implies exponential clustering in the ground state of such systems. 1
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 17 (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
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 10 (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
Recent progress in quantum spin systems
 Markov Processes Rel. Fields, arXiv:mathph/0512020
"... This paper is dedicated to the memory of John T. Lewis. Some recent developments in the theory of quantum spin systems are reviewed. 1 ..."
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Cited by 4 (4 self)
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This paper is dedicated to the memory of John T. Lewis. Some recent developments in the theory of quantum spin systems are reviewed. 1
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 3 (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
LiebRobinson Bounds and Existence of the Thermodynamic Limit for a Class of Irreversible Quantum Dynamics
, 2011
"... Dedicated to Robert A. Minlos at the occasion of his 80th birthday Abstract. We prove LiebRobinson bounds and the existence of the thermodynamic limit for a general class of irreversible dynamics for quantum lattice systems with timedependent generators that satisfy a suitable decay condition in s ..."
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Cited by 2 (0 self)
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Dedicated to Robert A. Minlos at the occasion of his 80th birthday Abstract. We prove LiebRobinson bounds and the existence of the thermodynamic limit for a general class of irreversible dynamics for quantum lattice systems with timedependent generators that satisfy a suitable decay condition in space. 1.
Spectral gap and decay of correlations in U(1)symmetric lattice systems
 in dimensions D
"... We consider manybody systems with a global U(1) symmetry on a class of lattices with the (fractal) dimensions D < 2 and their zero temperature correlations whose observables behave as a vector under the U(1) rotation. For a wide class of the models, we prove that if there exists a spectral gap a ..."
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Cited by 2 (1 self)
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We consider manybody systems with a global U(1) symmetry on a class of lattices with the (fractal) dimensions D < 2 and their zero temperature correlations whose observables behave as a vector under the U(1) rotation. For a wide class of the models, we prove that if there exists a spectral gap above the ground state, then the correlation functions have a stretched exponentially decaying upper bound. This is an extension of the McBryanSpencer method at finite temperatures to zero temperature. The class includes quantum spin and electron models on the lattices, and our method also allows finite or infinite (quasi)degeneracy of the ground state. The resulting bounds rule out the possibility of the corresponding magnetic and electric longrange order. KEY WORDS: Spectral gap; U(1) symmetry; absence of continuous symmetry breaking, decay of correlations, low dimensional systems.
The 1D Area Law and the Complexity of Quantum States: A combinatorial approach
"... Abstract—The classical description of quantum states is in general exponential in the number of qubits. Can we get polynomial descriptions for more restricted sets of states such as ground states of interesting subclasses of local Hamiltonians? This is the basic problem in the study of the complexit ..."
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Cited by 1 (1 self)
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Abstract—The classical description of quantum states is in general exponential in the number of qubits. Can we get polynomial descriptions for more restricted sets of states such as ground states of interesting subclasses of local Hamiltonians? This is the basic problem in the study of the complexity of ground states, and requires an understanding of multiparticle entanglement and quantum correlations in such states. Area laws provide a fundamental ingredient in the study of the complexity of ground states, since they offer a way to bound in a quantitative way the entanglement in such states. Although they have long been conjectured for many body systems in arbitrary dimensions, a general rigorous was only recently proved in Hastings ’ seminal paper [1] for 1D systems. In this paper, we give a combinatorial proof of the 1D area law for the special case of frustration free systems, improving by an exponential factor the scaling in terms of the inverse spectral gap and the dimensionality of the particles. The scaling in terms of the dimension of the particles is a potentially important issue in the context of resolving the 2D case and higher dimensions, which is one of the most important open questions in Hamiltonian complexity. Our proof is based on a reformulation of the detectability lemma, introduced by us in the context of quantum gap amplification [2]. We give an alternative proof of the detectability lemma, which is not only simpler and more intuitive than the original proof, but also removes a key restriction in the original statement, making it more suitable for this new context. We also give a one page proof of Hastings ’ proof that the correlations in the ground states of gapped Hamiltonians decay exponentially with the distance, demonstrating the simplicity of the combinatorial approach for those problems. Keywordscomponent; formatting; style; styling;
Quantum Spin Systems after DLS1978
, 2005
"... In their 1978 paper, Dyson, Lieb, and Simon (DLS) proved the existence of Néel order at positive temperature for the spinS Heisenberg antiferromagnet on the ddimensional hypercubic lattice when either S ≥ 1 and d ≥ 3 or S = 1/2 and d is sufficiently large. This was the first proof of spontaneous b ..."
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In their 1978 paper, Dyson, Lieb, and Simon (DLS) proved the existence of Néel order at positive temperature for the spinS Heisenberg antiferromagnet on the ddimensional hypercubic lattice when either S ≥ 1 and d ≥ 3 or S = 1/2 and d is sufficiently large. This was the first proof of spontaneous breaking of a continuous symmetry in a quantum model at finite temperature. Since then the ideas of DLS have been extended and adapted to a variety of other problems. In this paper I will present an overview of the most important developments in the study of the Heisenberg model and related quantum lattice systems since 1978, including but not restricted to those directly related to the paper by DLS.