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Quantum Field Theory of Many-Body Systems
, 2004
"... condensation Extended objects, such as strings and membranes, have been studied for many years in the context of sta-tistical physics. In these systems, quantum effects are typically negligible, and the extended objects can be treated classically. Yet it is natural to wonder how strings and membrane ..."
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Cited by 155 (4 self)
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condensation Extended objects, such as strings and membranes, have been studied for many years in the context of sta-tistical physics. In these systems, quantum effects are typically negligible, and the extended objects can be treated classically. Yet it is natural to wonder how strings and membranes behave in the quantum regime. In this chapter, we will investigate the properties of one dimensional, string-like, objects with large quan-tum fluctuations. Our motivation is both intellectual curiosity and (as we will see) the connection between quantum strings and topological/quantum orders in condensed matter systems. It is useful to organize our discussion using the analogy to the well understood theory of quantum par-ticles. One of the most remarkable phenomena in quantum many-particle systems is particle condensation. We can think of particle condensed states as special ground states where all the particles are described by the same quantum wave function. In some sense, all the symmetry breaking phases examples of particle con-densation: we can view the order parameter that characterizes a symmetry breaking phase as the condensed wave function of certain “effective particles. ” According to this point of view, Landau’s theory [Landau (1937)] for symmetry breaking phases is really a theory of “particle ” condensation. The theory of particle condensation is based on the physical concepts of long range order, symmetry
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 85 (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 non-Abelian 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 non-Abelian 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.
Adiabatic quantum computation is equivalent to standard quantum computation
- SIAM Journal on Computing
"... Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the convention ..."
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Cited by 80 (12 self)
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Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models provides a new vantage point from which to tackle the central issues in quantum computation, namely designing new quantum algorithms and constructing fault tolerant quantum computers. In particular, by translating the main open questions in the area of quantum algorithms to the language of spectral gaps of sparse matrices, the result makes these questions accessible to a wider scientific audience, acquainted with mathematical physics, expander theory and rapidly mixing Markov chains. 1
Non-abelian anyons and topological quantum computation
- Reviews of Modern Physics
"... Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are partic ..."
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Cited by 56 (0 self)
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Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as Non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles,
Sparse-Graph Codes for Quantum Error-Correction
- IEEE TRANSACTIONS ON INFORMATION THEORY
, 2004
"... We present sparse-graph codes appropriate for use in quantum error-correction. Quantum error-correcting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparsegraph codes keep the number ..."
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Cited by 49 (0 self)
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We present sparse-graph codes appropriate for use in quantum error-correction. Quantum error-correcting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparsegraph codes keep the number of quantum interactions associated with the quantum error-correction process small: a constant number per quantum bit, independent of the blocklength. Third, sparse-graph codes often offer great flexibility with respect to blocklength and rate. We believe some of the codes we present are unsurpassed by previously published quantum error-correcting codes.
On exotic modular tensor categories
- Commun. Contemp. Math
"... Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular S-matrix S and −S has both topological and physical significance, so in our convention there are a tot ..."
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Cited by 37 (12 self)
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Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular S-matrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 non-trivial prime UMTCs are given in Section 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Section 6. 1.
Unpaired majorana fermions in quantum wires
, 2000
"... Certain one-dimensional Fermi systems have an energy gap in the bulk spectrum while boundary states are described by one Majorana operator per boundary point. A finite system of length L possesses two ground states with an energy difference proportional to exp(−L/l0) and different fermionic parities ..."
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Cited by 35 (3 self)
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Certain one-dimensional Fermi systems have an energy gap in the bulk spectrum while boundary states are described by one Majorana operator per boundary point. A finite system of length L possesses two ground states with an energy difference proportional to exp(−L/l0) and different fermionic parities. Such systems can be used as qubits since they are intrinsically immune to decoherence. The property of a system to have boundary Majorana fermions is expressed as a condition on the bulk electron spectrum. The condition is satisfied in the presence of an arbitrary small energy gap induced by proximity of a 3-dimensional p-wave superconductor, provided that the normal spectrum has an odd number of Fermi points in each half of the Brillouin zone (each spin component counts separately).
A magnetic model with a possible Chern-Simons phase
- Commun. Math. Phys
"... A rather elementary family of local Hamiltonians H◦,ℓ,ℓ = 1,2,3,..., is described for a 2−dimensional quantum mechanical system of spin = 1 2 particles. On the torus, the ground state space G◦,ℓ is essentially infinite dimensional but may collapse under “perturbation ” to an anyonic system with a co ..."
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Cited by 30 (3 self)
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A rather elementary family of local Hamiltonians H◦,ℓ,ℓ = 1,2,3,..., is described for a 2−dimensional quantum mechanical system of spin = 1 2 particles. On the torus, the ground state space G◦,ℓ is essentially infinite dimensional but may collapse under “perturbation ” to an anyonic system with a complete mathematical description: the quantum double of the SO(3)−Chern-Simons modular functor at q = e 2πi/ℓ+2 which we call DEℓ. The Hamiltonian H◦,ℓ defines a quantum loop gas. We argue that for ℓ = 1 and 2, G◦,ℓ is unstable and the collapse to Gǫ,ℓ ∼ = DEℓ can occur truly by perturbation. For ℓ ≥ 3 G◦,ℓ is stable and in this case finding Gǫ,ℓ ∼ = DEℓ must require either ǫ> ǫℓ> 0, help from finite system size, surface roughening (see section 3), or some other trick, hence the initial use of quotes “ ”. A hypothetical phase diagram is included in the introduction. The effect of perturbation is studied algebraically: the ground state G◦,ℓ of H◦,ℓ is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state Gǫ,ℓ described by a quotient algebra. By classification, this implies Gǫ,ℓ ∼ = DEℓ. The fundamental point is that nonlinear structures
