Carlo A. Trugenberger

Chern-Simons Theory in the Schrodinger Representation

Abstract We quantize the (2 + 1)-dimensional Chern-Simons theory in the functional Schrodinger representation. The realization of gauge transformations on states involves a 1-cocycle. We determine this cocycle; we show how solving the Gauss law constraint in the non-Abelian theory requires quantizing the parameter that normalizes the action; we trivialize the 1-cocycle with a spatially non-local cochain related to a 2-dimensional fermion determinant and we find the physical states that satisfy the Gauss law constraint. The quantum holonomy of physical states involves a contribution that is missed when the constraint is solved before quantization. We compute this quantity for the Abelian theory in Minkowski space, where it exhibits an interesting group theoretic structure. (In a note added in proof the corresponding non-Abelian computation is presented.) Also we consider coupling to external sources and offer yet another derivation of the anomalous statistics and spin of the charge and flux carrying particles —a calculation which is especially simple in the functional Schrodinger representation.

Infinite Symmetry in the Quantum Hall Effect

Abstract Free planar electrons in a uniform magnetic field are shown to possess the dynamical symmetry of area-preserving diffeomorphisms (W-infinity algebra). Intuitively, this is a consequence of gauge invariance, which forces dynamics to depend only on the flux. The infinity of generators of this symmetry act within each Landau level, which is infinite dimensional in the thermodynamic limit. The incompressible ground states corresponding to completely filled Landau levels (integer quantum Hall effect) possess a dynamical symmetry, since they are left invariant by an infinite subset of generators. This geometrical characterization of incompressibility also holds for fractional fillings of the lowest level (simplest fractional Hall effect) in the presence of Haldanes effective two-body interactions. Although these modify the symmetry algebra, the corresponding incompressible ground states proposed by Laughlin are again symmetric with respect to the modified infinite algebra.

Probabilistic Quantum Memories

Typical address-oriented computer memories cannot recognize incomplete or noisy information. Associative (content-addressable) memories solve this problem but suffer from severe capacity shortages. I propose a model of a quantum memory that solves both problems. The storage capacity is exponential in the number of qbits and thus optimal. The retrieval mechanism for incomplete or noisy inputs is probabilistic, with postselection of the measurement result. The output is determined by a probability distribution on the memory which is peaked around the stored patterns closest in Hamming distance to the input.

Quantum Pattern Recognition

AbstractI review and expand the model of quantum associative memory that I have recently proposed. In this model binary patterns of n bits are stored in the quantum superposition of the appropriate subset of the computational basis of n qbits. Information can be retrieved by performing an input-dependent rotation of the memory quantum state within this subset and measuring the resulting state. The amplitudes of this rotated memory state are peaked on those stored patterns which are closest in Hamming distance to the input, resulting in a high probability of measuring a memory pattern very similar to it. The accuracy of pattern recall can be tuned by adjusting a parameter playing the role of an effective temperature. This model solves the well-known capacity shortage problem of classical associative memories, providing a large improvement in capacity. PACS: 03.67.-a

Conformal symmetry and universal properties of quantum Hall states

Abstract The low-lying excitations of a quantum Hall state on a disk geometry are edge excitations. Their dynamics is governed by a conformal field theory on the cylinder defined by the disk boundary and the time variable. We give a simple and detailed derivation of this conformal field theory for integer filling, starting from the microscopic dynamics of (2 + 1)-dimensional non-relativistic electrons in Landau levels. This construction can be generalized to describe Laughlins fractional Hall states via chiral bosonization, thereby making contact with the effective Chern-Simons theory approach. The conformal field theory dictates the finite-size effects in the energy spectrum. An experimental or numerical verification of these universal effects would provide a further confirmation of Laughlins theory of incompressible quantum fluids.The low-lying excitations of a quantum Hall state on a disk geometry are edge excitations. Their dynamics is governed by a conformal field theory on the cylinder defined by the disk boundary and the time variable. We give a simple and detailed derivation of this conformal field theory for integer filling, starting from the microscopic dynamics of

Stable hierarchical quantum Hall fluids as W1+∞ minimal models

(2+1)

Phase transitions in quantum pattern recognition.

-dimensional non-relativistic electrons in Landau levels. This construction can be generalized to describe Laughlins fractional Hall states via chiral bosonization, thereby making contact with the effective Chern-Simons theory approach. The conformal field theory dictates the finite-size effects in the energy spectrum. An experimental or numerical verification of these universal effects would provide a further confirmation of Laughlins theory of incompressible quantum fluids.

Large N Limit in the Quantum Hall Effect

Abstract In this paper, we pursue our analysis of the W1+∞ symmetry of the low-energy edge excitations of incompressible quantum Hall fluids. These excitations are described by (1 + 1)-dimensional effective field theories, which are built by representations of the W1+∞ algebra. Generic W1+∞ theories predict many more fluids than the few, stable ones found in experiments. Here we identify a particular class of W1+∞ theories, the minimal models, which are made of degenerate representations only - a familiar construction in conformal field theory. The W1+∞ minimal models exist for specific values of the fractional Hall conductivity, which nicely fit the experimental data and match the results of the Jain hierarchy of quantum Hall fluids. We thus obtain a new hierarchical construction, which is based uniquely on the concept of quantum incompressible fluid and is independent of Jains approach and hypotheses. Furthermore, a surprising non-abelian structure is found in the W1+∞ minimal models: they possess neutral quark-like excitations with SU(m) quantum numbers and non-abelian fractional statistics. The physical electron is made of anyon and quark excitations. We discuss some properties of these neutral excitations which could be seen in experiments and numerical simulations.

Exact multi-anyon wave functions in a magnetic field☆

With the help of quantum mechanics one can formulate a model of associative memory with optimal storage capacity. This model is generalized by introducing a parameter playing the role of an effective temperature. The corresponding thermodynamics provides criteria to tune the efficiency of quantum pattern recognition. It is shown that the associative memory undergoes a phase transition from a disordered, high-temperature phase with no correlation between input and output to an ordered, low-temperature phase with minimal input-output Hamming distance.

Topological excitations in compact Maxwell-Chern-Simons theory.

Abstract The Laughlin states for N interacting electrons at the plateaus of the fractional Hall effect are studied in the thermodynamic limit of large N . It is shown that this limit leads to the semiclassical regime for these states, thereby relating their stability to their semiclassical nature. The equivalent problem of two-dimensional plasmas is solved analytically, to leading order for N →∞, by the saddle-point approximation — a two-dimensional extension of the method used in random matrix models of quantum gravity and gauge theories. To leading order, the Laughlin states describe classical droplets of fluids with uniform density and sharp boundaries, as expected from the Laughlin “plasma analogy”. In this limit, the dynamical W ∞-symmetry of the quantum Hall states expresses the kinematics of the area-preserving deformations of incompressible liquid droplets.