Spin networks, quantum automata and link invariants

The spin network simulator model represents a bridge between (generalized) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFTs). The key tool is provided by the fiber space structure underlying the model which exhibits combinatorial properties closely related to SU (2) state sum models, widely employed in discretizing TQFTs and quantum gravity in low spacetime dimensions.

We show that the fundamental time-reversal invariant (TRI) insulator exists in $4+1$ dimensions, where the effective-field theory is described by the $(4+1)$-dimensional Chern-Simons theory and the topological properties of the electronic structure are classified by the second Chern number. These topological properties are the natural generalizations of the time reversal-breaking quantum Hall insulator in $2+1$ dimensions. The TRI quantum spin Hall insulator in $2+1$ dimensions and the topological insulator in $3+1$ dimensions can be obtained as descendants from the fundamental TRI insulator in $4+1$ dimensions through a dimensional reduction procedure. The effective topological field theory and the ${Z}_{2}$ topological classification for the TRI insulators in $2+1$ and $3+1$ dimensions are naturally obtained from this procedure. All physically measurable topological response functions of the TRI insulators are completely described by the effective topological field theory. Our effective topological field theory predicts a number of measurable phenomena, the most striking of which is the topological magnetoelectric effect, where an electric field generates a topological contribution to the magnetization in the same direction, with a universal constant of proportionality quantized in odd multiples of the fine-structure constant $\ensuremath{\alpha}={e}^{2}∕\ensuremath{\hbar}c$. Finally, we present a general classification of all topological insulators in various dimensions and describe them in terms of a unified topological Chern-Simons field theory in phase space.

It is a long-standing question to extend the definition of 3-dimensional Chern-Simons theory to one which associates values to 1-manifolds with boundary and to 0-manifolds. We provide a solution in case the gauge group is a torus. We also develop from different points of view an associated 4-dimensional invertible topological field theory which encodes the anomaly of Chern-Simons. Finite gauge groups are also revisited, and we describe a theory of path integrals as a general construction for a certain class of finite topological field theories. Topological pure gauge theories in lower dimension are presented as a warm-up.

We show that recently proposed generalized Chern-Simons action can be identified with the field theory action of a topological point particle. We find the crucial correspondence which makes it possible to derive the field theory actions from a special version of the generalized Chern-Simons actions. We provide arguments that the general coordinate invariance in the target space and the flat connection condition as a topological field theory can be accommodated in a very natural way. We propose series of new gauge invariant observables.

This graduate/research level text describes in a unified fashion the statistical mechanics of random walks, random surfaces and random higher dimensional manifolds with an emphasis on the geometrical aspects of the theory and applications to the quantisation of strings, gravity and topological field theory. With chapters on random walks, random surfaces, two- and higher dimensional quantum gravity, topological quantum field theories and Monte Carlo simulations of random geometries, the text provides a self-contained account of quantum geometry from a statistical field theory point of view. The approach uses discrete approximations and develops analytical and numerical tools. Continuum physics is recovered through scaling limits at phase transition points and the relation to conformal quantum field theories coupled to quantum gravity is described. The most important numerical work is covered, but the main aim is to develop mathematically precise results that have wide applications. Many diagrams and references are included.

Monopoles and topological field theory

In this book we shall review the systematic approaches and applications of a theory which investigates situations where Feynman path integrals of physical systems can be evaluated exactly leading to a complete understanding of the quantum physics. These mathematical formalisms are in large part motivated by the symmetries present in integrable systems and topological quantum field theories which make these latter examples exactly solvable problems. Besides providing conceptual understandings of the solvability features of these special classes of problems, this framework yields geometric approaches to evaluating the quantum spectrum of generic quantum mechanical and quantum field theoretical partition functions. The techniques that we shall present here in fact motivate an approach to studying generic physical problems by relating their properties to those of integrable and topological field theories. In doing so, we shall therefore also review some of the more modern quantum field theoretical and mathematical ideas which have been at the forefront of theoretical physics over the past two decades.KeywordsPartition FunctionPath IntegralEquivariant CohomologyCoadjoint OrbitSupersymmetric Quantum MechanicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Entanglement and decoherence are two intimately related phenomena that lie at the heart of quantum information. The EPR-Bell states are the most basic quantum systems, which form the complete entangled basis of the two-particle Hilbert space, and which are important in many quantum computation schemes. The bit flip, phase flip, and bit-phase flip noise channels represent all the possible errors in quantum computation. Thus, the quantum decoherence characteristic of maximally and partially Bell entangled states under different noisy channels is studied using the concurrence. Our work is useful for linear-optical quantum computation and quantum communication schemes whose building blocks are entangled state.

Topological quantum field theory (TQFT) is a powerful tool to describe homologies, which normally involve complexes and a variety of maps/morphisms, what makes a functional integration approach with a sum over a single kind of map seemingly problematic. In TQFT this problem is overcome by exploiting the rich set of zero modes of BRST operators, which appear sufficient to describe complexes. We explain what this approach looks like for the important class of Khovanov-Rozansky (KR) cohomologies, which categorify the observables (Wilson lines or knot polynomials) in 3D Chern-Simons theory. We develop a construction of odd differential operators, associated with all link diagrams, including tangles with open ends. These operators become nilpotent only for a diagram with no external legs, but even for open tangles one can develop a factorization formalism, which preserves Reidemeister/topological invariance—the symmetry of the problem. This technique seems much more “physical” than the conventional language of homological algebra and should have many applications to various problems beyond Chern-Simons theory. We also hope that this language will provide efficient algorithms, and finally allow one to computerize the calculation of KR cohomologies—for closed diagrams and for open tangles.

Topological quantum field theories provide some of the most interesting examples for the usefulness of path integrals. One of the best known of these examples was discovered in [32] where one particular topological quantum field theory, Chern-Simons theory, was studied and the so-called “Wilson loop observables” (WLOs) were computed explicitly. These WLOs are heuristic path integrals and the interesting thing about the expressions obtained in [32] is that they involve highly non-trivial link invariants like the Jones polynomial, the HOMFLY polynomial, and the Kauffman polynomial, cf. [26, 161. A more thorough study of the WLOs by other methods [18, 8, 5, 171 later led to a breakthrough in knot theory, the discovery of the universal Vassiliev invariant [27]. Unfortunately, it has not yet been possible to establish the aforementioned connection between path integrals and knot polynomials at a rigorous level. In the special case, however, where the base manifold M of the Chern-Simons model considered is of product form the situation looks much more promising and as we will show in the present paper it is reasonable to expect that, at least for some of these special manifolds M, it will eventually be possible

Entanglement entropy is an important quantity in field theory, but its definition poses some challenges. The naive definition involves an extension of quantum field theory in which one assigns Hilbert spaces to spatial sub-regions. For two-dimensional topological quantum field theory we show that the appropriate extension is the open-closed topological quantum field theory of Moore and Segal. With the addition of one additional axiom characterizing the “entanglement brane” we show how entanglement calculations can be cast in this framework. We use this formalism to calculate modular Hamiltonians, entanglement entropy and negativity in two-dimensional Yang-Mills theory and relate these to singularities in the modular ow. As a byproduct we find that the negativity distinguishes between the “log dim R” edge term and the “Shannon” edge term. We comment on the possible application to understanding the Bekenstein-Hawking entropy in two-dimensional gravity.

In a recent article Turaev and Viro [TV] have constructed nontrivial “quantum” invariants of closed compact 3-manifolds M 3 in the form of state sums (called partition functions in statistical physics and vacuum functionals in quantum field theory) associated with the quantized universal enveloping algebra Uq(sl(2, C)) when q is a complex root of unity of a certain degree 2r > 4. The state sum is first defined for a given triangulation X of M and then shown to be independent of the triangulation X of M thus giving rise to a well defined invariant of M, which thus depends on q. This result may be viewed as a rigorous mathematical construction of what is called a topological quantum field theory. In fact, in the language of physicists, a triangulation corresponds to the introduction of a high-energy cut-off. Now topological quantum field theories have trivial dynamics, in other words they only deal with the vacuum sector, are scale invariant and more generally independent of any metrics. Invariance under subdivision is just the statement that the associated renormalization group transformation is trivial. This result suggests that the familiar techniques from algebraic topology should become useful to construct and discuss other topological quantum field theories. Moreover, similar combinatorial techniques could become helpful in discussing particle structures of low dimensional quantum field theories, whose vacuum sector is given in terms of a topological quantum field theory.

Defects in [formula omitted] coset, [formula omitted] topological field theory and discrete Fourier–Mukai transform

Embedded graph invariants in Chern-Simons theory

The proposal of fault-tolerant quantum computations, which promise to dramatically improve the operation of quantum computers and to accelerate the development of the compact hardware for them, is based on topological quantum field theories, which rely on the existence in Nature of physical systems described by a Lagrangian containing a non-Abelian (NA) topological term. These are solid-state systems having two-dimensional electrons, which are coupled to magnetic-flux-quanta vortexes, forming complex particles, known as anyons. Topological quantum computing (TQC) operations thus represent a physical realization of the mathematical operations involving NA representations of a braid group Bn, generated by a set of n localized anyons, which can be braided and fused using a “tweezer” and controlled by a detector. For most of the potential TQC material systems known so far, which are 2D-electron–gas semiconductor structure at high magnetic field and a variety of hybrid superconductor/topological-material heterostructures, the realization of anyon localization versus tweezing and detecting meets serious obstacles, chief among which are the necessity of using current control, i.e., mobile particles, of the TQC operations and high density electron puddles (containing thousands of electrons) to generate a single vortex. Here we demonstrate a novel system, in which these obstacles can be overcome, and in which vortexes are generated by a single electron. This is a ~ 150 nm size many electron InP/GaInP2 self-organized quantum dot, in which molecules, consisting of a few localized anyons, are naturally formed and exist at zero external magnetic field. We used high-spatial-resolution scanning magneto-photoluminescence spectroscopy measurements of a set of the dots having five and six electrons, together with many-body quantum mechanical calculations to demonstrate spontaneous formation of the anyon magneto-electron particles (eν) having fractional charge ν = n/k, where n = 1–4 and k = 3–15 are the number of electrons and vortexes, respectively, arranged in molecular structures having a built-in (internal) magnetic field of 6–12 T. Using direct imaging of the molecular configurations we observed fusion and braiding of eν-anyons under photo-excitation and revealed the possibility of using charge sensing for their control. Our investigations show that InP/GaInP2 anyon-molecule QDs, which have intrinsic transformations of localized eν-anyons compatible with TQC operations and capable of being probed by charge sensing, are very promising for the realization of TQC.