Free Fermion Theory Research Articles

From quantum gravity to quantum field theory via noncommutative geometry

A link between canonical quantum gravity and fermionic quantum field theory is established in this paper. From a spectral triple construction, which encodes the kinematics of quantum gravity, we construct semi-classical states which, in a semi-classical limit, give a system of interacting fermions in an ambient gravitational field. The emergent interaction involves flux tubes of the gravitational field. In the additional limit, where all gravitational degrees of freedom are turned off, a free fermionic quantum field theory emerges.

Fermion mass mixing and vacuum triviality in the renormalization group procedure for effective particles

Renormalization group procedure for effective particles is applied to the model quantum theory of free fermions to which one adds an interaction in the form of a mass mixing term. If one used a standard approach based on the instant form of dynamics, the theory would suffer from a generic vacuum problem caused by a divergent production of virtual quanta out of a bare vacuum and it would require an adjustment of its degrees of freedom to the added interaction term before quantization, considered a means of avoiding the quantum vacuum problem. In the effective particle approach, the quantum vacuum problem is dealt with instead by using the front form of dynamics, where the pair production is excluded by momentum conservation. The corresponding Hamiltonian includes mass parameters through constraint equations while the required quantum field operators are constructed independently of all mass parameters, including the parameters that appear in the added mass mixing interaction term. Then the masses and states of physical fermions emerge at an end of the non-perturbative calculation that is carried out entirely in one and the same interacting quantum theory with a trivial vacuum and no quantization adjustment. An a priori infinite set of renormalization group equations for all momentum modes of fermion fields is reduced to just one equation for a two-by-two mass squared matrix, thanks to 7 kinematical symmetries of the front form (the instant form has only 6). For strong mass mixing interactions, the fermion model solutions qualitatively differ from the analogous, earlier found boson model solutions by the absence of tachyons.

Electron spin resonance shifts inS=1antiferromagnetic chains

We discuss electron spin resonance (ESR) shifts in spin-1 Heisenberg antiferromagnetic chains with a weak single-ion anisotropy based on several effective field theories, the O(3) nonlinear sigma model (NLSM) in the Haldane phase, free fermion theories around the lower and the upper critical fields. In the O(3) NLSM, the single-ion anisotropy corresponds to a composite operator which creates two magnons at the same time and position. Therefore, even inside a parameter range where free magnon approximation is valid, we have to take interactions among magnons into account. Though the O(3) NLSM is only valid in the Haldane phase, an appropriate translation of Faddeev-Zamolodchikov operators of the O(3) NLSM to fermion operators enables one to treat ESR shifts near the lower critical field in a similar manner to discussions in Haldane phase. We present that our theory gives quantitative agreements with recent ESR experimental results on an spin-1 chain compounds NDMAP.

Light states in Chern-Simons theory coupled to fundamental matter

Motivated by developments in vectorlike holography, we study SU(N) Chern-Simons theory coupled to matter fields in the fundamental representation on various spatial manifolds. On the spatial torus T^2, we find light states at small `t Hooft coupling \lambda=N/k, where k is the Chern-Simons level, taken to be large. In the free scalar theory the gaps are of order \sqrt {\lambda}/N and in the critical scalar theory and the free fermion theory they are of order \lambda/N. The entropy of these states grows like N Log(k). We briefly consider spatial surfaces of higher genus. Based on results from pure Chern-Simons theory, it appears that there are light states with entropy that grows even faster, like N^2 Log(k). This is consistent with the log of the partition function on the three sphere S^3, which also behaves like N^2 Log(k). These light states require bulk dynamics beyond standard Vasiliev higher spin gravity to explain them.

Two flavor massless Schwinger model on a torus at a finite chemical potential

We study the thermodynamics of the two flavor massless Schwinger model on a torus at a finite chemical potential. We show that the physics only depends on the isospin chemical potential, and there are marked deviations from a free fermion theory. We argue that spatial inhomogeneities can develop in the system at very low temperatures.

Intersecting Quantum Gravity with Noncommutative Geometry - a Review

We review applications of noncommutative geometry in canonical quantum gravity. First, we show that the framework of loop quantum gravity includes natural noncommutative structures which have, hitherto, not been explored. Next, we present the construction of a spectral triple over an algebra of holonomy loops. The spectral triple, which encodes the kinematics of quantum gravity, gives rise to a natural class of semiclassical states which entail emerging fermionic degrees of freedom. In the particular semiclassical approximation where all gravitational degrees of freedom are turned off, a free fermionic quantum field theory emerges. We end the paper with an extended outlook section.

Classical probabilities for Majorana and Weyl spinors

We construct a map between the quantum field theory of free Weyl or Majorana fermions and the probability distribution of a classical statistical ensemble for Ising spins or discrete bits. More precisely, a Grassmann functional integral based on a real Grassmann algebra specifies the time evolution of the real wave function q τ ( t ) for the Ising states τ. The time dependent probability distribution of a generalized Ising model obtains as p τ ( t ) = q τ 2 ( t ) . The functional integral employs a lattice regularization for single Weyl or Majorana spinors. We further introduce the complex structure characteristic for quantum mechanics. Probability distributions of the Ising model which correspond to one or many propagating fermions are discussed explicitly. Expectation values of observables can be computed equivalently in the classical statistical Ising model or in the quantum field theory for fermions.

Boundary states and edge currents for free fermions

We calculate the ground state current densities for 2+1 dimensional free fermion theories with local, translationally invariant boundary states. Deformations of the bulk wave functions close to the edge and boundary states both may cause edge current divergencies, which have to cancel in realistic systems. This yields restrictions on the parameters of quantum field theories which can arise as low energy limits of solid state systems. Some degree of Lorentz invariance for boosts parallel to the boundary can be recovered, when the cutoff is removed.

Quantum Brownian motion on a triangular lattice and Fermi-Bose equivalence: an application of boundary state formulation

We discuss the Bose-Fermi equivalence in the quantum Brownian motion (QBM) on a triangular lattice, mapping the action for the QBM into a string theory action with a periodic boundary tachyon potential. We construct new Klein factors which are more appropriate than the conventional ones to deal with the quantum field theories defined on a two dimensioanl space-time with boundaries. Using the Fermi-Bose equivalence with the new Klein factors, we show that the model for the quantum Bownian motion on a triangular lattice is equivalent to the Thirring model with boundary terms, which are quadratic in fermion field operators, in the off-critical regions and to a SU(3) × SU(3) free fermion theory with quadratic boundary terms at the critical point.

Supersymmetric gauge theories, intersecting branes and free fermions

We show that various holomorphic quantities in supersymmetric gauge theories can be conveniently computed by configurations of D4-branes and D6-branes. These D-branes intersect along a Riemann surface that is described by a holomorphic curve in a complex surface. The resulting I-brane carries two-dimensional chiral fermions on its world-volume. This system can be mapped directly to the topological string on a large class of non-compact Calabi-Yau manifolds. Inclusion of the string coupling constant corresponds to turning on a constant B-field on the complex surface, which makes this space non-commutative. Including all string loop corrections the free fermion theory is elegantly formulated in terms of holonomic D-modules that replace the classical holomorphic curve in the quantum case.

Two-dimensional thermofield bosonization II: Massive fermions

We consider the perturbative computation of the N-point function of chiral densities of massive free fermions at finite temperature within the thermofield dynamics approach. The infinite series in the mass parameter for the N-point functions are computed in the fermionic formulation and compared with the corresponding perturbative series in the interaction parameter in the bosonized thermofield formulation. Thereby we establish in thermofield dynamics the formal equivalence of the massive free fermion theory with the sine-Gordon thermofield model for a particular value of the sine-Gordon parameter. We extend the thermofield bosonization to include the massive Thirring model.

SLE( [formula omitted]) and boundary Coulomb gas

We consider the Coulomb gas model on the upper half-plane with different boundary conditions, namely, Dirichlet, Neumann and mixed. We relate this model to SLE( κ , ρ ) theories. We derive a set of conditions connecting the total charge of the Coulomb gas, the boundary charges, the parameters κ and ρ. Also we study a free fermion theory in presence of a boundary and show with the same methods that it would lead to logarithmic boundary changing operators.

Anyons in an exactly solved model and beyond

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 Z 2 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.

Exact treatment of D0 brane decay in two‐dimensional string theory

AbstractWe study the decay of the unstable D0‐branes in the $\hat{c}=1$ noncritical string theory with the help of the duality with free fermionic field theory.

BRST approach to Lagrangian construction for fermionic massless higher spin fields

We develop the BRST approach to Lagrangian formulation for all massless half-integer higher spin fields on an arbitrary dimensional flat space. General procedure of Lagrangian construction describing the dynamics of fermionic field with any spin is given. It is shown that in fermionic case the higher spin field model is a reducible gauge theory and the order of reducibility grows with the value of spin. No off-shell constraints on the fields and the gauge parameters are used. We prove that in four dimensions after partial gauge fixing the Lagrangian obtained can be transformed to Fang–Fronsdal form however, in general case, it includes the auxiliary fields and possesses the more gauge symmetries in compare with Fang–Fronsdal Lagrangian. As an example of general procedure, we derive the new Lagrangian for spin 5 / 2 field containing all set of auxiliary fields and gauge symmetries of free fermionic higher spin field theory.

STRING HAMILTONIAN FROM GENERALIZED YANG–MILLS GAUGE THEORY IN TWO DIMENSIONS

Two-dimensional SU (N) Yang–Mills theory is known to be equivalent to a string theory, as found by Gross in the large N limit, using the 1/N expansion. Later it was found that even a generalized YM theory leads to a string theory of the Gross type. In the standard YM theory case, Douglas and others found the string Hamiltonian describing the propagation and the interactions of states made of strings winding on a cylindrical space–time. We address the problem of finding a similar Hamiltonian for the generalized YM theory. As in the standard case we start by writing the theory as a theory of free fermions. Performing a bosonization, we express the Hamiltonian in terms of the modes of a bosonic field, that are interpreted as in the standard case as creation and destruction operators for states of strings winding around the cylindrical space–time. The result is similar to the standard Hamiltonian, but with new kinds of interaction vertices.

Conformal four point functions and the operator product expansion

Various aspects of the four point function for scalar fields in conformally invariant theories are analysed. This depends on an arbitrary function of two conformal invariants u,v. A recurrence relation for the function corresponding to the contribution of an arbitrary spin field in the operator product expansion to the four point function is derived. This is solved explicitly in two and four dimensions in terms of ordinary hypergeometric functions of variables z,x which are simply related to u,v. The operator product expansion analysis is applied to the explicit expressions for the four point function found for free scalar, fermion and vector field theories in four dimensions. The results for four point functions obtained by using the AdS/CFT correspondence are also analysed in terms of functions related to those appearing in the operator product discussion.

HIGHER-DERIVATIVE TWO-DIMENSIONAL MASSIVE FERMION THEORIES

We consider the canonical quantization of a generalized two-dimensional massive fermion theory containing higher odd-order derivatives. The requirements of Lorentz invariance, hermiticity of the Hamiltonian and absence of tachyon excitations suffice to fix the mass term, which contains a derivative coupling. We show that the basic quantum excitations of a higher-derivative theory of order 2N+1 consist of a physical usual massive fermion, quantized with positive metric, plus 2N unphysical massless fermions, quantized with opposite metrics. The positive-metric Hilbert subspace, which is isomorphic to the space of states of a massive free fermion theory, is selected by a subsidiary-like condition. Employing the standard bosonization scheme, the equivalent boson theory is derived. The results obtained are used as a guideline to discuss the solution of a theory including a current–current interaction.

Schwinger terms in the fully quantized 1 + 1 dimensional model – exact solution

We calculate the equal-time commutator of two fermionic currents within the framework of the 1+1 dimensional fully quantized theory, describing the interaction of fermions with a vector boson. It is shown that the interaction does not change the result obtained within the theory of free fermions.

Free strings and superstrings on the [formula omitted] spaces

We construct a free bosonic, fermionic and supersymmetric field theories on the M C n spaces for critical dimensions, i.e., n=2 p . This procedure allows us to exhibit the underlying geometry of these spaces. In particular, these spaces are shown to be in correspondence with the n real spheres S n−1 .