Phases of the Dirac determinant, Abelian Chern-Simons terms and Berry's phases in the field theoretic description of graphene
E. M. Santangelo
备注：
6 pages, 1 figure, to appear in Proceedings of the XIII Conference Selected Problems in Mathematical and Theoretical Physics, Dubna, Russia, June 23-27, 2008
This talk presents a study of massless relativistic Dirac fields in three Euclidean dimensions, at finite temperature and density, in the presence of a uniform electromagnetic background. Apart from explaining the behavior of Hall's conductivity for graphene, our results show a direct relationship between the selection of a phase for the Dirac determinant and the generation (or lack thereof) of Berry's phases and Chern-Simons terms.
In 1984 Michael Berry discovered that an isolated eigenstate of an adiabatically changing periodic Hamiltonian $H(t)$ acquires a phase, called the Berry phase. We show that under very general assumptions the adiabatic approximation of the phase of the zeta-regularized determinant of the imaginary-time Schrodinger operator with periodic Hamiltonian is equal to the Berry phase.
We present a quantized non-Abelian Berry phase for time reversal invariant systems such as quantum spin Hall effect. Ordinary Berry phase is defined by an integral of Berry's gauge potential along a loop (an integral of the Chern-Simons one-form), whereas we propose that a similar integral but over five dimensional parameter space (an integral of the Chern-Simons five-form) is suitable to define a non-Abelian Berry phase. We study its global topological aspects and show that it is indeed quantized into two values. We also discuss its close relationship with the nonperturbative anomalies.
The three dimensional abelian fermionic determinant of a two component massive spinor in flat euclidean space-time is resetted to a pure Chern-Simons action through a nonlinear redefinition of the gauge field.
We study anomalous transport phenomena induced by phase-space Berry curvature. For that purpose we construct a kinetic theory in $2d$ phase space when all abelian Berry curvatures are nonzero. We derive anomalous currents by calculating the complete form of the Poisson brackets of phase space coordinates. Then we construct the low-energy effective theory to reproduce the anomalous currents obtained from the kinetic theory. Such an effective theory is given by the Chern-Simons theory in $1+2d$ dimensions. Some implications of the Chern-Simons theory are also discussed.
The partition function of the Chern-Simons theory on the three-sphere with the unitary group $U(N)$ provides a one-matrix model. The corresponding $N$-particle system can be mapped to the determinantal point process whose correlation kernel is expressed by using the Stieltjes-Wigert orthogonal polynomials. The matrix model and the point process are regarded as $q$-extensions of the random matrix model in the Gaussian unitary ensemble and its eigenvalue point process, respectively. We prove the convergence of the $N$-particle system to an infinite-dimensional determinantal point process in $N \…
A refined expression for the Faddeev-Popov determinant is derived for gauge theories quantised around a reducible classical solution. We apply this result to Chern-Simons perturbation theory on compact spacetime 3-manifolds with quantisation around an arbitrary flat gauge field isolated up to gauge transformations, pointing out that previous results on the finiteness and formal metric-independence of perturbative expansions of the partition function continue to hold.
Berry connection is conventionally defined as a static gauge field in the Brillouin zone. Here we show that for three-dimensional (3d) time-reversal invariant superconductors, a generalized Berry gauge field behaves as a fluctuating field of a Chern-Simons gauge theory. The gapless nodal lines in the momentum space play the role of Wilson loop observables, while their linking and knot invariants modify the gravitational theta angle. This angle induces a topological gravitomagnetoelectric effect where a temperature gradient induces a rotational energy flow. We also show how topological strings …
We study the adiabatic pumping of the Chern-Simons axion (CSA) coupling along a parametric loop characterized by a non-zero second Chern number $C^{(2)}$ from the viewpoint of the hybrid Wannier representation, in which the Wannier charge centers (WCCs) are visualized as sheets defined over a projected 2D Brillouin zone. We derive a new formula for the CSA coupling, expressing it as an integral involving Berry curvatures and potentials defined on the WCC sheets. We show that a loop characterized by a non-zero $C^{(2)}$ requires a series of sheet-touching events at which $2π$ quanta of Berry cu…
We construct the kinetic theory in ($1+2d$)-dimensional phase space and time when all abelian and nonabelian phase-space Berry curvatures are nonzero. Then we calculate anomalous transports induced by the Berry curvatures on the basis of the kinetic theory. As an example, we study anomalous charge and spin transports induced by the SU($2$) Berry curvatures. We also derive the topological effective theory to reproduce the transports in insulators calculated from the kinetic theory. Such an effective theory is given by the nonabelian phase space Chern-Simons theory.
We study 1-loop effects for massless Dirac fields in two spatial dimensions, coupled to homogeneous electromagnetic backgrounds, both at zero and at finite temperature and density. In the case of a purely magnetic field, we analyze the relationship between the invariance of the theory under large gauge transformations, the appearance of Chern-Simons terms and of different Berry's phases. In the case of a purely electric background field, we show that the effective Lagrangian is independent of the chemical potential and of the temperature. More interesting: we show that the minimal conductivity…
Quantum mechanics on sphere $S^{n}$ is studied from the viewpoint that the Berry's connection has to appear as a topological term in the effective action. Furthermore we show that this term is the Chern-Simons term of gauge variables that correspond to the extra degrees of freedom of the enlarged space.
We outline a novel chiral kinetic theory framework for systematic computations of the Chiral Magnetic Effect (CME) in ultrarelativistic heavy-ion collisions. The real part of the fermion determinant in the QCD effective action is expressed as a supersymmetric world-line action of spinning, colored, Grassmanian point particles in background gauge fields, with equations of motion that are covariant generalizations of the Bargmann-Michel-Telegdi and Wong equations. Berry's phase is obtained in a consistent non-relativistic adiabatic limit. The chiral anomaly, in contrast, arises from the phase of…
We study the moduli space of a super Chern-Simons theory on a manifold with the topology ${\bf R}\times §$, where $§$ is a compact surface. The moduli space is that of flat super connections modulo gauge transformations on $§$, and we study in detail the case when $§$ is atorus and the supergroup is $OSp(m|2n)$. The bosonic moduli space is determined by the flat connections for the maximal bosonic subgroup $O(m)\times Sp(2n)$, while the fermionic moduli appear only for special parts of the bosonic moduli space, which are determined by a vanishing determinant of a matrix associated to the boson…
Topological insulators are substances which are bulk insulators but which carry current via special "topologically protected" edge states. The understanding of long range topological order in these systems is built around the idea of a Berry connection, which is a gauge connection obtained from the phase of the electron wave function transported over momentum space rather than coordinate space. The phase of a closed Wilson loop of the Berry connection around the Brillouin zone defines a topological order parameter which labels discrete flux vacua. The conducting states are surface modes on the…
Topological insulators in odd dimensions are characterized by topological numbers. We prove the well-known relation between the topological number given by the Chern character of the Berry curvature and the Chern-Simons level of the low energy effective action for a general class of Hamiltonians bilinear in the fermion with general U(1) gauge interactions including non-minimal couplings by an explicit calculation. A series of Ward-Takahashi identities are crucial to relate the Chern-Simons level to a winding number, which could then be directly reduced to Chern character of Berry curvature by …
We present new proposals for the representation of a Chern-Simons term on the lattice. In the first part, such a term is constructed from the fermion determinant, and in the second part directly from the Abelian gauge term. In both cases, the parity transformation is modified on the lattice without affecting its continuum limit.
We study ${\mathcal{N}}=2$ supersymmetric $U(N)$ Chern-Simons with $N_{f}$ fundamental and $N_{f}$ antifundamental chiral multiplets of mass $m$ in the complete parameter space spanned by $(g,\,m,\,N,\,N_{f})$, where $g$ denotes the coupling constant. In particular, we analyze the matrix model description of its partition function, both at finite $N$ using the method of orthogonal polynomials together with Mordell integrals and, at large $N$ with fixed $g$, using the theory of Toeplitz determinants. We show for the massless case that there is an explicit realization of the Giveon-Kutasov duali…
We study matrix integration over the classical Lie groups $U(N),Sp(2N),O(2N)$ and $O(2N+1)$, using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz$\pm$Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large $N$ the partition functions, Wilson loops and Hopf links of Chern-Simons theory on $S^{3}$ with the aforementioned symmetry groups. The identities found for the general models translate in this…
We consider the finite-volume spectra of QCD in the chiral limit of massless up and down quarks and massive strange quarks in the baryon number sectors $B = 0$ and $B = 1$ for different values of the isospin. Spontaneous symmetry breaking gives rise to rotor spectra, as the chiral order parameter precesses through the vacuum manifold. Baryons of different isospin influence the motion of the order parameter through non-trivial Berry phases and associated abstract monopole fields. Our investigation provides detailed insights into the dynamics of spontaneous chiral symmetry breaking in QCD on a t…
Hitchin has shown that the moduli space ${\mathcal M}$ of the dimensionally reduced self-dual Yang-Mills equations has a hyperKähler structure. In this paper we first explicitly show the hyperKähler structure, the details of which is missing in Hitchin's paper. We show here that ${\mathcal M}$ admits three pre-quantum line bundles, corresponding to the three symplectic forms. We use Quillen's determinant line bundle construction and show that the Quillen curvatures of these prequantum line bundles are proportional to each of the symplectic forms mentioned above. The prequantum line bundles are…
Topological insulators in odd dimensions are characterized by topological numbers. We prove the well-known relation between the topological number given by the Chern character of the Berry curvature and the Chern-Simons level of the low energy effective action for a general class of Hamiltonians bilinear in the fermion with general U(1) gauge interactions including non-minimal couplings by an explicit calculation. A series of Ward-Takahashi identities are crucial to relate the Chern-Simons level to a winding number, which could then be directly reduced to Chern character of Berry curvature by …
In this paper we evaluate the induced Lorentz and CPT violating Chern-Simons term in the QED action at finite temperature. We do this using the method of derivative expansion of fermion determinants. Also, we use the imaginary-time formalism to calculate the temperature dependence of the Chern-Simons term
We study $QED_3$ with magnetic-like defects using the Julia-Toulouse condensation mechanism (JTM). By a careful treatment of the symmetries we suggest a geometrical interpretation for distinct debatable issues in the MCS-monopole system: (i) the induction of the non-conserved electric current together with the Chern-Simons term (CS), (ii) the deconfinement transition and, (iii) the computation of the fermionic determinant in the presence of Dirac string singularities. The JTM leads to proper interpretation of the non-conserved current as originating from Dirac brane symmetry breaking. The mech…
We develop an equivalence between two Hilbert spaces: (i) the space of states of $U(1)^n$ Chern-Simons theory with a certain class of tridiagonal matrices of coupling constants (with corners) on $T^2$; and (ii) the space of ground states of strings on an associated mapping torus with $T^2$ fiber. The equivalence is deduced by studying the space of ground states of $SL(2,Z)$-twisted circle compactifications of $U(1)$ gauge theory, connected with a Janus configuration, and further compactified on $T^2$. The equality of dimensions of the two Hilbert spaces (i) and (ii) is equivalent to a known id…
A classical identity due to Giambelli in representation theory states that the character in any representation is expressed as a determinant whose components are characters in the hook representation constructed from all the combinations of the arm and leg lengths of the original representation. Previously it was shown that the identity persists in taking, for each character, the matrix integration in the super Chern-Simons matrix model in the grand canonical ensemble. We prove here that this Giambelli compatibility still holds in the deformation of the fractional-brane background.
In topological field theories determinants of maps with negative as well as positive eigenvalues arise. We give a generalisation of the zeta-regularisation technique to derive expressions for the phase and scaling-dependence of these determinants. For theories on odd-dimensional manifolds a simple formula for the scaling dependence is obtained in terms of the dimensions of certain cohomology spaces. This enables a non-perturbative feature of Chern-Simons gauge theory to be reproduced by path-integral methods.
We introduce a supersymmetric Chern-Simons theory whose low energy physics is that of the fractional quantum Hall effect. The supersymmetry allows us to solve the theory analytically. We quantise the vortices and, by relating their dynamics to a matrix model, show that their ground state wavefunction is in the same universality class as the Laughlin state. We further construct coherent state representations of the excitations of a finite number of vortices. These are quasi-holes. By an explicit computation of the Berry phase, without resorting to a plasma analogy, we show that these excitation…
A Fermi surface threaded by a Berry phase can be described by the Wess-Zumino-Witten (WZW) term. After gauging, it produces a five-dimensional Chern-Simons term in the action. We show how this Chern-Simons term captures the essence of the Abelian, non-Abelian, and mixed gravitational anomalies in describing both in- and off-equilibrium phenomena. In particular we derive a novel contribution to the Chiral Vortical Effect that arises when a temperature gradient is present. We also discuss the issue of universality of the anomalous currents.
There is considerable evidence, based on large $N_c$ chiral dynamics, holographic QCD, and Monte Carlo studies, that the QCD vacuum is permeated by discrete quasivacua separated by domain walls across which the local value of the topological $θ$ parameter jumps by $\pm2π$. In the 2-dimensional $CP^{N-1}$ sigma model, a pointlike charge is a domain wall, and $θ$ describes the background electric flux and the polarization of charged pairs in the vacuum. We show that the screening process, and the role of $θ$ as an order parameter describing electric polarization, are naturally formulated in term…
In this paper we investigate properties of Chern-Simons theory coupled to massive fermions in the large N limit. We demonstrate that at low temperatures the system is in a Fermi liquid state whose features can be systematically compared to the standard phenomenological theory of Landau Fermi liquids. This includes matching microscopically derived Landau parameters with thermodynamic predictions of Landau Fermi liquid theory. We also calculate the exact conductivity and viscosity tensors at zero temperature and finite chemical potential. In particular we point out that the Hall conductivity of …
Employing the Foldy-Wouthuysen transformation it is demonstrated straightforwardly that the first and second Chern numbers are equal to the coefficients of the 2+1 and 4+1 dimensional Chern-Simons actions which are generated by the massive Dirac fermions coupled to the Abelian gauge fields. A topological insulator model in 2+1 dimensions is discussed and by means of a dimensional reduction approach the 1+1 dimensional descendant of the 2+1 dimensional Chern-Simons theory is presented. Field strength of the Berry gauge field corresponding to the 4+1 dimensional Dirac theory is explicitly derive…
We study novel type of contributions to the partition function of the Maxwell system defined on a small compact manifold such as torus. These new terms can not be described in terms of the physical propagating photons with two transverse polarizations. Rather, these novel contributions emerge as a result of tunnelling events when transitions occur between topologically different but physically identical vacuum winding states. These new terms give an extra contribution to the Casimir pressure. The infrared physics in the system can be described in terms of the topological auxiliary non-propagat…
The abelian Chern-Simons theory is considered on a cylindrical spacetime $\mathbb{R} \times D$, in a not necessarily flat Lorentzian background. As in the flat bulk case with planar boundary, we find that also on the radial boundary of a curved background a Kaç-Moody algebra exists, with the same central charge as in the flat case, which henceforth depends neither on the bulk metric nor on the geometry of the boundary. The holographically induced theory on the 2D boundary is topologically protected, in the sense that it describes a Luttinger liquid, no matter which the bulk metric is. The main…
In this article are computed magnetic solutions of Einstein Maxwell Chern-Simons theory coupled to a dilaton-like scalar field. These solutions are computed by applying a space-time duality suggested by the author to known electric solutions of the same theory. As a redundancy check for the space-time duality it is explicitly shown that the magnetic configurations obtained are, as expected, solutions of the equations of motion. The magnetic configurations have metric determinant $\sqrt{-g}\sim r^p$ for the range of the parameter $p\in]-\infty,+\infty[/\{-1\}$ and are interpreted either as magn…
Applying the machinery of random matrix theory and Toeplitz determinants we study the level $k$, $U(N)$ Chern-Simons theory coupled with fundamental matter on $S^2\times S^1$ at finite temperature $T$. This theory admits a discrete matrix integral representation, i.e. a unitary discrete matrix model of two-dimensional Yang-Mills theory. In this study, the effective partition function and phase structure of the Chern-Simons matter theory, in a special case with an effective potential namely the Gross-Witten-Wadia potential, are investigated. We obtain an exact expression for the partition funct…
2+1 dimensional topological insulator described by the Kane-Mele model in the presence of Rashba spin-orbit interaction is considered. The effective action of the external fields coupled to electromagnetic and spin degrees of freedom is accomplished within this model. The Hamiltonian methods are adopted to provide the coefficients appearing in the action. It is demonstrated straightforwardly that the coefficients of the Chern-Simons terms are given by the first Chern number attained through the related non-Abelian Berry gauge field. The effective theory which we obtain is in accord with the ex…
The purpose of these lectures is to describe the basic theoretical structures underlying the rich and beautiful physics of the quantum Hall effect. The focus is on the interplay between microscopic wavefunctions, long-distance effective Chern-Simons theories, and the modes which live on the boundary. The notes are aimed at graduate students in any discipline where $\hbar=1$. A working knowledge of quantum field theory is assumed. Contents: 1. The Basics (Landau levels and Berry phase). 2. The Integer Quantum Hall Effect. 3. The Fractional Quantum Hall Effect. 4. Non-Abelian Quantum Hall States…
We give a derivation of the Verlinde formula for the $G_{k}$ WZW model from Chern-Simons theory, without taking recourse to CFT, by calculating explicitly the partition function $Z_{Σ\times S^{1}}$ of $Σ\times S^{1}$ with an arbitrary number of labelled punctures. By a suitable gauge choice, $Z_{Σ\times S^{1}}$ is reduced to the partition function of an Abelian topological field theory on $Σ$ (a deformation of non-Abelian BF and Yang-Mills theory) whose evaluation is straightforward. This relates the Verlinde formula to the Ray-Singer torsion of $Σ\times S^{1}$. We derive the $G_{k}/G_{k}$ mod…
Topological field theories emerge at low energy in strongly-correlated condensed matter systems and appear in the context of planar gravity. In particular, the study of Chern-Simons terms gives rise to the concept of flux attachment when the gauge field is coupled to matter, yielding flux-charge composites. Here we investigate the generation of flux attachment in a Bose-Einstein condensate in the presence of non-linear synthetic gauge potentials. In doing so, we identify the U(1) Chern-Simons gauge field as a singular density-dependent gauge potential, which in turn can be expressed as a Berry…
We study the partition function of the three-dimensional N=6 U(N)_k x U(N+M)_{-k} superconformal Chern-Simons matter theory known as the ABJ theory. We prove that the ABJ partition function on S^3 is exactly the same as a formula recently proposed by Awata, Hirano and Shigemori. While this formula was previously obtained by an analytic continuation from the L(2,1) lens space matrix model, we directly derive this by using a generalization of the Cauchy determinant identity. We also give an interpretation for the formula from brane picture.
We derive the full Wess-Zumino-Witten term of a gauged chiral lagrangian in D=4 by starting from a pure Yang-Mills theory of gauged quark flavor in a flat, compactified D=5. The theory is compactified such that there exists a B_5 zero mode, and supplemented with quarks that are ``chirally delocalized'' with q_L (q_R) on the left (right) boundary (brane). The theory then necessarily contains a Chern-Simons term (anomaly flux) to cancel the fermionic anomalies on the boundaries. The constituent quark mass represents chiral symmetry breaking and is a bilocal operator in D=5 of the form: \bar{q}_L…
We study globally supersymmetric 3d gauge theories on curved manifolds by describing the coupling of 3d topological gauge theories, with both Yang-Mills and Chern-Simons terms in the action, to background topological gravity. In our approach the Seifert condition for manifolds supporting global supersymmetry is elegantly deduced from the topological gravity BRST transformations. A cohomological characterization of the geometrical moduli which affect the partition function is obtained. In the Seifert context Chern-Simons topological (framing) anomaly is BRST trivial. We compute explicitly the c…
We derive an effective action for Dirac fermions coupled to O(3) non-linear sigma-model (NLSM) through the Yukawa-type interaction. The nonperturbative (global) quantum anomaly of this model results in a Hopf term for the effective NLSM. We obtain this term using the ``embedding'' of the CP$^1$ model into the CP$^M$ generalization of the model which makes the quantum anomaly perturbative. This perturbative anomaly is calculated by means of a gradient expansion of a fermionic determinant and is given by the Chern-Simons term for an auxiliary gauge field.
We discuss non-Abelian bosonization of two and three dimensional fermions using a path-integral framework in which the bosonic action follows from the evaluation of the fermion determinant for the Dirac operator in the presence of a vector field. This naturally leads to the Wess-Zumino-Witten action for massless two-dimensional fermions and to a Chern-Simons action for very massive three dimensional fermions. One advantage of our approach is that it allows to derive the exact bosonization recipe for fermion currents in a systematic way.
We derive from a microscopic model the effective theory of nematic order in a system with a spontaneous quantum anomalous Hall effect in two dimensions. Starting with a model of two-component fermions (a spinor field) with a quadratic band crossing and short range four-fermion marginally relevant interactions we use a 1/N expansion and bosonization methods to derive the effective field theory for the hydrodynamic modes associated with the conserved currents and with the local fluctuations of the nematic order parameter. We focus on the vicinity of the quantum phase transition from the isotropi…
