Fermionic Field Theory Research Articles

Derivation of the Time-Reversal Anomaly for (2+1)-Dimensional Topological Phases.

We prove an explicit formula conjectured recently by Wang and Levin for the anomaly of time-reversal symmetry in (2+1)-dimensional fermionic topological quantum field theories. The crucial step is to determine the cross-cap state in terms of the modular S matrix and T^{2} eigenvalues, generalizing the recent analysis by Barkeshli etal. in the bosonic case.

Point-particle effective field theory III: relativistic fermions and the Dirac equation

We formulate point-particle effective field theory (PPEFT) for relativistic spin-half fermions interacting with a massive, charged finite-sized source using a first-quantized effective field theory for the heavy compact object and a second-quantized language for the lighter fermion with which it interacts. This description shows how to determine the near-source boundary condition for the Dirac field in terms of the relevant physical properties of the source, and reduces to the standard choices in the limit of a point source. Using a first-quantized effective description is appropriate when the compact object is sufficiently heavy, and is simpler than (though equivalent to) the effective theory that treats the compact source in a second-quantized way. As an application we use the PPEFT to parameterize the leading energy shift for the bound energy levels due to finite-sized source effects in a model-independent way, allowing these effects to be fit in precision measurements. Besides capturing finite-source-size effects, the PPEFT treatment also efficiently captures how other short-distance source interactions can shift bound-state energy levels, such as due to vacuum polarization (through the Uehling potential) or strong interactions for Coulomb bound states of hadrons, or any hypothetical new short-range forces sourced by nuclei.

Universal bounds for large determinants from non-commutative Hölder inequalities in fermionic constructive quantum field theory

Efficiently bounding large determinants is an essential step in non-relativistic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength [Formula: see text] of the interparticle interaction. We provide, for large determinants of fermionic covariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one-particle Hamiltonians. We find the smallest universal determinant bound to be exactly [Formula: see text]. In particular, the convergence of perturbation series at [Formula: see text] of any fermionic quantum field theory is ensured if the matrix entries (with respect to some fixed orthonormal basis) of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use Hölder inequalities for general non-commutative [Formula: see text]-spaces derived by Araki and Masuda [Positive cones and [Formula: see text]-spaces for von Neumann algebras, Publ. RIMS[Formula: see text] Kyoto Univ. 18 (1982) 339–411].

Functional integral method in quantum field theory of Dirac fermions in graphene

The purpose of this work is to elaborate the functional integral method in quantum field theory of Dirac fermions in the Dirac fermion gas of a graphene single layer at vanishing absolute temperature. The starting point to be assumed as the fundamental principle of the theory is the explicit expression of the action functional of this system. The efficient mathematical tool to be used in the study is the generating functional containing the Grassmann parameters anticommuting with the Dirac fermion field operators.The analytical expression of the generating functional of free Dirac fermion system is exactly derived and efficiently used in the study of 2n-point Green functions of free Dirac fermions. Then the celebrated Hubbard–Stratonovich transformation is applied to rewrite the functional integral of the interacting system of Dirac fermions in a new form expressing in terms of a scalar Hermitian quantum field describing the collective excitations in the interacting Dirac fermion gas and related to the graphene plasmons.

Positions of the magnetoroton minima in the fractional quantum Hall effect

The multitude of excitations of the fractional quantum Hall state are very accurately understood, microscopically, as excitations of composite fermions across their Landau-like $\Lambda$ levels. In particular, the dispersion of the composite fermion exciton, which is the lowest energy spin conserving neutral excitation, displays filling-factor-specific minima called "magnetoroton" minima. Simon and Halperin employed the Chern-Simons field theory of composite fermions [Phys. Rev. B {\bf 48}, 17368 (1993)] to predict the magnetoroton minima positions. Recently, Golkar \emph{et al.} [Phys. Rev. Lett. {\bf 117}, 216403 (2016)] have modeled the neutral excitations as deformations of the composite fermion Fermi sea, which results in a prediction for the positions of the magnetoroton minima. Using methods of the microscopic composite fermion theory we calculate the positions of the roton minima for filling factors up to 5/11 along the sequence $s/(2s+1)$ and find them to be in reasonably good agreement with both the Chern-Simons field theory of composite fermions and Golkar \emph{et al.}'s theory. We also find that the positions of the roton minima are insensitive to the microscopic interaction in agreement with Golkar \emph{et al.}'s theory. As a byproduct of our calculations, we obtain the charge and neutral gaps for the fully spin polarized states along the sequence $s/(2s\pm 1)$ in the lowest Landau level and the $n=1$ Landau level of graphene.

Fermions as generalized Ising models

We establish a general map between Grassmann functionals for fermions and probability or weight distributions for Ising spins. The equivalence between the two formulations is based on identical transfer matrices and expectation values of products of observables. The map preserves locality properties and can be realized for arbitrary dimensions. We present a simple example where a quantum field theory for free massless Dirac fermions in two-dimensional Minkowski space is represented by an asymmetric Ising model on a euclidean square lattice.

Dynamics of local symmetry correlators for interacting many-particle systems.

Recently [P. A. Kalozoumis et al. Phys. Rev. Lett. 113, 050403 (2014)] the concept of local symmetries in one-dimensional stationary wave propagation has been shown to lead to a class of invariant two-point currents that allow to generalize the parity and Bloch theorem. In the present work, we establish the theoretical framework of local symmetries for higher-dimensional interacting many-body systems. Based on the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, we derive the equations of motion of local symmetry correlators which are off-diagonal elements of the reduced one-body density matrix at symmetry related positions. The natural orbital representation yields equations of motion for the convex sum of the local symmetry correlators of the natural orbitals as well as for the local symmetry correlators of the individual orbitals themselves. An alternative integral representation with a unique interpretation is provided. We discuss special cases, such as the bosonic and fermionic mean field theory, and show in particular that the invariance of two-point currents is recovered in the case of the non-interacting one-dimensional stationary wave propagation. Finally we derive the equations of motion for anomalous local symmetry correlators which indicate the breaking of a global into a local symmetry in the stationary non-interacting case.

Nonperturbative emergence of the Dirac fermion in a strongly correlated composite Fermi liquid

The classic composite fermion field theory (Ref. 1) builds up an excellent framework to uniformly study important physical objects and globally explain anomalous experimental phenomena in fractional quantum Hall physics while there are also inherent weaknesses. We present a nonperturbative emergent Dirac fermion theory from this strongly correlated composite fermion field theory, which overcomes these serious long-standing shortcomings. The particle-hole symmetry of Dirac equation resolves this particle-hole symmetry enigma in the composite fermion field theory. With the help of presented numerical data, we show that for main Jain's sequences of fractional quantum Hall effects, this emergent Dirac fermion theory is most likely nonperturbatively stable.

Symplectic fermions and a quasi-Hopf algebra structure on [formula omitted

We consider the (finite-dimensional) restricted quantum group U‾qsℓ(2) at q=i. We show that U‾isℓ(2) does not allow for a universal R-matrix, even though U⊗V≅V⊗U holds for all finite-dimensional representations U,V of U‾isℓ(2). We then give an explicit coassociator Φ and a universal R-matrix R such that U‾isℓ(2) becomes a quasi-triangular quasi-Hopf algebra.Our construction is motivated by the two-dimensional chiral conformal field theory of symplectic fermions with central charge c=−2. There, a braided monoidal category, SF, has been computed from the factorisation and monodromy properties of conformal blocks, and we prove that Rep(U‾isℓ(2),Φ,R) is braided monoidally equivalent to SF.

Quantum holonomy theory

We present quantum holonomy theory, which is a non‐perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a ‐algebra that involves holonomy‐diffeo‐morphisms on a 3‐dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi‐classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure over the space of Ashtekar connections, to define a scalar curvature operator, from which we obtain a candidate for a Hamilton operator. We show that the classical Hamilton constraint of general relativity emerges from this in a semi‐classical limit and we then compute the operator constraint algebra. Also, we find states in the kinematical Hilbert space on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in a semi‐classical limit and thus provides a connection to fermionic quantum field theory. Finally, an almost‐commutative algebra emerges from the holonomy‐diffeomorphism algebra in the same limit.

Hamiltonian truncation approach to quenches in the Ising field theory

In contrast to lattice systems where powerful numerical techniques such as matrix product state based methods are available to study the non-equilibrium dynamics, the non-equilibrium behaviour of continuum systems is much harder to simulate. We demonstrate here that Hamiltonian truncation methods can be efficiently applied to this problem, by studying the quantum quench dynamics of the 1+1 dimensional Ising field theory using a truncated free fermionic space approach. After benchmarking the method with integrable quenches corresponding to changing the mass in a free Majorana fermion field theory, we study the effect of an integrability breaking perturbation by the longitudinal magnetic field. In both the ferromagnetic and paramagnetic phases of the model we find persistent oscillations with frequencies set by the low-lying particle excitations not only for small, but even for moderate size quenches. In the ferromagnetic phase these particles are the various non-perturbative confined bound states of the domain wall excitations, while in the paramagnetic phase the single magnon excitation governs the dynamics, allowing us to capture the time evolution of the magnetisation using a combination of known results from perturbation theory and form factor based methods. We point out that the dominance of low lying excitations allows for the numerical or experimental determination of the mass spectra through the study of the quench dynamics.

Higher spin holography with Galilean symmetry in general dimensions

We construct Schrödinger-like solutions of the Vasiliev higher spin theory in dimension. Symmetries of such solutions and the linearized equation of motion for the scalar on such backgrounds are analyzed. We further propose Galilean invariant bosonic and fermionic field theories that could be dual to the two parity invariant higher spin theories on the Schrödinger-like background respectively. The discussion is phrased mainly in D = 4 dimension, while similar constructions follow straightforwardly in higher dimensions.

Charged entanglement entropy of local operators

In this work we consider the time evolution of charged Renyi entanglement entropies after exciting the vacuum with local fermionic operators. In order to explore the information contained in charged Renyi entropies, we perform computations of their excess due to the operator excitation in 2d CFT, free fermionic field theories in various dimensions as well as holographically. In the analysis we focus on the dependence on the entanglement charge, the chemical potential and the spacetime dimension. We find that excesses of charged (Renyi) entanglement entropy can be interpreted in terms of charged quasiparticles. Moreover, we show that by appropriately tuning the chemical potential charged Renyi entropies can be used to extract entanglement in a certain charge sector of the excited state.

Quantum quenches in free field theory: universal scaling at any rate

Quantum quenches display universal scaling in several regimes. For quenches which start from a gapped phase and cross a critical point, with a rate slow compared to the initial gap, many systems obey Kibble-Zurek scaling. More recently, a different scaling behaviour has been shown to occur when the quench rate is fast compared to all other physical scales, but still slow compared to the UV cutoff. We investigate the passage from fast to slow quenches in scalar and fermionic free field theories with time dependent masses for which the dynamics can be solved exactly for all quench rates. We find that renormalized one point functions smoothly cross over between the regimes.

Chiral magnetic effect in ZrTe5

A magnetotransport study of zirconium pentatelluride now reveals evidence for a chiral magnetic effect, a striking macroscopic manifestation of the quantum and relativistic nature of Weyl semimetals. The chiral magnetic effect is the generation of an electric current induced by chirality imbalance in the presence of a magnetic field. It is a macroscopic manifestation of the quantum anomaly1,2 in relativistic field theory of chiral fermions (massless spin 1/2 particles with a definite projection of spin on momentum)—a remarkable phenomenon arising from a collective motion of particles and antiparticles in the Dirac sea. The recent discovery3,4,5,6 of Dirac semimetals with chiral quasiparticles opens a fascinating possibility to study this phenomenon in condensed matter experiments. Here we report on the measurement of magnetotransport in zirconium pentatelluride, ZrTe5, that provides strong evidence for the chiral magnetic effect. Our angle-resolved photoemission spectroscopy experiments show that this material’s electronic structure is consistent with a three-dimensional Dirac semimetal. We observe a large negative magnetoresistance when the magnetic field is parallel with the current. The measured quadratic field dependence of the magnetoconductance is a clear indication of the chiral magnetic effect. The observed phenomenon stems from the effective transmutation of a Dirac semimetal into a Weyl semimetal induced by parallel electric and magnetic fields that represent a topologically non-trivial gauge field background. We expect that the chiral magnetic effect may emerge in a wide class of materials that are near the transition between the trivial and topological insulators.

Quantum entanglement of fermionic local operators

In this paper we study the time evolution of (Renyi) entanglement entropies for locally excited states in four dimensional free massless fermionic field theory. Locally excited states are defined by being acted by various local operators on the ground state. Their excesses are defined by subtracting (Renyi) entanglement entropy for the ground state from those for locally excited states. They finally approach some constant if the subsystem is given by half of the total space. They have spin dependence. They can be interpreted in terms of quasi-particles.

Fixed-point structure of low-dimensional relativistic fermion field theories: Universality classes and emergent symmetry

We investigate a class of relativistic fermion theories in 2<d<4 space-time dimensions with continuous chiral U(Nf)xU(Nf) symmetry. This includes a number of well-studied models, e.g., of Gross-Neveu and Thirring type, in a unified framework. Within the limit of pointlike interactions, the RG flow of couplings reveals a network of interacting fixed points, each of which defines a universality class. A subset of fixed points are "critical fixed points" with one RG relevant direction being candidates for critical points of second-order phase transitions. Identifying invariant hyperplanes of the RG flow and classifying their attractive/repulsive properties, we find evidence for emergent higher chiral symmetries as a function of Nf. For the case of the Thirring model, we discover a new critical flavor number that separates the RG stable large-Nf regime from an intermediate-Nf regime in which symmetry-breaking perturbations become RG relevant. This new critical flavor number has to be distinguished from the chiral-critical flavor number, below which the Thirring model is expected to allow spontaneous chiral symmetry breaking, and its existence offers a resolution to the discrepancy between previous results obtained in the continuum and the lattice Thirring models. Moreover, we find indications for a new feature of universality: details of the critical behavior can depend on additional "spectator symmetries" that remain intact across the phase transition. Implications for the physics of interacting fermions on the honeycomb lattice, for which our theory space provides a simple model, are given.

Boundary effects in bosonic and fermionic field theories

The dynamics of quantum field theories on bounded domains requires the introduction of boundary conditions on the quantum fields. We address the problem from a very general perspective by using charge conservation as a fundamental principle for scalar and fermionic quantum field theories. Unitarity arises as a consequence of the choice of charge preserving boundary conditions. This provides a powerful framework for the analysis of global geometrical and topological properties of the space of physical boundary conditions. Boundary conditions which allow the existence of edge states can only arise in theories with a mass gap which is also a physical requirement for topological insulators.

Universality in fast quantum quenches

We expand on the investigation of the universal scaling properties in the early time behaviour of fast but smooth quantum quenches in a general $d$-dimensional conformal field theory deformed by a relevant operator of dimension $\Delta$ with a time-dependent coupling. The quench consists of changing the coupling from an initial constant value $\lambda_1$ by an amount of the order of $\delta \lambda$ to some other final value $\lambda_2$, over a time scale $\delta t$. In the fast quench limit where $\delta t$ is smaller than all other length scales in the problem, $ \delta t \ll \lambda_1^{1/(\Delta-d)}, \lambda_2^{1/(\Delta-d)}, \delta \lambda^{1/(\Delta-d)}$, the energy (density) injected into the system scales as $\delta{\cal E} \sim (\delta \lambda)^2 (\delta t)^{d-2\Delta}$. Similarly, the change in the expectation value of the quenched operator at times earlier than the endpoint of the quench scales as $<{\cal O}_\Delta> \sim \delta \lambda (\delta t)^{d-2\Delta}$, with further logarithmic enhancements in certain cases. While these results were first found in holographic studies, we recently demonstrated that precisely the same scaling appears in fast mass quenches of free scalar and free fermionic field theories. As we describe in detail, the universal scaling refers to renormalized quantities, in which the UV divergent pieces are consistently renormalized away by subtracting counterterms derived with an adiabatic expansion. We argue that this scaling law is a property of the conformal field theory at the UV fixed point, valid for arbitrary relevant deformations and insensitive to the details of the quench protocol. Our results highlight the difference between smooth fast quenches and instantaneous quenches where the Hamiltonian abruptly changes at some time.

Non-Gaussian fixed points in fermionic field theories without auxiliary Bose fields

The functional equation governing the renormalization flow of fermionic field theories is investigated in \(d\) dimensions without introducing auxiliary Bose fields on the example of the Gross–Neveu and the Nambu–Jona-Lasinio model. The UV-safe fixed points and the eigenvectors of the renormalization group equations linearized around them are found in the local potential approximation. The results are compared carefully with those obtained with partial bosonization. The results do not receive any correction in the next-to-leading order approximation of the gradient expansion of the effective action.