Exploiting the split property of quantum field theories (QFTs), a notion of von Neumann entropy associated to pairs of spatial subregions has been recently proposed both in the holographic context -- where it has been argued to be related to the entanglement wedge cross section -- and for general QFTs. We argue that the definition of this "reflected entropy" can be canonically generalized in a way which is particularly suitable for orbifold theories -- those obtained by restricting the full algebra of operators to those which are neutral under a global symmetry group. This turns out to be given by the full-theory reflected entropy minus an entropy associated to the expectation value of the "twist" operator implementing the symmetry operation. Then we show that the reflected entropy for Gaussian fermion systems can be simply written in terms of correlation functions and we evaluate it numerically for two intervals in the case of a two-dimensional Dirac field as a function of the conformal cross-ratio. Finally, we explain how the aforementioned twist operators can be constructed and we compute the corresponding expectation value and reflected entropy numerically in the case of the $\mathbb{Z}_2$ bosonic subalgebra of the Dirac field.
We consider the reflected entropy and the associated entanglement spectrum for free fermions reduced to two intervals in 1+1 dimensions. Working directly in the continuum theory the reflected entropy can be extracted from the spectrum of a singular integral equation whose kernel is determined by the known free fermion modular evolved correlation function. We find the spectrum numerically and analytically in certain limits. For intervals that almost touch the reflected entanglement spectrum approaches the spectrum of the thermal density matrix. This suggests that the reflected entanglement spec…
It is well-known that staggered fermions do not necessarily satisfy the same global symmetries as the continuum theory. We analyze the mechanism behind this phenomenon for arbitrary dimension and gauge group representation. For this purpose we vary the number of lattice sites between even and odd parity in each single direction. Since the global symmetries are manifest in the lowest eigenvalues of the Dirac operator, the spectral statistics and also the symmetry breaking pattern will be affected. We analyze these effects and compare our predictions with Monte-Carlo simulations of naive Dirac o…
I describe the interplay between Minkowski and Euclidean signature gamma matrices, Majorana fermions, and discrete and continuous symmetries in all spacetime dimensions.
This paper deals with the dispersionless KP hierarchy from the point of view of quasi-classical limit. Its Lax formalism, W-infinity symmetries and general solutions are shown to be reproduced from their counterparts in the KP hierarchy in the limit of $\hbar \to 0$. Free fermions and bosonized vertex operators play a key role in the description of W-infinity symmetries and general solutions, which is technically very similar to a recent free fermion formalism of $c=1$ matrix models.
We study the global symmetries of naive lattices Dirac operators in QCD-like theories in any dimension larger than two. In particular we investigate how the chosen number of lattice sites in each direction affects the global symmetries of the Dirac operator. These symmetries are important since they do not only determine the infra-red spectrum of the Dirac operator but also the symmetry breaking pattern and, thus, the lightest pseudo-scalar mesons. We perform the symmetry analysis and discuss the possible zero modes and the degree of degeneracy of the lattice Dirac operators. Moreover we expli…
We continue our study of reflected entropy, $R(A,B)$, for Gaussian systems. In this paper we provide general formulas valid for free scalar fields in arbitrary dimensions. Similarly to the fermionic case, the resulting expressions are fully determined in terms of correlators of the fields, making them amenable to lattice calculations. We apply this to the case of a $(1+1)$-dimensional chiral scalar, whose reflected entropy we compute for two intervals as a function of the cross-ratio, comparing it with previous holographic and free-fermion results. For both types of free theories we find that …
We extend to a non-Hermitian fermionic quantum field theory with PT symmetry our previous discussion of second quantization, discrete symmetry transformations, and inner products in a scalar field theory [arXiv:2006.06656]. For illustration, we consider a prototype model containing a single Dirac fermion with a parity-odd, anti-Hermitian mass term. In the phase of unbroken PT symmetry, this Dirac fermion model is equivalent to a Hermitian theory under a similarity transformation, with the non-Hermitian nature of the model residing only in the spinor structure, whereas the algebra of the creati…
We obtain a formula for the determinant of a block Toeplitz matrix associated with a quadratic fermionic chain with complex coupling. Such couplings break reflection symmetry and/or charge conjugation symmetry. We then apply this formula to compute the Renyi entropy of a partial observation to a subsystem consisting of $X$ contiguous sites in the limit of large $X$. The present work generalizes similar results due to Its, Jin, Korepin and Its, Mezzadri, Mo. A striking new feature of our formula for the entanglement entropy is the appearance of a term scaling with the logarithm of the size of $…
Higher spin extensions of nonabelian gauge symmetries for both free fermionic model and WZNW model are considered on a classical level. It is characteristic property of the WZNW model that the higher spin currents which correspond to linear realized higher spin extended affine algebra do not form an invariant space. An extended invariant current space is obtained which allows us to gauge the symmetry.
Field theoretic models possessing a global internal fermionic shift symmetry are considered. When such a symmetry is realized locally, spin 3/2 fields appear naturally as gauge fields. Implementation of the gauging procedure requires not only the usual replacement of ordinary derivatives by covariant derivatives containing the spin 3/2 fields, but also the inclusion of additional monomials. The Higgs mechanism and the high energy Nambu-Goldstone fermion equivalence theorem are explicitly demonstrated.
We construct a purely fermionic system with spontaneously broken supersymmetry that shares the common feature with a fracton phase of matter. Our model is gapless due to the Nambu-Goldstone mechanism. It shows a ground-state degeneracy with the "Area-law" entropy due to fermionic subsystem symmetries. In the strongly coupled limit, it becomes a variant of the Nicolai model, and the ground-state degeneracy shows the "Volume-law" entropy. Gauging the fermionic subsystem symmetry has an t'Hooft anomaly by itself, but the would-be gauged theory may possess a fermionic defect that is immobile in ce…
The higher spin symmetry for both Dirac and Majorana massless free fermionic field models are considered. An infinite Lie algebra which is a linear realization of the higher spin extension of the cross products of the Virasoro and affine Kac-Moody algebras is obtained. The corresponding current algebra is closed which is not the case of annalogous current algebra in the WZNW model. The gauging procedure for the higher spin symmetry is given also.
Starting from Wigner's symmetry representation theorem, we give a general account of discrete symmetries (parity P, charge conjugation C, time-reversal T), focusing on fermions in Quantum Field Theory. We provide the rules of transformation of Weyl spinors, both at the classical level (grassmanian wave functions) and quantum level (operators). Making use of Wightman's definition of invariance, we outline ambiguities linked to the notion of classical fermionic Lagrangian. We then present the general constraints cast by these transformations and their products on the propagator of the simplest a…
The term "particle-hole symmetry" is beset with conflicting meanings in contemporary physics. Conceived and written from a condensed-matter standpoint, the present paper aims to clarify and sharpen the terminology. In that vein, we propose to define the operation of "particle-hole conjugation" as the tautological algebra automorphism that simply swaps single-fermion creation and annihilation operators, and we construct its invariant lift to the Fock space. Particle-hole symmetries then arise for gapful or gapless free-fermion systems at half filling, as the concatenation of particle-hole conju…
In $n$-dimensional spacetimes ($n>3$), there exists an internal gauge symmetry of the Palatini action with a cosmological constant that is the natural generalization of the so-called "local translations" of three-dimensional general relativity. We report the extension of this symmetry to include the minimal coupling of Yang-Mills and fermion fields to the Palatini action with a cosmological constant. We show that, as in the case of three-dimensional local translations, the extended symmetry depends on the energy-momentum tensor of the corresponding matter field and, for fermions, it contains a…
A non-Abelian gauge field with a topological action is coupled to a spin 3/2 Majorana spinor. The symmetries of this model are analyzed using the Dirac constraint formalism. These symmetries include a Fermionic symmetry and the algebra of these symmetries closes; it is not the algebra of supergravity. The action is invariant without the need to introduce auxiliary fields.
In this paper, two things are done. First, we analyze the compatibility of Dirac fermions with the hidden duality symmetries which appear in the toroidal compactification of gravitational theories down to three spacetime dimensions. We show that the Pauli couplings to the p-forms can be adjusted, for all simple (split) groups, so that the fermions transform in a representation of the maximal compact subgroup of the duality group G in three dimensions. Second, we investigate how the Dirac fermions fit in the conjectured hidden overextended symmetry G++. We show compatibility with this symmetry …
Non-Invertible Symmetries and Boundaries for Two-Dimensional Fermions
INFN, Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；Abdus Salam Centre for Theoretical Physics, Imperial College London（伦敦帝国理工学院阿卜杜斯·萨拉姆理论物理中心）；SISSA（的里雅斯特国际高等研究学院）
A framework based on a class of abelian gauge symmetries is proposed in which the masses of only the third generation quarks and leptons arise at the tree level. The fermions of the first and second families receive their masses through radiative corrections induced by the new gauge bosons in the loops. It is shown that the class of abelian symmetries which can viably implement this mechanism are flavour non-universal in nature. Taking the all-fermion generalization of the well-known leptonic $L_μ-L_τ$ and $L_e - L_μ$ symmetries, we construct an explicit renormalizable model based on two $U(1)…
The fermionic fields constructed from Elko have several unexpected properties. They satisfy the Klein-Gordon but not the Dirac equation and are of mass dimension one instead of three-half. Starting with the Klein-Gordon Lagrangian, we initiate a careful study of the symmetries and interactions of these fermions and their higher-spin generalisations. We find, although the fermions are of mass dimension one, the four-point fermionic self-interaction violates unitarity at high-energy. Therefore, it cannot be a fundamental interaction of the theory. It follows that for the spin-half fermions, the …
Inspired by the tachyon-free non-supersymmetric heterotic SO(16)xSO(16) string we consider a special class of non-supersymmetric field theories: Those that can be obtained from supersymmetric field theories by supersymmetry breaking twists. We argue that such theories, like their supersymmetric counter parts, may still possess some fermionic symmetries as left-overs of the super gauge transformations and have special one-loop non-renormalization properties due to holomorphicity. In addition, we extend the supergraph techniques to these theories to calculate some explicit supersymmetry-breaking…
Usually, a left-moving fermion in d=1+1 dimensions reflects off a boundary to become a right-moving fermion. This means that, while overall fermion parity $(-1)^F$ is conserved, chiral fermion parity for left- and right-movers individually is not. Remarkably, there are boundary conditions that do preserve chiral fermion parity, but only when the number of Majorana fermions is a multiple of 8. In this paper we classify all such boundary states for $2N$ Majorana fermions when a $U(1)^N$ symmetry is also preserved. The fact that chiral-parity-preserving boundary conditions only exist when $2N$ is…
Scattering discussion due to Double Dirac Equation in Quaternionic version of relativistic quantum mechanics has been studied in this paper in details. In such a quantum mechanics Dirac equation in presence vector and scalar potential has been considered. Then a Quaternionic double Dirac delta potential comes to our considered system which causes to scatter the particles. Scattering states of the particles have been derived as well as reflected and transmission coefficients are calculated.
In this article we derive the full interacting effective actions for supersymmetric D-branes in arbitrary bosonic type II supergravity backgrounds. The actions are presented in terms of component fields up to second order in fermions. As one expects, the actions are built from the supercovariant derivative operator and the $κ$-symmetry projector. The results take a compact and elegant form exhibiting $κ$-symmetry, as well as supersymmetry in a background with Killing spinors. We give the explicit transformation rules for these symmetries in all cases, including the M2-brane. As an example, we …
The connection between symmetries and conservation laws as made by Noether's theorem is extended to the context of causal variational principles and causal fermion systems. Different notions of continuous symmetries are introduced. It is proven that these symmetries give rise to corresponding conserved quantities, expressed in terms of so-called surface layer integrals. In a suitable limiting case, the Noether-like theorems for causal fermion systems reproduce charge conservation and the conservation of energy and momentum in Minkowski space. Thus the conservation of charge and energy-momentum…
The problem of fermions in 1+1 dimensions in the presence of a pseudoscalar Coulomb potential plus a mixing of vector and scalar Coulomb potentials which have equal or opposite signs is investigated. We explore all the possible signs of the potentials and discuss their bound-state solutions for fermions and antifermions. We show the relation between spin and pseudospin symmetries by means of charge-conjugation and $γ^{5}$ chiral transformations. The cases of pure pseudoscalar and mixed vector-scalar potentials, already analyzed in previous works, are obtained as particular cases. The results p…
This is a short review on selected theory developments on Tensor Network (TN) states for strongly correlated systems. Specifically, we briefly review the effect of symmetries in TN states, fermionic TNs, the calculation of entanglement Hamiltonians from Projected Entangled Pair States (PEPS), and the relation between the Multi-scale Entanglement Renormalization Ansatz (MERA) and the AdS/CFT or gauge/gravity duality. We stress the role played by entanglement in the emergence of several physical properties and objects through the TN language. Some recent results along these lines are also discus…
This manuscript is devoted to introduce a gauge theory of the Lorentz Group based on the ambiguity emerging in dealing with isometric diffeo-morphism-induced Lorentz transformations. The behaviors under local transformations of fermion fields and spin connections (assumed to be ordinary world vectors) are analyzed in flat space-time and the role of the torsion field, within the generalization to curved space-time, is briefly discussed. The fermion dynamics is then analyzed including the new gauge fields and assuming time-gauge. Stationary solutions of the problem are also analyzed in the non-r…
We investigate the fermionic sector of a given theory, in which massive and charged Dirac fermions interact with an Abelian gauge field, including a non standard contribution that violates both Lorentz and CPT symmetries. We offer an explicit calculation in which the radiative corrections due to the fermions seem to generate a Chern-Simons-like effective action. Our results are obtained under the general guidance of dimensional regularization, and they show that there is no room for Lorentz and CPT violation in both commutative and noncommutative spacetime.
In this paper, the basis states of the minimal left ideals of the complex Clifford algebra $C\ell(8)$ are shown to contain three generations of Standard Model fermion states, with full Lorentzian, right and left chiral, weak isospin, spin, and electrocolor degrees of freedom. The left adjoint action algebra of $C\ell(8)\cong\mathbb{C}(16)$ on its minimal left ideals contains the Dirac algebra, weak isopin and spin transformations. The right adjoint action algebra on the other hand encodes the electrocolor symmetries. These results extend earlier work in the literature that shows that the eight…
We study the conformal bootstrap for 4-point functions of fermions $\langle ψ_i ψ_j ψ_k ψ_{\ell} \rangle$ in parity-preserving 3d CFTs, where $ψ_i$ transforms as a vector under an $O(N)$ global symmetry. We compute bounds on scaling dimensions and central charges, finding features in our bounds that appear to coincide with the $O(N)$ symmetric Gross-Neveu-Yukawa fixed points. Our computations are in perfect agreement with the $1/N$ expansion at large $N$ and allow us to make nontrivial predictions at small $N$. For values of $N$ for which the Gross-Neveu-Yukawa universality classes are relevan…
Exact Chiral Symmetries of 3+1D Hamiltonian Lattice Fermions
Walter Burke Institute for Theoretical Physics, Caltech（加州理工学院沃尔特·伯克理论物理研究所）；Department of Physics, Caltech（加州理工学院物理系）；Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California, Los Angeles（加州大学洛杉矶分校物理学与天文学系马尼·L·巴乌米克理论物理研究所）
Non-invertible Symmetries in Weyl Fermions, and Applications to Fermion-Boundary Scattering Problem
Department of Physics, Nanjing University（南京大学物理系）；School of Quantum and Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学量子学院和理论科学研究所）
We demonstrate that the gauge-fixed Lagrangian of the Christ-Lee model respects four fermionic symmetries, namely; (anti-)BRST symmetries, (anti-)co-BRST symmetries within the framework of BRST formalism. The appropriate anticommutators amongst the fermionic symmetries lead to a unique bosonic symmetry. It turns out that the algebra obeyed by the symmetry transformations (and their corresponding conserved charges) is reminiscent of the algebra satisfied by the de Rham cohomological operators of differential geometry. We also provide the physical realizations of the cohomological operators in t…
In a 4D four-fermion model we study the dynamical restoration of Lorentz and CPT symmetries at finite temperature. We evaluate the gap equation both at zero and at finite temperature and observe that, depending on the parameters of the theory, there is a critical temperature at which the Lorentz and CPT symmetries are restored.
We compute the chiral symmetries of the Lagrangian for confining "vector-like" gauge theories with massless fermions in $d$-dimensional Minkowski space and, under a few reasonable assumptions, determine the form of the quadratic fermion condensates which arise through spontaneous breaking of these symmetries. We find that for each type (complex, real, or pseudoreal) of representation of the gauge group carried by the fermions, the chiral symmetries of the Lagrangian, as well as the residual symmetries after dynamical breaking, exactly follow the pattern of Bott periodicity as the dimension cha…
We study boundary states for Dirac fermions in d=1+1 dimensions that preserve Abelian chiral symmetries, meaning that the left- and right-moving fermions carry different charges. We derive simple expressions, in terms of the fermion charge assignments, for the boundary central charge and for the ground state degeneracy of the system when two different boundary conditions are imposed at either end of an interval. We show that all such boundary states fall into one of two classes, related to SPT phases supported by (-1)^F, which are characterised by the existence of an unpaired Majorana zero mod…
We present a fully many-body formulation of topological invariants for various topological phases of fermions protected by antiunitary symmetry, which does not refer to single particle wave functions. For example, we construct the many-body $\mathbb{Z}_2$ topological invariant for time-reversal symmetric topological insulators in two spatial dimensions, which is a many-body counterpart of the Kane-Mele $\mathbb{Z}_2$ invariant written in terms of single-particle Bloch wave functions. We show that an important ingredient for the construction of the many-body topological invariants is a fermioni…
We review the geodesic motion of pseudo-classical spinning particles in curved spaces. Investigating the generalized Killing equations for spinning spaces, we express the constants of motion in terms of Killing-Yano tensors. Passing from the spinning spaces to the Dirac equation in curved backgrounds we point out the role of the Killing-Yano tensors in the construction of the Dirac-type operators. The general results are applied to the case of the four-dimensional Euclidean Taub-Newman-Unti-Tamburino space. The gravitational and axial anomalies are studied for generalized Euclidean Taub-NUT me…
Discrete symmetries are studied in warped space-times with one extra dimension. In particular, we analyze the compatibility of five- and four-dimensional charge conjugation, parity, time reversal and the orbifold symmetry Z_2 with localization of fermions on the four-dimensional brane-world and Lorentz invariance. We then show that, when a suitable topological scalar field (the ``kink'') is included, fermion localization is a consequence of (five-dimensional) CPT invariance.
Local symmetries and extensive ground-state degeneracy of a 1D supersymmetric fermionic chain
C. N. Yang Institute for Theoretical Physics, Stony Brook University（纽约州立大学石溪分校杨振宁理论物理研究所）；Department of Physics and Astronomy, Stony Brook University（纽约州立大学石溪分校物理与天文学系）；Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University（纽约州立大学石溪分校物理与天文学系核理论中心）；Department of Physics, University of Massachusetts Boston（马萨诸塞大学波士顿分校物理系）
We show that three generations of leptons and quarks with unbroken Standard Model gauge symmetry $SU(3)_c\times U(1)_{em}$ can be described using the algebra of complexified sedenions $\mathbb{C}\otimes\mathbb{S}$. A primitive idempotent is constructed by selecting a special direction, and the action of this projector on the basis of $\mathbb{C}\otimes\mathbb{S}$ can be used to uniquely split the algebra into three complex octonion subalgebras $\mathbb{C}\otimes \mathbb{O}$. These subalgebras all share a common quaternionic subalgebra. The left adjoint actions of the 8 $\mathbb{C}$-dimensional…
Two dimensional QCD coupled to fermions in the adjoint representation of the gauge group $SU(N)$, a useful toy model of QCD strings, is supersymmetric for a certain ratio of quark mass and gauge coupling constant. Here we study the theory in the vicinity of the supersymmetric point; in particular we exhibit the algebraic structure of the model and show that the mass splittings as one moves away from the supersymmetric point obey a universal relation of the form ${M_i}^2(B)-{M_i}^2(F)=M_iδm+O(δm^3)$. We discuss the connection of this relation to string and quark model expectations and verify it…
Following up on our earlier work where we showed how to amend a scheme originally proposed by M. Gell-Mann to identify the 48 spin 1/2 fermions of N=8 supergravity that remain after complete breaking of N=8 supersymmetry with the 3x16 quarks and leptons of the Standard Model, we further generalize the construction to account for the full SU(3)_c x SU(2)_w x U(1)_Y assignments, with an additional family symmetry SU(3)_f. Our proposal relies in an essential way on embedding the SU(8) R-symmetry of N=8 supergravity into the (infinite-dimensional) `maximal compact' subgroup K(E10) of the conjectur…
The Vafa-Witten arguments on the realization of parity and flavour symmetries in the QCD vacuum do not apply to two lattice regularizations of QCD which are able to reproduce the chiral anomaly: Wilson fermions and Ginsparg-Wilson fermions. We show here how using the last regularization one can get, from first principles, that the more standard order parameters for these symmetries take a vanishing vacuum expectation value for any number of flavours, if quarks have a non-vanishing mass.
