Fermionic Space Research Articles

Fermionic shift symmetries in (anti) de Sitter space

We study extended shift symmetries that arise for fermionic fields on anti-de Sitter (AdS) space and de Sitter (dS) space for particular values of the mass relative to the curvature scale. We classify these symmetries for general mixed-symmetry fermionic fields in arbitrary dimension and describe how fields with these symmetries arise as the decoupled longitudinal modes of massive fermions as they approach partially massless points. For the particular case of AdS4, we look for non-trivial Lie superalgebras that can underly interacting theories that involve these fields. We study from this perspective the minimal such theory, the Akulov-Volkov theory on AdS4, which is a non-linear theory of a spin-1/2 Goldstino field that describes the spontaneous breaking of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 1 supersymmetry on AdS4 down to the isometries of AdS4. We show how to write the nonlinear supersymmetry transformation for this theory using the fermionic ambient space formalism. We also study the Lie superalgebras of candidate multi-field examples and rule out the existence of a supersymmetric special galileon on AdS4.

Random finite noncommutative geometries and topological recursion

In this paper, we investigate a model for quantum gravity on finite noncommutative spaces using the theory of blobbed topological recursion. The model is based on a particular class of random finite real spectral triples {(\mathcal{A}, \mathcal{H}, D, \gamma, J)} , called random matrix geometries of type {(1,0)} , with a fixed fermion space {(\mathcal{A}, \mathcal{H}, \gamma, J)} and a distribution of the form {e^{- \mathcal{S} (D)}\, {\mathrm{d}} D} over the moduli space of Dirac operators. The action functional {\mathcal{S} (D)} is considered to be a sum of terms of the form {\prod_{i=1}^s \mathrm{Tr} ({D^{n_i}})} for arbitrary {s \geqslant 1} . The Schwinger–Dyson equations satisfied by the connected correlators {W_n} of the corresponding multi-trace formal 1-Hermitian matrix model are derived by a differential geometric approach. It is shown that the coefficients {W_{g,n}} of the large N expansion of {W_n} ’s enumerate discrete surfaces, called stuffed maps, whose building blocks are of particular topologies. The spectral curve {( {\Sigma, \omega_{0,1}, \omega_{0,2}} )} of the model is investigated in detail. In particular, we derive an explicit expression for the fundamental symmetric bidifferential {\omega_{0,2}} in terms of the formal parameters of the model.

Ising field theory in a magnetic field: φ3 coupling at T > Tc

We study the “three particle coupling” {Gamma}_{11}^1left(xi right) , in 2d Ising Field Theory in a magnetic field, as the function of the scaling parameter ξ := h/(−m)15/8, where m ∼ Tc − T and h ∼ H are scaled deviation from the critical temperature and scaled external field, respectively. The “φ3 coupling” {Gamma}_{11}^1 is defined in terms of the residue of the 2 → 2 elastic scattering amplitude at its pole associated with the lightest particle itself. We limit attention to the High-Temperature domain, so that m is negative. We suggest “standard analyticity”: {left({Gamma}_{11}^1right)}^2 , as the function of u := ξ2, is analytic in the whole complex u-plane except for the branch cut from – ∞ to – u0 ≈ – 0.03585, the latter branching point – u0 being associated with the Yang-Lee edge singularity. Under this assumption, the values of {Gamma}_{11}^1 at any complex u are expressed through the discontinuity across the branch cut. We suggest approximation for this discontinuity which accounts for singular expansion of {Gamma}_{11}^1 near the Yang-Lee branching point, as well as its known asymptotic at u → +∞. The resulting dispersion relation agrees well with known exact data, and with numerics obtained via Truncated Free Fermion Space Approach. This work is part of extended project of studying the S-matrix of the Ising Field Theory in a magnetic field.

Clifford odd and even objects offer description of internal space of fermions and bosons, respectively, opening new insight into the second quantization of fields

In a long series of works the author has demonstrated that the model named the spin-charge-family theory offers the explanation for all in the standard model assumed properties of the fermion and boson fields, as well as for many of their so far observed properties if the space-time is ≥ (13 + 1) while fermions interact with gravity only. In this talk, I briefly report on the so far achievements of the theory. The main contribution demonstrates the offer of the Clifford odd and even objects for the description of the internal spaces of fermion (Clifford odd) and boson (Clifford even) fields, which is opening up a new understanding of the second quantization postulates for the fermion and boson fields: The “basis vectors” determined by the Clifford odd objects demonstrate all the properties of the internal space of fermions and transfer their anti-commutativity to their creation and annihilation operators, while the “basis vectors” determined by the Clifford even objects demonstrate all the properties of the internal space of boson fields and transfer their commutativity to their creation and annihilation operators. The toy model with d = (5 + 1) illustrates the statements.

2D Ising Field Theory in a magnetic field: the Yang-Lee singularity

We study Ising Field Theory (the scaling limit of Ising model near the Curie critical point) in pure imaginary external magnetic field. We put particular emphasis on the detailed structure of the Yang-Lee edge singularity. While the leading singular behavior is controlled by the Yang-Lee fixed point (= minimal CFT mathcal{M} 2/5), the fine structure of the subleading singular terms is determined by the effective action which involves a tower of irrelevant operators. We use numerical data obtained through the “Truncated Free Fermion Space Approach” to estimate the couplings associated with two least irrelevant operators. One is the operator Toverline{T} , and we use the universal properties of the Toverline{T} deformation to fix the contributions of higher orders in the corresponding coupling parameter α. Another irrelevant operator we deal with is the descendant L_4 overline{L} _4ϕ of the relevant primary ϕ in mathcal{M} 2/5. The significance of this operator is that it is the lowest dimension operator which breaks integrability of the effective theory. We also establish analytic properties of the particle mass M (= inverse correlation length) as the function of complex magnetic field.

Gravitational dark matter: Free streaming and phase space distribution

Gravitational dark matter (DM) is the simplest possible scenario that has recently gained interest in the early Universe cosmology. In this scenario, DM is assumed to be produced from the scattering of inflatons through the gravitational interaction during reheating. Gravitational production from the radiation bath will be ignored as our analysis shows it to be suppressed for a wide range of reheating temperatures $({T}_{\mathrm{re}})$. Ignoring any other internal parameters except the DM mass $({m}_{Y})$ and spin, a particular inflation model such as $\ensuremath{\alpha}$ attractor, with a specific scalar spectral index $({n}_{s})$ has been shown to uniquely fix the DM mass of the present Universe. For fermion type DM, we found the mass ${m}_{f}$ should be within $({10}^{4}--{10}^{13})\text{ }\text{ }\mathrm{GeV}$, and for boson/vector type DM, the mass ${m}_{s}/{m}_{X}$ turned out to be within $(1.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}/9.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}--{10}^{13})\text{ }\text{ }\mathrm{GeV}$. Interestingly, if the inflaton equation of state ${\ensuremath{\omega}}_{\ensuremath{\phi}}\ensuremath{\rightarrow}1/3$, the DM mass also approaches towards unique value, ${m}_{f}\ensuremath{\sim}{10}^{10}\text{ }\text{ }\mathrm{GeV}$ and ${m}_{s}({m}_{X})\ensuremath{\sim}{10}^{3}(8\ifmmode\times\else\texttimes\fi{}{10}^{3})\text{ }\text{ }\mathrm{GeV}$. We further analyzed the phase space distribution ${f}_{Y}$ and free streaming length ${\ensuremath{\lambda}}_{\mathrm{fs}}$ of these gravitationally produced DM. ${f}_{Y}$, which is believed to encode important information about DM, contains a characteristic primary peak at the initial time where the gravitational production is maximum for both fermionic and bosonic DM. Apart from this fermionic phase space, the distribution function contains an additional peak near the inflaton and fermion mass equality (${m}_{Y}\ensuremath{\sim}{m}_{\ensuremath{\phi}}$) arising for ${\ensuremath{\omega}}_{\ensuremath{\phi}}>5/9$. Furthermore, the height of this additional peak turned out to be increasing with decreasing ${T}_{\mathrm{re}}$ and at some point, dominates over the primary one. Since reheating is a causal process and DM is produced during this phase, gravitational instability forming small-scale DM structures during this period will encode those phase space information and be observed at present. The crucial condition of forming such a small-scale DM structure during reheating suggests that the ${\ensuremath{\lambda}}_{\mathrm{fs}}$ must be less than the length scale associated with the mode reentering the Hubble radius at the end of reheating ${\ensuremath{\lambda}}_{\mathrm{re}}$, which has been analyzed in detail. We further estimate in detail the range of scales within which the above condition will be satisfied for different masses of scalar, fermionic, and vector DM. Finally, we end by stating the fact that all our results are observed to be insensitive on the parameter $\ensuremath{\alpha}$ of the inflaton potential within the allowed range set by the latest Planck and BICEP/Keck results.

Hidden Kac-Moody structures in the fermionic sector of five-dimensional supergravity

We study the supersymmetric quantum dynamics of the cosmological models obtained by reducing $D=5$ supergravity to one timelike dimension. This consistent truncation has fourteen bosonic degrees of freedom, while the quantization of the homogeneous gravitino field leads to a $2^{16}$--dimensional fermionic Hilbert space. We construct a consistent quantization of the model in which the wave function of the Universe is a $2^{16}$--component spinor %\textcolor{red}{of Spin(24,8)} depending on fourteen continuous coordinates, which satisfies eight Dirac-like wave equations (supersymmetry constraints) and one Klein-Gordon-like equation (Hamiltonian constraint). The fermionic part of the quantum Hamiltonian is built from operators that generate a $2^{16}$-dimensional representation of the (infinite-dimensional) maximally compact sub-algebra $K(G_2^{++})$ of the rank-4 hyperbolic Kac--Moody algebra $G_2^{++}$. The (quartic-in-fermions) squared-mass term $\widehat \mu^2$ entering the Klein-Gordon-like equation has several remarkable properties: (i) it commutes with the generators of $K(G_2^{++})$; and (ii) it is a quadratic polynomial in the fermion number $N_F \sim \overline\Psi \Psi$, and a symplectic fermion bilinear $C_F \sim \Psi C\Psi$. Some aspects of the structure of the solutions of our model are discussed, and notably the Kac-Moody meaning of the operators describing the reflection of the wave function on the fermion-dependent potential walls ("quantum fermionic Kac-Moody billiard").

Deformation quantization for systems with second-class constraints in deformed fermionic phase space

In order to quantize systems involving second-class constraints, one should use Dirac bracket instead of Poisson bracket. Furthermore, one can specify a star product in which the term linear in [Formula: see text] is proportional to the Dirac bracket. In this way an oscillator system in a deformed fermionic phase space is analyzed and the corresponding energy level and Wigner functions are evaluated according to scheme of deformation quantization. We also study the entanglement entropy induced by the deformation of the fermionic phase space.

Critical Field Theories with OSp(1|2M) Symmetry

In the paper [L. Fei etal., J. High Energy Phys. 09 (2015) 076JHEPFG1029-847910.1007/JHEP09(2015)076] a cubic field theory of a scalar field σ and two anticommuting scalar fields, θ and θ[over ¯], was formulated. In 6-ε dimensions it has a weakly coupled fixed point with imaginary cubic couplings where the symmetry is enhanced to the supergroup OSp(1|2). This theory may be viewed as a "UV completion" in 2<d<6 of the nonlinear sigma model with hyperbolic target space H^{0|2} described by a pair of intrinsic anticommuting coordinates. It also describes the q→0 limit of the critical q-state Potts model, which is equivalent to the statistical mechanics of spanning forests on a graph. In this Letter we generalize these results to a class of OSp(1|2M) symmetric field theories whose upper critical dimensions are d_{c}(M)=2[(2M+1)/(2M-1)]. They contain 2M anticommuting scalar fields, θ^{i}, θ[over ¯]^{i}, and one commuting one, with interaction g(σ^{2}+2θ^{i}θ[over ¯]^{i})^{(2M+1)/2}. Ind_{c}(M)-ε dimensions, we find a weakly coupled IR fixed point at an imaginary value of g. We propose that these critical theories are the UV completions of the sigma models with fermionic hyperbolic target spaces H^{0|2M}. Of particular interest is the quintic field theory with OSp(1|4) symmetry, whose upper critical dimension is 10/3. Using this theory, we make a prediction for the critical behavior of the OSp(1|4) lattice system in three dimensions.

Wavefunction structure in quantum many-fermion systems with k-body interactions: conditional q-normal form of strength functions

For finite quantum many-particle systems modeled with say m fermions in N single particle states and interacting with k-body interactions (k ⩽ m), the wavefunction structure is studied using random matrix theory. The Hamiltonian for the system is chosen to be H = H 0(t) + λV(k) with the unperturbed H 0(t) Hamiltonian being a t-body operator and V(k) a k-body operator with interaction strength λ. Representing H 0(t) and V(k) by independent Gaussian orthogonal ensembles of random matrices in t and k fermion spaces, respectively, the first four moments, in m-fermion spaces, of the strength functions F κ (E) are derived; the strength functions contain all of the information about wavefunction structure. With E denoting the H energies or eigenvalues and κ denoting unperturbed basis states with energy E κ , the F κ (E) give the spreading of the κ states over the eigenstates E. It is shown that the first four moments of F κ (E) are essentially the same as that of the conditional q-normal distribution given in (Szabowski 2010 Electron. J. Probab. 15 1296). This naturally gives asymmetry in F κ (E) with respect to E as E κ increases and also the peak value changes with E κ . Thus, the wavefunction structure in quantum many-fermion systems with k-body interactions in general follows the conditional q-normal distribution.

The attributes of the Spin-Charge-Family theory giving hope that the theory offers the next step beyond the Standard Model

The assumptions of the standard model are still waiting for an explanation. The spin-charge-family theory is promising in offering not only the explanation for the standard model postulates for quarks and leptons and vector and scalar gauge fields, but also for the cosmological observations, like there are the appearance of the dark matter, the matter-antimatter asymmetry, making several predictions. This theory assumes that the internal space of fermions (spins, handedness and all the charges) are described by the Clifford algebra in d > (13 + 1)-dimensional space, representing in d = (3 + 1) all by the standard model required properties of quarks and leptons and antiquarks and antileptons, with families included. Fermions interact with gravity only (the vielbeins and the two kinds of the spin connection fields), manifesting in d = (3 + 1) as all the vector gauge fields and the scalar gauge fields (higgs scalars and Yukawa couplings). In this talk I overview shortly the achievements of the spin-charge-family theory so far, explaining in particular the new way of the second quantization of fermions, offered by the description of the internal space of fermions with the anticommuting Clifford algebra objects of the odd character.

Octagon with finite bridge: free fermions and determinant identities

We continue the study of the octagon form factor which helps to evaluate a class of four-point correlation functions in mathcal{N} = 4 SYM theory. The octagon is characterised, besides the kinematical parameters, by a “bridge” of ℓ propagators connecting two nonadjacent operators. In this paper we construct an operator representation of the octagon with finite bridge as an expectation value in the Fock space of free complex fermions. The bridge ℓ appears as the level of filling of the Dirac sea. We obtain determinant identities relating octagons with different bridges, which we derive from the expression of the octagon in terms of discrete fermionic oscillators. The derivation is based on the existence of a previously conjectured similarity transformation, which we find here explicitly.

Longitudinal magnetization dynamics in the quantum Ising ring: A Pfaffian method based on correspondence between momentum space and real space.

As perhaps the most studied paradigm for a quantum phase transition, the periodic quantum Ising chain is exactly solvable via the Jordan-Wigner transformation followed by a Fourier transform that diagonalizes the model in the momentum space of spinless fermions. Although the above procedures are well-known, there remain some subtle points to be clarified regarding the correspondence between the real-space and momentum-space representations of the finite-size quantum Ising ring, especially those related to fermion parities. In this work, we establish the relationship between the two fully aligned ferromagnetic states in real space and the two degenerate momentum-space ground states of the classical Ising ring, with the former being a special case of the factorized ground states of the more general XYZ model on the frustration-free hypersurface. Based on this observation, we then provide a Pfaffian formula for calculating real-time dynamics of the parity-breaking longitudinal magnetization with the system initially prepared in one of the two ferromagnetic states and under translationally invariant drivings. The formalism is shown to be applicable to large systems with the help of online programs for the numerical computation of the Pfaffian, thus providing an efficient method to numerically study, for example, the emergence of discrete time crystals in related systems.

AN APPLICATION OF THE DEMOCRATIC MAPPING IN REALISTIC SYSTEMS

The democratic mapping is used for the calculation of low lying states of nuclei in the sd and fp shells. In addition to demonstrating the applicability of the method in realistic cases where many non-degenerate levels are present, the method allows for the ranking of the various bosons according to their importance as building blocks of low lying states. It is proven that the s and d bosons are the most important building blocks, followed by the d' and g bosons. Thus one of the basic assumptions of the Interacting Boson Model (IBM) is proven to be correct. Very good agreement between the boson calculation and the shell model results is obtained for A = 20 nuclei when 12 bosons are taken into account, while an even larger number of bosons is required to reproduce the low-lying states of the A = 44 nuclei. In order to obtain equally good results with a smaller number of bosons one needs to introduce effective boson hamiltonians which correspond to truncated fermion spaces.

Pin groups in general relativity

There are eight possible Pin groups that can be used to describe the transformation behaviour of fermions under parity and time reversal. We show that only two of these are compatible with general relativity, in the sense that the configuration space of fermions coupled to gravity transforms appropriately under the space-time diffeomorphism group.

The mechanical strain induced anomalous de Haas–van Alphen effect on graphene

When a uniform magnetic field is applied to graphene, the magnetisation and magnetic susceptibility both exhibit oscillations periodic in 1/B, i.e., de Haas–van Alphen (dHvA) effect. Such magnetic dHvA oscillations reflect the fundamental reorganisation of carrier states into Landau levels as a canonical response of graphene to the applied magnetic field. We predict here that, remarkably, in graphene, dHvA oscillations can occur in the complete absence of magnetic field. These zero-field dHvA oscillations are driven by mechanical strain which, in the space of the low-energy Dirac fermions, acts as a gauge field, i.e. a uniform strain pseudomagnetic field with magnitudes up to hundreds of tesla, far beyond the practical magnetic field, thereby providing a wider range of modulation than the traditional dHvA effect.

Entropy, purity, and fidelity in Majorana phase space

Majorana phase-space representations for fermions map fermionic many-body physics into a distribution over one of the Cartan symmetric spaces of Lie group theory. The representation is in terms of $2M\ifmmode\times\else\texttimes\fi{}2M$ complex antisymmetric matrices, which generate the Gaussian Majorana operators. Here we show how this expansion can be utilized to calculate quantities arising in quantum thermodynamics and quantum information. Purity and the linear entropy are calculated, as well as the quantum fidelity between two general fermionic states, with numerical examples for pure states. We describe the geometrical properties of the phase space, and show that the overlap between two Gaussian Majorana states depends on the product of their antisymmetric matrices. Fermionic phase space is divided up into two orthogonal subspaces of different number parity, whose matrix representations differ by an orthogonal reflection in the phase space.

Quenching dissipative quantum Ising chain: an exact result for nonequilibrium dynamics

To create a more comprehensive understanding of nonequilibrium dynamics of open quantum many-body systems, we visit an exactly solvable example, that is a quenched transverse-field Ising chain coupled to Markovian baths, which act locally on the Jordan–Wigner fermionic space. Performing explicit calculations on the heat transfer, transverse magnetization, and kink density, we find that the imbalance of two opposite damping mechanisms play a crucial role in constructing a nontrivial nonequilibrium steady state accompanied with a dissipative quantum phase transition, also that the competition between unitary drive and decoherence does not necessarily lead to a quasi-stationary state or prethermalization under certain ordinary relaxation.

Exceptional quantum geometry and particle physics II

We continue the study undertaken in [13] of the relevance of the exceptional Jordan algebra J38 of hermitian 3×3 octonionic matrices for the description of the internal space of the fundamental fermions of the Standard Model with 3 generations. By using the suggestion of [30] (properly justified here) that the Jordan algebra J28 of hermitian 2×2 octonionic matrices is relevant for the description of the internal space of the fundamental fermions of one generation, we show that, based on the same principles and the same framework as in [13], there is a way to describe the internal space of the 3 generations which avoids the introduction of new fundamental fermions and where there is no problem with respect to the electroweak symmetry.

Exploring the possibilities of dynamical quantum phase transitions in the presence of a Markovian bath

We explore the possibility of dynamical quantum phase transitions (DQPTs) occurring during the temporal evolution of a quenched transverse field Ising chain coupled to a particle loss type of bath (local in Jordan-Wigner fermion space) using two versions of the Loschmidt overlap (LO), namely, the fidelity induced LO and the interferometric phase induced LO. The bath, on the one hand, dictates the dissipative evolution following a sudden quench and on the other, plays a role in dissipative mixed state preparation in the later part of the study. During a dissipative evolution following a sudden quench, no trace of DQPTs are revealed in both the fidelity and the interferometric phase approaches; however, remarkably the interferometric phase approach reveals the possibility of inter-steady state DQPTs in passage from one steady state to the other when the system is subjected to a quench after having reached the first steady state. We further probe the occurrences of DQPTs when the system evolves unitarily after being prepared in a mixed state of engineered purity by ramping the transverse field in a linear fashion in the presence of the bath. In this case though the fidelity approach fails to indicate any DQPT, the interferometric approach indeed unravels the possibility of occurrence of DQPTs which persists even up to a considerable loss of purity of the engineered initial state as long as a constraint relation involving the dissipative coupling and ramping time (rate) is satisfied. This constraint relation also marks the boundary between two dynamically inequivalent phases; in one the LO vanishes for the critical momentum mode (and hence DQPTs exist) while in the other no such critical mode can exist and hence the LO never vanishes.