Grassmann Fields Research Articles

Exact WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms

Quantum deformed potentials arise naturally in quantum mechanical systems of one bosonic coordinate coupled to ${N}_{f}$ Grassmann valued fermionic coordinates, or to a topological Wess-Zumino term. These systems decompose into sectors with a classical potential plus a quantum deformation. Using exact Wentzel-Kramers-Brillouin, we derive exact quantization condition and its median resummation. The solution of median resummed form gives physical Borel-Ecalle resummed results, as we show explicitly in quantum deformed double- and triple-well potentials. Despite the fact that instantons have finite actions, for generic quantum deformation, they do not contribute to the energy spectrum at leading order in semiclassics. For certain quantized quantum deformations, where the alignment of levels to all order in perturbation theory occurs, instantons contribute to the spectrum. If deformation parameter is not properly quantized, their effect disappears, but higher-order effects in semiclassics survive. In this sense, we classify saddle contributions as fading and robust. Finally, for quantum deformed triple-well potential, we demonstrate the perturbative/nonperturbative relation, by computing period integrals and Mellin transform.

Competing chiral d -wave superconductivity and magnetic phases in the strong-coupling Hubbard model on the honeycomb lattice

We address the competing superconducting and magnetic phases on the honeycomb lattice, in a field-theoretic approach suitable to yield a low-energy perturbative theory for the strong-coupling limit of the Hubbard model, both at half filling and in the low and high hole-doped regimes. The effective low-lying Hamiltonian is presented in terms of charge (Grassmann fields) and spin [SU(2) gauge fields] degrees of freedom. We analyze the competing phases by calculating the ground-state energy, electronic spectrum, and other observables associated with the $s$- and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$-wave superconducting phases, doped antiferromagnetic, and doped ferromagnetic states. We find that, while the antiferromagnetic order has the lowest ground-state energy for low hole doping near half filling, a dominant superconducting state with chiral ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$-wave symmetry emerges in the vicinity of the Van Hove singularity in the high hole-doped regime, with the presence of a quantum first-order transition accompanied by spatial phase separation. We highlight that advances in the understanding of chiral superconducting states on the honeycomb lattice are relevant to a number of doped compounds including the graphene monolayer system.

Fermionic wave functions and Grassmann fields as possible sources of dark energy

We study a cosmological model with a fermionic field which can be interpreted as a source of dark energy in the universe. Two different approaches were considered, the first one with a massless fermionic field represented by a standard wave-function and the second one where a massive field is a Grassmann variable. The first case naturally reduces to a XCDM model with a constant equation of state parameter, while the last case reproduces a w(z)CDM model for a massive field, and in the massless limit, the intrinsic Grassmannian property of the field leads always to a vacuum equation of state parameter, irrespective the specific form of the potential. Both cases leads to a dark energy contribution of the fermionic sector. The models are totally compatible with recent cosmological data from Supernovae, BAO and Hubble parameter measurements. A brief study of linear evolution of density perturbations shows that some of the small scale problems related to standard model can be at least alleviated.

Supersymmetries in nonequilibrium Langevin dynamics.

Stochastic phenomena are often described by Langevin equations, which serve as a mesoscopic model for microscopic dynamics. It has been known since the work of Parisi and Sourlas that reversible (or equilibrium) dynamics present supersymmetries (SUSYs). These are revealed when the path-integral action is written as a function not only of the physical fields, but also of Grassmann fields representing a Jacobian arising from the noise distribution. SUSYs leave the action invariant upon a transformation of the fields that mixes the physical and the Grassmann ones. We show that contrary to common belief, it is possible to extend the known reversible construction to the case of arbitrary irreversible dynamics, for overdamped Langevin equations with additive white noise-provided their steady state is known. The construction is based on the fact that the Grassmann representation of the functional determinant is not unique, and can be chosen so as to present a generalization of the Parisi-Sourlas SUSY. We show how such SUSYs are related to time-reversal symmetries and allow one to derive modified fluctuation-dissipation relations valid in nonequilibrium. We give as a concrete example the results for the Kardar-Parisi-Zhang equation.

Stochastic equation of motion approach to fermionic dissipative dynamics. I. Formalism.

In this work, we establish formally exact stochastic equation of motion (SEOM) theory to describe the dissipative dynamics of fermionic open systems. The construction of the SEOM is based on a stochastic decoupling of the dissipative interaction between the system and fermionic environment, and the influence of environmental fluctuations on the reduced system dynamics is characterized by stochastic Grassmann fields. Meanwhile, numerical realization of the time-dependent Grassmann fields has remained a long-standing challenge. To solve this problem, we propose a minimal auxiliary space (MAS) mapping scheme with which the stochastic Grassmann fields are represented by conventional c-number fields along with a set of pseudo-levels. This eventually leads to a numerically feasible MAS-SEOM method. The important properties of the MAS-SEOM are analyzed by making connection to the well-established time-dependent perturbation theory and the hierarchical equations of motion theory. The MAS-SEOM method provides a potentially promising approach for the accurate and efficient simulation of fermionic open systems at ultra-low temperatures.

Stochastic equation of motion approach to fermionic dissipative dynamics. II. Numerical implementation

This paper provides a detailed account of the numerical implementation of the stochastic equation of motion (SEOM) method for the dissipative dynamics of fermionic open quantum systems. To enable direct stochastic calculations, a minimal auxiliary space (MAS) mapping scheme is adopted, with which the time-dependent Grassmann fields are represented by c-number noises and a set of pseudo-operators. We elaborate on the construction of the system operators and pseudo-operators involved in the MAS-SEOM, along with the analytic expression for the particle current. The MAS-SEOM is applied to study the relaxation and voltage-driven dynamics of quantum impurity systems described by the single-level Anderson impurity model, and the numerical results are benchmarked against those of the highly accurate hierarchical equations of motion method. The advantages and limitations of the present MAS-SEOM approach are discussed extensively.

Bilinear approach to Kuperschmidt super-KdV type equations

Hirota bilinear form and soliton solutions for the super-KdV (Korteweg–de Vries) equation of Kuperschmidt (Kuper–KdV) are given. It is shown that even though the collision of supersolitons is more complicated than in the case of the supersymmetric KdV equation of Manin–Radul, the asymptotic effect of the interaction is simpler. As a physical application it is shown that the well-known FPU problem, having a phonon-mediated interaction of some internal degrees of freedom expressed through Grassmann fields, transforms to the Kuper–KdV equation in a multiple-scale approach.

Grassmann phase space methods for fermions. II. Field theory

In both quantum optics and cold atom physics, the behaviour of bosonic photons and atoms is often treated using phase space methods, where mode annihilation and creation operators are represented by c-number phase space variables, with the density operator equivalent to a distribution function of these variables. The anti-commutation rules for fermion annihilation, creation operators suggests the possibility of using anti-commuting Grassmann variables to represent these operators. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of Grassmann phase space methods in quantum-atom optics to treat fermionic systems is rather rare, though fermion coherent states using Grassmann variables are widely used in particle physics.This paper presents a phase space theory for fermion systems based on distribution functionals, which replace the density operator and involve Grassmann fields representing anti-commuting fermion field annihilation, creation operators. It is an extension of a previous phase space theory paper for fermions (Paper I) based on separate modes, in which the density operator is replaced by a distribution function depending on Grassmann phase space variables which represent the mode annihilation and creation operators. This further development of the theory is important for the situation when large numbers of fermions are involved, resulting in too many modes to treat separately. Here Grassmann fields, distribution functionals, functional Fokker–Planck equations and Ito stochastic field equations are involved. Typical applications to a trapped Fermi gas of interacting spin 1/2 fermionic atoms and to multi-component Fermi gases with non-zero range interactions are presented, showing that the Ito stochastic field equations are local in these cases. For the spin 1/2 case we also show how simple solutions can be obtained both for the untrapped case and for an optical lattice trapping potential.

Protostring scattering amplitudes

We calculate some tree level scattering amplitudes for a generalization of the protostring, which is a novel string model implied by the simplest string bit models. These bit models produce a lightcone worldsheet which supports $s$ integer moded Grassmann fields. In the generalization we supplement this Grassmann worldsheet system with $d=24-s$ transverse coordinate worldsheet fields. The protostring corresponds to $s=24$ and the bosonic string to $s=0$. The interaction vertex is a simple overlap with no operator insertions at the break/join point. Assuming that $s$ is even we calculate the multi-string scattering amplitudes by bosonizing the Grassmann fields, each pair equivalent to one compactified bosonic field, and applying Mandelstam's interacting string formalism to a system of $s/2$ compactified and $d$ uncompactified bosonic worldsheet fields. We obtain all amplitudes for open strings with no oscillator excitations and for closed strings with no oscillator excitations and zero winding number. We then study in detail some simple special cases. Multi-string processes with maximal helicity violation have much simplified amplitudes. We also specialize to general four string amplitudes and discuss their high energy behavior. Most of these models are not covariant under the full Lorentz group $O(d+1,1)$. The exceptions are the bosonic string whose Lorentz group is $O(25,1)$ and the protostring whose Lorentz group is $O(1,1)$. The models in between only enjoy an $O(1,1)\times O(d)$ spacetime symmetry.

Grassmann phase space theory for fermions

A phase space theory for fermions has been developed using Grassmann phase space variables which can be used in numerical calculations for cold Fermi gases and for large fermion numbers. Numerical calculations are feasible because Grassmann stochastic variables at later times are related linearly to such variables at earlier times via c‐number stochastic quantities. A Grassmann field version has been developed making large fermion number applications possible. Applications are shown for few mode and field theory cases.

Dynamical symmetry breaking in a 2D electron gas with a spectral node

We study a disordered 2D electron gas with a spectral node in a vicinity of the node. After identifying the fundamental dynamical symmetries of this system, the spontaneous breaking of the latter by a Grassmann field is studied within a nonlinear sigma model approach. This allows us to reduce the average two-particle Green's function to a diffusion propagator with a random diffusion coefficient. The latter has non-degenerate saddle points and is treated by the conventional self-consistent Born approximation. This leads to a renormalized chemical potential and a renormalized diffusion coefficient, where the DC conductivity increases linearly with the density of quasiparticles. Applied to the special case of Dirac fermions, our approach provides a comprehensive description of the minimal conductivity at the Dirac node as well as for the V-shape conductivity inside the bands.

Grassmann phase space theory and the Jaynes–Cummings model

The Jaynes–Cummings model of a two-level atom in a single mode cavity is of fundamental importance both in quantum optics and in quantum physics generally, involving the interaction of two simple quantum systems—one fermionic system (the TLA), the other bosonic (the cavity mode). Depending on the initial conditions a variety of interesting effects occur, ranging from ongoing oscillations of the atomic population difference at the Rabi frequency when the atom is excited and the cavity is in an n-photon Fock state, to collapses and revivals of these oscillations starting with the atom unexcited and the cavity mode in a coherent state. The observation of revivals for Rydberg atoms in a high-Q microwave cavity is key experimental evidence for quantisation of the EM field. Theoretical treatments of the Jaynes–Cummings model based on expanding the state vector in terms of products of atomic and n-photon states and deriving coupled equations for the amplitudes are a well-known and simple method for determining the effects.In quantum optics however, the behaviour of the bosonic quantum EM field is often treated using phase space methods, where the bosonic mode annihilation and creation operators are represented by c-number phase space variables, with the density operator represented by a distribution function of these variables. Fokker–Planck equations for the distribution function are obtained, and either used directly to determine quantities of experimental interest or used to develop c-number Langevin equations for stochastic versions of the phase space variables from which experimental quantities are obtained as stochastic averages. Phase space methods have also been developed to include atomic systems, with the atomic spin operators being represented by c-number phase space variables, and distribution functions involving these variables and those for any bosonic modes being shown to satisfy Fokker–Planck equations from which c-number Langevin equations are often developed. However, atomic spin operators satisfy the standard angular momentum commutation rules rather than the commutation rules for bosonic annihilation and creation operators, and are in fact second order combinations of fermionic annihilation and creation operators. Though phase space methods in which the fermionic operators are represented directly by c-number phase space variables have not been successful, the anti-commutation rules for these operators suggest the possibility of using Grassmann variables—which have similar anti-commutation properties. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of phase space methods in quantum optics to treat fermionic systems by representing fermionic annihilation and creation operators directly by Grassmann phase space variables is rather rare. This paper shows that phase space methods using a positive P type distribution function involving both c-number variables (for the cavity mode) and Grassmann variables (for the TLA) can be used to treat the Jaynes–Cummings model. Although it is a Grassmann function, the distribution function is equivalent to six c-number functions of the two bosonic variables. Experimental quantities are given as bosonic phase space integrals involving the six functions. A Fokker–Planck equation involving both left and right Grassmann differentiations can be obtained for the distribution function, and is equivalent to six coupled equations for the six c-number functions.The approach used involves choosing the canonical form of the (non-unique) positive P distribution function, in which the correspondence rules for the bosonic operators are non-standard and hence the Fokker–Planck equation is also unusual. Initial conditions, such as those above for initially uncorrelated states, are discussed and used to determine the initial distribution function. Transformations to new bosonic variables rotating at the cavity frequency enable the six coupled equations for the new c-number functions–that are also equivalent to the canonical Grassmann distribution function–to be solved analytically, based on an ansatz from an earlier paper by Stenholm. It is then shown that the distribution function is exactly the same as that determined from the well-known solution based on coupled amplitude equations.In quantum–atom optics theories for many atom bosonic and fermionic systems are needed. With large atom numbers, treatments must often take into account many quantum modes—especially for fermions. Generalisations of phase space distribution functions of phase space variables for a few modes to phase space distribution functionals of field functions (which represent the field operators, c-number fields for bosons, Grassmann fields for fermions) are now being developed for large systems. For the fermionic case, the treatment of the simple two mode problem represented by the Jaynes–Cummings model is a useful test case for the future development of phase space Grassmann distribution functional methods for fermionic applications in quantum–atom optics.

Quantum diffusion in two-dimensional random systems with particle–hole symmetry

We study the scattering dynamics of an n-component spinor wavefunction in a random environment on a two-dimensional lattice. If the particle–hole symmetry of the Hamiltonian is spontaneously broken the dynamics of the quantum particles becomes diffusive on large scales. The latter is described by a non-interacting Grassmann field, indicating a special kind of asymptotic freedom on large scales in d = 2.

Exact Spin Liquid Ground States of the Quantum Dimer Model on the Square and Honeycomb Lattices

We study a generalized quantum hard-core dimer model on the square and honeycomb lattices, allowing for first and second neighbor dimers. At generalized Rokhsar-Kivelson points, the exact ground states can be constructed, and ground-state correlation functions can be equated to those of interacting (1+1)-dimensional Grassmann fields. When the concentration of second neighbor dimers is small, the ground-state correlations are shown to be short ranged corresponding to a (gaped) spin liquid phase. On a 2-torus, the ground states exhibit fourfold topological degeneracy. On a finite cylinder we have found a dramatic even-odd effect depending on the circumference and propose that this can be used as a numerical diagnostic of gapped spin-liquid phases, more generally.

The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice ℤ3

We consider an Euclidean supersymmetric field theory in ℤ3 given by a supersymmetric Φ4 perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green’s function of a (stable) Lévy random walk in ℤ3. The Green’s function depends on the Lévy-Khintchine parameter $\alpha ={3+\varepsilon \over 2}$ with 0<α<2. For $\alpha ={3\over 2}$ the Φ4 interaction is marginal. We prove for $\alpha -{3\over 2}={\varepsilon \over 2}>0$ sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green’s function of a (weakly) self-avoiding Lévy walk in ℤ3 is a second moment (two point correlation function) of the supersymmetric measure governing this model. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding Lévy walk in ℤ3.

Spin glass and ferromagnetism in disordered cerium compounds

The competition between spin glass, ferromagnetism and Kondo effect is analyzed here in a Kondo lattice model with an intersite random coupling ${J}_{ij}$ between the localized magnetic moments given by a generalization of the Mattis model [D. J. Mattis, Phys. Lett. 56A, 421 (1977)], which represents an interpolation between ferromagnetism and a highly disordered spin glass. Functional integral techniques with Grassmann fields have been used to obtain the partition function. The static approximation and the replica symmetric ansatz have also been used. The solution of the problem is presented as a phase diagram giving $T∕J$ vs ${J}_{K}∕J$, where $T$ is the temperature, ${J}_{K}$ and $J$ are the strengths of the intrasite Kondo and the intersite random couplings, respectively. If ${J}_{K}∕J$ is small, when temperature is decreased, there is a second-order transition from a paramagnetic to a spin glass phase. For lower $T∕J$, a first-order transition appears between the spin glass phase and a region where there are Mattis states which are thermodynamically equivalent to the ferromagnetism. For very low $T∕J$, the Mattis states become stable. On the other hand, it is found as solution a Kondo state for large ${J}_{K}∕J$ values. These results can improve the theoretical description of the well-known experimental phase diagram of ${\mathrm{CeNi}}_{1\ensuremath{-}x}{\mathrm{Cu}}_{x}$.

Quantum critical point in the spin glass–antiferromagnetism competition in Kondo latticesystems

A theory is proposed to describe the competition among antiferromagnetism(AF), spin glasses (SG) and the Kondo effect. The model describestwo Kondo sublattices with an intrasite Kondo interaction strengthJK and an interlattice quantum Ising interaction in the presence of a transverse fieldΓ. The interlattice coupling is a random Gaussian distributed variable (with average−2J0/N andvariance 32J2/N) while the Γ field is introduced as a quantum mechanism to produce spin flipping. The path integralformalism is used to study this fermionic problem where the spin operators arerepresented by bilinear combinations of Grassmann fields. The disorder is treated withinthe framework of the replica trick. The free energy and the order parametersof the problem are obtained by using the static ansatz and by choosing bothJ0/J and Γ/J≈(JK/J)2 to allow, as previously, a better comparison with the experimentalfindings. The results indicate the presence of a SG solution at lowJK/J and fortemperature T<Tf (Tf is the freezingtemperature). When JK/J isincreased, a mixed phase AF+SG appears, then an AF solution and finally a Kondo state is obtained for high values ofJK/J.Moreover, the behaviours of the freezing and Néel temperatures are also affected by the relationship betweenJK and thetransverse field Γ. The first presents a slight decrease while the second decreases towards a quantum criticalpoint (QCP). The obtained phase diagram has the same sequence as the experimental one forCe2Au1−xCoxSi3, ifJK is assumed toincrease with x, and in addition it also shows a qualitative agreement concerning the behaviour of thefreezing and the Néel temperatures.

Simulations of Script N = 2 super Yang-Mills theory in two dimensions

We present results from lattice simulations of ${\cal N}=2$ super Yang-Mills theory in two dimensions. The lattice formulation we use was developed in \cite{2dpaper} and retains both gauge invariance and an exact (twisted) supersymmetry for any lattice spacing. Results for both U(2) and SU(2) gauge groups are given. We focus on supersymmetric Ward identities, the phase of the Pfaffian resulting from integration over the Grassmann fields and the nature of the quantum moduli space.

The spin glass-antiferromagnetism competition in Kondo-lattice systems in the presence of a transverse applied magnetic field

A theory is proposed to describe the competition among antiferromagnetism (AF), spin glass (SG) and Kondo effect. The model describes two Kondo sublattices with an intrasite Kondo interaction strength J K and a random Gaussian interlattice interaction in the presence of a transverse field Γ . The Γ field is introduced as a quantum mechanism to produce spin flipping and the random coupling has average - 2 J 0 / N and variance 32 J 2 / N . The path integral formalism with Grassmann fields is used to study this fermionic problem, in which the disorder is treated within the framework of the replica trick. The free energy and the order parameters are obtained using the static ansatz. In this many parameters problem, we choose J 0 / J ≈ ( J K / J ) 2 and Γ / J ≈ ( J K / J ) 2 to allow a better comparison with the experimental findings. The obtained phase diagram has not only the same sequence as the experimental one for Ce 2 Au 1 - x Co x Si 3 , but mainly, it also shows a qualitative agreement concerning the behavior of the freezing temperature and the Neel temperature which decreases until a quantum critical point (QCP).

Landau gauge Jacobian and BRST symmetry

We propose a generalisation of the Faddeev–Popov trick for Yang–Mills fields in the Landau gauge. The gauge-fixing is achieved as a genuine change of variables. In particular, the Jacobian that appears is the modulus of the standard Faddeev–Popov determinant. We give a path integral representation of this in terms of auxiliary bosonic and Grassmann fields extended beyond the usual set for standard Landau gauge BRST. The gauge-fixing Lagrangian density appearing in this context is local and enjoys a new extended BRST and anti-BRST symmetry though the gauge-fixing Lagrangian density in this case is not BRST exact.