Functional Integral Formalism Research Articles

Functional integral for ultracold fermionic atoms

We develop a functional integral formalism for ultracold gases of fermionic atoms. It describes the BEC–BCS crossover and involves both atom and molecule fields. Beyond mean field theory we include the fluctuations of the molecule field by the solution of gap equations. In the BEC limit, we find that the low temperature behavior is described by a Bogoliubov theory for bosons. For a narrow Feshbach resonance these bosons can be associated with microscopic molecules. In contrast, for a broad resonance the interaction between the atoms is approximately pointlike and microscopic molecules are irrelevant. The bosons represent now correlated atom pairs or composite “dressed molecules”. The low temperature results agree with quantum Monte Carlo simulations. Our formalism can treat with general inhomogeneous situations in a trap. For not too strong inhomogeneities the detailed properties of the trap are not needed for the computation of the fluctuation effects—they enter only in the solutions of the field equations.

The Metric Operator and the Functional Integral Formulation of Pseudo-Hermitian Quantum Mechanics

Pseudo-Hermitian quantum theories are those in which the Hamiltonian H satisfies H† = ηHη-1, where η = e-Q is a positive-definite Hermitian operator, rather than the usual H† = H. In the operator formulation of such theories the standard Hilbert-space metric must be modified by the inclusion of η in order to ensure their probabilistic interpretation. With possible generalizations to quantum field theory in mind, it is important to ask how the functional integral formalism for pseudo-Hermitian theories differs from that of standard theories. It turns out that here Q plays quite a different role, serving primarily to implement a canonical transformation of the variables. It does not appear explicitly in the expression for the vacuum generating functional. Instead, the relation to the Hermitian theory is encoded via the dependence of Z on the external source j(t). These points are illustrated and amplified in various versions of the Swanson model, a non-Hermitian transform of the simple harmonic oscillator.

Aspects of hadron physics

Detailed investigations of the structure of hadrons are essential for understanding how matter is constructed from the quarks and gluons of Quantum chromodynamics (QCD), and amongst the questions posed to modern hadron physics, three stand out. What is the rigorous, quantitative mechanism responsible for confinement? What is the connection between confinement and dynamical chiral symmetry breaking? And are these phenomena together sufficient to explain the origin of more than 98% of the mass of the observable universe? Such questions may only be answered using the full machinery of nonperturbative relativistic quantum field theory. This contribution provides a perspective on progress toward answering these key questions. In so doing it will provide an overview of the contemporary application of Dyson-Schwinger equations in Hadron Physics. The presentation assumes that the reader is familiar with the concepts and notation of relativistic quantum mechanics, with the functional integral formulation of quantum field theory and with regularisation and renormalisation in its perturbative formulation.

Density-induced quantum phase transitions in the ground state of triplet superconductors

We consider the possibility of quantum phase transitions in the ground state of triplet superconductors where particle density is the tuning parameter. For definiteness, we focus on the case of one band quasi-one-dimensional triplet superconductors but many of our conclusions regarding the nature of the transition are quite general. Within the functional integral formulation, we calculate the electronic compressibility and superfluid density tensor as a function of the particle density for various triplet order-parameter symmetries and find that these quantities are nonanalytic when a critical value of the particle density is reached.

Functional-integral based perturbation theory for the Malthus-Verhulst process

We apply a functional-integral formalism for Markovian birth and death processes to determine asymptotic corrections to mean-field theory in the Malthus-Verhulst process (MVP). Expanding about the stationary mean-field solution, we identify an expansion parameter that is small in the limit of large mean population, and derive a diagrammatic expansion in powers of this parameter. The series is evaluated to fifth order using computational enumeration of diagrams. Although the MVP has no stationary state, we obtain good agreement with the associated {\it quasi-stationary} values for the moments of the population size, provided the mean population size is not small. We compare our results with those of van Kampen's $\Omega$-expansion, and apply our method to the MVP with input, for which a stationary state does exist.

Time-dependent Ginzburg-Landau theory for atomic Fermi gases near the BCS-BEC crossover

We construct a time-dependent Ginzburg-Landau (TDGL) theory for the superfluid atomic Fermi-gases near the Feshbach resonance from the fermion-boson model on the basis of the functional integral formalism. We show that the GL coefficients in the TDGL equation are complex numbers except in the BEC limit. The complex TDGL equation describes both damping and propagating dynamics, which leads to very rich superfluid dynamics near the Feshbach resonance. We predict multiple plane-wave modes corresponding to the in-phase and out-of-phase like oscillations between fermionic and bosonic superfluids and a galaxylike spiral pattern for a vortex.

Condition of quasi-periodicity in imaginary time as a constraint at functional integration and the time-dependent ZZ-correlator of the XX Heisenberg magnet

We treat the functional integration approach for calculation of longitudinal correlation functions of the XY Heisenberg magnet in a constant homogeneous magnetic field. Generating functionals of the correlators are defined in the form of functional integrals over anti-commuting variables. The peculiarity of the functional integrals consists of the fact that integration variables depend on the imaginary time quasi-periodically. The corresponding boundary conditions are accounted for as constraints which reduce the integration domain. For the XX Heisenberg magnet, an application of the approach is given in the case of a two-point correlator of the third components of spins with an explicit dependence on time. Bibliography: 24 titles.

Universality in phase transitions for ultracold fermionic atoms

We describe the gas of ultracold fermionic atoms by a functional integral for atom and molecule fields. The crossover from Bose-Einstein condensation (BEC) to BCS-type superfluidity shows universal features in terms of a concentration parameter for the ratio between scattering length and average interatomic distance. We discuss the relevance of the Yukawa coupling between atoms and molecules, establish an exact narrow resonance limit, and show that renormalized quantities are independent of the Yukawa coupling for the broad resonance and BCS and BEC limits. Within our functional integral formalism we compute the atom scattering in vacuum and the molecular binding energy. This connects the universal concentration parameter to the magnetic field of a given experiment. Beyond mean-field theory we include the fluctuations of the molecule field and the renormalization effects for the atom-molecule coupling. We find excellent agreement with the observed fraction of bare molecules in $^{6}\mathrm{Li}$ and qualitative agreement with the condensate fraction in $^{6}\mathrm{Li}$ and $^{40}\mathrm{K}$. In addition to the phase diagram and condensate fraction we compute the correlation length for molecules, the in-medium scattering length for molecules and atoms, and the sound velocity.

Excitation Spectrum of Spin-1 Bosonic Atoms in Mott InsulatingPhase

The effective action for spin-1 bosonic atom in an opticallattice is derived. The quasiparticle and quasihole dispersionsare calculated for different cases by using a functional integralformalism. For all cases, the excitation spectra are analyzed. All thequasiparticle and quasihole excitations start with a gap.

Post-Gaussian Effective Potential of Double sine-Gordon Field

In the framework of the functional integral formalism, wecalculate the effective potential of the double sine-Gordon (DsG) model up to thesecond order with an optimized expansion and the Coleman'snormal-ordering prescription. Within the range of convergence, wemake a comparison among the classical and the effective potentialof the first and second order. The numerical analysis shows thatthe DsG post-Gaussian EP possesses some fine global properties and makes a substantial and a concordant quantum correction to thefeatures of the classical potential.

The Thirring–Wess model revisited: a functional integral approach

We consider the Wess–Zumino–Witten theory to obtain the functional integral bosonization of the Thirring–Wess model with an arbitrary regularization parameter. Proceeding a systematic of decomposing the Bose field algebra into gauge-invariant- and gauge-non-invariant field subalgebras, we obtain the local decoupled quantum action. The generalized operator solutions for the equations of motion are reconstructed from the functional integral formalism. The isomorphism between the QED 2 ( QCD 2) with broken gauge symmetry by a regularization prescription and the Abelian (non-Abelian) Thirring–Wess model with a fixed bare mass for the meson field is established.

Equivalence of Many-Gluon Green’s Functions in the Duffin-Kemmer-Petieu and Klein-Gordon-Fock Statistical Quantum Field Theories

We prove the equivalence of many-gluon Green functions in Duffin-Kemmer-Petieu and Klein-Gordon-Fock statistical quantum field theories. The proof is based on the functional integral formulation for the statistical generating functional in a finite-temperature quantum field theory. As an illustration, we calculate one-loop polarization operators in both theories and show that their expressions indeed coincide.

Bose representation for a strongly coupled nonequilibrium fermionic superfluid in a time-dependent trap

Using the functional integral formulation of a nonequilibrium quantum many-body theory we develop a regular description of a Fermi system with a strong attractive interaction in the presence of an external time-dependent potential. In the strong coupling limit this fermionic system is equivalent to a noequilibrium dilute Bose gas of diatomic molecules. We also consider a nonequilibrim strongly coupled Bardeen-Cooper-Schrieffer (BCS) theory and show that it reduces to the full nonlinear time-dependent Gross-Pitaevski (GP) equation, which determines an evolution of the condensate wave function.

Source optimization for image fidelity and throughput

In this study we first review existing source optimization methods. Lack of rigorous formulations motivates discussion of the optimization objectives and constraints. We stress the importance of using weighted and so-called Sobolev norms. Second, we state the main optimization problem as a set of the optimization objectives in a form of functional norm integrals to maximize image fidelity and source smoothness. We reduce this to a non-negative least square (NNLS) problem, which is solved by standard numerical methods. Third, we analyze solutions for important practical cases: alternating phase shifting applied to regular and semiregular pattern of contact holes, two types of SRAM cells with design rules from 130 to 250 nm, and a complex semidense contact layer pattern. Finally, we show how constraint optimization can be used to smooth strong off-axis quadrupole illuminations to achieve better image fidelity for some selected layout patterns.

Generalized Tomonaga-Schwinger equation from the Hadamard formula

A generalized Tomonaga-Schwinger equation, holding on the entire boundary of a finite spacetime region, has recently been considered as a tool for studying particle scattering amplitudes in background-independent quantum field theory. The equation has been derived using lattice techniques under assumptions on the existence of the continuum limit. Here I show that in the context of continuous Euclidean field theory the equation can be directly derived from the functional integral formalism, using a technique based on Hadamard's formula for the variation of the propagator.

Bosonization and generalized Mandelstam soliton operators

The generalized massive Thirring model (GMT) with three fermion species is bosonized in the context of the functional integral and operator formulations and shown to be equivalent to a generalized sine-Gordon model (GSG) with three interacting soliton species. The generalized Mandelstam soliton operators are constructed and the fermion-boson mapping is established through a set of generalized bosonization rules in a quotient positive definite Hilbert space of states. Each fermion species is mapped to its corresponding soliton in the spirit of particle/soliton duality of Abelian bosonization. In the semi-classical limit one recovers the so-called SU(3) affine Toda model coupled to matter fields (ATM) from which the classical GSG and GMT models were recently derived in the literature. The intermediate ATM like effective action possesses some spinors resembling the higher grading fields of the ATM theory which have non-zero chirality. These fields are shown to disappear from the physical spectrum, thus providing a bag model like confinement mechanism and leading to the appearance of the massive fermions (solitons). The ordinary MT/SG duality turns out to be related to each SU(2) sub-group. The higher rank Lie algebra extension is also discussed.

Equivalence of Many-Photon Green's Functions in the Duffin–Kemmer–Petiau and Klein–Gordon–Fock Statistical Quantum Field Theories

Using the functional integral formalism for the statistical generating functional in the statistical (finite temperature) quantum field theory, we prove the equivalence of many-photon Green's functions in the Duffin–Kemmer–Petiau and Klein–Gordon–Fock statistical quantum field theories. As an illustration, we calculate the one-loop polarization operators in both theories and demonstrate their coincidence.

Functional integral bosonization for an impurity in a Luttinger liquid

We use a functional integral formalism developed earlier for the pure Luttinger liquid (LL) to find an exact representation for the electron Green function of the LL in the presence of a single backscattering impurity in the low-temperature limit. This allows us to reproduce results (well known from the bosonization techniques) for the suppression of the electron local density of states (LDOS) at the position of the impurity and for the Friedel oscillations at finite temperature. In addition, we have extracted from the exact representation an analytic dependence of LDOS on the distance from the impurity and shown how it crosses over to that for the pure LL.

Decoherence and dissipation of an open quantum system with a squeezed and frequency-modulated heat bath

The functional integral formalism is used to provide a unified treatment of the effect of squeezing of the bath and frequency modulation of the system–bath coupling on the decoherence and dissipation properties of an open quantum system. The system chosen is a standard one consisting of a particle in a harmonic oscillator potential interacting with a bath of harmonic oscillators by a coupling of the position–position kind. Using an Ohmic bath and the high-temperature limit, the coefficients of the master equation describing dissipation and decoherence are obtained and analyzed.

Existence of Baryons, Baryon Spectrum and Mass Splitting in Strong Coupling Lattice QCD

We consider a functional integral formulation for one-flavor lattice Quantum Chromodynamics in d=2,3 space dimensions and imaginary time, and work in the regime of the small hopping parameter , and zero plaquette coupling. Following the standard construction, this model exhibits positivity which is used to obtain the underlying physical Hilbert space . Then, using a Feynman-Kac formalism, we write the correlation functions for the model as functional integrals over the space of Grassmannian (fermionic) fields for one quark specie and the SU(3) gauge fields. We determine the energy-momentum spectrum associated with gauge invariant local baryon (anti-baryon) fields which are composites of three quark (anti-quark) fields. With the associated correlation functions, we establish a Feynman-Kac formula, and a spectral representation for the Fourier transform of the two-point functions. This representation allows us to show that baryons and anti-baryons arise as tightly bound, bound states of three (anti-)quarks. Labelling the components of the baryon fields by s=3/2,1/2,-1/2,-3/2, we show that the baryon and anti-baryon mass spectrum only depends on |s|, and the associated masses are given by M s = −3lnκ+r s (κ), where r s (κ) is real analytic in κ, for each d=2,3. The mass splitting is M 3/2 −M 1/2 =18κ6, for d=2 and, if any, is at least of (κ7), for d=3. In the subspace o ⊂ generated by an odd number of fermions, the baryon and anti-baryon energy-momentum dispersion curves are isolated up to near the baryon-meson threshold −5lnκ (upper gap property), identical and are determined up to (κ5). The symmetries of coordinate reflections, spatial lattice rotations, parity and charge conjugation are established for the correlation functions, and are shown to be implemented on by unitary (anti-unitary, for time reversal) operators.