Coherent State Path Integral Research Articles

Coherent-state path integral versus coarse-grained effective stochastic equation of motion: From reaction diffusion to stochastic sandpiles.

We derive and study two different formalisms used for nonequilibrium processes: the coherent-state path integral, and an effective, coarse-grained stochastic equation of motion. We first study the coherent-state path integral and the corresponding field theory, using the annihilation process A+A→A as an example. The field theory contains counterintuitive quartic vertices. We show how they can be interpreted in terms of a first-passage problem. Reformulating the coherent-state path integral as a stochastic equation of motion, the noise generically becomes imaginary. This renders it not only difficult to interpret, but leads to convergence problems at finite times. We then show how alternatively an effective coarse-grained stochastic equation of motion with real noise can be constructed. The procedure is similar in spirit to the derivation of the mean-field approximation for the Ising model, and the ensuing construction of its effective field theory. We finally apply our findings to stochastic Manna sandpiles. We show that the coherent-state path integral is inappropriate, or at least inconvenient. As an alternative, we derive and solve its mean-field approximation, which we then use to construct a coarse-grained stochastic equation of motion with real noise.

Towards the entanglement negativity of two disjoint intervals for a one dimensional free fermion

We study the moments of the partial transpose of the reduced density matrix of two intervals for the free massless Dirac fermion. By means of a direct calculation based on a coherent state path integral, we find an analytic form for these moments in terms of the Riemann theta function. We show that moments of arbitrary order are equal to the same quantities for the compactified boson at the self-dual point. These equalities also imply the non-trivial result that the negativity of the free fermion and the self-dual boson are equal.

Coherent-state overcompleteness, path integrals, and weak values

In the Hilbert space of a quantum particle the standard coherent-state resolution of unity is written in terms of a phase-space integration of the outer product zz. Because no pair of coherent states is orthogonal, one can represent the closure relation in non-standard ways, in terms of a single phase-space integration of the “unlike” outer product z′z, z′≠z. We show that all known representations of this kind have a common ground and that our reasoning extends to spin coherent states. These unlike identities make it possible to write formal expressions for a phase-space path integral, where the role of the Hamiltonian H is played by a weak energy value Hweak. Therefore, in this context, we can speak of weak values without any mention to measurements. The quantity Hweak appears as the ruler of the phase-space dynamics in the semiclassical limit.

Rabi Oscillations in a Two-Level Atomic System with a Pseudo-Hermitian Hamiltonian: A Path Integral Approach

The time development of a two-level system with a pseudo-Hermitian Hamiltonian is studied using the spin coherent state path integral. The propagator is first written in the standard form by replacing the spin by a unit vector aligned along the polar and azimuthal directions (θ,φ). Then it is determined exactly using perturbation methods. The metric of the pseudo-Hermitian Hamiltonian and its bi-orthonormal basis of system states are deduced.

Corrections to the continuous time semiclassical coherent state path integral

By returning to the underlying discrete time formalism, we relate spurious results in coherent state path integral calculations to the high frequency structure of their propagators. We show how to modify the standard expressions for thermodynamic quantities to yield correct results. These expressions are relevant to a broad range of physical problems, from the thermodynamics of Bose lattice gases to the dynamics of spin systems.

Pseudo-Hermitian Interaction Between an Oscillator and a Spin-1/2 Particle in an External Magnetic Field: A Path Integral Approach

We use the coherent-state path integral and an angular model for the spin to solve the pseudo-hermitian interaction between an oscillator and a spin-1/2 particle in an external magnetic field. The propagators are given explicitly as a perturbation series and are summed up exactly. The energy spectrum and the bi-orthonormal basis of states are deduced.

Evolution dynamics of a model for gene duplication under adaptive conflict.

We present and solve the dynamics of a model for gene duplication showing escape from adaptive conflict. We use a Crow-Kimura quasispecies model of evolution where the fitness landscape is a function of Hamming distances from two reference sequences, which are assumed to optimize two different gene functions, to describe the dynamics of a mixed population of individuals with single and double copies of a pleiotropic gene. The evolution equations are solved through a spin coherent state path integral, and we find two phases: one is an escape from an adaptive conflict phase, where each copy of a duplicated gene evolves toward subfunctionalization, and the other is a duplication loss of function phase, where one copy maintains its pleiotropic form and the other copy undergoes neutral mutation. The phase is determined by a competition between the fitness benefits of subfunctionalization and the greater mutational load associated with maintaining two gene copies. In the escape phase, we find a dynamics of an initial population of single gene sequences only which escape adaptive conflict through gene duplication and find that there are two time regimes: until a time t single gene sequences dominate, and after t double gene sequences outgrow single gene sequences. The time t is identified as the time necessary for subfunctionalization to evolve and spread throughout the double gene sequences, and we show that there is an optimum mutation rate which minimizes this time scale.

Macroscopic quantum spin tunneling with two interacting spins

We study the simple Hamiltonian, $H=\ensuremath{-}K({S}_{1z}^{2}+{S}_{2z}^{2})+\ensuremath{\lambda}{\stackrel{P\vec}{S}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{P\vec}{S}}_{2}$, of two large, coupled spins which are taken equal, each of total spin $s$ with $\ensuremath{\lambda}$ the exchange coupling constant. The exact ground state of this simple Hamiltonian is not known for an antiferromagnetic coupling corresponding to the $\ensuremath{\lambda}>0$. In the absence of the exchange interaction, the ground state is fourfold degenerate, corresponding to the states where the individual spins are in their highest weight or lowest weight states, $|\ensuremath{\uparrow},\ensuremath{\uparrow}\ensuremath{\rangle},|\ensuremath{\downarrow},\ensuremath{\downarrow}\ensuremath{\rangle},|\ensuremath{\uparrow},\ensuremath{\downarrow}\ensuremath{\rangle},|\ensuremath{\downarrow},\ensuremath{\uparrow}\ensuremath{\rangle}$, in obvious notation. The first two remain exact eigenstates of the full Hamiltonian. However, we show that the two states $|\ensuremath{\uparrow},\ensuremath{\downarrow}\ensuremath{\rangle},|\ensuremath{\downarrow},\ensuremath{\uparrow}\ensuremath{\rangle}$ organize themselves into the combinations $|\ifmmode\pm\else\textpm\fi{}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|\ensuremath{\uparrow},\ensuremath{\downarrow}\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\rangle})$, up to perturbative corrections. For the antiferromagnetic case, we show that the ground state is nondegenerate, and we find the interesting result that for integer spins the ground state is $|+\ensuremath{\rangle}$ and the first excited state is the antisymmetric combination $|\ensuremath{-}\ensuremath{\rangle}$ while for half odd integer spin, these roles are exactly reversed. The energy splitting, however, is proportional to ${\ensuremath{\lambda}}^{2s}$, as expected by perturbation theory to the $2s$th order. We obtain these results through the spin coherent state path integral.

Geometric phase accompanying SU(2) coherent states for quantum polarized light

The geometric phase is studied for quantum polarized light described by an SU(2) coherent state. This is revealed by using quantum interference. The SU(2) coherent state is constructed from the quantized Stokes parameters. The quantum interferometry is arranged by the evolution operator implemented by the Hamiltonian inspired from the birefringence of the Faraday and Kerr effect. The geometric phase is given by the coherent state path integral. In particular, by considering a resonant Hamiltonian, the geometric phase is extracted from the total factor that consists of both geometric and dynamical phases. This extraction forms the basis of an experimental way to detect the geometric phase.

SU(2) COHERENT STATE PATH INTEGRALS LABELED BY A FULL SET OF EULER ANGLES: BASIC FORMULATION

We develop a basic formulation of the spin (SU(2)) coherent state path integrals based not on the conventional highest or lowest weight vectors but on arbitrary fiducial vectors. The coherent states, being defined on a 3-sphere, are specified by a full set of Euler angles. They are generally considered as states without classical analogues. The overcompleteness relation holds for the states, by which we obtain the time evolution of general systems in terms of the path integral representation; the resultant Lagrangian in the action has a monopole-type term à la Balachandran et al. as well as some additional terms, both of which depend on fiducial vectors in a simple way. The process of the discrete path integrals to the continuous ones is clarified. Complex variable forms of the states and path integrals are also obtained. During the course of all steps, we emphasize the analogies and correspondences to the general canonical coherent states and path integrals that we proposed some time ago. In this paper we concentrate on the basic formulation. The physical applications as well as criteria in choosing fiducial vectors for real Lagrangians, in relation to fictitious monopoles and geometric phases, will be treated in subsequent papers separately.

Renormalization Group: Applications in Statistical Physics

These notes provide a concise introduction to important applications of the renormalization group (RG) in statistical physics. After reviewing the scaling approach and Ginzburg-Landau theory for critical phenomena, Wilson's momentum shell RG method is presented, and the critical exponents for the scalar Phi^4 model are determined to first order in an eps expansion about d_c = 4. Subsequently, the technically more versatile field-theoretic formulation of the perturbational RG for static critical phenomena is described. It is explained how the emergence of scale invariance connects UV divergences to IR singularities, and the RG equation is employed to compute the critical exponents for the O(n)-symmetric Landau-Ginzburg-Wilson theory. The second part is devoted to field theory representations of non-linear stochastic dynamical systems, and the application of RG tools to critical dynamics. Dynamic critical phenomena in systems near equilibrium are efficiently captured through Langevin equations, and their mapping onto the Janssen-De Dominicis response functional, exemplified by the purely relaxational models with non-conserved (model A) / conserved order parameter (model B). The Langevin description and scaling exponents for isotropic ferromagnets (model J) and for driven diffusive non-equilibrium systems are also discussed. Finally, an outlook is presented to scale-invariant phenomena and non-equilibrium phase transitions in interacting particle systems. It is shown how the stochastic master equation associated with chemical reactions or population dynamics models can be mapped onto imaginary-time, non-Hermitian `quantum' mechanics. In the continuum limit, this Doi-Peliti Hamiltonian is represented through a coherent-state path integral, which allows an RG analysis of diffusion-limited annihilation processes and phase transitions from active to inactive, absorbing states.

Exact spin coherent state path integral for a two-level atom in gravitational fields

The movement of a two-level atom interacting with an electromagnetic wave while subject to gravity is studied using path-integral formalism. The propagator is first written in a standard form, ∫[Formula: see text](path) exp(i/ℏ)S(path), by replacing the spin with a unit vector aligned along the polar and azimuthal directions to determine the propagator exactly. Thus, the exact wave functions of the system are deduced.

Generalization of the coherent-state path integrals and systematic derivation of semiclassical propagators

The coherent path integral is generalized such that the identity operator represented in a complete (actually overcomplete) set of the coherent states with the ``time-variable'' exponents are inserted between two consecutive short-time propagators. Since such a complete set of any given exponent can constitute the identity operator, the exponent may be varied from time to time without loss of generality as long as it is set common to all the Gaussians. However, a finite truncation of the coherent state expansion should result in different values of the propagator depending on the choice of the exponents. Furthermore, approximation methodology to treat with the exact propagator can also depend on this choice, and thereby many different semiclassical propagators may emerge from these combinations. Indeed, we show that the well-known semiclassical propagators such as those of Van Vleck, Herman-Kluk, Heller's thawed Gaussian, and many others can be derived in a systematic manner, which enables one to comprehend these semiclassical propagators from a unified point of view. We are particularly interested in our generalized form of the Herman-Kluk propagator, since the relative accuracy of this propagator has been well established by Kay, and since, nevertheless, its derivation was not necessarily clear. Thus our generalized Herman-Kluk propagator replaces the classical Hamiltonian with a Gaussian averaged quantum Hamiltonian, generating non-Newtonian trajectories. We perform a numerical test to assess the quality of such a family of generalized Herman-Kluk propagators and find that the original Herman-Kluk gives an accurate result. The reason why this has come about is also discussed.

Phase structure of the anisotropic antiferromagnetic Heisenberg model on a layered triangular lattice: Spiral state and deconfined spin liquid

In the present paper, we study a spin-1/2 antiferromagnetic (AF) Heisenberg model on layered anisotropic triangular lattice and obtain its phase structure. We use the Schwinger bosons for representing spin operators and also a coherent-state path integral for calculating physical quantities. Finite-temperature properties of the system are investigated by means of the numerical Monte-Carlo simulations. A detailed phase diagram of the system is obtained by calculating internal energy, specific heat, spin correlation functions, etc. There are AF N\'eel, paramagnetic, and spiral states. Turning on the plaquette term (i.e., the Maxwell term on a lattice) of an emergent U(1) gauge field that flips a pair of parallel spin-singlet bonds, we found that there appears a phase that is regarded as a deconfined spin-liquid state, though ``transition'' to this phase from the paramagnetic phase is not of second order but a crossover. In that phase, the emergent gauge boson is a physical gapless excitation coupled with spinons. These results support our previous study on an AF Heisenberg model on a triangular lattice at vanishing temperature.

A path integral for the master constraint of loop quantum gravity

In the present paper, we start from the canonical theory of loop quantum gravity and the master constraint program. The physical inner product is expressed by using the group averaging technique for a single self-adjoint master constraint operator. By the standard technique of skeletonization and the coherent state path integral, we derive a path-integral formula from the group averaging for the master constraint operator. Our derivation in this paper suggests there exists a direct link connecting the canonical loop quantum gravity with a path-integral quantization or a spin-foam model of general relativity.

Coherent-State Path Integrals and Brownian Motions

A stochastic approach to the rigorous foundation of the coherent-state (phase-space) path integral is given. Stochastic integrals and some generalizations of the Feynman–Kac theorem are used for this purpose. In this approach, quantum mechanics is described in terms of the Fock–Bargmann representation; a classical Hamiltonian is related to the corresponding quantum Hamiltonian on the Fock–Bargmann space, seen as a Hilbert subspace of \(L^{2}({\bf R}^{2})\). The coherent-state path integral is realized as a conditional expectation of a stochastic process defined by the exponential of the Fisk–Stratonovich integral of the fundamental 1-form along a path of Brownian motion on the phase space \({\bf R}^{2}\).

Exact solution of the Bose-Hubbard model on the Bethe lattice

The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent state path integral, leads in the large connectivity limit to the mean field treatment of Fisher et al. [Phys. Rev. B {\bf 40}, 546 (1989)] at the leading order, and to the bosonic Dynamical Mean Field Theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B {\bf 77}, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions.

A note on the limiting procedures for path integrals

Limiting procedures for the Feynman type path integral are considered. The evolution operator is approximated with operators corresponding to the exponential of the Hamiltonian's symbol. The proof of convergence uses Chernoff's theory and its extension on the class of stable operators. The approach is especially adequate for the generalized coherent states path integral, where decomposition of a Hamiltonian into the sums used in the approach based on Trotter's formula may be unnatural.

Coherent state path integral and super‐symmetry for condensates composed of bosonic and fermionic atoms

AbstractA super‐symmetric coherent state path integral on the Keldysh time contour is considered for bosonic and fermionic atoms which interact among each other with a common short‐ranged two‐body potential. We investigate the symmetries of Bose‐Einstein condensation for the equivalent bosonic and fermionic constituents with the same interaction potential so that a super‐symmetry results between the bosonic and fermionic components of super‐fields. Apart from the super‐unitary invariance U(L | S) of the density terms, we specialize on the examination of super‐symmetries for pair condensate terms. Effective equations are derived for anomalous terms which are related to the molecular‐ and BCS‐ condensate pairs. A Hubbard‐Stratonovich transformation from ‘Nambu’‐doubled super‐fields leads to a generating function with super‐matrices for the self‐energy whose manifold is given by the orthosympletic super‐group Osp(S,S | 2L). A nonlinear sigma model follows from the spontaneous breaking of the ortho‐symplectic super‐group Osp(S,S | 2L) to the coset decomposition Osp(S,S | 2L) \ U(L | S)⊗ U(L | S). The invariant subgroup U(L | S) for the vacuum or background fields is represented by the density terms in the self‐energy whereas the super‐matrices on the coset space Osp(S,S | 2L) \ U(L | S) describe the anomalous molecular and BCS‐ pair condensate terms. A change of integration measure is performed for the coset decomposition Osp(S,S | 2L) \ U(L | S) ⊗ U(L | S), including a separation of density and anomalous parts of the self‐energy with a gradient expansion for the Goldstone modes. The independent anomalous fields in the actions can be transformed by the inverse square root $\hat{G}_{Osp\backslash U}^{-1/2}$ of the metric tensor of Osp(S,S | 2L) \ U(L | S) so that the non‐Euclidean integration measure with super‐Jacobi‐determinant $[{\rm SDET} (\hat{G}_{Osp\backslash U})]^{1/2}$ can be removed from the coherent state path integral and Gaussian‐like integrations remain. The variations of the independent coset fields in the effective actions result in classical field equations for a nonlinear sigma model with the anomalous terms. The dynamics of the eigenvalues of the coset matrices is determined by Sine‐Gordon equations which have a similar meaning for the dynamics of the molecular‐ and BCS‐pair condensates as the Gross‐Pitaevskii equation for the coherent wave function in BEC phenomena.

Ensemble averaged coherent state path integral for disordered bosons with a repulsive interaction (Derivation of mean field equations)

AbstractWe consider bosonic atoms with a repulsive contact interaction in a trap potential for a Bose‐Einstein condensation (BEC) and additionally include a random potential. The ensemble averages for two models of static (I) and dynamic (II) disorder are performed and investigated in parallel. The bosonic many body systems of the two disorder models are represented by coherent state path integrals on the Keldysh time contour which allow exact ensemble averages for zero and finite temperatures. These ensemble averages of coherent state path integrals therefore present alternatives to replica field theories or super‐symmetric averaging techniques. Hubbard‐Stratonovich transformations (HST) lead to two corresponding self‐energies for the hermitian repulsive interaction and for the non‐hermitian disorder‐interaction. The self‐energy of the repulsive interaction is absorbed by a shift into the disorder‐self‐energy which comprises as an element of a larger symplectic Lie algebra sp(4M) the self‐energy of the repulsive interaction as a subalgebra (which is equivalent to the direct product of M × sp(2); ‘M’ is the number of discrete time intervals of the disorder‐self‐energy in the generating function ). After removal of the remaining Gaussian integral for the self‐energy of the repulsive interaction, the first order variations of the coherent state path integrals magnified image result in the exact mean field or saddle point equations, solely depending on the disorder‐self‐energy matrix. These equations can be solved by continued fractions and are reminiscent to the `Nambu‐Gorkov' Green function formalism in superconductivity because anomalous terms or pair condensates of the bosonic atoms are also included into the selfenergies. The derived mean field equations of the models with static (I) and dynamic (II) disorder are particularly applicable for BEC in d = 3 spatial dimensions because of the singularity of the density of states at vanishing wavevector. However, one usually starts out from restricted applicability of the mean field approach for d = 2; therefore, it is also pointed out that one should consider different HST's in d = 2 spatial dimensions with the block diagonal densities as ‘hinge’ functions and that one has to introduce a coset decomposition Sp (4M)\U (2M) into densities and anomalous terms of the total disorder‐self‐energy sp(4M) for deriving a nonlinear sigma model.