Abstract We 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 algebrasp(4M) the self‐energy of the repulsive interaction as a subalgebra (which is equivalent to the direct product ofM×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 ind= 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 ford= 2; therefore, it is also pointed out that one should consider different HST's ind= 2 spatial dimensions with the block diagonal densities as ‘hinge’ functions and that one has to introduce a coset decompositionSp(4M)\U(2M) into densities and anomalous terms of the total disorder‐self‐energysp(4M) for deriving a nonlinear sigma model.
Abstract We 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 algebrasp(4M) the self‐energy of the repulsive interaction as a subalgebra (which is equivalent to the direct product ofM×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 ind= 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 ford= 2; therefore, it is also pointed out that one should consider different HST's ind= 2 spatial dimensions with the block diagonal densities as ‘hinge’ functions and that one has to introduce a coset decompositionSp(4M)\U(2M) into densities and anomalous terms of the total disorder‐self‐energysp(4M) for deriving a nonlinear sigma model.
A coherent state path integral is considered for bosons with an ensemble average of a random potential and with an additional, repulsive interaction in the context of BEC under inclusion of specially prepared disorder. The essential normalization of the coherent state path integral, as a generating function of observables, is obtained from the non-equilibrium time contour for 'forward' and 'backward' propagation so that a time contour metric has to be taken into account in the ensemble average with the random potential. Therefore, the respective symmetries for the derivation of a nonlinear sigma model follow from the involved time contour metric which leads to a coset decomposition Sp(4)/U(2) x U(2) of the symplectic group Sp(4) with the subgroup U(2) for the unitary invariance of the density-related vacuum or ground state; the corresponding spontaneous symmetry breaking gives rise to anomalous- or 'Nambu'-doubled field degrees of freedom within self-energy matrices which are finally regarded by remaining coset matrices. The notion of a 'return probability', according to the original 'Anderson-localization', is thus naturally contained within coherent state path integrals of a non-equilibrium contour time for equivalent 'forward' and 'backward' propagation.
Abstract A 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 invarianceU(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‐groupOsp(S,S| 2L). A nonlinear sigma model follows from the spontaneous breaking of the ortho‐symplectic super‐groupOsp(S,S| 2L) to the coset decompositionOsp(S,S| 2L) \U(L|S)⊗U(L|S). The invariant subgroupU(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 spaceOsp(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 decompositionOsp(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 rootG^Osp\U−1/2 of the metric tensor ofOsp(S,S| 2L) \U(L|S) so that the non‐Euclidean integration measure with super‐Jacobi‐determinant[SDET(G^Osp\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.
Abstract On the basis of the attractive Hubbard model with a Gaussian random potential we describe superconductivity in the presence of quenched disorder. A dynamic path‐integral formalism with an 8‐component Keldysh‐Nambu spinor is shown to be convenient for solving the averaging problem without the recourse to the replica trick. The effective action is bosonized in terms of collective variables, namely local Q‐matrices and real space pair fields. Saddle point values are calculated in two simplified limits.
Weakly interacting Bose gases in a disorder environment have long been a challenging topic in the field of solid-state physics due to the intriguing interplay between superfluidity and localization. In this thesis we investigate theoretically the equilibrium properties of one- and three-dimensional harmonically trapped dirty Bose-Einstein condensates (BECs). In particular we focus on the decomposition of the total particle density into three components, namely, the condensate density, the thermal density, and the density of fragmented local Bose-Einstein condensates within the respective minima of the random potential landscape. In order to determine the different density components we develop a Hartree-Fock mean-field theory for a weakly interacting trapped BEC in a random environment, then specialize it to contact interaction and delta-correlated disorder. This non-perturbative theory is worked out on the basis of the replica method, which represents a well- established tool to deal with disorder problems. With this we calculate the corresponding free energy at finite temperature and derive from it the underlying self-consistency equations for the respective density components. Then, as a first step, we investigate a quasi one-dimensional Bose-Einstein condensed gas in a harmonic trapping potential with an additional delta- correlated disorder potential at zero temperature. To this end we solve the self-consistency equations within the Thomas-Fermi approximation and find the emergence of a Bose-glass region, where the condensate vanishes. We corroborate this analysis by an elaborate numerical treatment, where the corresponding one-dimensional time-independent Gross-Pitaevskii equation is numerically solved and disorder-ensemble averages of the condensate wave function are performed. The variance of the condensate wave function quantifies the number of bosons which condense in the local minima of the random potential. For weak disorder these mini-condensates turn out to occur preferentially at the border of the condensate, while for intermediate disorder strength this happens in the trap center. Additionally we use a variational ansatz in order to describe analytically the numerically observed redistribution of the fragmented mini-condensates with increasing disorder strength. In close analogy to the one-dimensional case we then treat the three-dimensional dirty Bose gas in an isotropic harmonic trap and a delta- correlated disorder potential at zero temperature using the corresponding self-consistency equations obtained via the Hartree-Fock mean-field theory within the Thomas-Fermi approximation. Additionally we use a variational ansatz, whose results turn out to coincide qualitatively with those obtained from the Thomas-Fermi approximation. Inparticular, a first order quantum phase transition from the superfluid phase, where the condensate density contributes to the total density, to the Bose-glass phase, where all particles are in the mini-condensates, is detected at a critical disorder strength, which agrees with findings in the literature. Furthermore, in a generaltriaxial harmonic trap, we investigate the geometric effect of different trap aspect ratios on the respective properties of the dirty BEC system. Finally, we consider the three- dimensional dirty BEC at finite temperature. This allows us to study the impact of both temperature and disorder fluctuations on the respective components of the density as well as their Thomas-Fermi radii. In particular, we find that the superfluid region, the Bose-glass region, and the thermal region coexist. Furthermore, depending on the system parameters, three phase transitions are detected, namely, one from the superfluid to the Bose-glass phase, one from the Boseglass to the thermal phase, and finally one directly from the superfluid to the thermal phase. All these results could be particularly useful for a quantitative analysis of world-wide on-going experiments with dirty bosons in quasi one-dimensional and three-dimensional harmonic traps.
Abstract Here we work out in detail a non-perturbative approach to the dirty boson problem, which relies on the Hartree–Fock theory and the replica method. For a weakly interacting Bose gas within a trapped confinement and a delta-correlated disorder potential at finite temperature, we determine the underlying free energy. From it we determine via extremization self-consistency equations for the three components of the particle density, namely the condensate density, the thermal density, and the density of fragmented local Bose–Einstein condensates within the respective minima of the random potential landscape. Solving these self-consistency equations in one and three dimensions in two other publications has revealed how these three densities change for increasing disorder strength.
We describe repulsively interacting Bose-Einstein condensates in spatially correlated disorder potentials of arbitrary dimension. The first effect of disorder is to deform the mean-field condensate. The second is that the quantum excitation spectrum and condensate population are affected. By a saddle-point expansion of the many-body Hamiltonian around the deformed mean-field ground state, we derive the fundamental quadratic Hamiltonian of quantum fluctuations. Importantly, a basis is used such that excitations are orthogonal to the deformed condensate. Via Bogoliubov-Nambu perturbation theory, we compute the effective excitation dispersion, including mean free paths and localization lengths. Corrections to the speed of sound and average density of states are calculated, due to correlated disorder in arbitrary dimensions, extending to the case of weak lattice potentials.