One-particle Correlation Functions Research Articles

Effective Action in Free Probability

Recent works have explored relations between classical and quantum statistical physics on the one hand and Voiculescu's theory of free probability on the other. Motivated by these results, the present work focuses on the notion of effective action, which is closely related to the large deviation rate function in classical probability and one-particle irreducible correlation functions in quantum field theories. The central aim is to understand how it can be defined and studied in free probability. In this respect, we introduce a suitable diagrammatic formalism.

Subdynamics of fluctuations in a many-particle quantum system

It is shown that the relevant part of an equilibrium two-time correlation function of operators related to a subsystem of S < N particles, selected from the whole system of N ≫ 1 quantum particles, obeys the exact completely closed (homogeneous) evolution equation. This equation differs from the conventional Nakajima–Zwanzig equation by the absence of the undesirable inhomogeneous term comprising all interparticle correlations at the initial time moment. The equation defines the subdynamics in a subspace of S selected particles and is obtained without any use of approximation like the Boltzmann ‘molecular chaos’, factorized initial state or Bogoliubov’s principle of weakening of initial correlations. In the second order in a small inter-particle interaction, this equation reduces to the Markovian evolution equation, which for a one-particle correlation function (S = 1) takes the form of the linear generalized quantum Boltzmann-like equation for a relevant part of a correlation function (one particle momentum distribution function) with the additional terms conditioned by initial correlations taken into account. This equation makes it possible to obtain the solution for the corresponding one-particle correlation function, which is valid for all timescales and reveals the influence of initial correlations on the evolution in time process. On a large timescale t ≫ t cor ∼ ℏ/k B T (decoherence time), the initial correlations cease to influence the damping process and the evolution enters a kinetic stage. However, the factor related to initial correlations survives on the large timescale, and can contribute to the kinetic coefficients.

Experimental extraction of the quantum effective action for a non-equilibrium many-body system

Far-from-equilibrium situations are ubiquitous in nature. They are responsible for a wealth of phenomena, which are not simple extensions of near-equilibrium properties, ranging from fluid flows turning turbulent to the highly organized forms of life. On the fundamental level, quantum fluctuations or entanglement lead to novel forms of complex dynamical behaviour in many-body systems for which a description as emergent phenomena can be found within the framework of quantum field theory. A central quantity in these efforts, containing all information about the measurable physical properties, is the quantum effective action. Though the problem of non-equilibrium quantum dynamics can be exactly formulated in terms of the quantum effective action, the solution is in general beyond capabilities of classical computers. In this work, we present a strategy to determine the non-equilibrium quantum effective action using analog quantum simulators, and demonstrate our method experimentally with a quasi one-dimensional spinor Bose gas out of equilibrium. Building on spatially resolved snapshots of the spin degree of freedom, we infer the quantum effective action up to fourth order in an expansion in one-particle irreducible correlation functions at equal times. We uncover a strong suppression of the irreducible four-vertex emerging at low momenta, which solves the problem of dynamics in the highly occupied regime far from equilibrium where perturbative descriptions fail. Similar behaviour in this non-pertubative regime has been proposed in the context of early-universe cosmology. Our work constitutes a new realm of large-scale analog quantum computing, where the high level of control of synthetic quantum systems provides the means for the solution of long-standing theoretical problems in high-energy and condensed matter physics with an experimental approach.

Low-density expansions for the homogeneous dipolar Bose gas at zero temperature

The low-density expansions for the energy, chemical potential, and condensate depletion of the homogeneous dilute dipolar Bose gas are obtained by regularizing the dipole-dipole interaction at long distances. It is shown that the leading term, proportional to the density, allows a simple physical interpretation and consistently describes the thermodynamic stability of the system. The long-range asymptotics are obtained analytically for the normal and anomalous one-particle correlation functions and the pair distribution function. We discuss the properties of the two-body scattering with zero relative momentum for the dipole-dipole interaction, in particular, we derive the asymptotics of the wave function and a correction to the scattering length for small values of the dipolar range. We show how the density expansions can be derived within the Bogoliubov model of weakly interacting particles without any divergence from the assumption of universality of the expansions at low densities.

Quantum critical properties of Bose–Hubbard models

The Mott insulator-to-superfluid transition exhibited by the Bose–Hubbard model on a two-dimensional square lattice occurs for any value of the chemical potential, but becomes critical at the tips of the so-called Mott lobes only. Employing a numerical approach based on a combination of high-order perturbation theory and hypergeometric analytic continuation we investigate how quantum critical properties manifest themselves in computational practice. We consider two-dimensional triangular lattices and three-dimensional cubic lattices for comparison, providing accurate parametrizations of the phase boundaries at the tips of the respective first lobes. In particular, we lend strong support to a recently suggested inequality which bounds the divergence exponent of the one-particle correlation function in terms of that of the two-particle correlation function, and which sharpens to an equality if and only if a system becomes critical.

Droplet formation in a one-dimensional system of attractive spinless fermions

A translation invariant one-dimensional system of spinless fermions with a finite-range attraction experiences a quantum phase transition to a phase-separated state. While being a conventional Luttinger liquid for a small interaction strength, spinless fermions form a droplet with the size smaller than the available one-dimensional volume when the interaction strength exceeds some critical value. A particularly remarkable signature of the droplet formation is the change in the lower edge of the many-body excitation spectrum. In the homogeneous phase, it has a Luttinger-liquid shape of periodic arcs on top of the shallow parabolic dispersion of the center-of-mass. When the interaction strength is increased, the arcs disappear completely as soon as the droplet is formed. We perform an exact diagonalization study of this system with the focus on the signatures of the quantum phase transition and the droplet properties. The one-particle and density-density correlation functions, the pressure, the sound velocity, and the droplet density are examined. The value of the critical interaction strength obtained from numerical data reasonably agrees with a simple mean-field analytical estimate. Due to the boson-fermion correspondence valid in one dimension, our results also hold for hard-core bosons with a finite-range attraction.

Nonequilibrium fluctuation-dissipation relations for one- and two-particle correlation functions in steady-state quantum transport.

We study the non-equilibrium (NE) fluctuation-dissipation (FD) relations in the context of quantum thermoelectric transport through a two-terminal nanodevice in the steady-state. The FD relations for the one- and two-particle correlation functions are derived for a model of the central region consisting of a single electron level. Explicit expressions for the FD relations of the Green's functions (one-particle correlations) are provided. The FD relations for the current-current and charge-charge (two-particle) correlations are calculated numerically. We use self-consistent NE Green's functions calculations to treat the system in the absence and in the presence of interaction (electron-phonon) in the central region. We show that, for this model, there is no single universal FD theorem for the NE steady state. There are different FD relations for each different class of problems. We find that the FD relations for the one-particle correlation function are strongly dependent on both the NE conditions and the interactions, while the FD relations of the current-current correlation function are much less dependent on the interaction. The latter property suggests interesting applications for single-molecule and other nanoscale transport experiments.

Structure of Spherical Electric Double Layers Containing Mixed Electrolytes: A Systematic Study by Monte Carlo Simulations and Density Functional Theory

The structure of spherical electric double layers in the presence of mixed electrolytes is studied using Monte Carlo simulation and density functional theory within the restricted primitive model. The macroion is modeled as an impenetrable charged hard sphere carrying a uniform surface charge density, surrounded by the small ions represented as charged hard spheres, and the solvent is taken as a dielectric continuum. The density functional theory uses a partially perturbative scheme, where the hard-sphere contribution to the one-particle correlation function is evaluated using weighted density approximation and the ionic interactions are calculated using a second-order functional Taylor expansion with respect to a bulk electrolyte. The Monte Carlo simulations have been performed in canonical ensemble. The system is studied at varying ionic concentrations, at different concentration ratios of mono- and multivalent counterions of mixed electrolytes, at different diameters of hard spheres, at different macroion radius, and at varying polyion surface charge densities. The theoretical predictions in terms of the density profiles and the mean electrostatic potential profiles are found to be in good agreement with the simulation results. This model study shows clear manipulations of ionic size and charge correlations in dictating a number of interesting phenomena relating to width of the diffuse layer and charge inversion under different parametric conditions.

Spin-charge Separation of the Luttinger Model after an Interaction Quench

A Luttinger model of spin-½ fermions is considered after the interaction is suddenly switched on at time t = 0. By means of the bosonization technique, we evaluate analytically the one-particle correlation functions in detail, mainly involving equal-time correlations and propagators. The critical exponent which governs the power-law behavior of equal-time correlations for this spinful non-equilibrium system is obtained. In comparison with the published results, the difference between critical exponents of correlations in spinful and spinless non-equilibrium systems is found and explained. Furthermore, it is found that the propagator exhibits different power-law behavior from other equal-time correlations in this non-equilibrium system.

ANDERSON LOCALIZATION VS. MOTT–HUBBARD METAL–INSULATOR TRANSITION IN DISORDERED, INTERACTING LATTICE FERMION SYSTEMS

We review recent progress in our theoretical understanding of strongly correlated fermion systems in the presence of disorder. Results were obtained by the application of a powerful nonperturbative approach, the dynamical mean-field theory (DMFT), to interacting disordered lattice fermions. In particular, we demonstrate that DMFT combined with geometric averaging over disorder can capture Anderson localization and Mott insulating phases on the level of one-particle correlation functions. Results are presented for the ground state phase diagram of the Anderson–Hubbard model at half-filling, both in the paramagnetic phase and in the presence of antiferromagnetic order. We find a new antiferromagnetic metal which is stabilized by disorder. Possible realizations of these quantum phases with ultracold fermions in optical lattices are discussed.

Quantum fluctuations of a Bose-Josephson junction in a quasi-one-dimensional ring trap

Using a Luttinger-liquid approach we study the quantum fluctuations of a Bose-Josephson junction, consisting of a Bose gas confined to a quasi-one-dimensional ring trap which contains a localized repulsive potential barrier. For an infinite barrier we study the one-particle and two-particle static correlation functions. For the one-body density-matrix we obtain different power-law decays depending on the location of the probe points with respect to the position of the barrier. This quasi-long-range order can be experimentally probed in principle using an interference measurement. The corresponding momentum distribution at small momenta is also shown to be affected by the presence of the barrier and to display the universal power-law behavior expected for an interacting one-dimensional fluid. We also evaluate the particle-density profile, and by comparing with the exact results in the Tonks-Girardeau limit we fix the nonuniversal parameters of the Luttinger-liquid theory. Once the parameters are determined from one-body properties, we evaluate the density-density correlation function, finding a remarkable agreement between the Luttinger-liquid predictions and the exact result in the Tonks-Girardeau limit, even at the length scale of the Friedel-like oscillations which characterize the behavior of the density-density correlation function at intermediate distance. Finally, for a large but finite barrier we use the one-body correlation function to estimate the effect of quantum fluctuations on the renormalization of the barrier height, finding a reduction in the effective Josephson coupling energy, which depends on the length of the ring and on the interaction strength.

Density matrix of a quantum field in a particle-creating background

We examine the time evolution of a quantized field in external backgrounds that violate the stability of vacuum (particle-creating backgrounds). Our purpose is to study the exact form of the final quantum state (the density operator at the final instant of time) that has emerged from a given arbitrary initial state (from a given arbitrary density operator at the initial time instant) in the course of evolution. We find a generating functional that allows one to obtain density operators for an arbitrary initial state. Averaging over states of the subsystem of antiparticles (particles), we obtain explicit forms of reduced density operators for the subsystem of particles (antiparticles). Analyzing one-particle correlation functions, we establish a one-to-one correspondence between these functions and the reduced density operators. It is shown that in the general case a presence of bosons (e.g., gluons) in the initial state increases the creation rate of the same type of bosons. We discuss the question (and its relation to the initial stage of quark–gluon plasma formation) whether a thermal form of one-particle distribution can appear even if the final state of the complete system is not in thermal equilibrium. In this respect, we discuss some cases when pair-creation by an electric-like field can mimic the one-particle thermal distribution. We apply our technics to some QFT problems in slowly varying electric-like backgrounds: electric, SU(3) chromoelectric, and metric. In particular, we analyze the time and temperature behavior of the mean numbers of created particles, provided that the effects of switching the external field on and off are negligible. It is demonstrated that at high temperatures and in slowly varying electric fields the rate of particle-creation is essentially time-dependent.

Structure of spherical electric double layers: A density functional approach

A density functional theory is presented for the structure of spherical electric double layers within the restricted primitive model, where the macroion is considered as a hard sphere having uniform surface charge density, the small ions as charged hard spheres, and the solvent is taken as a dielectric continuum. The theory is partially perturbative as the hard-sphere contribution to the one-particle correlation function is evaluated using suitably averaged weighted density and the ionic part is obtained through a second-order functional Taylor expansion around the uniform fluid. The theory is in quantitative agreement with Monte Carlo simulation for the density profiles and the zeta potentials over a wide range of macroion sizes and electrolyte concentrations. The theory is able to provide interesting insights about the layering and the charge inversion phenomena occurring at the interface.

Time-dependent evolution of two coupled Luttinger liquids

We consider two disconnected Luttinger liquids which are coupled at $t=0$ through chiral density-density interactions. Both for $t<0$ and $t \geq 0$ the system is exactly solvable by means of bosonization and this allows to evaluate analytically the time-dependence of correlation functions. We find that in the long-time limit the critical exponent governing the one-particle correlation function differs from the exponent dictated by the equilibrium ground state of the coupled system. We also discuss how this reflects on some physical quantities which are accessible in real experiments.

Effect of Suddenly Turning on Interactions in the Luttinger Model

The evolution of correlations in the exactly solvable Luttinger model (a model of interacting fermions in one dimension) after a suddenly switched-on interaction is analytically studied. When the model is defined on a finite-size ring, zero-temperature correlations are periodic in time. However, in the thermodynamic limit, the system relaxes algebraically towards a stationary state which is well described, at least for some simple correlation functions, by the generalized Gibbs ensemble recently introduced by Rigol et al. (cond-mat/0604476). The critical exponent that characterizes the decay of the one-particle correlation function is different from the known equilibrium exponents. Experiments for which these results can be relevant are also discussed.

Detecting Correlation Functions of Ultracold Atoms through Fourier Sampling of Time-of-Flight Images

We propose a detection method for ultracold atoms which allows reconstruction of the full one-particle and two-particle correlation functions from the measurements. The method is based on Fourier sampling of the time-of-flight images through two consecutive impulsive Raman pulses. For applications of this method, we discuss a few examples, including detection of phase separation between superfluid and Mott insulators, various types of spin or superfluid orders, entanglement, exotic or fluctuating orders.

Stochastic Dynamics and Hierarchy for the Boltzmann Equation with Arbitrary Differential Scattering Cross Section

The stochastic dynamics for point particles that corresponds to the Boltzmann equation with arbitrary differential scattering cross section is constructed. We derive the stochastic Boltzmann hierarchy the solutions of which outside the hyperplanes of lower dimension where the point particles interact are equal to the product of one-particle correlation functions, provided that the initial correlation functions are products of one-particle correlation functions. A one-particle correlation function satisfies the Boltzmann equation. The Kac dynamics in the momentum space is obtained.

Phase theory and critical exponents for the Tomonaga–Luttinger model with harmonic confinement

A phase operator formulation for a recent model of interacting one-dimensional fermions in a harmonic trap is developed. The resulting theory is similar to the corresponding approach for the Luttinger model with open boundary conditions (OBC). However, in place of the spatial coordinate z, a dimensionless variable u defined on the unit circle appears as argument of the phase fields and u is nonlinearly related to z. Furthermore, form factors appear which reflect the harmonic trap geometry. The theory is applied to calculate one-particle correlation functions. In a properly defined thermodynamic limit, bulk and boundary critical exponents are calculated for the static two-point correlation function and the dynamic local correlation function. The local spectral density is also considered. The critical exponents found are in agreement with those known for OBC with the exception of the boundary scaling exponent Δ⊥.

Nonunitary spin-charge separation in a one-dimensional fermion gas.

In this Letter we report exact results on the infrared asymptotics of the one-particle dynamical correlation function of the gas of impenetrable spin 1/2 fermions at infinitesimal temperature. The correlation function shows signs of spin-charge separation with scaling behavior in the charge part and exponential decay as a function of the space coordinate in the spin part. Surprisingly, the anomalous dimensions in the charge part do not correspond to any unitary conformal field theory. We find that the fermion spectral weight has a power law divergency at low energy with the anomalous exponent -1/2.

Zero temperature correlation functions for the impenetrable fermion gas

We calculate the long time and distance asymptotics of the one-particle correlation functions in the model of impenetrable spin 1/2 fermions in 1+1 dimensions. We consider the spin disordered zero temperature regime, which occurs when the limit $T\to0$ is taken at a positive chemical potential. The asymptotic expressions are found from the asymptotic solution of the matrix Riemann-Hilbert problem related to the determinant representation of the correlation functions.