In this thesis, cold gases as well as the electron gas are analyzed with two expansion techniques that have, in essence, the same aim: To describe parameter regimes where the few-body aspect dominates over the collective many-body behavior.The Operator Product Expansion (OPE) of quantum field theory is used to extract high-frequency properties of the correlation function that determines the radio-frequency (RF) response of a two-dimensional Fermi gas with zero-range interactions.Exact sum rules that relate the zeroth and first moment of the radio-frequency spectrum to the particle number and the contact of the gas, are derived from these properties.A quantum cluster (virial-) expansion is employed to study the properties of the spectral function of the two-dimensional Fermi gas at small to moderate degeneracy.From the spectral function, observables like the density of states and the momentum distribution are calculated and are found to be in excellent agreement with Tan's universal relations.In the second part of the thesis, the cluster expansion is applied to both bosonic and fermionic three-dimensional quantum gases.Virial coefficients, contacts and momentum distribution are calculated up to third order in the fugacity.This includes three-particle correlations exactly.The results are are in excellent agreement with known universal relations.In the third part of this thesis, an OPE is applied to the problem of the interacting electron gas in both two and three dimensions.A power-law decay of the momentum distribution and the static structure factor at high momenta is found, compatible with earlier findings by J. Kimball.
In this thesis, cold gases as well as the electron gas are analyzed with two expansion techniques that have, in essence, the same aim: To describe parameter regimes where the few-body aspect dominates over the collective many-body behavior.The Operator Product Expansion (OPE) of quantum field theory is used to extract high-frequency properties of the correlation function that determines the radio-frequency (RF) response of a two-dimensional Fermi gas with zero-range interactions.Exact sum rules that relate the zeroth and first moment of the radio-frequency spectrum to the particle number and the contact of the gas, are derived from these properties.A quantum cluster (virial-) expansion is employed to study the properties of the spectral function of the two-dimensional Fermi gas at small to moderate degeneracy.From the spectral function, observables like the density of states and the momentum distribution are calculated and are found to be in excellent agreement with Tan's universal relations.In the second part of the thesis, the cluster expansion is applied to both bosonic and fermionic three-dimensional quantum gases.Virial coefficients, contacts and momentum distribution are calculated up to third order in the fugacity.This includes three-particle correlations exactly.The results are are in excellent agreement with known universal relations.In the third part of this thesis, an OPE is applied to the problem of the interacting electron gas in both two and three dimensions.A power-law decay of the momentum distribution and the static structure factor at high momenta is found, compatible with earlier findings by J. Kimball.
The thesis addresses the fundamental interplay between indistinguishability and interactions in few- and many-body quantum systems, treated in two separate approaches, both analytic in nature, using semiclassical methods. The few-body sector, considering a strictly fixed number of particles, is addressed within the framework of first quantization. By sticking to descriptions as (micro-)canonical ensembles the approach overcomes the inappropriate use of virial expansions. It is based on cluster expansions where quantum statistics are treated exactly and interparticle forces are described non-perturbatively. After a full formal development the cluster expansions are supplemented with short-time dynamical information that is equated with the description of smooth spectral properties or equivalently temperatures above strong quantum degeneracy, which, in the single-particle context, is the essential ingredient to what is known as semiclassical Weyl expansions. This reduction allows for the fully analytical description of universal features of large classes of systems by either truncating to a specific class of allowed collision processes, i.e., two-body collisions, or by extracting higher-order interaction kernels from related integrable models. The consistent combination with scaling considerations, minimal ground-state information and strong-coupling expansions leads to a toolbox providing most of the thermodynamic and spectral properties of one-dimensional systems with short-range interactions, all in terms of the lowest-order interaction kernels. The analytical results are in excellent qualitative and quantitative agreement with extensive numerical simulations for both integrable and non-integrable models of experimental relevance in cold atom physics. The many-body sector is in contrast treated in the second quantized formulation of quantum mechanics making use of the well-known collection of semiclassical methods that are subsumed under the name periodic orbit theory. As these methods apply in a highly system specific manner, the attractive Bose gas on a ring is chosen as exemplary model to show a case where the semiclassical quantization of mean-field dynamics, here considered as the classical analogue, can be used to obtain analytical non-trivial information on correlation-related many-body effects, especially related to quantum critical behavior in this model. After a simplification by truncating to the lowest single-particle momentum modes the consistent application of Einstein-Brillouin-Keller quantization yields a description valid for large numbers of bosons. Besides analytically reproducing numerical spectra to high accuracy, the approach enables one to extract asymptotic scaling laws for level spacings around quantum criticality, representing properties. These (finite) energy differences are intrinsically tied to inter-particle correlations and therefore elude a more conventional description by means of a mean-field theory extended with Bogoliubov quasi particles. An application to the scrambling time of the system, related to a pseudo quantum butterfly effect that stems from a local instability, underlines the importance of the method in the otherwise inaccessible regime of extreme particle numbers.
Universal relations that hold for any state provide powerful constraints on systems consisting of fermions with two spin states interacting with a large scattering length. In radio-frequency (rf) spectroscopy, the mean shift in the rf frequency and the large-frequency tail of the rf transition rate are proportional to the contact, which measures the density of pairs with small separations. We show that these universal relations can be derived and extended by using the short-time operator product expansion of quantum field theory. This is a general method for identifying aspects of many-body physics that are controlled by few-body physics.
The virial expansion characterizes the high-temperature approach to the quantum-classical crossover in any quantum many-body system. Here, we calculate the virial coefficients up to the fifth order of Fermi gases in one, two, and three dimensions, with attractive contact interactions, as relevant for a variety of applications in atomic and nuclear physics. To that end, we discretize the imaginary-time direction and calculate the relevant canonical partition functions. In coarse discretizations, we obtain analytic results featuring relationships between the interaction-induced changes $\mathrm{\ensuremath{\Delta}}{b}_{3}, \mathrm{\ensuremath{\Delta}}{b}_{4}$, and $\mathrm{\ensuremath{\Delta}}{b}_{5}$ as functions of $\mathrm{\ensuremath{\Delta}}{b}_{2}$, the latter being exactly known in many cases by virtue of the Beth-Uhlenbeck formula. Using automated-algebra methods, we push our calculations to progressively finer discretizations and extrapolate to the continuous-time limit. We find excellent agreement for $\mathrm{\ensuremath{\Delta}}{b}_{3}$ with previous calculations in all dimensions and we formulate predictions for $\mathrm{\ensuremath{\Delta}}{b}_{4}$ and $\mathrm{\ensuremath{\Delta}}{b}_{5}$ in one and two dimensions. We also provide, for a range of couplings, the subspace contributions $\mathrm{\ensuremath{\Delta}}{b}_{31}, \mathrm{\ensuremath{\Delta}}{b}_{22}, \mathrm{\ensuremath{\Delta}}{b}_{41}$, and $\mathrm{\ensuremath{\Delta}}{b}_{32}$, which determine the equation of state and static response of polarized systems at high temperature. As a performance check, we compare the density equation of state and Tan contact with quantum Monte Carlo calculations, diagrammatic approaches, and experimental data where available. Finally, we apply Pad\'e and Pad\'e-Borel resummation methods to extend the usefulness of the virial coefficients to approach and in some cases go beyond the unit-fugacity point.
Understanding quantum many-body systems with interactions is an important problem common to various fields of physics, such as atomic physics, condensed matter physics, nuclear physics, and particle physics.Among others, ultracold atoms have provided ideal grounds to study strongly interacting systems and to explore exotic quantum states because of their high controllability.We can widely control the strengths of the interactions from weak to strong as well as the geometry of the systems.In order to deepen our understanding of quantum manybody systems with interactions, we in this thesis investigate two types of one-dimensional (1D) systems realizable with ultracold atoms; bosons and spinless fermions near two-body resonances and bosons near a three-body resonance.One-dimensional systems near two-body resonances have been intensively studied because, in the homogeneous cases, they are exactly solvable systems.The energy spectra and thermodynamic quantities exactly obtained by the Bethe ansatz have greatly contributed to deepening our comprehension of many-body systems in 1D.On the other hand, the exact computations of correlation functions, which encode information about excitations of the systems, are in general much more complicated even in the integrable systems.Motivated by this background, we derive a series of exact relations for correlation functions in bosons and fermions near two-body resonances.These relations called universal relations originate from the universal properties of the resonant systems: Their properties depend on the interactions only through the scattering lengths characterizing the low-energy scattering.The universal relations are strong constraints on the systems because the relations hold for any particle number, scattering length, temperature, and with or without a trapping potential.The relations include the asymptotic behaviors of correlation functions in high-energy regime as well as the energy relations, in which the energies are expressed in term of the momentum distributions.The universal relations involve two-and three-body contacts, which are the integrals of local pair and triad correlations, respectively.Firstly we derive, in both bosons and fermions, universal relations for static correlation functions such as static structure factors and momentum distributions within the first-quantized formalisms.The power laws of the static correlation functions for large momentum and the energy relations are obtained.The coefficients in these power-law behaviors are proportional to the two-body contact.We clarify that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one.Secondly, we derive universal relations for dynamic correlation functions for bosons.Using the operator product expansion (OPE) in the field theoretical formalism, we obtain the largeenergy-momentum behaviors of the dynamic structure factor, the dynamic current correlation, and the single-particle spectral density as well as the quasiparticle energy and the scattering rate in high-energy regime.While the behaviors of the dynamic structure factor and the current correlation are proportional to the two-body contact, that of the single-particle spectral density is proportional to the number density.Because of the interrelation between 1D bosons and fermions near two-body resonances, the results of the dynamic structure factor and the current i
The study of strongly correlated quantum gases in two dimensions has important ramifications for understanding many intriguing phenomena in solid materials, such as high-Tc superconductivity and the fractional quantum-Hall effect. However, theoretical methods are plagued by the existence of significant quantum fluctuations. Here, we present two- and three-body exact solutions for both fermions and bosons trapped in a two-dimensional harmonic potential with an arbitrary s-wave scattering length. These few-particle solutions link in a natural way to the high-temperature properties of many-particle systems via a quantum virial expansion. As a concrete example, using the energy spectrum of few fermions, we calculate the second and third virial coefficients of a strongly interacting Fermi gas in two dimensions, and consequently investigate its high-temperature thermodynamics. Our thermodynamic results may be useful for ongoing experiments on two-dimensional Fermi gases. These exact results also provide an unbiased benchmark for quantum Monte Carlo simulations of two-dimensional Fermi gases at high temperatures.
In this thesis ultracold quantum gases are studied with field theoretical methods. The quantum mechanical three-body problem is solved exactly and the recently experimentally observed universality of the three-body parameter is explained. By means of the functional renormalization group (fRG) the four-body problem is analyzed and it is shown that no additional four-body parameter has to be introduced. The second part of this thesis studies a single impurity immersed in a majority Fermi sea. We develop a new, numerical fRG technique which allows the computation of Green’s functions with full frequency and momentum dependence. Our results are in excellent agreement with recent experiments and exemplify the intricate interplay between few- and many-body physics in ultracold quantum gases.