Final experimental achievement of Bose-Binstein condensation of trapped al- kali atoms boosted a renewed interest in the properties of weakly interacting Bose gases.The low-temperature behavior of such systems has been established years ago and became a standard subject of numerous textbooks.However, attempts to consistently extend the treatment to finite-temperature domain are still rare.In order to bridge this gap, we developed an approach, which allows one to estimate the physical parameters of weakly interacting Bose gases from first principles.We show that our method captures all essential physics in the whole range of tem- peratures except of those close to the critical.The source of the latter drawback turns out to be common to all schemes which rely on perturbative expansions.The issue can supposedly be dealt with by the use of the alternative framework, based on the n-particle irreducible effective action functional techniques.This approach attracts a lot of attention nowadays due to its unique resummation properties and its ability to deal with the non-equilibrium problems consistently.However, it is known to violate the gauge invariance when applied to high energy physics problems.We resolve the question by constructing a new modificcl effcctive theory, which not only preserves all the merits of the base formalism, but is also free from the gauge-dependence issues, thus making the entire approach internally consistent.Breakdown of perturbation expansions 63 Renormalization group approach 63 1.10.5 Calculations of the critical temperature [ 63 ot Jt, 38
Final experimental achievement of Bose-Binstein condensation of trapped al- kali atoms boosted a renewed interest in the properties of weakly interacting Bose gases.The low-temperature behavior of such systems has been established years ago and became a standard subject of numerous textbooks.However, attempts to consistently extend the treatment to finite-temperature domain are still rare.In order to bridge this gap, we developed an approach, which allows one to estimate the physical parameters of weakly interacting Bose gases from first principles.We show that our method captures all essential physics in the whole range of tem- peratures except of those close to the critical.The source of the latter drawback turns out to be common to all schemes which rely on perturbative expansions.The issue can supposedly be dealt with by the use of the alternative framework, based on the n-particle irreducible effective action functional techniques.This approach attracts a lot of attention nowadays due to its unique resummation properties and its ability to deal with the non-equilibrium problems consistently.However, it is known to violate the gauge invariance when applied to high energy physics problems.We resolve the question by constructing a new modificcl effcctive theory, which not only preserves all the merits of the base formalism, but is also free from the gauge-dependence issues, thus making the entire approach internally consistent.Breakdown of perturbation expansions 63 Renormalization group approach 63 1.10.5 Calculations of the critical temperature [ 63 ot Jt, 38
This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at T=0. The perturbative expansion of these properties is essentially an expansion in the gas parameter na3, where a is the s-wave scattering length and n is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include 1/N techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01
Quantitative numerical analyses of interacting dilute Bose-Einstein condensates are most often based on semiclassical approximations. Since the complex-valued field-theoretic action of the Bose gas does not offer itself to the direct application of standard Monte Carlo techniques, simulations beyond their scope by now almost exclusively rely on quantum-mechanical techniques. Here we explore an alternative approach based on a Langevin-type sampling in an extended state space, known as the complex Langevin (CL) algorithm. While the use of the CL technique has a long-standing history in high-energy physics, in particular in the simulation of QCD at finite baryon density, applications to ultracold atoms are still in their infancy. Here we examine the applicability of the CL approach for a one- and two-component, three-dimensional nonrelativistic Bose gas in thermal equilibrium, below and above the Bose-Einstein phase transition. By comparison with analytic descriptions at the Gaussian level, including Bogoliubov and Hartree-Fock theory, we find that the method allows computing reliably and efficiently observables in the regime of experimentally accessible parameters. Close to the transition, quantum corrections lead to a shift of the critical temperature which we reproduce within the numerical range known in the literature. With this work, we aim to provide a first test of CL as a potential out-of-the-box tool for the simulation of experimentally realistic situations, including trapping geometries and multicomponent (or multiple species) models.
We compute thermodynamic properties of dilute Bose gases using non-perturbative approximations of the two-particle irreducible (2PI) effective action. It is shown how to systematically renormalise the self-consistent descriptions beyond conventional Gaussian approximations such as Hartree-Fock-Bogoliubov theory. This allows us to determine the condensate depletion from low to high temperatures, including its critical behaviour at the phase transition. While the universal anomalous dimension at criticality is vanishing for Gaussian approximations, we determine its non-zero value at next-to-leading order of a self-consistent expansion in the number of field components.
We present a detailed derivation of the quantum and quantum–thermal effective action for non-relativistic systems, starting from the single-particle case and extending to the Gross–Pitaevskii (GP) field theory for weakly interacting bosons. In the single-particle framework, we introduce the one-particle-irreducible (1PI) effective action formalism, taking explicitly into account the choice of the initial quantum state, its saddle-point plus Gaussian-fluctuation approximation, and its finite-temperature extension via Matsubara summation, yielding a clear physical interpretation in terms of zero-point and thermal contributions to the Helmholtz free energy. The formalism is then applied to the GP action, producing the 1PI effective potential at zero and finite temperature, including beyond-mean-field Lee–Huang–Yang and thermal corrections. We discuss the gapless and gapped Bogoliubov spectra, their relevance to equilibrium and non-equilibrium regimes, and the role of regularization. Applications include the inclusion of an external potential within the local density approximation, the derivation of finite-temperature Josephson equations, and the extension to D-dimensional systems, with particular attention to the zero-dimensional limit. This unified approach provides a transparent connection between microscopic quantum fluctuations and effective macroscopic equations of motion for Bose–Einstein condensates.
In this chapter, we introduce the analytical machinery to investigate the properties of many-body systems perturbatively. Specifically, employing the “φ 4 -theory” as an example, we learn how to describe systems that are not too far from a known reference state by perturbative means. Diagrammatic methods are introduced as a tool to efficiently implement perturbation theory at large orders. The new concepts are then applied to the analysis of various properties of the weakly interacting electron gas. In previous chapters we have emphasized repeatedly that the majority of many-particle problems cannot be solved in closed form. Therefore, in general, one is compelled to think about approximation strategies. One promising ansatz leans on the fact that, when approaching the low-temperature physics of a many-particle system, we often have some idea, however vague, of its preferred states and/or its low-energy excitations. One may then set out to explore the system by using these prospective ground state configurations as a working platform. For example, one might expand the Hamiltonian in the vicinity of the reference state and check that, indeed, the residual “perturbations” acting in the low-energy sector of the Hilbert space are weak and can be dealt with by some kind of approximate expansion. Consider, for example, the quantum Heisenberg magnet. In dimensions higher than one, an exact solution of this system is out of the question. However, we know (or, more conservatively, “expect”) that, at zero temperature, the spins will be frozen into configurations aligned along some (domain-wise) constant magnetisation axes.