Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Path integral formalism of Bose Einstein condensation and superconductivity

Tom Kim

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

I set up the path integral formalism to study interacting bosonic gases, binary boson-fermion mixtures, and fermionic superconductors. The large-N theory is applied to bosonic interactions. The corresponding auxiliary field theory is applied to fermionic interactions. Using the thermodynamic quantities obtained by evaluating the path integral, I map out the phase boundaries. For fermionic superconductors, the path integral method reproduces well-known methods that were developed previously.

Related papers

Thermodynamics and structural transition of binary atomic Bose-Fermi mixtures in box or harmonic potentials: A path-integral study

Tom Kim, Chih-Chun Chien

Experimental realizations of a variety of atomic binary Bose-Fermi mixtures have brought opportunities for studying composite quantum systems with different spin statistics. The binary atomic mixtures can exhibit a structural transition from a mixture into phase separation as the boson-fermion interaction increases. By using a path-integral formalism to evaluate the grand partition function and the thermodynamic grand potential, we obtain the effective potential of binary Bose-Fermi mixtures. Thermodynamic quantities in a broad range of temperatures and interactions are also derived. The structural transition can be identified as a loop of the effective potential curve, and the volume fraction of phase separation can be determined by the lever rule. For $^{6}\mathrm{Li}\ensuremath{-}^{7}\mathrm{Li}$ and $^{6}\mathrm{Li}\ensuremath{-}^{41}\mathrm{K}$ mixtures, we present the phase diagrams of the mixtures in a box potential at zero and finite temperatures. Due to the flexible densities of atomic gases, the construction of phase separation is more complicated when compared to conventional liquid or solid mixtures where the individual densities are fixed. For harmonically trapped mixtures, we use the local density approximation to map out the finite-temperature density profiles and present typical trap structures, including the mixture, partially separated phases, and fully separated phases.

arXiv:1802.06510•Physical Review A•Mar 29, 2018•6 citations

Effects of disorder on bosonic superfluids

Nicola Dragoni

In this work we showed and review a modern approach to treat disorder in bosonic superfluids, within the framework of path-integral approach to quantum field theory. The methods we present are focused on the evaluation of physical quantities of interest, like the condensate depletion and normal-fluid particle density, using correlators evaluated in harmonic approximation. We treated different kinds of disorder and presented an original extension of the theory to the case of particles interacting with a finite-range potential, instead of the contact interaction. We also showed how this formalism can be applied to the problem of flux lines in type-II sueprconductors.

•Sep 1, 2018•0 citations

Coherent-state path integrals in quantum thermodynamics

Luca Salasnich, Cesare Vianello

In these notes, we elucidate some subtle aspects of coherent-state path integrals, focusing on their application to the equilibrium thermodynamics of quantum many-particle systems. These subtleties emerge when evaluating path integrals in the continuum, either in imaginary time or in Matsubara-frequency space. Our central message is that, when handled with due care, the path integral yields results identical to those obtained from the canonical Hamiltonian approach. We illustrate this through a pedagogical treatment of several paradigmatic systems: the bosonic and fermionic harmonic oscillators, the single-site Bose-Hubbard and Hubbard models, the weakly-interacting Bose gas with finite-range interactions, and the BCS superconductor with finite-range interactions.

arXiv:2511.11547•SciPost Physics Lecture Notes•Feb 18, 2026•1 citation

Path Integral Approach to the BCS Theory

Lukong Cornelius Fai

This chapter explains an accurate theory of interacting Fermi mixtures with spin imbalance that shows novel features and a far richer phase diagram compared to the balanced counterpart. Ultracold quantum gases of fermionic atoms are at the crossroads of both experimental and theoretical physicists and present the possibility of experimentally studying various pairing phenomena due to the remarkable experimental control in these gases. This has related to many fundamental discoveries and, in particular, to the realization of the crossover from a Bardeen, Cooper and Schrieffer (BCS) superfluid of loosely bound Cooper pairs to a Bose-Einstein condensate (BEC) of tightly bound molecules, the so-called the BEC-BCS crossover. The BCS theory is one of the greatest theoretical advances in physics in the second half of the twentieth century and was first fully formulated in 1957 by Bardeen, Cooper and Schrieffer.

CRC Press eBooks•Jun 20, 2019•0 citations

Path integral molecular dynamics for thermodynamics and Green’s function of ultracold spinor bosons

Y. C. Yu, Shujuan Liu, Hongwei Xiong +1 more

Most recently, the path integral molecular dynamics has been successfully used to consider the thermodynamics of single-component identical bosons and fermions. In this work, the path integral molecular dynamics is developed to simulate thermodynamics, Green's function, and momentum distribution of two-component bosons in three dimensions. As an example of our general method, we consider the thermodynamics of up to 16 bosons in a three-dimensional harmonic trap. For noninteracting spinor bosons, our simulation shows a bump in the heat capacity. As the repulsive interaction strength increases, however, we find the gradual disappearance of the bump in the heat capacity. We believe that this simulation result can be tested by ultracold spinor bosons with optical lattices and magnetic-field Feshbach resonance to tune the inter-particle interaction. We also calculate Green's function and momentum distribution of spinor bosons. Our work facilitates the exact numerical simulation of spinor bosons, whose property is one of the major problems in ultracold Bose gases.

arXiv:2207.07653•The Journal of Chemical Physics•Jul 14, 2022•8 citations

Path integral Monte Carlo simulations for fermion systems : Pairing in the electron-hole plasma

John B. Shumway, David M. Ceperley

We review the path integral method wherein quantum systems are mapped with Feynman's path integrals onto a classical system of "ring-polymers" and then simulated with the Monte Carlo technique. Bose or Fermi statistics correspond to possible "cross-linking" of polymers. As proposed by Feynman, superfluidity and Bose condensation result from macroscopic exchange of bosons. To map fermions onto a positive probability distribution, one must restrict the paths to lie in regions where the fermion density matrix is positive. We discuss a recent application to the two-component electron-hole plasma. At low temperature excitons and bi-excitons form. We have used nodal surfaces incorporating paired fermions and see evidence of a Bose condensation in the energy, specific heat and superfluid density. In the restricted path integral picture, pairing appears as intertwined electron-hole paths. Bose condensation occurs when these intertwined paths wind around the periodic boundaries.

arXiv:cond-mat/9909434•Journal de Physique IV (Proceedings)•Mar 1, 2000•14 citations
I set up the path integral formalism to study interacting bosonic gases, binary boson-fermion mixtures, and fermionic superconductors. The large-N theory is applied to bosonic interactions. The corresponding auxiliary field theory is applied to fermionic interactions. Using the thermodynamic quantities obtained by evaluating the path integral, I map out the phase boundaries. For fermionic superconductors, the path integral method reproduces well-known methods that were developed previously.
Experimental realizations of a variety of atomic binary Bose-Fermi mixtures have brought opportunities for studying composite quantum systems with different spin statistics. The binary atomic mixtures can exhibit a structural transition from a mixture into phase separation as the boson-fermion interaction increases. By using a path-integral formalism to evaluate the grand partition function and the thermodynamic grand potential, we obtain the effective potential of binary Bose-Fermi mixtures. Thermodynamic quantities in a broad range of temperatures and interactions are also derived. The structural transition can be identified as a loop of the effective potential curve, and the volume fraction of phase separation can be determined by the lever rule. For $^{6}\mathrm{Li}\ensuremath{-}^{7}\mathrm{Li}$ and $^{6}\mathrm{Li}\ensuremath{-}^{41}\mathrm{K}$ mixtures, we present the phase diagrams of the mixtures in a box potential at zero and finite temperatures. Due to the flexible densities of atomic gases, the construction of phase separation is more complicated when compared to conventional liquid or solid mixtures where the individual densities are fixed. For harmonically trapped mixtures, we use the local density approximation to map out the finite-temperature density profiles and present typical trap structures, including the mixture, partially separated phases, and fully separated phases.
In this work we showed and review a modern approach to treat disorder in bosonic superfluids, within the framework of path-integral approach to quantum field theory. The methods we present are focused on the evaluation of physical quantities of interest, like the condensate depletion and normal-fluid particle density, using correlators evaluated in harmonic approximation. We treated different kinds of disorder and presented an original extension of the theory to the case of particles interacting with a finite-range potential, instead of the contact interaction. We also showed how this formalism can be applied to the problem of flux lines in type-II sueprconductors.
In these notes, we elucidate some subtle aspects of coherent-state path integrals, focusing on their application to the equilibrium thermodynamics of quantum many-particle systems. These subtleties emerge when evaluating path integrals in the continuum, either in imaginary time or in Matsubara-frequency space. Our central message is that, when handled with due care, the path integral yields results identical to those obtained from the canonical Hamiltonian approach. We illustrate this through a pedagogical treatment of several paradigmatic systems: the bosonic and fermionic harmonic oscillators, the single-site Bose-Hubbard and Hubbard models, the weakly-interacting Bose gas with finite-range interactions, and the BCS superconductor with finite-range interactions.
This chapter explains an accurate theory of interacting Fermi mixtures with spin imbalance that shows novel features and a far richer phase diagram compared to the balanced counterpart. Ultracold quantum gases of fermionic atoms are at the crossroads of both experimental and theoretical physicists and present the possibility of experimentally studying various pairing phenomena due to the remarkable experimental control in these gases. This has related to many fundamental discoveries and, in particular, to the realization of the crossover from a Bardeen, Cooper and Schrieffer (BCS) superfluid of loosely bound Cooper pairs to a Bose-Einstein condensate (BEC) of tightly bound molecules, the so-called the BEC-BCS crossover. The BCS theory is one of the greatest theoretical advances in physics in the second half of the twentieth century and was first fully formulated in 1957 by Bardeen, Cooper and Schrieffer.
Most recently, the path integral molecular dynamics has been successfully used to consider the thermodynamics of single-component identical bosons and fermions. In this work, the path integral molecular dynamics is developed to simulate thermodynamics, Green's function, and momentum distribution of two-component bosons in three dimensions. As an example of our general method, we consider the thermodynamics of up to 16 bosons in a three-dimensional harmonic trap. For noninteracting spinor bosons, our simulation shows a bump in the heat capacity. As the repulsive interaction strength increases, however, we find the gradual disappearance of the bump in the heat capacity. We believe that this simulation result can be tested by ultracold spinor bosons with optical lattices and magnetic-field Feshbach resonance to tune the inter-particle interaction. We also calculate Green's function and momentum distribution of spinor bosons. Our work facilitates the exact numerical simulation of spinor bosons, whose property is one of the major problems in ultracold Bose gases.
We review the path integral method wherein quantum systems are mapped with Feynman's path integrals onto a classical system of "ring-polymers" and then simulated with the Monte Carlo technique. Bose or Fermi statistics correspond to possible "cross-linking" of polymers. As proposed by Feynman, superfluidity and Bose condensation result from macroscopic exchange of bosons. To map fermions onto a positive probability distribution, one must restrict the paths to lie in regions where the fermion density matrix is positive. We discuss a recent application to the two-component electron-hole plasma. At low temperature excitons and bi-excitons form. We have used nodal surfaces incorporating paired fermions and see evidence of a Bose condensation in the energy, specific heat and superfluid density. In the restricted path integral picture, pairing appears as intertwined electron-hole paths. Bose condensation occurs when these intertwined paths wind around the periodic boundaries.