Regularization of the zero-range interaction potential
When three particles interact only via the zero-range unitary potential, the total density matrix features a singularity at short three-body distances. This phenomenon, called the Thomas collapse (Thomas, 1935), may be overcome by a three-body short-range regularization.
Let us consider three distinguishable identical particles, mutually interacting with the zero-range potential, at very high temperature. For distances higher than the de Broglie thermal wave-length th (that may be set to an arbitrarily small value by increasing the temperature), the total interaction correction factor of a configuration will be, within the pair-product approximation (see Section 1.2.2):
g(r12; r12; )g(r23; r23; )g(r31; r31; ).
In the former section, we saw that in absence of a regularization, three bosons will collapse into a state of infinitely negative energy and zero extension for all values of a, although there is no pair bound state at unitarity (and for a < 0). It is thus reasonable to expect that with the three-body hard-core regularization, the same kind of effective interaction as that leading to the Thomas collapse will induce the existence of a three-body bound state, a surprising situation known as the Efimov effect. In this section, I use the path-integral formalism to give qualitative arguments for the emergence of three-body bound states at unitarity.
As we saw in Section 3.1.1, at unitarity, in absence of deeply-bound states, two particles never bind. Let us consider two bosons at a distance r, interacting via a unitary interaction potential. We add a third boson whose scattering length a with the two former ones is large, finite and positive, and we look for the condition under which this third boson stabilizes the system into a bound state. This situation, depicted in Fig. 3.3, is identical to that studied in Maji et al. (2010), although my arguments are based on the statistical weights of configurations, not on scaling considerations. For practical reasons, we use the square-well finite-range potential of Section 3.1.1 with a range r0 much smaller than all other length scales. As the energy is bounded by 3V0, this potential has the advantage of featuring no three-body collapse. However, we shall keep in mind that the limit r0 ! 0 (at fixed scattering length) should be taken only together with a short-distance regularization such as the three-body hard core discussed in Section 3.2.1.
Phase diagram of the unitary Bose gas
Now that we saw that the high-temperature behaviour of the unitary Bose gas is very well described in terms of its virial expansion up to the third order, I describe the lowtemperature physics of the unitary Bose gas. As we will see in this section, at low temperature and for small values of R0, the unitary Bose gas undergoes a first-order phase transition to a new quantum phase, held together by the same effects as Efimov trimers, the superfluid Efimov liquid.
Transition to the Efimov liquid phase
In order to identify a first-order phase transition, we observe the joint distribution of the distance of the centre of mass of two particles to the trap centre, rij = kxi + xjk=2 and of their pair distance rij = kxj xik as a function of the temperature.
Simple model for the transition to the Efimov liquid phase
To obtain an approximate theoretical description of the normal-gas-to-Efimov liquid coexistence line, we need a simple model both of the normal gas and of the superfluid Efimov liquid. We already showed that the virial expansion up to the third coefficient is a very good model for the former (see Section 4.1.2). In order to build this simple model for the coexistence line, we devised a simple theoretical model of the Efimov liquid, both from previous work achieved in the field of unitary bosons, and from our observations.
Table of contents :
1 Bosons in the path-integral formalism
1.1 Ideal Bose gas
1.1.1 Free density matrix and permutations
1.1.2 Path integrals
1.2 Interacting Bose gas
1.2.1 Trotter approximation
1.2.2 Pair-product approximation
1.2.3 Interaction boxes
Appendix 1.A Metropolis algorithm
2 The repulsive weakly-interacting Bose gas
2.1 The mean-field weakly-interacting Bose gas
2.1.1 Scattering length
2.1.2 Trapped bosons
2.2 Beyond-mean-field corrections
2.2.1 The Lee–Huang–Yang equation of state
2.2.2 Path-integral simulation and comparison to experiments .
2.2.3 Numerical and experimental grand-canonical equation of state .
3 Efimov trimers in the path-integral formalism
3.1 Unitary limit
3.1.1 Square-well interaction potential
3.1.2 Zero-range interaction potential
3.2 Efimov effect
3.2.1 Regularization of the zero-range interaction potential
3.2.2 Path-integral argument
3.2.3 Numerical simulation of a single Efimov trimer
3.2.4 Universality of Efimov trimers
Appendix 3.A Path-integral argument for the dimer state
Appendix 3.B Calculation of the pair-product zero-range correction factor
Appendix 3.C Hyperspherical coordinates
4 The unitary Bose gas
4.1 High-temperature equation of state
4.1.1 Many-body simulation
4.1.2 Equation of state
4.2 Phase diagram of the unitary Bose gas
4.2.1 Transition to the Efimov liquid phase
4.2.2 Simple model for the transition to the Efimov liquid phase
4.2.3 Homogeneous phase diagram
4.3 Unitary liquid in experimental systems
4.3.1 Experimentally accessible regions of the phase diagram
4.3.2 Experimental observables
Appendix 4.A Path integrals and momentum distribution
4.A.1 Off-diagonal density matrix and momentum distribution
4.A.2 Momentum distribution of two interacting bosons
4.A.3 Simple algorithm to obtain the momentum distribution