A unique feature of ultracold atoms is the separation of length scales, r_{0}≪k_{F}^{-1}, where k_{F} and r_{0} are the Fermi momentum characterizing the average particle distance and the range of interaction between atoms, respectively. For s-wave scattering, Shina Tan discovered that such diluteness leads to universal thermodynamic relations governed by contact. Here, we show that the concept of contact can be generalized to an arbitrary partial-wave scattering. Contact of all partial-wave scatterings forms a contact spectrum, which establishes universal thermodynamic relations with notable differences from those in the presence of s-wave scattering alone. Such a contact spectrum is particularly useful for characterizing many-body correlations in atomic quantum Hall states (QHSs). It has an interesting connection with a special bipartite entanglement spectrum of QHSs and enables an intrinsic probe of atomic QHSs using short-range two-body correlations.
A central quantity of importance for ultracold atoms is contact, which measures two-body correlations at short distances in dilute systems. It appears in universal relations among thermodynamic quantities, such as large momentum tails, energy and dynamic structure factors, through the renowned Tan relations. However, a conceptual question remains open as to whether or not contact can signify phase transitions that are insensitive to short-range physics. Here we show that, near a continuous classical or quantum phase transition, contact exhibits a variety of critical behaviours, including scaling laws and critical exponents that are uniquely determined by the universality class of the phase transition, and a constant contact per particle. We also use a prototypical exactly solvable model to demonstrate these critical behaviours in one-dimensional strongly interacting fermions. Our work establishes an intrinsic connection between the universality of dilute many-body systems and universal critical phenomena near a phase transition.
Contact has been well established as an important quantity to govern dilute quantum systems, in which the pairwise correlation at short distance traces a broad range of thermodynamic properties. So far, most studies have focused on contact in individual angular momentum channels. Here we point out that, to have a complete description of the pairwise correlation in a general dilute quantum systems, contact should be defined as a matrix. Whereas the diagonal terms of such a matrix include contact of all partial wave scatterings, the off-diagonal terms characterize the coherence of the asymptotic pairwise wave function in the angular momentum space and determine important thermodynamic quantities including the momentum distribution. The contact matrix allows physicists to access unexplored connections between short-range correlations and macroscopic quantum phenomena. As an example, we show the direct connection between the contact matrix and order parameters of a superfluid with mixed partial waves.
The length scale separation in dilute quantum gases in quasi-one-dimensional or quasi-two-dimensional traps has spatially divided the system into two distinct regimes. Whereas universal relations defined in strict one or two dimensions apply in a scale that is much larger than the characteristic length of the transverse confinements, physical observables in the short distances are inevitably governed by three-dimensional contacts. Here, we show that
p-wave contacts defined in different length scales are intrinsically connected by a universal relation, which depends on a simple geometric factor of the transverse confinements. While this universal relation is derived for one of the
p-wave contacts, it establishes a concrete example of how dimensional crossover interplays with contacts and universal relations for arbitrary partial wave scatterings.
We study the universal relations and normal-phase thermodynamics of a two-component ultracold Fermi gas with coexisting
s- and
p-wave interactions. Due to the orthogonality of two-body wave functions of different scattering channels, the universal thermodynamic relations of the system appear to be direct summations of contributions from each partial-wave scattering channel. These universal relations are dictated by a set of contacts, which can be associated with either
s- or
p-wave interactions. Interestingly, due to the interplay of
s- and
p-wave interactions on the many-body level, the contacts, and hence all the relevant thermodynamic quantities, behave differently from those with only
s- or
p-wave interactions. These are manifest in our numerical calculations based on second-order virial expansions for
40K atoms under typical experimental parameters. A particularly interesting finding is that, due to the coexistence of
s- and
p-wave scatterings, the interaction energy of the repulsive branch features abrupt changes across the
p-wave resonances. Our results can be readily checked experimentally for
40K atoms near the
198Gp-wave Feshbach resonance, where multiple partial-wave scatterings naturally coexist.
Abstract We derive the full set of universal relations for spin-polarized Fermi gases with p -wave interaction in two dimensions, simply using the short-range asymptotic behavior of fermion-pair wave functions. For p -wave interactions, an additional contact related to the effective range needs to be introduced, besides the one related to the scattering volume. Since the subleading tail ( k −4 ) of the large-momentum distribution cannot fully be captured by the contacts defined by the adiabatic relations, an extra term resulted from the center-of-mass motions of the pairs gives rise to an additional divergence in the kinetic energy of the system, besides those related to the contacts defined. We show in Tan’s energy theorem that if only two-body correlations are taken into account, all these divergences are reasonably removed, leading to a finite internal energy of the system. In addition, we find that all the other universal relations, such as the high-frequency behavior of the radio-frequency response, short-range behavior of the pair correlation function, generalized virial theorem, and pressure relation, remain unaffected by the center-of-mass motions of the pairs, and are fully governed by the contacts defined by the adiabatic relations. Our results confirm the feasibility of generalizing the contact theory for higher-partial-wave scatterings, and could readily be confirmed in current experiments with ultracold 40 K and 6 Li atoms.
The contact parameter in unitary Fermi gases governs the short-range correlations and high-momentum properties of the system. We perform accurate quantum Monte Carlo calculations with highly optimized trial functions to precisely determine this parameter at
T=0, demonstrate its universal application to a variety of observables, and determine the regions of momentum and energy over which the leading short-range behavior is dominant. We derive Tan's expressions for the contact parameter using just the short-range behavior of the ground-state many-body wave function, and use this behavior to calculate the two-body distribution function, one-body density matrix, and the momentum distribution of unitary Fermi gases; providing a precise value of the contact parameter that can be compared to experiments.