Universal Relations for a Fermi Gas Close to a p-Wave Interaction Resonance

Exact results of the one-dimensional repulsive Hubbard model

We present analytical results of the fundamental properties of the one-dimensional (1D) Hubbard model with a repulsive interaction. The new model results with arbitrary external fields include: (I) using the exact solutions of the Bethe ansatz equations of the Hubbard model, we first rigorously calculate the gapless spin and charge excitations, exhibiting exotic features of fractionalized spinons and holons. We then investigate the gapped excitations in terms of the spin string and thek-Λstring bound states at arbitrary driving fields, showing subtle differences in spin magnons and chargeη-pair excitations. (II) For a high-density and high spin magnetization region, i.e. near the quadruple critical point, we further analytically obtain the thermodynamic properties, dimensionless ratios and scaling functions near quantum phase transitions. (III) Importantly, we give the general scaling functions at quantum criticality for arbitrary filling and interaction strength. These can directly apply to other integrable models. (IV) Based on the fractional excitations and the scaling laws, the spin-incoherent Luttinger liquid (SILL) with only the charge propagation mode is elucidated by the asymptotic of the two-point correlation functions with the help of conformal field theory. We also, for the first time, obtain the analytical results of the thermodynamics for the SILL. (V) Finally, to capture deeper insights into the Mott insulator and interaction-driven criticality, we further study the double occupancy and propose its associated contact and contact susceptibilities, through which an adiabatic cooling scheme based upon quantum criticality is proposed. In this scenario, we build up general relations among arbitrary external- and internal-potential-driven quantum phase transitions, providing a comprehensive understanding of quantum criticality. Our methods offer rich perspectives of quantum integrability and offer promising guidance for future experiments with interacting electrons and ultracold atoms, both with and without a lattice.

Temperature-Dependent Contact of Weakly Interacting Single-Component Fermi Gases and Loss Rate of Degenerate Polar Molecules

Motivated by the experimental realization of single-component degenerate Fermi gases of polar ground state KRb molecules with intrinsic two-body losses [L. De Marco et al., A degenerate Fermi gas of polar molecules, Science 363, 853 (2019).SCIEAS0036-807510.1126/science.aau7230], this work studies the finite-temperature loss rate of single-component Fermi gases with weak interactions. First, we establish a relationship between the two-body loss rate and the p-wave contact. Second, we evaluate the contact of the homogeneous system in the low-temperature regime using p-wave Fermi liquid theory and in the high-temperature regime using the second-order virial expansion. Third, conjecturing that there are no phase transitions between the two temperature regimes, we smoothly interpolate the results to intermediate temperatures. It is found that the contact is constant at temperatures close to zero and increases first quadratically with increasing temperature and finally-in agreement with the Bethe-Wigner threshold law-linearly at high temperatures. Fourth, applying the local-density approximation, we obtain the loss-rate coefficient for the harmonically trapped system, reproducing the experimental KRb loss measurements within a unified theoretical framework over a wide temperature regime without fitting parameters. Our results for the contact are not only applicable to molecular p-wave gases but also to atomic single-component Fermi gases, such as ^{40}K and ^{6}Li.

Maximum Energy Growth Rate in Dilute Quantum Gases

In this Letter we study how fast the energy density of a quantum gas can increase in time, when the interatomic interaction characterized by the s-wave scattering length a_{s} is increased from zero with arbitrary time dependence. We show that, at short time, the energy density can at most increase as sqrt[t], which can be achieved when the time dependence of a_{s} is also proportional to sqrt[t], and especially, a universal maximum energy growth rate can be reached when a_{s} varies as 2sqrt[ℏt/(πm)]. If a_{s} varies faster or slower than sqrt[t], it is, respectively, proximate to the quench process and the adiabatic process, and both result in a slower energy growth rate. These results are obtained by analyzing the short time dynamics of the short-range behavior of the many-body wave function characterized by the contact, and are also confirmed by numerically solving an example of interacting bosons with time-dependent Bogoliubov theory. These results can also be verified experimentally in ultracold atomic gases.

Heuristic machinery for thermodynamic studies of SU(N) fermions with neural networks

The power of machine learning (ML) provides the possibility of analyzing experimental measurements with a high sensitivity. However, it still remains challenging to probe the subtle effects directly related to physical observables and to understand physics behind from ordinary experimental data using ML. Here, we introduce a heuristic machinery by using machine learning analysis. We use our machinery to guide the thermodynamic studies in the density profile of ultracold fermions interacting within SU(N) spin symmetry prepared in a quantum simulator. Although such spin symmetry should manifest itself in a many-body wavefunction, it is elusive how the momentum distribution of fermions, the most ordinary measurement, reveals the effect of spin symmetry. Using a fully trained convolutional neural network (NN) with a remarkably high accuracy of ~94% for detection of the spin multiplicity, we investigate how the accuracy depends on various less-pronounced effects with filtered experimental images. Guided by our machinery, we directly measure a thermodynamic compressibility from density fluctuations within the single image. Our machine learning framework shows a potential to validate theoretical descriptions of SU(N) Fermi liquids, and to identify less-pronounced effects even for highly complex quantum matter with minimal prior understanding.

Role of Effective Range in the Bulk Viscosity of Resonantly Interacting s- and p-Wave Fermi Gases

We investigate the role of the effective range on the bulk viscosity of s- and p-wave Fermi gases. At resonance, the presence of the effective range breaks the scale invariance of the system, and hence results in a nonzero bulk viscosity. However, we show that the effective range plays a very different role in the two cases. In the s-wave case, the role of the effective range is perturbative, and its contribution to the bulk viscosity vanishes in the limit of zero effective range. On the other hand, the effective range in p-wave Fermi gases leads to a nonzero bulk viscosity, even in the zero-range limit. We employ a general diagrammatic approach to compute the bulk viscosity spectral function that includes the effects of the effective range. We then compute the analytic expressions for the spectral function in the high temperature limit, at low and high frequencies. We also derive the sum rules for the bulk viscosity spectral function for both s- and p-wave gases.

Universal relations for ultracold reactive molecules

The realization of ultracold polar molecules in laboratories has pushed physics and chemistry to new realms. In particular, these polar molecules offer scientists unprecedented opportunities to explore chemical reactions in the ultracold regime where quantum effects become profound. However, a key question about how two-body losses depend on quantum correlations in interacting many-body systems remains open so far. Here, we present a number of universal relations that directly connect two-body losses to other physical observables, including the momentum distribution and density correlation functions. These relations, which are valid for arbitrary microscopic parameters, such as the particle number, the temperature, and the interaction strength, unfold the critical role of contacts, a fundamental quantity of dilute quantum systems, in determining the reaction rate of quantum reactive molecules in a many-body environment. Our work opens the door to an unexplored area intertwining quantum chemistry; atomic, molecular, and optical physics; and condensed matter physics.

Fermi-Liquid Description of a Single-Component Fermi Gas with p-Wave Interactions

We study the Fermi liquid properties of a single component Fermi gas with p-wave interactions. In the weak repulsive limit, we obtain exact perturbative expansions for the ground state energy, the chemical potential, and the effective mass of the Landau quasiparticle up to second order in scattering volume a. We also calculated the corresponding Landau functions and Landau parameters and show that they satisfy the general Fermi liquid identities. Using the Landau transport equation, we show that undamped zero sound only appears in the second order in scattering volume, in contrast to the s-wave case.

Local conservation laws in ultracold Fermi systems with time-dependent interaction potential

In the context of an ultracold Fermi gas, we derive conservation laws for mass, energy and momentum based on a generalized nonlocal Boltzmann equation with gradient corrections in the collision integral. The corrections are expressed in terms of effective collision duration, particle displacement and changes of total momentum and energy. Their origin is in the in-medium T matrix. Using variations of the optical theorem, we show that in the collision integral the particle-hole symmetry can be recast into a form of collision symmetry amenable to semiclassical simulation. Pauli-blocked collisions are distinguished from Bose-stimulated nondissipative ones; the latter are not present in the absence of gradient corrections. Consolidating with the microscopic theory, we extract local conservation laws for a general time-dependent interaction potential, and demonstrate how both types of collisions affect densities and flows of conserving quantities. Comparison is made with the approach of Nozières and Schmitt-Rink in the limit of thermal equilibrium. Under approximations used for normal-state ultracold Fermi gases interacting via Feshbach resonances we demonstrate the effect of the collision delay on the shear viscosity.

Contact Theory for Spin-Orbit-Coupled Fermi Gases

We develop the contact theory for spin-orbit-coupled Fermi gases. By using a perturbation method, we derive analytically the universal two-body behavior at short distance, which does not depend on the short-range details of interatomic potentials. We find that two new scattering parameters need to be introduced because of spin-orbit coupling, besides the traditional s- and p-wave scattering length (volume) and effective ranges. This is a general and unique feature for spin-orbit-coupled systems. Consequently, two new adiabatic energy relations with respect to the new scattering parameters are obtained, in which a new contact is involved because of spin-orbit coupling. In addition, we derive the asymptotic behavior of the large-momentum distribution, and find that the subleading tail is corrected by the new contact. This work paves the way for exploring the profound properties of spin-orbit-coupled many-body systems, according to two-body solutions.

Concept of a Contact Spectrum and Its Applications in Atomic Quantum Hall States

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.

Universal High-Momentum Asymptote and Thermodynamic Relations in a Spinless Fermi Gas with a Resonant p-Wave Interaction

We investigate universal relations in a spinless Fermi gas near a p-wave Feshbach resonance, and show that the momentum distribution n_{k} has an asymptote proportional to k^{-2} with the proportionality constant-the p-wave contact-scaling with the number of closed-channel molecules. We prove the adiabatic sweep theorem for a p-wave resonance which reveals the thermodynamic implication of the p-wave contact. In contrast to the unitary Fermi gas in which Tan's contact is universal, the p-wave contact depends on the short-range details of the interaction.