Exact relations for quantum-mechanical few-body and many-body problems with short-range interactions in two and three dimensions

Abstract

We derive relations between various observables for N particles with zero-range or short-range interactions, in continuous space or on a lattice, in two or three dimensions, in an arbitrary external potential. Some of our results generalise known relations between large-momentum behavior of the momentum distribution, short-distance behavior of the pair correlation function and of the one-body density matrix, derivative of the energy with respect to the scattering length or to time, and the norm of the regular part of the wavefunction; in the case of finite-range interactions, the interaction energy is also related to dE/da. The expression relating the energy to a functional of the momentum distribution is also generalised, and is found to break down for Efimov states with zero-range interactions, due to a subleading oscillating tail in the momentum distribution. We also obtain new expressions for the derivative of the energy of a universal state with respect to the effective range, the derivative of the energy of an efimovian state with respect to the three-body parameter, and the second order derivative of the energy with respect to the inverse (or the logarithm in the two-dimensional case) of the scattering length. The latter is negative at fixed entropy. We use exact relations to compute corrections to exactly solvable three-body problems and find agreement with available numerics. For the unitary gas, we compare exact relations to existing fixed-node Monte-Carlo data, and we test, with existing Quantum Monte Carlo results on different finite range models, our prediction that the leading deviation of the critical temperature from its zero range value is linear in the interaction effective range r_e with a model independent numerical coefficient.

51 pages, 5 figures. Split into three articles: Phys. Rev. A 83, 063614 (2011) [arXiv:1103.5157]; Phys. Rev. A 86, 013626 (2012) [arXiv:1204.3204]; Phys. Rev. A 86, 053633 (2012) [ arXiv:1210.1784]

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• Generalized Hydrodynamics in the 1D Bose gas: theory and experiments
• Stability of a unitary Bose gas
• Evidence for universal relations describing a gas with $p$-wave interactions
• Observation of a smooth polaron-molecule transition in a degenerate Fermi gas
• Two-body and Three-body Contacts for Identical Bosons near Unitarity
• Universal Scaling Laws in the Dynamics of a Homogeneous Unitary Bose Gas
• Mobile impurity in a Bose-Einstein condensate and the orthogonality catastrophe
• Concept of contact spectrum and its applications in atomic quantum Hall states
• Lattice methods for strongly interacting many-body systems
• Observation of quantum depletion in a nonequilibrium exciton-polariton condensate
• Breakdown of Tan's relation in lossy one-dimensional Bose gases
• Strongly correlated Bose gases
• Polarons and dressed molecules near narrow Feshbach resonances
• Dynamics of three-body correlations in quenched unitary Bose gases
• Lindblad Equation for the Inelastic Loss of Ultracold Atoms
• Efimov Physics in Quenched Unitary Bose Gases
• Three-body recombination in heteronuclear mixtures at finite temperature
• Large-momentum distribution of a polarized Fermi gas and p-wave contacts
• Third virial coefficient of the unitary Bose gas
• Field-theoretical aspects of one-dimensional Bose and Fermi gases with contact interactions
• Ground-state properties of unitary bosons: from clusters to matter
• Efimov correlations in strongly interacting Bose gases
• Bose-Einstein condensation of Efimovian triples in the unitary Bose gas
• Universal Tan relations for quantum gases in one dimension
• Mass-imbalanced Three-Body Systems in Two Dimensions
• Critical behaviours of contact near phase transitions
• Comparative study of one-dimensional Bose and Fermi gases with contact interactions from the viewpoint of universal relations for correlation functions
• Thermodynamics, contact and density profiles of the repulsive Gaudin-Yang model
• Cumulant theory of the unitary Bose gas: Prethermal and Efimovian dynamics
• Contact matrix in dilute quantum systems
• The third virial coefficient of a two-component unitary Fermi gas across an Efimov-effect threshold
• Single-Particle Momentum Distributions of Efimov States in Mixed-Species Systems
• Ballistic quench-induced correlation waves in ultracold gases
• Effective-range dependence of resonant Fermi gases
• Few-body correlations in two-dimensional Bose and Fermi ultracold mixtures
• Correlation functions of one-dimensional strongly interacting two-component gases
• The Generalized Nuclear Contact and its Application to the Photoabsorption Cross-Section
• Tan's two-body contact across the superfluid transition of a planar Bose gas
• Scattering hypervolume for ultracold bosons from weak to strong interactions
• Analytical treatment of the interaction quench dynamics of two bosons in a two-dimensional harmonic trap
• Pattern formation in one-dimensional polaron systems and temporal orthogonality catastrophe
• Universal relations and normal phase of an ultracold Fermi gas with coexisting $s$- and $p$-wave interactions
• Heteronuclear Rydberg molecules
• Bunching, clustering, and the buildup of few-body correlations in a quenched unitary Bose gas
• Excitation picture of an interacting Bose gas
• Stationary and dynamical properties of two harmonically trapped bosons in the crossover from two dimensions to one
• 2D dilute Bose mixture at low temperatures
• Universal relations for dipolar quantum gases
• Ground-state properties of dilute one-dimensional Bose gas with three-body repulsion
• Dimensional Effects on the Momentum distribution of Bosonic Trimer States
• $s$-wave Contacts of Quantum Gases in Quasi-one and Quasi-two Dimensions
• Universal and non-universal effective $N$-body interactions for ultracold harmonically-trapped few-atom systems
• Universal Short Range Correlations in Bosonic Helium Clusters
• Many Bosons in a Narrow Magnetic Feshbach Resonance
• Dynamical fermionization in a one-dimensional Bose-Fermi mixture
• Emergent s-wave interactions between identical fermions in quasi-one-dimensional geometries
• High-momentum tail and universal relations of a Fermi gas near a Raman-dressed Feshbach resonance
• Contact formalism for coupled channels
• Quantum critical behavior and thermodynamics of the repulsive one-dimensional Hubbard model in a magnetic field
• Tan's contact as an indicator of completeness and self-consistency of a theory
• Shear viscosity and spin sum rules in strongly interacting Fermi gases
• Momentum distribution in the unitary Bose gas from first principles
• P-wave contacts of quantum gases in quasi-one-dimensional and quasi-two-dimensional traps
• Mean-square radii in mixed-species systems in two dimensions
• Universal relations for a spin-polarized Fermi gas in two dimensions
• Symmetry oscillations in strongly interacting one-dimensional mixtures
• Universal relations for spin-orbit-coupled Fermi gases in two and three dimensions
• Universal relations for hybridized $s$- and $p$-wave interactions from spin-orbital coupling
• Universal relations and normal-state properties of a Fermi gas with laser-dressed mixed-partial-wave interactions
• On the survival of the quantum depletion of a condensate after release from a magnetic trap
• High-momentum oscillating tails of strongly interacting 1D gases in a box
• Thermal fading of the $1/k^4$-tail of the momentum distribution induced by the hole anomaly
• Evolution of the unitary Bose gas for broad to narrow Feshbach resonances
• Bound states of a light atom and two heavy dipoles in two dimensions
• Non-relativistic trace anomaly and equation of state in dense fermionic matter
• Critical behavior of Tan's contact for bosonic systems with a fixed chemical potential
• Dynamical excitation processes and correlations of three-body two-dimensional mixtures
• Momentum reconstruction and contact of the one-dimensional Bose-Fermi mixture
• Ground-state properties of a dilute two-dimensional Bose gas
• Emergent $s$-wave interactions in orbitally active quasi-two-dimensional Fermi gases
• Resonance triplet dynamics in the quenched unitary Bose gas
• Three-body contact for fermions. I. General relations
• Large-momentun tail of one-dimensional Fermi gases with spin-orbit coupling
• Contact parameters in two dimensions for general three-body systems
• Quantum Monte Carlo studies of a trimer scaling function with microscopic two- and three-body interactions
• Structural properties of bosonic He clusters with N=2-10 for different potential models at the physical point and at unitarity
• Nuclear three-body short-range correlations in coordinate space
• Synergy between deep neural networks and the variational Monte Carlo method for small ${}^{4}H{e}_N$ clusters
• Single particle momentum distributions for three-bosons in two and three dimensions and dimensional crossover
• Tan's adiabatic sweep theorem from the variational theorem for the scattering length
• Universal Relations for Fermions with Large Scattering Length
• Efimov Effect in Long-range Quantum Spin Chains
• Hybrid-pair superfluidity in a strongly driven Fermi gas
• Two-Body Contact Dynamics in a Bose Gas near a Fano-Feshbach Resonance
• Reflections on dipolar quantum fluids
• Strong-coupling RPA theory of a Bose gas near the superfluid--Mott-insulator transition: universal thermodynamics and two-body contact
• Exact Thermodynamics For Weakly Interacting Normal-Phase Quantum Gases: Equations of State For All Partial Waves
• How to measure the free energy and partition function from atom-atom correlations
• Losses in interacting quantum gases: ultra-violet divergence and its regularization
• Two-body contact of a Bose gas near the superfluid--Mott-insulator transition
• The variational theorem for the scattering length in low dimensions and its applications to universal systems
• Adiabatic sweep theorem for three-dimensional dipolar Bose gases
• The role of the effective range in strongly-interacting few-body systems
• Ab initio study of the neutron and Fermi polarons on the lattice
• Probing non-Gaussian correlations through entanglement generation in a many-body quantum system
• Interaction quenches in Bose gases studied with a time-dependent hypernetted-chain Euler-Lagrange method
• Thermodynamics at zero temperature: inequalities for the ground state of a quantum many-body system
• Equivalence of dissipative and dissipationless dynamics of interacting quantum systems with its application to the unitary Fermi gas