Few-body route to one-dimensional quantum liquids

Abstract

Gapless many-body quantum systems in one spatial dimension are universally described by the Luttinger liquid effective theory at low energies. Essentially, only two parameters enter the effective low-energy description, namely the speed of sound and the Luttinger parameter. These are highly system dependent and their calculation requires accurate non-perturbative solutions of the manybody problem. Here, we present a simple method that only uses collisional information to extract the low-energy properties of these systems. Our results are in remarkable agreement with available results for integrable models and from large scale Monte Carlo simulations of one-dimensional helium and hydrogen isotopes. Moreover, we estimate theoretically the critical point for spinodal decomposition in one-dimensional helium-4, and show that the exponent governing the divergence of the Luttinger parameter near the critical point is exactly 1/2, in excellent agreement with Monte Carlo simulations.PACS numbers: 67.10. 73.21.Hb,34.50.Cx,64., Introduction . Interacting quantum systems in one spatial dimension, long ago considered toy models far away from the three-dimensional reality, now hold the status of physically relevant theories. Advances in the transversal confinement of trapped ultracold atomic gases [1,2], the realisation of carbon nanotubes by rolling up sheets of graphene [3,4], or helium isotopes adsorbed in nanopores [5,6], make it possible to investigate manybody quantum physics in wire geometries with unprecedented level of control. Most one-dimensional systems, whether weakly or strongly interacting, are universally described by the Luttinger liquid effective field theory at low energies [7], and by its recently developed nonlinear counterpart at higher energies [8]. Essentially, many correlation functions, and the excitation spectrum, have universal behaviours, and the non-universal parameters -the Luttinger parameter and speed of soundare the only system-dependent quantities of interest. To extract these, however, one needs to either invoke perturbation theory, only valid for weak interactions, or to solve the many-body problem numerically "exact" using Monte Carlo [6, 9-11] for continuous or DMRG methods [12] for lattice models, or quasi-analytically for integrable models via the Bethe ansatz [8,13]. In this Letter, we develop a simple, yet highly non-perturbative method, that uses only two-body scattering information to extract the speed of sound and Luttinger parameter of strongly interacting many-body quantum systems in one dimension. To show the reliability of our theory, we study all the stable isotopes of helium and spin-polarised hydrogen, and tritium, when tightly confined to one dimension, using realistic molecular potentials, which are stronglyinteracting and intractable with perturbative methods, and compare our results to the Monte Carlo data of references [9][10][11]. Using similar methods, we also study the liquid phase of 4 He, which is not a Luttinger liquid.

Thermodynamic behavior of a one-dimensional Bose gas at low temperature

“…( 4) and Eq. ( 25), one can calculate also the first correction, at large γ, to the sound velocity [36,39]:…”

Bose-Fermi dualities for arbitrary one-dimensional quantum systems in the universal low energy regime

“…This allows to choose the most convenient particle statistics, depending on how weakly or strongly coupled they are. For example, spinless bosons with effectively attractive twobody interactions and repulsive three-body interactions can form one-dimensional quantum droplets [53], and therefore the same is true for spinless fermions (whose low-energy interactions have been recently manipulated by means of p-wave Feshbach resonances [30]) with their dual Hamiltonian. Quantum droplets of two-component bosons [79], which have been created and observed in three dimensions [80,81], are also predicted to occur in one dimension [82][83][84][85].…”

“…V, then it is possible to assert that at low energies fermions and bosons are completely equivalent and, therefore, the description of universal low-energy physics in one dimension can be done from either fermionic or bosonic side, whichever is most convenient for the particular application. For instance, 4 He atoms tightly confined to (quasi-) one dimension, have a strongly repulsive (infinite) two-body core at short distances [51,52] but the two-body scattering length is very large and positive [53]. The existence of a short-distance hard core makes this system of 4 He atoms be equivalent to fermions with the same Hamiltonian [38], suggesting that the lowenergy physics at low densities should be described by a fermionic effective field theory (EFT).…”

“…However, the large scattering length means that the fermionic EFT is in the strong-coupling (attractive) regime [15,39], making the description of the system highly non-perturbative. The use of the more convenient soft-core bosonic EFT, which is weakly-coupled for large scattering lenghts, including three-body forces that stabilize its liquid phase [53], can only be theoretically justified if the fermionic and bosonic representations of the low-energy EFT are equivalent.…”

Universality and tails of long-range interactions in one dimension

“…The effective interactions discussed above are all concerned with low-energy scattering. In one spatial dimension, however, we have shown [47] that Luttinger liquids whose constituents are scalar particles may depend very little on low-energy interactions except for extremely low densities, since the relevant energy scale for two-particle collisions is twice the Fermi energy. When realistic interactions are involved, such as Born-Oppenheimer potentials, which have Van der Waals tails, and especially in multichannel problems, it can be considerably easier to numerically diagonalise the two-body Hamiltonian in a finite box rather than solving the Schrödinger or Lippmann-Schwinger equations in infinite space.…”