We show that a system of three species of one-dimensional fermions, with an attractive three-body contact interaction, features a scale anomaly directly related to the anomaly of two-dimensional fermions with two-body contact forces. We show, furthermore, that those two cases (and their multispecies generalizations) are the only nonrelativistic systems with contact interactions that display a scale anomaly. While the two-dimensional case is well known and has been under study both experimentally and theoretically for years, the one-dimensional case presented here has remained unexplored. For the latter, we calculate the impact of the anomaly on the equation of state, which appears through the generalization of Tan's contact for three-body forces, and determine the pressure at finite temperature. In addition, we show that the third-order virial coefficient is proportional to the second-order coefficient of the two-dimensional two-body case.
We show that a system of three species of one-dimensional fermions, with an attractive three-body contact interaction, features a scale anomaly directly related to the anomaly of two-dimensional fermions with two-body contact forces. We show, furthermore, that those two cases (and their multispecies generalizations) are the only nonrelativistic systems with contact interactions that display a scale anomaly. While the two-dimensional case is well known and has been under study both experimentally and theoretically for years, the one-dimensional case presented here has remained unexplored. For the latter, we calculate the impact of the anomaly on the equation of state, which appears through the generalization of Tan's contact for three-body forces, and determine the pressure at finite temperature. In addition, we show that the third-order virial coefficient is proportional to the second-order coefficient of the two-dimensional two-body case.
A system of two-species, one-dimensional fermions, with an attractive two-body interaction of the derivative-delta type, features a scale anomaly. In contrast to the well-known two-dimensional case with contact interactions, and its one-dimensional cousin with three-body interactions (studied recently by some of us and others), the present case displays dimensional transmutation featuring a power-law rather than a logarithmic behavior. We use both the Schrödinger equation and quantum field theory to study bound and scattering states, showing consistency between both approaches. We show that the expressions for the reflection (R) and the transmission (T) coefficients of the renormalized, anomalous derivative-delta potential are identical to those of the regular delta potential. The second-order virial coefficient is calculated analytically using the Beth–Uhlenbeck formula, and we make comments about the proper ϵB→0 (where ϵB is the bound-state energy) limit. We show the impact of the quantum anomaly (which appears as the binding energy of the two-body problem, or equivalently as Tan's contact) on the equation of state and on other universal relations. Our emphasis throughout is on the conceptual and structural aspects of this problem.
A system of three-species fermions in one spatial dimension (1D) with a contact three-body interaction is known to display a scale anomaly. This anomaly is identical to that of a two-dimensional (2D) system of two-species fermions. The exact relation between those two systems, however, is limited to the two-particle sector of the 2D case and the three-particle sector of the 1D case. Here, we implement a nonperturbative Monte Carlo approach, based on the worldline representation, to calculate the thermodynamics and static response of three-species fermions in 1D, thus tackling the many-body sector of the problem. We determine the energy, density, and pressure equations of state and the compressibility and magnetic susceptibility for a wide range of temperatures and coupling strengths. We compare our results with the third-order virial expansion and interpret the classical-quantum crossover as the onset of trimer degrees of freedom.
In ultracold Fermi gases, the spatial dimension and the number of particles involved in interactions largely determine the behavior of the system. By exploiting harmonic trapping potentials and multiple hyperfine states, experiments wield a high degree of control over both of these parameters. In this work, we show that a nonrelativistic, one-dimensional, three-component Fermi system with three-body contact interactions exhibits a quantum anomaly, breaking the scale invariance that exists for this system classically. The exact solution of this three-body problem, wherein a bound trimer forms at arbitrarily small attractive couplings, features a correspondence to the two-dimensional, two-body Fermi system, through which we determine the first non-trivial virial coefficient. We further develop a numerical technique for solving the few-body problem for up to six particles and find that the effective interaction between two trimers is repulsive. To extend our analysis to the many-body problem, we apply the worm algorithm of quantum Monte Carlo and present results for various thermodynamic quantities. We find that for increasing attractive couplings, the system undergoes a Fermi-Fermi crossover from a set of three ideal Fermi gases to one weakly repulsive Fermi gas consisting of massive composite trimers. Finally, we adapt the three-body interaction to a two-dimensional lattice model, where we find strong evidence for a second-order phase transition with an associated reduction in a discrete global symmetry. While the exact experimental parameters we require have not yet been physically realized, our work provides a comprehensive first set of predictions for this model.
Using ab initio lattice methods, we calculate the finite temperature thermodynamics of homogeneous two-dimensional spin-1/2 fermions with attractive short-range interactions. We present results for the density, pressure, compressibility, and quantum anomaly (i.e., Tan's contact) for a wide range of temperatures (mostly above the superfluid phase, including the pseudogap regime) and coupling strengths, focusing on the unpolarized case. Within our statistical and systematic uncertainties, our prediction for the density equation of state differs quantitatively from the prediction by Luttinger-Ward theory in the strongly coupled region of parameter space, but otherwise agrees well with it. We also compare our calculations with the second- and third-order virial expansion, with which they are in excellent agreement in the low-fugacity regime.
In this paper we explore the transport properties of three-component Fermi gases confined to one spatial dimension, interacting via a three-body interaction, in the high temperature limit. At the classical level, the three-body interaction is scale invariant in one dimension. However, upon quantization, an anomaly appears which breaks the scale invariance. This is very similar to the physics of two-component fermions in two spatial dimensions, where the two-body interaction is also anomalous. Previous studies have already hinted that the physics of these two systems are intimately related. Here we expand upon those studies by examining the thermodynamic properties of this anomalous one-dimensional system in the high temperature limit. We show there is an exact mapping between the traditional two-body anomalous interaction in two dimensions, to that of three-body interaction in one dimension. This result is valid in the high temperature limit, where the thermodynamics can be understood in terms of few-body correlations.
Motivated by advances in the manipulation and detection of ultracold atoms with multiple internal degrees of freedom, we present a finite-temperature lattice Monte Carlo calculation of the density and pressure equations of state, as well as Tan's contact, of attractively interacting SU(4)-and SU(6)-symmetric fermion systems in one spatial dimension.We also furnish a nonperturbative proof of a universal relation whereby quantities computable in the SU(2) case completely determine the virial coefficients of the SU(N f ) case.These one-dimensional systems are appealing because they can be experimentally realized in highly constrained traps and because of the dominant role played by correlations.The latter are typically nonperturbative and are crucial for understanding ground states and quantum phase transitions.While quantum fluctuations are typically overpowered by thermal ones in one and two dimensions at any finite temperature, we find that quantum effects do leave their imprint in thermodynamic quantities.Our calculations show that the additional degrees of freedom, relative to the SU(2) case, provide a dramatic enhancement of the density and pressure (in units of their noninteracting counterparts) in a wide region around vanishing βµ, where β is the inverse temperature and µ the chemical potential.As shown recently in experiments, the thermodynamics we explore here can be measured in a controlled and precise fashion in highly constrained traps and optical lattices.Our results are a prediction for such experiments in one dimension with atoms of high nuclear spin.