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Physical Review E

Ground state of interacting quantum magnetic dipoles: Transition from a ferromagnetic Fermi liquid to an antiferromagnetic solid

Sudhanshu Shekhar Jha, S. D. Mahanti

Published Nov 3, 2010• 2 citations
Publisher

Abstract

We show that the ground state of a system of magnetic dipoles, with no electric charge, is a ferromagnetic quantum Fermi liquid at high densities, driven by the dipolar exchange energy. As in the system of classical point dipoles, the direct dipole energy is zero in this case. With decreasing density, there is a transition to an antiferromagnetic lattice state. An addition of short range hard core repulsive potential will arrest the infinite density collapse of the ferromagnetic state, and possible melting of the low density antiferromagnetic lattice state.

DOI
10.1103/physreve.82.052101

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We show that the ground state of a system of magnetic dipoles, with no electric charge, is a ferromagnetic quantum Fermi liquid at high densities, driven by the dipolar exchange energy. As in the system of classical point dipoles, the direct dipole energy is zero in this case. With decreasing density, there is a transition to an antiferromagnetic lattice state. An addition of short range hard core repulsive potential will arrest the infinite density collapse of the ferromagnetic state, and possible melting of the low density antiferromagnetic lattice state.
We obtain the best upper bound for the ground-state energy of a system of chargeless fermions of mass m, spin s=1/2 , and magnetic moment mus[over ] as a function of its density in the fully spin-polarized Hartree-Fock determinantal state, specified by a prolate spheroidal plane-wave single-particle occupation function n_(k[over ]) , by minimizing the total energy E at each density with respect to the variational spheroidal deformation parameter beta(2),0< or =beta(2)< or =1 . We find that at high densities, this spheroidal ferromagnetic state is the most likely ground state of the system, but it is still unstable towards the infinite-density collapse. This optimized ferromagnetic state is shown to be a stable ground state of the dipolar system at high densities, if one has an additional repulsive short-range hardcore interaction of sufficient strength and nonvanishing range.
We consider the ground state of a system of chargeless fermions, such as neutrinos, of mass m and magnetic moment p interacting through long-range magnetic dipole interaction, within the framework of a Hartree-Fock variational approach. At high densities the uniform paramagnetic state becomes unstable towards a ferromagnetic state with quadrupolar deformation of the Fermi surface. The exchange energy which is attractive dominates the repulsive kinetic energy. If we let the density be a variable, then above a certain density the system will collapse to an infinite density state unless another short-range interaction stops the collapse. In the case of large deformations, the possibility of a purely dipolar deformation exists.
We study a two-component mixture of fermionic dipoles in two dimensions at zero temperature, interacting via a purely repulsive 1/r31/{r}^{3}1/r3 potential. This model can be realized with ultracold atoms or molecules when their dipole moments are aligned in the confinement direction orthogonal to the plane. We characterize the unpolarized mixture by means of the diffusion Monte Carlo technique. Computing the equation of state, we identify the regime of validity for a mean-field theory based on a low-density expansion and compare our results with the hard-disk model of repulsive fermions. At high density, we address the possibility of itinerant ferromagnetism, namely, whether the ground state can be fully polarized in the fluid phase. Within the fixed-node approximation, we show that the accuracy of Jastrow-Slater trial wave functions, even with the typical two-body backflow correction, is not sufficient to resolve the relevant energy differences. By making use of the iterative-backflow improved trial wave functions, we observe no signature of a fully polarized ground state up to the freezing density.
We will take the low- T Fermi liquid state of Section 11.2.2 in dimensions d ≥ 2 (or its spinful generalization) and examine the nature of its instabilities to other ground states of a dense gas of fermions. Possibilities include ferromagnets, states in which there is spin or charge density wave order (to be defined more precisely below), or various types of superconductors. All of these cases are of considerable practical importance and have numerous experimental applications. A theoretical treatment of the quantum transition between a Fermi liquid and a magnetically or charge ordered state was given in a paper by Hertz, although many important points were anticipated in earlier work. We shall present Hertz's basic arguments in Section 12.1 for the case of a transition between a Fermi liquid and a spin density wave state. We shall not treat the other cases here and will, instead, refer the reader to the literature. There are a number of reasons for this neglect: (i) Many aspects of these transitions are not fully understood (we will note some below) and are the subject of considerable debate in the literature; it is therefore inappropriate to include them in this introductory treatment. (ii) We shall only consider systems in spatial dimensions d ≥ 2 here (the d = 1 case requires a separate treatment appropriate to Tomonaga–Luttinger liquids and will be addressed in Chapter 14). […]
We study by quantum Monte Carlo simulations the low-temperature phase diagram of dipolar bosons confined to one dimension, with dipole moments aligned along the direction of particle motion. A hard-core repulsive potential of varying range ($\ensuremath{\sigma}$) is added to the dipolar interaction, in order to ensure stability of the system against collapse. In the $\ensuremath{\sigma}\ensuremath{\rightarrow}0$ limit the physics of the system is dominated by the potential energy and the ground state is quasicrystalline; as $\ensuremath{\sigma}$ is increased the attractive part of the interaction weakens and the equilibrium phase evolves from quasicrystalline to a nonsuperfluid liquid. At a critical value ${\ensuremath{\sigma}}_{c}$, the kinetic energy becomes dominant and the system undergoes a quantum phase transition from a self-bound liquid to a gas. In the gaseous phase with $\ensuremath{\sigma}\ensuremath{\rightarrow}{\ensuremath{\sigma}}_{c}$, at low density attractive interactions bring the system into a ``weak'' superfluid regime. However, gas-liquid coexistence also occurs, as a result of which the topologically protected superfluid regime is not approached.
The liquid and crystal phases of a single-component Fermi gas with dipolar interactions are investigated using quantum Monte Carlo methods in two spatial dimensions and at zero temperature. The dipoles are oriented by an external field perpendicular to the plane of motion, resulting in a purely repulsive 1/r(3) interaction. In the liquid phase we calculate the equation of state as a function of the interaction strength and other relevant properties characterizing the Fermi-liquid behavior: effective mass, discontinuity at the Fermi surface, and pair correlation function. In the high density regime we calculate the equation of state of the Wigner crystal phase and the critical density of the liquid to solid quantum phase transition. Close to the freezing density we also search for the existence of a stripe phase, but such a phase is never found to be energetically favorable.