Quantum Field Theory of Nematic Transitions in Spin-Orbit-Coupled Spin-1 Polar Bosons

Abstract

We theoretically study an ultracold gas of spin-1 polar bosons in a one-dimensional continuum, which are subject to linear and quadratic Zeeman fields and a Raman induced spin orbit coupling. Concentrating on the regime in which the background fields can be treated perturbatively, we analytically solve the model in its low-energy sector; i.e., we characterize the relevant phases and the quantum phase transitions between them. Depending on the sign of the effective quadratic Zeeman field ε, two superfluid phases with distinct nematic order appear. In addition, we uncover a spin-disordered superfluid phase at strong coupling. We employ a combination of renormalization group calculations and duality transformations to access the nature of the phase transitions. At ε=0, a line of spin-charge separated pairs of Luttinger liquids divides the two nematic phases, and the transition to the spin-disordered state at strong coupling is of the Berezinskii-Kosterlitz-Thouless type. In contrast, at ε≠0, the quantum critical theory separating nematic and strong coupling spin-disordered phases contains a Luttinger liquid in the charge sector that is coupled to a Majorana fermion in the spin sector (i.e., the critical theory at finite ε maps to a quantum critical Ising model that is coupled to the charge Luttinger liquid). Because of an emergent Lorentz symmetry, both have the same logarithmically diverging velocity. We discuss the experimental signatures of our findings that are relevant to ongoing experiments in ultracold atomic gases of ^{23}Na.

Strongly interacting spin-orbit coupled Bose-Einstein condensates in one dimension

“…However, with nonzero SOC Kc is reduced below 1 and the long range asymptotics is given by Czz cos(2πρ0x)x −2K which is seen as enhanced oscillations for large x. spin-orbit coupling and quadratic Zeeman field. Complementary to the previous study at weak coupling [50], we here concentrated on the limit when interaction effects are stronger than the kinetic energy. Our main finding, which is supported by the excellent agreement between analytics and numerics, is that in this regime the spinliquid gap is substantial and therefore the perturbative inclusion of symmetry breaking terms is insufficient to restore the algebraic nematic order.…”

“…In this paper we focus on the strong coupling regime g 0 , g 2 t, where the interaction energies parametrically exceed the bandwidth. For clarity and completeness, we also discuss analytical results in the opposite limit [50] to provide a complete understanding of the problem. The analytical strong coupling calculations [51] are derived in the dilute limit of small superfluid density corresponding to 0 < µ + 2t t and perturbatively in δH [Eq.…”

“…The main result for this is that the model remains "stuck" in the spin liquid phase despite tuning the degeneracy lifting quadratic Zeeman and transverse fields, if we stay in the non-trivial regime at which many-body effects are dominant. To demonstrate this we first analyze the nematic order parameter N zz − N yy , which should vanish linearly as q → 0 in the spin liquid phase [50]. In addition, we use the entanglement entropy to determine the number of gapless modes and show that it is independent of the fields.…”

Orientational Order Parameters for Arbitrary Quantum Systems

“…In recent years, there has been an increasing interest in applying the concept of liquid‐crystalline phases to quantum‐mechanical systems. [ 23–57 ] Nematic order is very important in spin systems, [ 32–40 ] where it is relevant for ultracold gases [ 32–34 ] and quantum computers. [ 58–60 ] A further source of quantum nematic order in fermionic systems are deformations of the Fermi surface, which have gained significant attention in the past years.…”

Interaction-induced velocity renormalization in magic angle twisted trilayer graphene

“…When Dirac electrons reside close to a phase transition and interact with critical bosonic modes, Lorentz symmetry is often emergent 35 . This implies that different bare velocities must renormalize to become equal, and the average velocity may vanish 36 , diverge 37 , or approach a finite value, as it occurs, for example, to the two Dirac velocities in spin 3/2 systems. [38][39][40] While in TTG devices, the vicinity to external gates implies effective short-range interactions, we highlight that velocity renormalization due to Coulomb interactions in suspended graphene was predicted theoretically [41][42][43] and subsequently confirmed experimentally [44][45][46] more than a decade ago.…”