Analogue spacetimes from nonrelativistic Goldstone modes in spinor condensates Justin H. Wilson,1, 2 Jonathan B. Curtis,3 and Victor M. Galitski3 1Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, NJ 08854 USA 2Institute of Quantum Information and Matter and Department of Physics, Caltech, CA 91125, USA 3Joint Quantum Institute and Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA (Dated: January 17, 2020) It is well established that linear dispersive modes in a flowing quantum fluid behave as though they are coupled to an Einstein-Hilbert metric and exhibit a host of phenomena coming from quantum field theory in curved space, including Hawking radiation. We extend this analogy to any nonrel- ativistic Goldstone mode in a flowing spinor Bose-Einstein condensate. In addition to showing the linear dispersive result for all such modes, we show that the quadratically dispersive modes couple to a special nonrelativistic spacetime called a Newton-Cartan geometry. The kind of spacetime (Einstein-Hilbert or Newton-Cartan) is intimately linked to the mean-field phase of the conden- sate. To illustrate the general result, we further provide the specific theory in the context of a pseudo-spin-1/2 condensate where we can tune between relativistic and nonrelativistic geometries. We uncover the fate of Hawking radiation upon such a transition: it vanishes and remains absent in the Newton-Cartan geometry despite the fact that any fluid flow creates a horizon for certain wave numbers. Finally, we use the coupling to different spacetimes to compute and relate various energy and momentum currents in these analogue systems. While this result is general, present day ex- periments can realize these different spacetimes including the magnon modes for spin-1 condensates such as 87Rb, 7Li, 14K (Newton-Cartan), and 23Na (Einstein-Hilbert). I. INTRODUCTION Goldstone Dispersion Analogue Lagrangian mode spacetime Type-I ! ∼ k Einstein-Hilbert Eq. (37) The marriage of quantum mechanics and general rel- 2 ativity is one of the greatest outstanding problems in Type-II ! ∼ k Newton-Cartan Eq. (43) modern physics. This is in part due to the fact that this theory would only become truly necessary under the TABLE I. Analogue spacetimes which appear for the differ- most extreme conditions|the singularity of a black-hole ent Goldstone modes in the presence of a background conden- or the initial moments after the big bang. As such, it sate flow. These spacetimes emerge as effective field theories is extremely difficult to theoretically describe, let alone governing the long-wavelength behavior. As we demonstrate physically probe. in this work, the emergent geometry is determined by the Despite the seeming intractability, some headway may flow profile of the background condensate. This is explicitly be made in the understanding of such extreme theories demonstrated in Sec. II D for the Type-I modes and Sec. II E for the Type-II modes, where we also provide an overview of by way of analogy. This idea traces back to Unruh, who the Newton-Cartan formalism. in 1981[1] suggested that a flowing quantum fluid could realize a laboratory scale analogue of a quantum field theory in a curved spacetime. Access to even the most full analysis of all Goldstone modes in a flowing spinor rudimentary quantum simulator for such a curved space- (or multicomponent) condensate. Spinor condensates time could provide valuable insights into this otherwise [24] have been studied in the context of analogue curved inaccessible regime. space before [25, 26]; however a full accounting of all gap- Since Unruh's initial proposal, many systems have less modes has not been done to the best of our knowl- been advanced as candidates for realizing analogue space- edge. The Goldstone modes which realize the Newton- times [2], including liquid helium [3{5], Bose-Einstein Cartan geometry exhibit a quadratic ! ∼ k2 dispersion, condensates [6{14], nonlinear optical media [15], elec- known as \Type-II" Goldstone modes [25, 26]. For ex- tromagnetic waveguides [16], magnons in spintronic de- ample, the spin wave excitations about an SU(2) symme- vices [17], semi-conductor microcavity polaritons [18], try breaking ferromagnetic mean-field are such a mode. arXiv:2001.05496v1 [cond-mat.quant-gas] 15 Jan 2020 Weyl semi-metals [19], and even in classical water Distinct from the linearly dispersing case (called \Type- waves [20]. Analogue gravity systems are no longer a I" modes), Newton-Cartan spacetimes implement local theoretical endeavor; recent experiments have realized Galilean invariance, as opposed to local Lorentz invari- the stimulated Hawking effect [21], and in the case of ance. These results are general and summarized in Ta- a Bose-Einstein condensate a spontaneous Hawking ef- ble I, where we give a general prescription for separat- fect [14]. ing out all Goldstone modes into either Type-I (linearly In this paper we introduce a new kind of analogue dispersing) or Type-II (quadratically dispersing) modes gravity system, one which exhibits Newton-Cartan ge- and assigning them either an Einstein-Hilbert or Newton- ometry [22, 23]. This geometry naturally arises from a Cartan spacetime geometry. 2 Newton-Cartan geometry was developed by Car- out, we take ~ = kB = 1 and our relativistic metrics tan [22] and refined by others [27] as a geometric formu- have signature (+ − − −). We also indicate spatial vec- lation and extension of Newtonian gravity. It has since tor with a boldface (e.g. r), while spacetime vectors are found application across different areas of physics, in- indicated without boldface (e.g. x = (t; r)). cluding in quantum Hall systems [23, 28, 29] and effec- tive theories near Lifshitz points [30, 31] with interest to the high-energy community with implications for quan- II. RELATIONSHIP BETWEEN SPACETIME tum gravity [32, 33]. We extend these applications here AND GOLDSTONE'S THEOREM to flowing condensates for the case of Type-II Goldstone modes. In this work we consider models of ultra-cold bosonic Heuristically, one may view the quadratic dispersion spinor quantum gases described by an N-component field 2 T relation ! ∼ jkj + ::: as the limit of a linear disper- variable Ψ(r; t) = [Ψ1; Ψ2;:::; ΨN ] residing in d spa- sion relation ! ∼ vjkj + ::: with vanishing group velocity tial dimensions (we do not make the distinction between v ! 0. In terms of the analogue spacetime, this corre- \spinor" and higher multiplet fields in this work). The sponds to an apparent vanishing of the speed of light. Lagrangian describing this system is taken to be of the As such, the formation of event horizons and their cor- general form responding Hawking radiation ought to be ubiquitous in i y−! y − 1 y y such spacetimes; however our results contradict this in- L = 2 (Ψ @t Ψ − Ψ @t Ψ) − 2m rΨ · rΨ − V (Ψ ; Ψ); (1) tuition. Specifically, we find that fields propagating in Newton-Cartan geometries exhibit an additional conser- where m is the mass of the atoms in the gas and V (Ψy; Ψ) vation law which precludes the emission of Hawking ra- is a general potential energy function that includes inter- diation. actions with an external potential as well as local inter- The immediate implication of this is that any Type-I particle interactions. Such a system may be realized by mode can have an effective event horizon and therefore cold-atoms, where in addition to the inter-particle in- a Hawking effect (similar things have been noticed for teractions external potentials such as a harmonic trap, specific other Type-I modes), and further, no Hawking optical lattice, or magnetic field may be present. For a effect can occur for Type-II modes, at least not without comprehensive review regarding the theory and experi- introducing quasiparticle interactions (which corresponds mental realization of spinor condensates see Ref. [24]. to going being a quadratic treatment of fluctuations). We consider the case where the Lagrangian exhibits invariance under an internal symmetry described by a Lie Finally, we discuss the relationship between trans- group G, according to which Ψ transforms via a linear port phenomena and gravitational metrics in our the- unitary representation R(G) such that the action S = ory [23, 28, 34, 35]. Specifically, we obtain the stress- R L dd+1x remains invariant. That is, tensor, energy flux, and momentum density for theories both with the Einstein-Hilbert and Newton-Cartan ge- Ψ(x) ! UΨ(x) ) S ! S 8U 2 R(G): (2) ometries. In particular, we relate the energy-momentum tensor calculated in an analogue Einstein-Hilbert geom- Recall that a Lie group G is generated by its correspond- etry to its nonrelativistic counterparts through the use ing Lie algebra g, and this has a representation of R(g) of Newton-Cartan geometry. This helps identify how the when acting on the field Ψ. For ease of calculations, we analogue Hawking effect results in nontrivial energy and use the mathematical convention that Lie algebras con- momentum currents in the underlying nonrelativistic sys- sist of anti-Hermitian elements. Hence, if A is an element tem. of R(g), then A = −Ay and the corresponding group el- The outline of the paper is as follows. Section II shows ement is eA = ((eA)−1)y. that in the presence of a flowing background conden- We pursue a semi-classical analysis of our system by sate Type-I and -II Goldstone modes couple to Einstein- first obtaining the classical equations of motion (i.e. the Hilbert (Section II D) and Newton-Cartan (Section II E) saddle-point of the action). Then we linearize the ac- geometries respectively. In Section III, we present a min- tion around the saddle-point, obtaining a description of imal model for these space-times and the phase transi- the symmetry-broken phases in terms of their Goldstone tion that connects them.