Analogue Spacetimes from Nonrelativistic Goldstone Modes in Spinor Condensates

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 nonrelativistic 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 condensate. 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 experiments 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 ω ∼ 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 ω ∼ 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 ω ∼ 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 effective 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 ω ∼ |k| + ... as the limit of a linear disper- variable Ψ(r, t) = [Ψ1, Ψ2,..., ΨN ] residing in d spa- sion relation ω ∼ v|k| + ... 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 †−→ †←− 1 † † such spacetimes; however our results contradict this in- L = 2 (Ψ ∂t Ψ − Ψ ∂t Ψ) − 2m ∇Ψ · ∇Ψ − 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 (Ψ†, 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 ∀U ∈ 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 = −A† and the corresponding group el- The outline of the paper is as follows. Section II shows ement is eA = ((eA)−1)†. 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. In Sec. III A we develop the modes. The primary point of our work is that this lin- Bogoliubov-de Gennes framework which we then use to earized action admits a simple description in terms of dif- analyze this system. In Sec. IV we apply this to a specific ferent emergent analogue spacetimes and depending on step-like flow geometry and show the effect of the geom- the nature of the saddle-point, this analogue spacetime etry on the emitted Hawking radiation. We then dis- may develop non-trivial curved geometry. cuss transport of energy and momentum in these differ- The rest of this section is organized as follows. We per- ent analogue spacetimes systems in Sec. V. We conclude form a quadratic fluctuation analysis in Section II A. In the paper in Section VI. Our two appendices include Ap- Section II B we review the proof of the Goldstone theo- pendix A where we put the full fluctuation calculation rem in non-relativistic settings [25, 26] and show how this of the Lagrangian and Appendix B where we review the allows us to classify Goldstone modes into Type-I and Hawking calculation for the phonon problem. Through- Type-II. Section II C then presents the full Lagrangian for 3 the Goldstone modes while Sections II D and II E make sense described below. Within the quadratic theory, this explicit the connection to curved space geometry. implies the fluctuations reside within a real vector space 2N N R ∼ C . The Goldstone modes σlΨ0(x) form a subspace of this manifold while the remaining basis elements A. Saddle-Point Expansion are generically massive and are written as ξn(x). We note that in general the basis elements are spacetime de- We begin by looking for saddle-points of the La- pendent simply because the mean-field is also spacetime grangian Eq. (1), the spinor Gross-Pitaevskii equation dependent. In order to make the notion of orthogonality precise we 1 ∂V N 2 lift the standard complex (C ) inner product onto our i∂tΨ = − ∇ Ψ + † . (3) 2m ∂Ψ real vector space R2N to obtain the real inner product g Suppose that we have found a mean-field solution to this defined by equation Ψ0(r, t) ≡ hΨ(r, t)i which describes the dy- 1 † † namics of a mean-field condensate (neglecting fluctua- g(ξ, χ) ≡ 2 (ξ χ + χ ξ). (6) tion back-reaction); for a general out-of-equilibrium sys- In terms of the Goldstone manifold and its complement, tem, the space-time dependence of Ψ0(r, t) may be non- trivial [8, 24, 36]. the variation δΨ(x) takes the compact form The presence of a non-zero mean-field solution Ψ0 spontaneously breaks the internal symmetry group G δΨ(x) = σ(x)Ψ0(x) + ξ(x), (7) down to a subgroup H ⊂ G. Let h be the Lie algebra that generates the subgroup H. This is defined by where we have defined the massive modes by the set of generators X ξ(x) = βn(x)ξn(x). (8) h = {τ ∈ g | τ Ψ0 = 0}. (4) n

We can form a complete basis for h = span{τk}. The We proceed to the expansion of the Lagrangian in original Lie algebra then separates into two sub-spaces; terms of the variation δΨ. First, we consider the po- g = h ⊕ hc, where hc is simply the complement of h. It tential. It is locally invariant under under G, so we can c is useful to form an explicit basis for h ≡ span{σl} so write c that g = span{τk} ∪ {σl} = span{σl, τk}. Formally, h is isomorphic to the quotient algebra g/h, and the basis V (Ψ†, Ψ) = V (Ψ†eσ(x), e−σ(x)Ψ). (9) elements σl are isomorphic to coset spaces. It is important to emphasize that, although in general Furthermore, we can use our expansion of Ψ(x) to obtain the mean-field Ψ0(x) may break the symmetry group G −σ −σ down to different subgroups H = H(x) at each spacetime e Ψ ≈ e [Ψ0 + σΨ0 + ξ] point, we do not consider this in full generality since it ≈ (1 − σ + 1 σ2)[Ψ + σΨ + ξ] (10) leads to a very complicated (but interesting) structure 2 0 0 1 2 involving a non-Abelian connection on the spacetime. ≈ Ψ0 + ξ − σξ − 2 σ Ψ0, However, we later consider flowing condensates which inhomogeneously break the U(1) subgroup of G. keeping terms up to quadratic order in fluctuations. This We now examine the quadratic fluctuations of the field allows us to expand the potential energy up to quadratic Ψ about the mean-field by expanding the Lagrangian in order (dropping the terms constant and linear in the vari- powers of δΨ(x) = Ψ(x) − Ψ0(x). This separates into ation) two distinct contributions; the massless Goldstone modes θl(x) which correspond to spontaneously broken symme- ∂V V (Ψ†, Ψ) = − · 1 σ2Ψ + σξ + c.c. tries, and massive fields βn(x) which describe all the re- ∂Ψ 2 0 maining modes. Each Goldstone mode corresponds to a ¯ 1 ∂2V ∂2V 1 ∂2V broken generator σl ∈ h acting on the mean-field conden- + ξ∗ξ∗ · + ξ∗ · · ξ + · ξξ, sate Ψ (x). These contribute to the fluctuation action as 2 ∂Ψ†∂Ψ† ∂Ψ†∂Ψ 2 ∂Ψ∂Ψ 0 (11) X (δΨ(x)) = θ (x)σ Ψ (x) ≡ σ(x)Ψ (x), (5) Goldstone l l 0 0 where all derivatives of the potential are understood as l being evaluated at the mean-field. The terms quadratic which serves to define the Goldstone matrix field σ(x). in ξ, ξ∗ represent massive terms, and the first line of The remaining degrees of freedom are generically massive Eq. (11) drops out when combined on-shell with simi- and are not amenable to a description in terms of the lar terms from the kinetic part of the Lagrangian. De- Lie algebra’s generators. It is advantageous to param- riving the full fluctuation Lagrangian is not instructive, eterize the fluctuations δΨ in terms of real fields with and has been relegated to Appendix A; the final result is massive terms orthogonal to the massless terms in the given below. 4

Focusing on the Goldstone modes, written in terms modes, which is done by going to the preferred basis of µ of the “angle fields” θl(x), the resulting Lagrangian for the matrix Pmn. fluctuations is given by To understand this, we return to the real vector space defined by the Goldstone mode manifold, which we label L = θ P µ (∂ θ ) + β Qµ (∂ θ ) fluc m mn µ n m mn µ n AR. That is, + (∂ θ )T jk (∂ θ ) + L (β , ∂ β ), (12) j n mn k n mass m µ m A = span {σ Ψ (x)}. (17) R R l 0 where we have instituted the Einstein summation convention. In this and the following, Roman indices i, j, k, . . . The real dimension DR of this subspace is simply equal to run over spatial dimensions while Greek indices µ, ν, . . . the number of broken generators. We can complexify this run over both temporal and spatial dimensions (with vector space by allowing for complex-valued coefficients µ = 0 = t the temporal index). The Roman indices A ≡ span {σ Ψ }. (18) n, m, . . . enumerate the different Goldstone modes or C C n 0 massive modes and are similarly summed. The terms µ µ jk It may be the case that two generators which are linearly Pmn, Qmn, and Tmn depend on both space and time, and are given by independent under real coefficients are linearly dependent when multiplied by complex coefficients. For this t i † reason, this vector space has an associated complex di- Pmn = 2 Ψ0[σn, σm]Ψ0, j 1 † † mension DC ≤ DR. The essence of the Goldstone mode Pmn = 4m (∂jΨ0[σm, σn]Ψ0 − Ψ0[σm, σn]∂jΨ0), theorem is that DR is the number of broken generators t † † Q = i(Ψ σnξm + ξ σnΨ0), and D is the number of modes, and these two quantities mn 0 m (13) C j 1 † † can be formally related by classifying each basis element Qmn = 2m (ξmσn∂jΨ0 − ∂jΨ0σnξm σlΨ0(x) ∈ AR due to whether iσnΨ0 ∈ AR or not. † † + Ψ0σn∂jξm − ∂jξmσnΨ0), To establish this we need to return to our real inner product g(·, ·). We can use the operation of multiplica- T jk = 1 δjkΨ†σ σ Ψ . mn 2m 0 n m 0 tion by i to define a symplectic bilinear form ω(·, ·) by

As mentioned previously, it is also important to keep i † † track of the massive modes in the full Lagrangian and ω(η, ξ) ≡ g(iη, ξ) = 2 (ξ η − η ξ). (19) we oﬀer that full analysis in Appendix A. The multiplication by i (acting on the basis vectors

σlΨ0(x)) can be restricted to the real vector space AR, B. Proof of the nonrelativistic Goldstone theorem which we deﬁne by the notation

i|A ≡ I : A → A . (20) Before proceeding to simplify the Lagrangian and de- R R R rive the curved space analogues, we need to understand Similarly, we define range I ≡ AII ⊂ AR as the range of I. and make use of the nonrelativistic Goldstone theo- The null space of I is then defined to be A and represents rem [25, 26], providing a complementary proof in the I states η ∈ AR which leave the real vector space upon process. multiplication by i. As a simple example, consider unit We consider the following ansatz for the mean-field T T vectorse ˆ1 = (1, 0) ande ˆ2 = (i, 0) . As elements of a p iϑ(x) † real vector space these are linearly independent, however Ψ0(x) = ρ(x)e χ, χ χ = 1, ∂µχ = 0. (14) ieˆ1 =e ˆ2 and so these are not linearly independent in Importantly the spinor structure given by χ is indepen- a complex vector space. In this case, we have DR = dent of space and time. The global U(1) symmetry im- 2,DC = 1 and range I = AR, null I = 0. However, if T T plies the phase and density obey a continuity relation eˆ1 = (1, 0) ande ˆ2 = (0, 1) then DR = 2 = DC and which can be conveniently written as range I = 0, null I = AR. The classification of basis elements may be accom- µ ∂µJ = 0, (15) plished by taking the real inner product of iη with the µ µ other elements of A—if this vanishes, then η is in the with the condensate four-current given by J = ρvs , kernel of I. But this is exactly given by the symplectic where the superfluid four-velocity field is vµ = (1, 1 ∇ϑ). s m bilinear form defined above so that This simplifies the term

µ i µ † AI ≡ null I = {η ∈ AR | ω(η, χ) = 0, ∀χ ∈ A}. (21) Pmn = − 2 J χ [σn, σm]χ, (16) which dictates which real ﬁelds θ are canonically conju- This condition can be simpliﬁed into a matrix condition n P gate to each other. In non-relativistic systems, the rela- if we note that we can let η = anσnΨ0 and χ = P n tionship between broken symmetry generators and Gold- m bmσmΨ0, so that stone modes is not one-to-one. Instead, we must sepa- i † rate out our modes into Type-I and Type-II Goldstone 0 = ω(η, χ) = − 2 anΨ0[σn, σm]Ψ0bm. (22) 5

I This relates the null-space of I to the null-space of the The modes represented by iσnΨ0 are exactly the massive † µ I matrix Ψ0[σn, σm]Ψ0 ∝ Pmn, the term appearing in our modes conjugate to σnΨ0 (by definition, they are not in Lagrangian which determines the canonically conjugate A and are thus not associated with a broken generator). pairs of modes. Using the rank-nullity theorem, we have At low energies (below the relevant mass gaps), massive modes that are not conjugate to Goldstone modes A = A ⊕ A . (23) R I II can be trivially integrated out and do not contribute in i † the IR. This then leaves the Goldstone modes, which are Since the matrix given by elements − 2 Ψ0[σn, σm]Ψ0 is real and antisymmetric, we can block-diagonalize the ma- gapless, and a few massive modes which are canonically trix with a special orthogonal transformation. Going to conjugate to the Type-I Goldstone modes. These massive this basis and using our ansatz for the flowing mean-field modes cannot be trivially integrated out and they are to √ Ψ = ρeiϑχ, the result is be included in the low-energy theory. Doing so amounts 0 I to adding the basis elements iσnΨ0 to our fluctuation i † i † − 2 Ψ0[σn, σm]Ψ0 = − 2 ρχ [σn, σm]χ manifold. AII AI C. Lagrangian for Goldstone Modes  0 λ1 0 0  −λ1 0 0 0 ··· 0 ···  0 0 0 λ  (24) We now employ this classification into Type-I and -II  2   0 0 −λ 0  modes to our benefit by using it to simplify the fluc-  2  = ρ  . .  , tuation Lagrangian. Recall that in this work we restrict  . ..    ourselves to flowing condensates which have a spatial tex-  0 0    ture to the phase mode (and thus inhomogeneously break . . . .. the global U(1) part of the symmetry group), but have a homogeneous and static spinor texture. For instance, with λ > 0. This defines a preferred basis for the broken 1 j one may consider a condensate of pseudo-spin- 2 atoms generators {σl} which we henceforth assume is the basis in its ferromagnetic phase which has a definite homoge- we are in. Note that in this basis A takes the form of a † 1 II neous magnetization hSzi = χ Szχ = 2 but a non-zero direct sum of decoupled symplectic forms. density and phase profile. As remarked earlier, this flow This matrix provides a natural way to break up the produces a non-zero spatial component for the Noether II µ generators. First, we can define σn and its conjugate gen- current Jµ(x). Going to the preferred basis of Pmn, ob- II i † II II erator σn via − 2 Ψ0[σn , σn ]Ψ0 = ρλn. This implies that tained in Sec. II B then yields the partitioning into the II II II II Goldstone modes given by {σIIΨ , σIIΨ , σI Ψ }. Let us σn Ψ0 = iσn Ψ0 (however σn 6= iσn ). Let nII be the num- n 0 n 0 n 0 ber of λj’s, so that dim(AII) = 2nII. As the coefficient remind the reader that Type-I modes are those for which 0 of the temporal derivative term in the Lagrangian, this iσnΨ0 cannot be written as a broken generator σnΨ0 and matrix tells us that the two Goldstone fields described by therefore, the associated real field comes with a massive σIIΨ (x) and σIIΨ (x) are canonically conjugate to each term in the Lagrangian. n 0 n 0 II I other and therefore describe the same mode, a Type-II The basis elements {σn Ψ0, σnΨ0} have the property Goldstone mode. Finally, let dim(A ) = n be dimen- that they are orthogonal in the conventional sense (e.g. I I † sion of the null-space of I. This is the number of Type-I η χ = 0). As a result of this, Goldstone modes; they represent modes which are canon- Ψ†σI σII Ψ = 0, ically conjugate to a massive mode. It is evident by the 0 n m 0 † II II rank-nullity result that −Ψ0σn σmΨ0 = λnδnmρ(x), (29) † I I 2nII + nI = DR (25) −Ψ0σnσmΨ0 = µnδnmρ(x), is the number of broken generators, while † I 2 where we have defined µn ≡ −χ (σn) χ > 0 and used † II 2 the fact that λn = −χ (σ ) χ > 0. nII + nI = DC (26) n In this basis, the field variation δΨ(x) may be de- is the number of Goldstone modes in the system. ¯ scribed by three real Goldstone fields θn, θn, and φn along With this particular grating into nII basis elements with the real massive field β via II I n σn Ψ0 and nI basis elements σnΨ0, we can rewrite our real vector space nII nI X II ¯ II X I σ = θnσn + θnσn + φnσn, II II I AR = span{σn Ψ0, σn Ψ0, σnΨ0}, (27) n=1 n=1 n (30) and similarly, we can write the complexified vector space XI ξ = β iσI Ψ + ··· , in two equivalent ways n n 0 n=1 II I A = span {σ Ψ0, σ Ψ0}, C C n n (28) where “··· ” represents other massive modes that can be II II I I trivially integrated out. In this basis, the coefficient P µ AC = span{σn Ψ0, σn Ψ0, σnΨ0, iσnΨ0}. mn 6 simplifies to term which we can separate out into its contribution to Type-I and Type-II fields µ µ Pmn = δnm¯ λnρ(x)vs , (31) jk 1 jk T |I = − δ ρ(x)µnδmn wherem ¯ is defined as the index of the conjugate field mn 2m (33) jk 1 jk to the field labeled by m. Similarly, we may simplify Tmn|II = − δ ρ(x)λnδmn µ 2m Qmn which connects Type-I Goldstone modes to their conjugate massive fields. We indeed find Notice that λn or µn multiplies all elements in the La- µ µ Qmn = 2δnmµnρ(x)vs , (32) grangian where that field (or its conjugate) appears, so we can simply absorb this constant into a re-definition of ¯ where the massive field with index m is indicated by the θn, θn, φn, and βn. Then, substituting the form of our I basis element iσmΨ0. Lastly, we have the kinetic energy fluctuations, the Lagrangian is

nI X µ 1 2 2 2 2 Lﬂuc = ρ(x) −2βnvs (x)∂µφn − 2m [(∇φn) + (∇βn) ] − 2mcn(x)βn n=1 nII X n µ ¯ −→ ¯ ←− 1 2 ¯ 2 o + ρ(x) −vs (x)(θn∂µθn − θn∂µθn) − 2m [(∇θn) + (∇θn) ] . (34) n=1

2 µ Since the basis for Type-I modes is not uniquely fixed by out via mc (x)β = −2vs ∂µφ. We get the resulting La- the canonical conjugate structure of Eq. (24), this leaves grangian, valid at long wavelengths and times us free to diagonalize the mass tensor produced by the " µ 2 # variation of the potential in Eq. (11). Doing so produces eff ρ(x) vs (x)∂µφ 2 2 LI = − (∇φ) . (36) the effective chemical potential terms, mcn(x). 2m c(x) We end this section with a note about the validity of this fluctuation Lagrangian: it can be seen that the over- This describes a scalar field propagating along geodesics all size of this action is set by the condensate density ρ(x), of an emergent space-time metric Gµν with which uniformly multiplies all terms. Thus, the conden- √ sate density ρ(x) acts to enforce the saddle-point in the eff 1 µν LI = 2 −GG ∂µφ∂ν φ, (37) sense that if it is large, the fluctuation contribution from µν Lfluc is suppressed. This tells us that our approach ought and G given by the line-element not be valid if either the condensate density is strongly ρ fluctuating or vanishing all-together, as might happen ds2 = [c2dt2 − (dx − vdt)2] = G dxµdxν . (38) c µν at finite temperatures or near e.g. the core of a vortex. Additionally, there may be breakdowns in smaller dimen- This was first observed by Unruh in Ref. [1] where he sional systems, where long-range order is prohibited by showed that metrics of the form given above can possess Mermin-Wagner [37–39]. Barring these considerations, non-trivial features including event-horizons. Indeed, the we proceed on to study the properties of the effective metric for a Schwarschild black hole can take a very simi- field theory described in Eq. (34). We first consider the lar form in certain coordinate systems. One of the central case where the Goldstone mode is Type-I, and then we results of this paper is the extension of this analogue to study the case of a Type-II mode. include the Type-II modes, which do not have emergent Lorentz invariance. This is shown below.

D. Type-I Goldstones: Relativistic Spacetime E. Type-II Goldstones: Non-relativistic Spacetime Consider an isolated Type-I Goldstone mode, with La- grangian We focus on a single Type-II Goldstone mode, for which there is no massive ﬁeld to integrate out. We are µ 1 2 2 LI = ρ(x)[−2βvs (x)∂µφ − 2m [(∇φ) + (∇β) ] left with the ﬂuctuation Lagrangian − 2mc2(x)β2], (35) µ ¯−→ ¯←− LII = ρ(x){−v (x)(θ∂µθ − θ∂µθ) we assume that mc2(x) is large enough to dominate over s 1 2 ¯ 2 the kinetic energy for β, so that β can be easily integrated − 2m [(∇θ) + (∇θ) ]}. (39) 7

To simplify things, we group the two real fields into one This gives us the geometric quantities complex field 2 2/d h = ρ , n0 = ρ , (45) ψ = θ + iθ,¯ (40) and hence so that we have 2/d nµ = [ρ , 0], i µ ∗−→ ∗←− 1 2 LII = ρ[ v (ψ ∂µψ − ψ ∂µψ) − |∇ψ| ]. (41) µ −2/d µ 2 s 2m v = ρ vs , (46) ij −2/d ij It turns out this too has a simple geometric description h = ρ δ . in terms of an emergent curved space-time. However, instead of being an “Einsteinian” geometry, the resulting One important aspect of Newton-Cartan geometry is description is in terms of a Newton-Cartan geometry [22, the notion of “torsion” [40]. Regarded as a differential µ 23, 28, 29, 35]. form, the clock one-form n = nµdx is in general not Newton-Cartan geometry consists of three key objects: an exact differential. This is seen by taking the exterior µ µν derivative, which defines the “torsion tensor” ω = dn. (nµ, v , h ). These are not all independent, but rather must satisfy the constraints Explicitly,

µ µν ωµν = ∂µnν − ∂ν nµ. (47) nµv = 1, nµh = 0. (42)

Also note that the indices on these objects are given as It is straightforward to see that in general, the torsion covariant and contravariant specifically and cannot be tensor in our geometry is non-zero; freely raised/lowered without the definition of a metric ω = ∂ n = ∂ ρ2/d. (48) tensor (which we describe how to construct in Sec. V). 0j j 0 j To understand the geometry these objects encode, we Were the torsion zero, we could define an absolute time begin with the fundamental object that enforces time’s coordinate T , from which we would get the clock one- special status within a nonrelativistic theory: nµ. As a form as n = dT . While the non-zero torsion implies one-form, nµ (colloquially, we call it the “clock” one- there is no such absolute time, we may confirm that the form) can be imagined as a series of surfaces (folia- more general condition tions), and when a spacetime displacement vector is con- tracted with it, it gives the elapsed time in a covari- n ∧ dn = 0 (49) ant manner. In conjunction with the clock one-form, we have the velocity field vµ, which must go forward a is satisfied. This is a necessary and sufficient condition for µ unit of time (hence the constraint nµv = 1) as a four- the foliation of spacetime into “space-like” sheets which velocity; flow along vµ causally connects spatial surfaces. are orthogonal to the flow of time [40]. As such, there is µν µν Lastly, the spatial metric h is degenerate (nµh = 0) still a notion of causality in this geometry. since it solely describes the geometry confined to the d- We conclude by commenting that the Newton-Cartan dimensional spatial foliations. While in what follows we geometry we find here is in fact intimately related to µν describe h emerging from intrinsic properties of the the gravitational field first considered by Luttinger in the fluid flow, it can also inherit extrinsic contributions (i.e. context of calculating heat transport [34]. In that limit Φ if the fluid is flowing on an actual curved manifold). nµ ∝ [e , 0], and so the gravitational potential (up to In the presence of this curved Newton-Cartan geome- scale factor in the logarithm) would be try, the Lagrangian for a massless scalar field takes the form 2 Φ = log(ρ). (50) √ d i µ ∗−→ ∗←− hµν ∗ L = n0 h[ v (ψ ∂µψ − ψ ∂µψ) − ∂µψ ∂ν ψ] (43) 2 2m Using this connection, quantities like energy current and where h = (| det hij|)−1. the stress-momentum tensor can be calculated as we dis- The Lagrangian of a Type-II Goldstone mode may be cuss in Sec. V. First, we explore a minimal realization of brought into this form. Relating Eq. (41) to Eq. (43), these geometries and the associated quantum phases in µ µν Sec. III as well as the fate of the Hawking effect across we can extract the geometric objects nµ, v , and h . We see that in our systems h00 = 0 = h0i, and that such a transition in Sec. IV. hij = h−1/dδij in d spatial dimensions. Therefore, we 0 know ni = 0; hence, n0v = 1. Relating terms, we have III. MINIMAL THEORETICAL MODEL √ h = ρ, √ In this section, we introduce a minimal model which i i (44) n0 hv = ρvs, exhibits a transition between an Einstein-Hilbert and (d−2)/(2d) n0h = ρ. Newton-Cartan spacetime. We begin by analyzing the 8 ground state within mean-field theory. Once this is understood, we study the behavior of fluctuations about the mean-field by employing a Bogoliubov-de Gennes (BdG) description. 1 The model is that of a pseudo-spin- 2 bosonic field T Ψ(x) = (Ψ↑(x), Ψ↓(x)) with the following Lagrangian density † 1 2 1 † 2 L = Ψ i∂t + ∇ + µ Ψ − g0 Ψ Ψ 2m 2 Ising SU(2) Easy-plane 1 † 2 − g3 Ψ σ3Ψ (51) Ferromagnet Ferromagnet Ferromagnet 2 where σj are the Pauli matrices for the pseudo-spin and FIG. 1. Illustration of the different ground-state Bloch-vector µ is the chemical potential, which controls the conserved manifolds as the parameter g3 is tuned. For g3 < 0 the † density of the bosons, ρ = Ψ Ψ. The coupling g0 > 0 ground state manifold consists of the north and south poles describes a U(2) = U(1) × SU(2) invariant repulsive and thus the system realizes an Ising ferromagnet, sponta- density-density contact interaction, as may be expected neously breaking the Z2 symmetry while maintaining the U(1) symmetry. For g = 0 the full SU(2) symmetry is realized and in a typical spinor BEC, while the g3 parameter intro- 3 duces anisotropy into the spin exchange interaction. The the ground-state manifold consists of the entire Bloch sphere. g coupling explicitly breaks the SU(2) symmetry down Thus, the system is a Heisenberg ferromagnet which sponta- 3 neously breaks the full SU(2) down to U(1) ⊂ SU(2). Finally, to U(1) ⊗ 2 comprised of rotations of the Bloch vector Z for g3 > 0 the ground state manifold consists of the equatorial by any angle about the z axis and reflections of the Bloch plane, rendering the system an XY (easy-plane) ferromagnet. vector through the xy mirror plane. Note that stability Thus, the initial symmetry is U(1) which is spontaneously requires that g3 > −g0. broken to the trivial group. Let us briefly comment that, while Lagrangian (51) is a perfectly valid model, a more natural set-up may be Phase Sound waves Spin waves realized by the more experimentally available spin-1 sys- 7 23 87 Ising Ferromagnet ω ∼ k Gapped tems such as condensed Li, Na, or Rb. All of these SU(2) Ferromagnet ω ∼ k ω ∼ k2 atoms are bosons which have a total hyperfine spin F = 1 Easy-plane Ferromagnet ω ∼ k ω ∼ k manifold [24]. In this case, the phase transition is between two phases which both respect the full SU(2) spin- rotation symmetry—the ferromagnetic phase and polar TABLE II. Goldstone modes associated to each phase shown (nematic) phase [36, 41]. In this case, rather than be- in Fig. 1. All phases have a Type-I Goldstone mode associ- ing driven by anisotropy, the transition is driven by the ated to the spontaneous breaking of the global U(1) phase, overall sign of the spin-exchange interaction. It turns corresponding to the conventional sound mode. Additionally, there may also be Goldstone modes associated with sponta- out that the different ground-state phases have different neous breaking of spin symmetries, leading to spin waves. In types of Goldstone modes and therefore exhibit different the Ising phase, the broken symmetry is discrete and there analogue spacetimes for the spin waves once condensate are no Goldstone modes. In the SU(2) invariant Heisenberg flow is introduced. The relevant coupling constant is the phase there is a Type-II Goldstone mode describing transverse spin-exchange coupling c2, which is given in terms of the fluctuations of the magnetization, while in the XY easy-plane scattering lengths by phase there is a Type-I Goldstone describing equatorial fluctuations of the magnetization. 4π a − a c = 2 0 . 2 m 3 For µ < 0 the ground state is trivial and there is no con- For 7Li and 87Rb,c < 0 while for 23Na c > 0 [24]. 2 2 densate. For µ > 0 there is Bose-Einstein condensation Thus, all else equal we can realize both the polar (ne- and the ground state is a BEC with a uniform condensate matic) phase (which occurs for c > 0) as well as the fer- 2 density which obeys the equation of state romagnetic phase (c2 < 0) by using two different species of trapped atom. All this is to say that, while Eq. (51) is ( µ , g3 > 0, † g0 not as easily realized experimentally, there may be more ρ = Ψ Ψ = µ , −g0 < g3 < 0. experimentally feasible models which realize the same g0−|g3| physics. We now move on to the analysis of the tech- A non-zero condensate density always spontaneously nically simpler model proposed above. break the overall U(1) phase symmetry. The correspond- The mean-field ground state of Eq. (51) is identified as ing Goldstone mode corresponds to the broken generator the homogeneous minimum of the energy density iσ0 = i1 where 1 is the 2 × 2 identity matrix. 1 2 1 2 Depending on the value of g3, additional symmetries V = g Ψ†Ψ + g Ψ†σ Ψ − µΨ†Ψ. 2 0 2 3 3 may be broken, resulting in the phase diagram illustrated 9 in Fig. 1. We write the condensed Ψ in the density-phase- where the columns refer, in order, to the generators 1 1 1 1 spinor representation as { 2 iσ0+ 2 iσ1, 2 iσ2, 2 iσ3}. In this case, we have one Type- √ II Goldstone mode associated with the two generators Ψ = ρeiΘχ, χ†χ = 1 (52) 1 1 { 2 iσ2, 2 iσ3} which exhibits a quadratic dispersion relation and hence realize the Newton-Cartan geometry in where χ yields the local magnetization density. It may the presence of inhomogeneous condensate flow. This is be parameterized in terms of one complex parameter ζ summarized in Table II. via XY phase.—We now move on to the case where g3 > 0. In this case there is an energy penalty associated with a 1 1 + ζ χ = , ζ ∈ . (53) non-zero z component of the magnetization and thus the p 2 1 − ζ C 2(1 + |ζ| ) ground state lies in the manifold defined by cos θ = 0 ⇒ θ = π/2. Thus, the ground state breaks the U(1) symme- Alternatively, it may be represented in the more canoni- try but remains invariant under reflections through the cal Euler angle representation as z = 0 plane. As such, the ground state resides in the symmetric space U(1) × / = U(1) ∼ S1, as depicted cos θ Z2 Z2 χ = 2 , ϕ ∈ [0, 2π) θ ∈ [0, π). in the right panel of Fig. 1. Without loss of generality sin θ eiϕ 2 we again take the Bloch vector to lie along the +x direc- We use both of these representations throughout. In tion. Thus, only two generators remain unbroken in the Lagrangian {i1, 1 iσ } and the mean-field breaks both of terms of ζ and θ, ϕ the anisotropic interaction is 2 3 them. We again refer to Eq. (24) to obtain ∗ 2 1 2 (ζ + ζ ) 1 2 2 t V = g ρ = g ρ cos θ. Pmn = 0. (55) 2 3 (1 + |ζ|2)2 2 3 Thus, there are no Type-II Goldstone modes in this sys- We now proceed to study the mean-field phase diagram tem, but instead two Type-I modes which are linearly of the ground state. dispersing and therefore exhibit an analogue Einstein- Ising phase.—We begin by considering the case of g3 < Hilbert spacetime, summarized in Table II. 0, i.e. the “Ising ferromagnet” phase. The interaction has i a U(1) × Z2 symmetry generated by 2 σ3 composed with inversion of the z component of the magnetization. In A. Bogoliubov-de Gennes Analysis this case it is energetically favorable for the Bloch vector to align with the z axis. This breaks the Z2 symmetry We now proceed to examine the fluctuations about and preserves U(1) so the ground state manifold is the the mean-field by obtaining and diagonalizing the symmetric space U(1) × Z2/U(1) ∼ Z2. This is depicted Bogoliubov-de Gennes equations of motion. To see how in the left-most panel of Fig. 1, which shows the ground- the analogue spacetime emerges we consider a mean-field state manifold for the spinor χ for various couplings. The condensate ψ0 which is inhomogeneous, but has a con- Goldstone modes associated with the broken-symmetry stant magnetization density. Taking the spin to point in ground-state, along with their dispersions are shown in the +x direction, we obtain Table II. As the ground-state manifold is discrete there ! is no additional Goldstone mode in this phase and we no √1 ψ = pρ(x)eiΘ(x)χ = pρ(x)eiΘ(x) 2 . (56) longer consider this portion of the phase diagram in this 0 0 √1 work. 2 Heisenberg phase.—When g3 = 0 the interaction term In this case, the mean-field describes a flowing conden- is isotropic and the model has the full SU(2) invariance. † sate with superfluid density ρ(x) = ψ0(x)ψ0(x) and su- The ground state then spontaneously break the SU(2) 1 perfluid velocity vs = m ∇Θ(x). Fluctuations about this symmetry down to U(1) so that the ground state mani- mean-field can be fully parameterized in terms of the two fold is the symmetric space SU(2)/U(1) ∼ S2—the full complex fields φ and ζ as Bloch sphere. This is illustrated in the middle panel of Fig. 1. Without loss of generality, we take the ground δΨ = (φσ0 + iζσ2) ψ0. (57) state magnetization to point along the positive x direc- To quadratic order, the Lagrangian from Eq. (51) decou- tion. Thus, ζ = 0 and χ = √1 (1, 1)T . Then the unbroken 2 ples into two quadratic BdG Lagrangians i generators are { (σ1 −1)} and the broken generators are 2 h 2 i i 1 i i i ∗ ∗ |∇φ| 1 ∗ 2 { 2 (σ1 + ), 2 σ2, 2 σ3, }. Using the formalism from Sec. II, Lφ = ρ 2 (φ Dtφ − φDtφ ) − 2m + 2 g0ρ(φ + φ ) , we find that the P matrix appearing in the Goldstone h i ∗ ∗ |∇ζ|2 1 ∗ 2i mode Lagrangian is Lζ = ρ 2 (ζ Dtζ − ζDtζ ) − 2m + 2 g3ρ(ζ + ζ ) , (58) 0 0 0 t 1 P = ρ 0 0 4  , (54) with Dt = ∂t + vs · ∇ the material derivative in the 1 0 − 4 0 frame co-moving with the superfluid flow. These two 10

Lagrangians are specific examples of the more general obeys a Galilean-invariant dispersion relation. This also Eq. (34). In particular, for g3 > 0 at long wavelengths results in an additional U(1) symmetry generated by τ3 we can apply the analysis of Sec. II D to obtain the which imposes a selection rule for the allowed Bogoliubov relativistic analogue spacetime. If on the other hand, transformations. In particular, there is no matrix ele- g3 = 0, then at long wavelengths we can apply the anal- ment which scatters a “particle-like” Bogoliubov quasi- ysis of Sec. II E to obtain the nonrelativistic Newton- particle into a “hole-like” particle. this process is the Cartan analogue spacetime. Nevertheless, it is instruc- one responsible for Hawking radiation and as such we tive to instead follow Ref. [42, 43], and directly employ find, counter-intuitively, that it is impossible to gener- the BdG equations when determining the consequences ate Hawking radiation in the Newton-Cartan spacetime of the changing spacetime structure. This is because the despite the fact that all flow velocities vs are now super- BdG equations provide us with a single unified descrip- sonic. This is explicitly demonstrated for the case of a tion with which we may capture both phases, as well as step-like horizon, which we analyze in the following sec- the transition between them. tion. The BdG equations are obtained as the Euler-Lagrange equations of Lagrangians Lφ, Lζ and are most transpar- ently expressed in terms of the Nambu spinors IV. STEP-LIKE HORIZON φ ζ In order to get a more quantitative understanding of Φ0 = ∗ , Φ3 = ∗ (59) φ ζ how the changing spacetimes affect observable physics, we imagine a specific flow profile and use the BdG equa- for condensate and spin wave fluctuations, respectively. tions to solve for the spin-wave scattering matrix. We We then find the BdG equations Kˆ Φ = 0, and Kˆ Φ = 0 0 3 3 imagine a quasi-one-dimensional stationary condensate 0, with the BdG differential operators flow with a superfluid density and velocity which obeys 1 ∂tρ = ∂tvs = 0. The continuity equation for the conden- Kˆ = τ (i∂ + iv · ∇) + ∇ · ρ∇τ − g ρ (τ + τ ) 0 3 t s 2mρ 0 0 0 1 sate then implies 1 Kˆ = τ (i∂ + iv · ∇) + ∇ · ρ∇τ − g ρ (τ + τ ) , ∂x(ρvs) = 0 ⇒ ρ(x)vs(x) = const. (61) 3 3 t s 2mρ 0 3 0 1 (60) The local speed of sound for the spin-waves (henceforth written in terms of the Nambu particle-hole Pauli matri- p simply written as c) is therefore c(x) = g3ρ(x)/m. ces τa. Let us emphasize that the only difference between To further simplify calculations, we consider the case ˆ ˆ K0 and K3 is the coupling constant appearing in front of a step-like profile for ρ(x), v(x) of the form of the τ0 + τ1 term. For sound waves it is g0, while for the spin waves it is g3. Thus, both Goldstone modes end ρl x < 0 ρ(x) = up coupling to the same background condensate density ρ x ≥ 0 and velocity, albeit with different speeds of sound. Sound r (62) − |vl| x < 0 waves end up propagating with the local group velocity v(x) = − |v | x ≥ 0. r r g0ρ(x) c0(x) = 2 2 m Note that continuity requires vlρl = vrρr ⇔ vlcl = vrcr. In this work we adopt the convention that v is negative, while the spin waves have the local group velocity so that the condensate flows from the right to the left. r With this set-up, we can employ the BdG techniques g3ρ(x) usually used for phonon modes to these spin waves [42, c3(x) = . m 43]. This step-like potential has the advantage that away Thus, we see that the coupling g allows us to inde- 3 from the jump, momentum eigenstates solve the BdG pendently tune the two speeds of sound relative to each equations, and the scattering matrix reduces to a simple other. plane-wave matching condition at the boundary. The de- For generic values of g > 0 and arbitrary condensate 3 tails of this procedure may be found, e.g. in Appendix B. flows we cannot find quantum numbers with which we Here we simply discuss the results of the calculation. We can diagonalize Kˆ . However, at the SU(2) symmetric 3 start by considering g > 0 to be large and then decrease point g = 0 we observe that the BdG kernel for spin 3 3 down to zero. As we do so, while keeping the flow profile waves obeys fixed, we pass through three regimes. 1 h i The first regime occurs for large g3 so that cl > |vl| Kˆ = τ (i∂ + iv · ∇) + ∇ · ρ∇τ ⇒ τ , Kˆ = 0. 3 3 t 2mρ 0 3 3 and cr > |vr|. Thus, there is no sonic horizon and no Hawking radiation. Since τ3 now commutes with the kernel, the two compo- Eventually as we continue decreasing g3 we enter the nents of the BdG spinor decouple and each independently regime where |vr| < cr but cl < |vl|. This exhibits a sonic 11 H

L supersonic subsonic subsonic the control parameter g3. We see the three distinct re-

2 to to to gions and importantly at g = 0 we see the Hawking

0.2 3 , supersonic supersonic subsonic y

t eﬀect vanish. i s

o To understand this eﬀect, we consider the dispersion n i relation of the waves away from the horizon, for which m 0.1 u momentum is a good quantum number. In the right and L left half-spaces we have the relations g n i k 4 w k

a 2 2 2 0.0 (ω − vαk) = cαk + , (64) H 2 0 1 2 3 4 4m 2 (cr/vr) where α = l, r for the left and right regions respectively. This relates the lab-frame frequency of a wave ω to the FIG. 2. The total luminosity due to the Hawking radiation for lab-frame momentum k. This dispersion relation is plot- a fixed density profile ρ(x) and velocity profile v(x). We see ted in Figs. 3 and 4. Due to the presence of a disconti- that there is no Hawking radiation when cr is sufficiently large nuity at x = 0 modes with different momenta mix and so that cl > vl (recall these are constrained by the continuity only ω can be fixed globally. Thus, the dispersion rela- equation). When cl < vl but cr > vr we get a region of tion is to be solved by finding the allowed momenta at subsonic flow that flows into a supersonic region and we begin each fixed lab-frame frequency. This amounts to finding seeing traditional Hawking radiation. As we further tune g3, the roots of a quartic polynomial with real coefficients, cr drops below vr and both regions become supersonic at low and as such there are always four solutions (which are frequencies. Evidently, there is still a channel for Hawking radiation emission as seen by the non-zero integrated flux. either real or complex conjugate pairs). The real momenta represent propagating modes while we later find However, as cr drops to zero this channel closes, vanishing precisely at the quantum phase transition into the Newton- that the complex roots describe evanescent modes local- Cartan geometry (cr = 0 = g3). In this plot, vl = 0.9, vr = ized around the horizon. 1.3, m = 10, and ρ(x)v(x) = 1. horizon at x = 0 and is thus accompanied by Hawking A. Subsonic-Supersonic Jump radiation. Finally, we reach the regime where |vl| > cl and |vr| > First, we consider the case of a jump between a sub- cr. This is a novel regime wherein both the interior and sonic and supersonic flow, depicted graphically in Fig. 3. exterior of the jump are supersonic. However, due to In this case, we recover the well-known result that there the non-linear Bogoliubov dispersion, there are still some is Hawking radiation emitted. The dispersion relation in short-wavelength modes for which one or both sides of each half-plane is plotted and intercepts with a constant the flow are not supersonic (this is due to the convex ω > 0 are found. These intercepts yield the momenta of dependence of the group-velocity on momentum). Thus the propagating modes in each region for the given fre- there is still Hawking radiation, however we find that as quency. Each curve is depicted with a color indicating we decrease g3 further, the total “flux” of modes which the sign of the group velocity in the co-moving frame, are emitted decreases until we recover the result that at which is what is used to distinguish between “particle- g3 = 0 there is no radiation at all. like” (red) and “hole-like” (blue), in accordance with the To see this, we define the “total number of Hawking BdG norm (see Appendix B and in particular Eq. (B3) modes” at a given frequency to be N(ω) (see Eqs. (B20) for definition). We see that the outgoing Hawking mode and (B22)). This is obtained by calculating the “Hawk- (combined with an evanescent piece at the horizon) is ing” element of the scattering matrix for the BdG equa- connected to three incoming waves, one of which is a tions. From N(ω) we can then define the total “luminos- negative norm state originating from the interior of the ity” [44] leaving the horizon by horizon. This particle-hole conversion processes is the origin of the Hawking effect, as this induces a Bogoli- Z ∞ ω ubov transformation which connects the vacuum of the LH = dω N(ω). (63) asymptotic past to a one-particle state in the asymptotic 0 2π future (and vice-versa). Note that in the conventional black hole case, N(ω) is We see that due to the convex non-linear Bogoliubov the number of photons at frequency ω seen at asymptotic dispersion relation, there is a maximum frequency of the infinity and thus this is simply the number flux per unit emitted Hawking radiation obtained by finding the lo- frequency of the radiation. cal maximum of the negative norm dispersion relation. The upshot is given by Fig. 2 which plots LH as a func- Above this frequency, the flow is no longer supersonic 2 2 tion of (cr/vr) = g3ρr/mvr . Thus, for fixed flow density since the group velocity of modes depends non-trivially and velocity, this is essentially plotting as a function of on the frequency. 12

Hawking region Hawking region Super-Hawking region

x=0 x x=0 x

Positive Norm Positive Norm Time Hawking Time Negative Norm Negative Norm Hawking Evanescent

FIG. 3. The Hawking effect for g3 such that cr > vr and FIG. 4. The Hawking effect for g3 such that cr < vr and cl < vl (sub-sonic to super-sonic). In this situation, one side cl < vl (super-sonic to super-sonic). With both regions flow- (left) flows faster than the speed of some excitations, and the ing faster than the speed of excitations (relative to the hori- other side (right) flows slower than the speed of any excita- zon), we still have a Hawking region, but now we also have a tion. The dashed line represents the constant lab frame en- “Super-Hawking” region where the positive and negative nor- ergy ω. The mode that carries away energy from the horizon malization modes from both regions can scatter between one is the “Hawking mode,” shown by the star marker. Tracing another. this mode back in time (bottom of figure), we find that it comes from a scattering process that includes positive (red) and negative (blue) norm states. It is the negative norm state to the left of the horizon that is responsible for particle cre- 0 ation in the Hawking channel. Notice that for frequencies c ) r = larger than those in the labeled “Hawking region,” there is no 1.0

( 8 Hawking eﬀect due to lack of negative energy modes to have 2 vr scattered from at earlier times. N g o l cr = 4 0.7 4vr B. Supersonic-Supersonic Jump

0.0 0.5 1.0 1.5 As we decrease g3 beyond a critical value the system enters a parameter regime where both sides of the 1 2 /( mvr ) jump are supersonic flows. In this case, the dispersion 2 relation still exhibits a Hawking-like region, as we see FIG. 5. Hawking flux N(ω) as a function of frequency for in Fig. 4. However, we also see a new region emerge the subsonic-to-supersonic case (red) and the supersonic-to- at low energies (labeled “super-Hawking” in the figure) supersonic case (blue). As we approach the Heisenberg sym- in which now both a positive and negative norm mode metric point g3 = 0, we find the Hawking flux disappears both can be scattered into. This opens a new channel in in its overall magnitude and singular behavior. The black the scattering matrix which leads to a reduction in the arrow indicates the onset of the “super-Hawking region” re- amplitude for scattering into the Hawking channel, as sponsible for the absence of the singular distribution. per generalized unitarity constraints. This is seen in Fig. 5, which compares N(ω) for the case of a subsonic- supersonic (red) and supersonic-supersonic jump (blue). on the right side of the horizon. Second, in this super- Both curves are qualitatively similar at high frequencies, Hawking region, the appearance of an outgoing negative- corresponding to the “Hawking” region of frequencies in norm mode provides an opportunity for the ingoing neg- Figure 4. On the other hand, we see that at low ω, when ative norm channel to avoid scattering into the positive we have subsonic-to-supersonic flow, N(ω) diverges in norm channel. We indeed find that the two channels be- the universal thermal manner, while in the supersonic- gin to decouple from each other, diminishing the amount to-supersonic regime, there is a noticeable change in be- of Hawking radiation that can be produced. havior between the Hawking and super-Hawking regimes, cutting off this low ω divergence. There are two effects occurring which are responsible C. Absence of Hawking Radiation for Type-II modes for decreasing the Hawking luminosity LH. First, in the Hawking region the incoming negative norm states now begin to more strongly backscatter into their correspond- This takes us directly into the point where g3 = 0, ing negative norm state, occupying the evanescent mode which exhibits the new Newton-Cartan spacetime ge- 13 ometry. One might expect that there should be some- thing akin to a Hawking effect since some modes “see” a horizon for any difference in |vl| and |vr|. However, this horizon does not translate into a Hawking effect. As explained earlier, at this point the BdG kernel Kˆ3 commutes with τ3. In terms of the BdG Lagrangian of Eq. (58), we find that there is now a new global U(1) symmetry ζ → eiϑζ. We can see explicitly from the BdG x=0 x analysis that this conserved charge density is given by Z 3 2 Time Positive Norm QBdG = d x ρ|ζ| . Evanescent Negative Norm On the other hand, by applying Noether’s theorem directly on the general Newton-Cartan action of Eq. (43), FIG. 6. For g3 = 0 in the Newton-Cartan geometry there in the limit where n0 is the only nonzero component of is an excitation number conservation that protects negative 0 nµ and the Lagrangian is independent of the x , we find norm states from scattering into positive norm states and as Z √ a result, if we scatter a negative norm state in what used to 3 2 be the “Hawking region,” we find it fully back scatters into QBdG = d r h|ψ| . (65) a negative norm state and leaks past the horizon only with an evanescent tail characteristic to a “classically forbidden” If we identify ψ = ζ and use the results of Eq. (46) we find region. that these two indeed match each other. In particular, Eq. (65) describes a conserved charge for the field ψ on a curved manifold given by hµν . Since, unlike the charge in Eq. (B3), this density is pos- Thus, it cannot be spontaneously excited from the in- itive definite it can be genuinely interpreted as the num- going vacuum. Ultimately, as the negative norm mode ber of BdG quasiparticles. This symmetry then imposes must be reflected, all the amplitude which initially went a selection rule on the scattering matrix which prohibits into the outgoing positive norm states when g3 > 0 is the scattering processes responsible for the Hawking pro- now transferred into the reflected negative norm state cess, which leads to a creation of BdG quasiparticles. and the evanescent tail. This is evident if we see that when g3 = 0, 1 i (∂ + v · ∇) + ∇ · ρ∇ ζ = 0, (66) t 2mρ V. TRANSPORT IN NEWTON-CARTAN and hence ζ and ζ∗ do not mix. Indeed, as Fig. 6 illus- GEOMETRY trates, though Hawking radiation is permissible by conservation of energy and momentum, as seen by the dispersion relation in Fig. 6, there is no permissible matrix In this section we take up the issue of energy transport element for any scattering process which mixes positive in systems exhibiting Newton-Cartan geometry. Build- and negative norm modes. Thus, at low frequencies (being on Luttinger’s work on computing heat transport via low the cutoff frequency on the right), negative norm coupling to a gravitational field [34], there has been a modes may be transmitted across the horizon but only well-established method of coupling systems to Newton- as outgoing negative norm modes. This is analogous to Cartan geometry in order to extract their heat transport the “super-Hawking” regime earlier, but since there is no properties [23, 28, 29, 35]. With these methods, we can begin with the results in Sec. II E and find the stress conversion between positive and negative norm modes, µν µ there is no Hawking radiation effect. tensor T , energy current , and momentum density Above the cutoff frequency on the right (in what we pµ. However, as we have mentioned previously, we can refer to as the “regular Hawking regime”), all negative also reformulate the relativistic Lagrangian in Sec. II D in norm modes incident from the interior of the horizon terms of a Newton-Cartan geometry with an additional must be reflected back. Even in this case, there is still a external field. Therefore, in the bulk of this section, we finite penetration of the negative norm state across the make that precise and use the energy transport machin- event horizon in the form of an evanescent mode which is ery to relate the relativistic stress-energy tensor of Type-I decaying away from the horizon, as originally predicted modes to its non-relativistic counterparts. in Ref. [42]. In fact, this evanescent tail is also present We begin by noting that the variations in the geometry when g3 > 0, but now it is not accompanied by any are not independent as they must satisfy the constraints µ other outgoing mode. Again, let us emphasize that this imposed by Newton-Cartan geometry that nµv = 1 and µν evanescent mode is associated with a negative norm mode nµh = 0. Parameterizing the variations so as to re- and therefore does not couple to positive norm modes. spect these constraints is done by introducing the per- 14

µ µν turbations δnµ, δu and δη such that A. Energy transport for Type-II modes µ µ λ µ δv = −v v δnλ + δu , (67) We proceed to vary the Newton-Cartan geometry in µν µ νλ ν µλ µν δh = −(v h + v h )δnλ − δη , action Eq. (43). This straightforwardly yields the mo- µ µν µ µν mentum density as where nµδu = 0, and nµδη = 0 so that δu and δη i α α are orthogonal to the clock one-form nµ. ¯ ¯ pµ = − 2 ψ(∂µ − nµv ∂α)ψ − ψ(∂µ − nµv ∂α)ψ . To find the full Lagrangian it is useful to formally de- (74) fine a non-degenerate metric in the full spacetime by The limit works out as expected: if we let nµ = (1, 0) and µ T gµν ≡ vµvν + hµν . (68) v = (1, 0) , only the spatial components survive and we obtain the momentum current for a non-relativistic Note that unlike relativistic metrics, this Newton-Cartan theory with conserved density |ψ|2. Next, we compute has no invariant distinction between space-like and time- the stress tensor, which describes the momentum flux. like separations (simultaneity is a global concept imposed We find µν by nµ). As g is non-degenerate, we may proceed to take µν i α ¯ ¯ µν the inverse which is defined by T = − 4 v ψ∂αψ − ψ∂αψ h 1 ¯ αµ βν αν βµ µν αβ αν ν + ∂αψ∂βψ(h h + h h − h h ) (75) gµαg = δµ, (69) 4m ν and the energy current as where δµ is the usual Kronecker delta. This also serves to define the inverse of the degenerate metric hµν by µ 1 ¯ α βµ β αµ µ αβ = − 2m (∂αψ)(∂βψ) v h + v h − v h . (76) gµν ≡ nµnν + hµν . (70) Both have sensible flat-space limits as well. Note that the constraints on the geometry then imply hµν obeys B. Energy transport for Type-I modes µσ µ µ h hσν = δν − v nν . (71) The right hand side essentially acts to project onto the For Type-I modes, an analogue relativistic theory manifold upon which hµν is not degenerate. These are emerges from a nonrelativistic theory, and in both the the “spatial” three-surfaces which are in some sense “iso- cases, we can compute energy densities, momentum den- temporal.” sities, and the stress-tensor. The objective of this section Introducing g is helpful in particular because we then is to compute how the quantities in the analogue relativistic system are related to their nonrelativistic coun- find that if take the determinant g = det(gµν ), we find √ √ terparts, motivated by the spacetime relations derived in that g = n h [45]. This is exactly the volume measure 0 Sec. II. of the Lagrangian Eq. (43). This assists in taking the We have shown the Type-I modes can be thought of variation as residing in a relativistic analogue spacetime, equipped √ √ µ 1 µν δ[ g] = g[v δnµ + 2 hµν δη ]. (72) with an analogue metric tensor Gµν . If we vary with respect to this tensor, we obtain a Lorentz-invariant stress- We can then use the variations to find the stress tensor µν µ energy-momentum tensor, T . Note Lorentz invariance Tµν , energy current , and momentum density pµ via constrains this to be symmetric, relating the energy cur- [35] rent and momentum densities to each other. Z √ On the other hand, we have shown that one can obtain δS = dd+1x g 1 T µν δη − µδn − p δuµ . (73) 2 µν µ µ the Type-I modes by gapping out one of the generators

µ of a Type-II mode. Thus, we can also consider vary- Due to the constraints on δu and δηµν , these values of µν ing the Newton-Cartan geometry that the Type-II mode pµ and T are not unique. In fact, we can make any sub- µν µν µ ν ν µ resides in before including a mass gap. This yields for stitution pµ → pµ +anµ or T → T +b v +b v . We µ µν us the Newton-Cartan stress tensor, momentum density, impose uniqueness by requiring pµv = 0 and T nν = 0. and energy current and provide for us a general rela- Lastly, one can derive continuity equations for these tionship between the relativistic energy-momentum ten- quantities by considering how these objects change under sor and the non-relativistic counterparts. a diffeomorphism (see Ref. [35]). First, we return to Eq. (36) and rewrite the Lagrangian We now compute these quantities for both the Type-I in terms of the Newton-Cartan geometry prior to inte- and Type-II modes. It is worth noting that these mod- grating out the massive mode (recall that unlike a Type- els describe the free propagation of Goldstone modes and II mode, a Type-I mode is canonically conjugate to a thus are in a sense “non-interacting.” By this, we mean massive mode). We obtain there are no additional terms due to interactions [46]. For Type-II modes, the resulting transport quantities are √ µ hµν L = g − 2βv ∂µφ − [∂µφ∂ν φ + ∂µβ∂ν β] known [23, 28, 29]. We briefly recapitulate this calcula- 2m 2 2 tion here. − 2mC (x)β , (77) 15 where c2 = ρ2/dC2 is the speed of sound of the Gold- is the momentum density measured by the comoving ob- stone mode (the factor of density essentially accounts for server as well. Relativistically, these are strictly related µν λ λµ the units of h ). If we integrate out the massive mode E = G Pµ. However, momentum is imposed by the β in the limit where we can neglect the dispersion (i.e. underlying non-relativistic field theory to be orthogonal λ at long wavelengths), we recover the Type-I relativistic to flow v pλ = 0. The form of pλ that accomplishes µ ν Lagrangian this includes the comoving energy density v Tµν v and µν αβ √ subtracts it off. Lastly, T is directly related to T g (vµ∂ φ)2 µν L = µ − hµν ∂ φ∂ φ . (78) projected to live only on spatial slices nµT = 0, again eff 2m C2 µ ν as imposed by the underlying non-relativistic theory. In effective, relativistic, analogue systems, there is a From this, we can identify the relativistic metric Gµν by preferred (lab) frame that is captured by the Newton- observing that this Lagrangian must be of the form in Cartan geometry (in particular nµ specifies the lab Eq. (6) such that frame’s “clock”). This preference is hidden in the high √ √ frequency dispersion of the type-I modes and, as we have µν g vµvν µν −GG = m C2 − h . (79) shown here, results in non-trivial momentum currents and stress-tensors. This yields an equation relating the relativistic metric to As a particular example, a Hawking flux against the the Newton-Cartan object and the gap of the massive flow in an analogue system should result in a real energy mode. We find and momentum current away from the analogue black − 2 2 hole. Far from the horizon (considering the effective G = (mC) d−1 C n n − h , µν µ ν µν 1+1D problem where the other two spatial dimensions 2 µ ν (80) µν d−1 v v µν G = (mC) C2 − h , are trivial) we obtain where d is the spatial dimension. As we can see, the rel- TH −TH Tµν = , (84) ativistic metric depends crucially on the potential C(x). −TH TH This is helpful since, on the one hand, we can easily for a constant TH [47] (for the radiation flowing to +∞). obtain the stress-energy tensor in the relativistic theory If we apply this to the above, and assume that at +∞ by varying δGµν . On the other hand, we can use the we have no velocity so that vµ = (v0, 0) and a flat hij = above formulae to connect this result to the actual stress ij 1/3 tensor and energy current/momentum density of the non- δ /h0 , we have relativistic model. In particular, xx 1 T = 2/3 TH, mh0 µν 2 µ νλ ν µλ µ ν λ λ v0 h (v0)2 −1/3 i d−1 v v (85) δG = (mC) [(v h + v h − 2 C2 v )δnλ = m TH C2 , h0 , 0, 0 , 1 µ ν ν µ λ µν 0 + C2 (v δλ + v δλ )δu + δη ]. (81) v pλ = mC2 TH[0, 1, 0, 0]. Thus, we can directly relate the relativistic energy- Importantly, we see that there is a finite energy current 1 momentum tensor Tµν to its non-relativistic counterparts and momentum p1 away from the horizon; there is no p0 µ by expanding component due to the constraint pµv = 0. While related Z √ to what is computed relativistically, these quantities are d+1 1 µν not exactly the same. δS = d x 2 −GTµν δG (82) in terms of the geometric objects in the NC geometry. VI. DISCUSSION AND CONCLUSIONS Doing so, we obtain

µν 1 µ µ αβ ν ν The primary result of this paper is establishing the con- T = 4 (δα − nαv )T (δβ − nβv ), m(mC) d−1 nection between the different types of Goldstone modes λ 1 µ λ and different types of analogue spacetimes, as summa- = 2 v Tµ , (83) m(mC) d−1 rized in Table I. This is done by revisiting the proof of 1 µ µ ν pλ = − mC2 (Tλµv − v Tµν v nλ). the non-relativistic Goldstone theorem and allowing for the possibility of an inhomogeneous mean-field solution. µν λ µν where the indices on T and Tµ are raised with G We then find that the conventional Type-I Goldstone while all Newton-Cartan objects use the metric gµν . Ig- modes come equipped with an Einstein-Hilbert metric noring the factors in front of these expressions, one can as appears in general relativity while Type-II Goldstone think of vν as a timelike vector with respect to the metric modes couple to a Newton-Cartan geometry. The geom- µ Gµν . In this case, v is directly related to the field of the etry itself is determined by the spacetime dependence of λ ν λ fluid flow and the object E ∝ v Tµ is the energy cur- symmetry-breaking mean-field—inhomogeneous symme- rent measured by an observer comoving with that flow try breaking ultimately produces the non-trivial space- ν (not the lab observer). By the same token Pλ ∝ Tλν v time metric. In this work we have restricted ourselves 16 to the case where only the overall U(1) symmetry is in- ken non-Abelian symmetry (including textures like spi- homogeneously broken. This corresponds to an overall ral magnetization, Bloch domain walls, and skyrmions) condensate flow. and synthetic gauge fields. The construction presented Another key result of this paper is establishing the con- here also considers just the quadratic excitations, but nection between quantum phase transitions and changes these Goldstone modes realize more complicated nonlin- in the nature of the spacetime. To elucidate this, we ear sigma models for which there is extra intrinsic ge- present a simple model where the analogue geometry ometry at play and would need to be incorporated into can be tuned by a single parameter. This drives a a full theory of these excitations. This new analogue quantum phase transition which accompanies the transi- also raises questions of the so-called back-reaction effects tion between the Einstein-Lorentz geometry and Newton- of quantum fields on the corresponding analogue space- Cartan geometry. As the phase transition is approached, time. This has been studied in the relativistic case [8, 49], the Hawking radiation produced by an event horizon and the non-relativistic case leaves us with the tantaliz- changes, as encapsulated in Fig. 2. One key result is ing prospect of a system with a dynamical Newtonian that the Newton-Cartan geometry exhibits no Hawking gravity. Finally, while in this work we exclusive focused radiation, even though all fluid flows are supersonic (the on the context of flowing spinor Bose-Einstein conden- group velocity of Goldstone modes vanishes at long wave- sates, the phenomenon should be more general. An in- lengths). teresting future direction to pursue would be to try and While Sec. III is a minimal theoretical model, the ex- extend these results to include more diverse platforms perimental system that most readily realizes these ge- including electrons in solid-state systems, liquid Helium, ometries are spin-1 condensates. In this case, for the superconductors, magnetic systems. The wide variety of scattering lengths a0 and a2 (for s-wave collisions into systems which exhibit symmetry-breaking means there is the spin-0 and spin-2 channels respectively), there are a wide variety of systems which might exhibit this ana- two phases that break the spin SU(2) symmetry: a0 > a2 logue spacetime and its consequences. gives a ferromagnetic phase with one Type-II magnon and a0 < a2 gives a polar phase (antiferromagnetic interactions) with two Type-I magnons. Upon flow, these two ACKNOWLEDGMENTS phases naturally realize the two different spacetimes described here. In fact, 7Li, 41K, and 87Rb realize the ferro- We would like to thank Gil Refael for crucial discus- magnetic phase [24] with 87Rb specifically already being sions which lead to this work. We also thank Andrey used for Hawking-like experiments with the phonon mode Gromov and Luca Delacrétazfor indispensable sugges- [14]. Additionally, 23Na realizes the polar phase and crit- tions. This work was supported the U. S. Army Re- ical spin superflow has been studied [48] (necessary for search Laboratory and the U. S. Army Research Office Hawking-like experiments). The magnon excitations in under contract number W911NF1810164, NSF DMR- these systems can be probed by observing correlations 1613029, and the Simons Foundation (J.B.C. and V.G.). in the spin-density, and the most basic proposal, would J.H.W. and V.G. performed part of this work at the be to establish the vanishing Hawking radiation in the Aspen Center for Physics, which is supported by Na- ferromagnetic phase. The progress in current spinor con- tional Science Foundation grant PHY-1607611. J.B.C. densate experiments highlights that these more exotic and V.G. performed part of this work at the Kavli In- analogue spacetimes may already be in reach. stitute for Theoretical Physics and thank KITP for hos- Finally, by considering the response of the Goldstone pitality and support. J.B.C. was supported in part by modes to variations in the analogue geometries, we relate the Heising-Simons Foundation, the Simons Foundation, the analogue stress-energy-momentum tensor in relativis- and National Science Foundation Grant No. NSF PHY- tic geometries directly to their non-relativistic counter- 1748958. J.H.W. thanks the Air Force Office for Scientific part. This is summarized by the equations below, which Research for support. shows how the metric tensor in both analogue spacetimes may be constructed from the underlying geometric objects of the Newton-Cartan geometry along with an ad- Appendix A: Calculating the fluctuation Lagrangian ditional field C = C(x): In this section, we put all of the algebra and La- gµν = nµnν + hµν , Non-relativistic, grangian manipulation that we left out of Section II. 2 (86) Gµν ∝ C nµnν − hµν , Relativistic. Our starting point is Eq. (1) upon substituting Ψ = Ψ0 + δΨ where Ψ0 solves the Euler-Lagrange equations We also provide a direct connection between the energy Eq. (3) and δΨ can be written in terms of broken gener- and momentum currents of an analogue relativistic sys- ators and massive fields Eq. (7). tem and the more fundamental Newton-Cartan geometry Most of the simplifying algebra comes from g(σΨ, ξ) = which describes the lab-frame. 0 and integration-by-parts. To facilitate the integration Within spinor Bose-Einstein condensates, there are by parts, all equalities are be understood to be up to a full other phenomena to include such as inhomogeneous bro- derivative. Furthermore, by construction the linear terms 17

† ˙ cancel, so we keep second-order terms only, indicated by Performing integration-by-parts on the Ψ0σξ term, we fluc= . get To deal with the term linear in derivatives, we use the object ←→ f ∂t g ≡ f(∂tg) − (∂tf)g, (A1) ˙ and for simplicity we sometimes replace ∂tf with f for i †←→ fluc i † ←→ † † time derivatives. We further take advantage of the Ein- 2 Ψ ∂t Ψ = − 2 Ψ0(σ ∂t σ)Ψ0 + i(Ψ0σ˙ ξ + ξ σ˙ Ψ0) stein summation convention (sum over indices is implied) + iΨ˙ †( 1 σ2Ψ + σξ) − i( 1 Ψ†σ2 − ξ†σ)Ψ˙ . (A3) for simplicity. The first term we investigate is 0 2 0 2 0 0

i †←→ ﬂuc i † ←→ 2 Ψ ∂t Ψ = − 2 Ψ0(σ ∂t σ)Ψ0 † ˙ † † ˙ + i(−Ψ0σξ + ξ σ˙ Ψ0 + ξ σΨ0) i † 2 ˙ i ˙ † 2 − 2 Ψ0σ Ψ0 + 2 Ψ0σ Ψ0. (A2)

The kinetic energy term takes the form

† ﬂuc 1 † 2 † † 1 † 2 † † ∂jΨ ∂jΨ = − 2 ∂jΨ0σ ∂jΨ0 − ∂jΨ0σ∂jσΨ0 − ∂jΨ0σ∂jξ − 2 ∂jΨ0σ ∂jΨ0 − Ψ0(∂jσ)σ∂jΨ0 + ∂jξ σ∂jΨ0 † † † † − Ψ0∂jσ∂jσΨ0 + ∂jξ ∂jξ − Ψ0∂jσ∂jξ + ∂jξ ∂jσΨ0. (A4)

1 † 2 We perform integration by parts on the two instances of − 2 ∂jΨ0σ ∂jΨ0 above in opposite ways to obtain

† ﬂuc 1 2 † 2 1 † 1 † 1 † 2 2 1 † 1 † ∂jΨ ∂jΨ = 2 ∇ Ψ0σ Ψ0 − 2 ∂jΨ0σ∂jσΨ0 + 2 ∂jΨ0(∂jσ)σΨ0 + 2 Ψ0σ ∇ Ψ0 − 2 Ψ0(∂jσ)σ∂jΨ0 + 2 Ψ0σ(∂jσ)∂jΨ0 † † † † † † − ∂jΨ0σ∂jξ + ∂jξ σ∂jΨ0 − Ψ0∂jσ∂jσΨ0 + ∂jξ ∂jξ − Ψ0∂jσ∂jξ + ∂jξ ∂jσΨ0. (A5)

† † If we further use integration by parts on −∂jΨ0σ∂jξ and ∂jξ σ∂jΨ0, we obtain (after some reordering)

† fluc † 1 † 1 † 1 † 1 † ∂jΨ ∂jΨ = −Ψ0∂jσ∂jσΨ0 − 2 ∂jΨ0σ∂jσΨ0 + 2 ∂jΨ0(∂jσ)σΨ0 − 2 Ψ0(∂jσ)σ∂jΨ0 + 2 Ψ0σ(∂jσ)∂jΨ0 † † † † † + ∂jξ ∂jξ + ∂jΨ0∂jσξ − ξ ∂jσ∂jΨ0 − Ψ0∂jσ∂jξ + ∂jξ ∂jσΨ0 2 † 1 2 1 † 2 2 + ∇ Ψ0( 2 σ Ψ0 + σξ) + ( 2 Ψ0σ − σξ)∇ Ψ0. (A6) We observe that, along with Eq. (11), the equation of motion cancels the last lines in Eqs. (A3) and (A6) with the first line of Eq. (11). All together, we can combine these equations to get the full fluctuation Lagrangian

ﬂuc i † ←→ † † 1 † 1 † 1 † 1 † L = − 2 Ψ0(σ ∂t σ)Ψ0 +i(Ψ0σ˙ ξ +ξ σ˙ Ψ0)− 2 ∂jΨ0σ∂jσΨ0 + 2 ∂jΨ0(∂jσ)σΨ0 − 2 Ψ0(∂jσ)σ∂jΨ0 + 2 Ψ0σ(∂jσ)∂jΨ0 † † † † † † † + ∂jξ ∂jξ + ∂jΨ0∂jσξ − ξ ∂jσ∂jΨ0 − Ψ0∂jσ∂jξ + ∂jξ ∂jσΨ0 − Ψ0∂jσ∂jσΨ0 + ∂jξ ∂jξ

1 ∂2V ∂2V 1 ∂2V − ξ∗ ξ∗ − ξ∗ ξ − ξ ξ . (A7) a † † b a † b a b 2 ∂Ψ ∂Ψ ∂Ψ ∂Ψ 2 ∂Ψa∂Ψb a b 0 a b 0 0

We can now expand our ﬂuctuations in terms of their ﬁelds σΨ0 = θnσnΨ0 and ξ = βnξn, and we obtain

ﬂuc i † 1 † † L = − 2 Ψ0[σm, σn]Ψ0θm∂tθn + 4m θm∂jθn(∂jΨ0[σm, σn]Ψ0 − Ψ0[σm, σn]∂jΨ0) † † 1 † † † † + iβn∂tθn(Ψ0σmξn + ξnσmΨ0) + 2m βm∂jθn(ξmσn∂jΨ0 − ∂jΨ0σnξm + Ψ0σn∂jξm − ∂jξmσnΨ0) 1 † + 2m Ψ0σnσmΨ0∂jθn∂jθm i † † i † † † † + 2 βm∂tβn(ξmξm − ξnξm) + 2 βnβm(ξm∂tξn − ∂tξmξn) + βm∂jβn(ξn∂jξm + ∂jξmξn) 2 2 2 1 † 1 † ∂ V ∗ T ∂ V † ∂ V − 2m ξnξm∂jβm∂jβn − 2 βnβm ξn † † ξm + ξn ξm + 2ξm † ξn . (A8) ∂Ψ ∂Ψ 0 ∂Ψ∂Ψ 0 ∂Ψ ∂Ψ 0 18

The ﬁrst three lines of Eq. (A8) lead to the Lagrangian presented in the text Eq. (12) while the last two lines represent the massive modes neglected in the main text.

T One can then easily check that once the full Lagrangian where Wω(r) = [Uω(r),Vω(r)] is a two-component in Eq. (34) is derived that the massive modes conjugate spinor which obeys the eigenvalue problem to Goldstone modes no longer have the term that goes as 1 βm∂µβn, only keeping the kinetic term and mass matrix ω + ivs · ∇ + ∇ · ρ∇τ3 − g3ρ (τ3 + iτ2) Wω(r) = 0. (which we diagonalize to find the type-I basis states). 2mρ (B6) We refer to [9, 42] for more details of solving this system. Appendix B: Bogoliubov Theory for Hawking What is important for our discussion are the details of Emission the dispersion relation, which are used to analyze the asymptotic scattering states at spatial infinity. As per Eq. (60), the magnon field (written in terms of We now focus on the case of a one-dimensional homo- ∗ T geneous flow. In this case both the momentum k and the complexified spinor Φ3(x) = (ζ, ζ ) ) obeys the BdG equation lab-frame frequency ω are good quantum numbers and obey the standard Bogoliubov dispersion relation (using 1 2 that mc = g3ρ) of iτ3Dˆ t + ∇ · ρ∇ − g3ρ (τ0 + τ1) Φ3(x) = 0, (B1) 2mρ s k2 2 written in terms of the co-moving frame material deriva- 2 2 ω = vsk ± c k + ≡ ω±(k), (B7) tive Dˆ t = ∂t + vs · ∇. 2m Before proceeding, there are two properties of this where the last equality is used to define the lab frequency equation that prove useful. First is the charge conju- ω (k). At a particular frequency ω > 0, we may deter- gation symmetry: if Υ solves Eq. (B1), then so does ± mine which scattering states are available by finding the ∗ Υ ≡ τ1Υ . (B2) real momenta k which obey ω = ω±(k). Considering a step-like variation in the flow, the flow In particular, this is important since the Nambu spinor profile is as given in Eq. (62). For x < 0 and x > 0 the so- should obey the self-conjugate property that Φ3 = ∗ T lutions to the BdG equations are still plane-waves which (ζ, ζ ) = Φ3. Thus, it is important that this is respected obey the Bogoliubov dispersion relation, albeit with dif- by the equations of motion, which we see it is. ferent parameters ρ and v. These two dispersion relations Furthermore, provided the density ρ(x) is time inde- are shown Figs. 3 and 4 for fixed values of the condensate pendent, we can define the conserved pseudo-scalar prod- velocities |vl| > |vr| and densities ρl, ρr. uct on the solution space Instead of the lab frame, we may measure frequency Z with respect to the frame co-moving with the fluid flow. (Υ , Υ ) ≡ ddr ρ(r)Υ†(r)τ Υ (r). (B3) 1 2 1 3 2 This is implemented by Doppler shifting to the (positive) comoving frequency This scalar product has a number of useful features including that the charge conjugation operation changes q 4 Ω(k) ≡ c2k2 + k , (B8) the sign, so that 4m2 so that ω (k) = vk ± Ω(k)(vk amounts to a Galilean (Υ1, Υ2) = −(Υ2, Υ1). (B4) ± boost). We use this pseudo-inner product to define a notion of For |v| < c (right dispersion in Fig. 3), there are only norm for solutions. Because of the τ3, this is not the usual two real-momenta at any positive frequency, which cor- d L2(R ) norm, and in fact is not a norm at all since it is respond to a right- and left-moving quasiparticle. For not positive semi-definite. There are non-trivial negative |v| > c (left dispersion in Fig. 3) a new scattering chan- norm states which we loosely refer to as “hole-like” states, nel opens whereby a wavepacket with negative free-fall in contrast to the “particle-like” solutions with positive frequency [ω−(k)] may have positive lab-frame frequency norm. As remarked earlier, hole-like solutions can be ω. related to particle-like solutions by charge conjugation We find the eigenfunctions for the step potential by since if Υ has negative norm we find employing matching equations at the step. These impose (Υ, Υ) < 0 ⇒ (Υ, Υ) > 0. the continuity requirements x=0+ To proceed further, we utilize the (assumed) time- [Wω(x)] − = 0 x=0 (B9) independence of the kernel to further separate the so- x=0+ lution Υ(x) = Υ(r, t) into energy eigenmodes [ρ∂xWω(x)]x=0− = 0. Z dω −iωt Additionally, we choose them to satisfy (Wω, W ω) = 0 Υ(x) = Wω(r)e , (B5) 2π and can be normalized such that (Wω,Wω) > 0 if ω = 19

ω+(k) (positive comoving frequency) and (Wω,Wω) < 0 All Wωα are orthogonal with respect to this inner prod- if ω = ω−(k) (negative comoving frequency). uct, and so a(Wωα) is either a creation or annihilation Combining all of this, we can express the full solution operator based on the sign of the norm. in terms of positive-frequency components only via The system may be exactly solved when the flow is homogeneous, in which case the momentum k is also a Z ∞ good quantum number. Assuming a solution of the form dω X −iωt Φ3(x, t) = A(Wωα)Wωα(x)e 0 2π ikx α Wω(x) = wke ∗ +iωt + A (Wωα)W ωαe , (B10) produces the momentum space eigenvalue problem 1 2 where the A(Wω,α) are the Fourier coefficients of the ex- ω − vk − k τ3 − g3ρ (τ3 + iτ2) wk = 0. (B13) pansion and α is a set of quantum numbers which are 2m used to label the different degenerate modes at each en- In principle, the momentum k depends in the energy ω, ergy ω > 0. At this point, we can second quantize the but we usually suppress this dependence for brevity. system and promote Υ to an operator. In such a case, To evaluate (Wωα,Wω0α0 ), we establish a couple of the operator equation looks like T facts. If we let wk = [uk, vk] , then we have

Z ∞ dω X k2 Υ(ˆ x, t) = a(W )W (x)e−iωt mc2v = ±Ω(k) − − mc2 u , (B14) 2π ωα ωα k k 0 α 2m † +iωt and hence + a (Wωα)W ωαe , (B11) 2 2 4 2 2 4 2 k 2 m c |vk| = m c ∓ 2Ω(k) mc + ∓ Ω(k) |uk| , where now a(Wωα) are operators satisfying 2m (B15) † 2 2 [a(Wωα), a (Wω0α0 )] = (Wωα,Wω0,α0 ). (B12) this relation between |uk| and |vk| allows us to evaluate

2 2ρ 2 k 2 0 (W ,W 0 0 ) = ±Ω(k) mc + ∓ Ω(k) |u | δ 0 δ[k (ω) − k 0 (ω )] ωα ω α m2c4 2m k αα α α (B16) 2 2ρ|vg| 2 k 2 0 = ±Ω(k) mc + ∓ Ω(k) |u | δ 0 δ(ω − ω ). m2c4 2m k αα

2 k2 The term in brackets mc + 2m − Ω(k) > 0, so the sign late the creation operators of the out-vacuum to the in- of the normalization depends exclusively on whether we vacuum have positive (+Ω(k)) or negative (−Ω(k)) comoving fre- † quency. The terms with negative comoving frequency a(WH) = αRa(WR,1)+αLa(WL,2)+βLa (WL,1). (B18) (or negative norm) are represented by the blue curves in Figs. 3 and 4. This implies that for WH at a particular frequency ω, We can now perform the Hawking calculation to deter- we can ﬁnd the number of Hawking modes leaving the mine the Bogoliubov transformation giving rise to exci- horizon by considering the expectation value tation production. This is presented ﬁrst in Fig. 3, where † 2 we consider a wavepacket moving away from the horizon h0in|a(WH) a(WH)|0ini = |βL| (WL,1,WL,1). (B19) to +∞ and frequency ω, this is the Hawking mode. If With the proper normalization and putting back in the we trace it back in time, it was related to a scattering dependence on frequency, the number of particles leaving process at the horizon itself, so in terms of three other the horizon at frequency ω is positive frequency modes

W = α W + α W + β W , (B17) 2 (WL,1(ω),WL,1(ω)) H R R,1 L L,2 L L,1 N(ω) = |βL(ω)| . (B20) (WH (ω),WH (ω)) where WH includes the far propagating right-moving mode along with the evanescent near horizon solution, This same analysis can be done for the supersonic-to- WR,1 is the left-moving mode on the right, and WL,(1,2) supersonic case presented in Fig. 4. For lack of a better are the right-moving modes on the left (counted left-to- term, we call the region where there are multiple positive right in Fig. 3). This immediately gives us how to re- and negative norm channels the “super-Hawking” region. 20

In this case, we have two modes in the Hawking process These equations can be similarly related to a Bogoliubov that need to be backwards scattered: one positive norm transformation, and we can ﬁnd the number of Hawking and the other negative norm. The result of the scattering particles leaving the horizon at frequency ω by consider- process is ing

WH = βRW R,1 + αRWR,2 + βLW L,1 + αLWL,2, 0 0 0 0 W H0 = αRW R,1 + βRWR,2 + αLW L,1 + βLWL,2. (B21)

2 (WL,1(ω),WL,1(ω)) 2 (WR,1(ω),WR,1(ω)) 0 2 (WL,2(ω),WL,2(ω)) 0 2 (WR,2(ω),WR,2(ω)) N(ω) = |βL(ω)| + |βR(ω)| + |β (ω)| + |β (ω)| . (WH (ω),WH (ω)) (WH (ω),WH (ω)) L (WH0 (ω),WH0 (ω)) R (WH0 (ω),WH0 (ω)) (B22) Despite there being more terms, there is generally less of a Hawking ﬂux due to a decoupling of the negative and positive norm channels as we can see in Fig. 2.

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