Quantum Hydrodynamics for Supersolid Crystals and Quasicrystals

Quantum hydrodynamics for supersolid crystals and quasicrystals

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As Published http://dx.doi.org/10.1103/physreva.99.063621

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Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW A 99, 063621 (2019)

Quantum hydrodynamics for supersolid crystals and quasicrystals

Vili Heinonen,1,* Keaton J. Burns,2 and Jörn Dunkel1,† 1Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA 2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA

(Received 3 August 2018; published 24 June 2019)

Supersolids are theoretically predicted quantum states that break the continuous rotational and translational symmetries of liquids while preserving superfluid transport properties. Over the last decade, much progress has been made in understanding and characterizing supersolid phases through numerical simulations for specific interaction potentials. The formulation of an analytically tractable framework for generic interactions still poses theoretical challenges. By going beyond the usually considered quadratic truncations, we derive a systematic higher-order generalization of the Gross-Pitaevskii mean-field model in conceptual similarity with the Swift- Hohenberg theory of pattern formation. We demonstrate the tractability of this broadly applicable approach by determining the ground-state phase diagram and the dispersion relations for the supersolid lattice vibrations in terms of the potential parameters. Our analytical predictions agree well with numerical results from direct hydrodynamic simulations and earlier quantum Monte Carlo studies. The underlying framework is universal and can be extended to anisotropic pair potentials with a complex Fourier-space structure.

DOI: 10.1103/PhysRevA.99.063621

I. INTRODUCTION surface crystals [22], and active fluids [23–25]. Whereas higher PDEs are often postulated as effective phenomenologi- Supersolids are superfluids in which the local density spon- cal descriptions of systems with crystalline or quasicrystalline taneously arranges in a state of crystalline order. The existence order [26], it turns out that such equations can be derived of supersolid quantum states was proposed in the late 1960s directly within the established GP framework. The resulting by Andreev, Lifshitz, and Chester [1,2] but the realization in mean-field theory yields analytical predictions that agree with the laboratory has proven extremely difficult [3]. Recent ex- direct quantum hydrodynamic and recent qMC simulations perimental breakthroughs in the control of ultracold quantum [12,27] and specify the experimental conditions for realizing gases [4–7] suggest that it may indeed be possible to design periodic supersolids and coexistence phases, as well as super- quantum matter that combines dissipationless flow with the solid states exhibiting quasicrystalline symmetry (Sec. V). intrinsic stiffness of solids. Important theoretical insights into the expected properties of supersolids and experimental II. MEAN-FIELD THEORY candidate systems have come from numerical mean-field calculations and quantum Monte Carlo (qMC) simulations As in the classical GP theory [13,14], we assume that a for specific particle interaction potentials [8–12]. What is system of spinless particles can be described by a complex still lacking, however, is a general analytically tractable scalar field (t, x) and that quantum fluctuations about the framework that allows the simultaneous characterization of mean density n(t, x) =||2 are negligible. Considering an whole classes of potentials as well as the direct computation isotropic pairwise interaction potential u(|x − x |), the total of ground-state phase diagrams and dispersion relations for potential energy density is given by a spatial convolution supersolid lattice vibrations in terms of the relevant potential integral which can be expressed as a sum over Fourier-mode parameters. contributions ∝ uˆ|nˆ|2, where hats denote Fourier transforms To help overcome such conceptual and practical limita- (see Appendix A for a detailed derivation). If u is isotropic tions, we introduce here a generalization of the classical with finite moments, then its Fourier expansion can be written = ∞ 2 j =| | Gross-Pitaveskii (GP) mean-field model [13,14]bydrawing asu ˆ(k) j=0 g2 jk [28], where k k is the modulus guidance from the Swift-Hohenberg theory [15], which uses of the wave vector, yielding the potential energy density 1 ∞ − j ∇2 j ∇2 fourth-order partial differential equations (PDEs) to describe 2 n j=0( 1) g2 j ( ) n, with denoting the spatial Lapla- pattern formation in Rayleigh-Bénard convection. Our ap- cian. The constant Fourier coefficients g2 j encode the spatial proach is motivated by the successful application of higher- structure of the potential. Variation of the total energy func- than-second-order PDE models to describe classical solidifi- tional with respect to yields the generalized GP equation cation phenomena [16–18], electrostatic correlations in con- (Appendix A) centrated electrolytes and ionic liquids [19,20], nonuniform ⎡ ⎛ ⎞⎤ 2 ∞ FFLO superconductors [21], symmetry breaking in elastic h¯ ih¯∂ = ⎣− ∇2 + ⎝ (−1)jg (∇2 ) j||2⎠⎦. (1) t 2m 2 j j=0

*[email protected] The classical GP model, corresponding to repulsive point † [email protected] interactions u = g0δ(x − x ), is recovered for g0 > 0 and

2469-9926/2019/99(6)/063621(16) 063621-1 ©2019 American Physical Society HEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019) g2 j = 0 otherwise. The authors of Ref. [29] studied the case the interaction parameter (Appendix C). In an infinite domain, g0, g2 > 0 and g2 j4 = 0, keeping partial information about the internal energy U[ρ] is completely parameterized by αu long-range (k → 0) hydrodynamic interactions by effectively and the average densityρ ¯ = ρdx/ dx. For the Rydberg- 2 −1 adding a surface energy term ∝|∇n| to the energy density. dressed BEC in Ref. [27], which has q0 = 1.87 μm , corre- However, as we see shortly, to describe supersolid states, the sponding to a hexagonal lattice spacing of 3.88 μm, one finds long-wavelength expansion has to be carried out at least to αu =−0.043 andρ ¯ ≈ 9.4 (Appendix G). order k4. To show this explicitly, it is convenient to express the III. GROUND-STATE STRUCTURE complex√ dynamics, (1), in real Madelung form [31]. Writing = n exp(iS) and defining the irrotational velocity field An advantageous aspect of the above framework is that ρ α , ρ v = (¯h/m)∇S,Eq.(1) can be recast as the quantum hydro- the ground-state structure of U[ ]inthe( u ¯ )-phase plane dynamic equations (Appendix C; see also [32]) can be explored both analytically and numerically in a fairly straightforward manner (Fig. 1). Standard linear stability analysis (Appendix D3) for uniform constant-density solutions ∂t n =−∇·(nv), (2) predicts a symmetry-breaking transition at αu =−1 when ρ¯ 1/8 and m(∂t + v ·∇)v =−∇(δU/δn), (3) 1 − 16¯ρ 4 α = whenρ> ¯ 1/8, (7) with the effective potential energy U[n] to order O(k ), u 64¯ρ2 h¯2 |∇n|2 g g g indicated by the thin dashed line in Fig. 1(a). Refined analyt- = + 0 2 + 2 |∇ |2 + 4 ∇2 2 . U dx n n ( n) (4) ical estimates for the ground-state phases can be obtained by 8m n 2 2 2 ρ = ρ + φ · considering the Fourier ansatz ¯ j j exp (iq j x). +∇2 2 The first term is the kinetic quantum potential [31] and the The pattern forming operator (1 ) penalizes modes with 2 | | = kinetic energy is K = (m/2) dx nv . For nonleaky boundary q j 1, suggesting that single-wavelength expansions can conditions, Eqs. (2) and (3) conserve the total particle number yield useful approximations for the ground-state solutions. N = dx n and energy E[n, v] = K[n, v] + U[n]. The quan- Conceptually similar studies for classical phase field models tum hydrodynamics defined by Eqs. (2)–(4) is Hamiltonian [18] imply that 2D and 1D close-packing structures realizing in terms of the momentum density mnv = hn¯ ∇S, thereby hexagonal and stripe patterns are promising candidates. The differing, for example, from generalized Ginzburg-Landau one-mode approximation ansatz for a hexagonal lattice reads theories for nonuniform FFLO superconductors [21] and 3 finite-temperature hydrodynamic approaches [1,33–35]. ρ = ρ¯ + φ [exp (iq · x) + c.c.], (8) , ≡ 0 j Local minima of E[n v] have zero flow, v 0, and hence j=1 the corresponding density fields must minimize U[n]. As- · suming short-range repulsion g0 > 0, we see that uniform where the lattice vectors q j form the “first star” with qi = 2 = − 2/ constant-density solutions minimize U[n] when g2 0 and q j q ,ifi j and q 2 otherwise (c.c. denotes complex g = 0 (Appendix F); this case was studied in Ref. [29]. If, conjugate terms). Similarly, the stripe phase is defined by 4 ρ = ρ + φ + however, we consider the more general class of pair inter- ¯ [ 0 exp (iqx1) c.c.]. These trial functions have to φ action potentials with g2 ≷ 0, then short-wavelength stabil- be minimized with respect to the amplitudes 0 and the 4 ity at order k requires that g4 > 0. This situation arises, magnitudes q of the reciprocal lattice vectors, which can be for example, for the two-dimensional (2D) Rydberg-dressed done analytically (Appendix D4). Our analytical calculations Bose-Einstein condensate (BEC) studied in Ref. [27], which predict four distinct pure ground-state types, which can be 2 2 2 has g0 = 0.189¯h /m, g2 =−0.113 (¯h /m) μm , and g4 = identified as uniform (U), supersolid (SS), normal solid– 2 4 / 0.016 (¯h /m) μm (Appendix G). Whereas for g2 > 0 roton like (NS), and droplet (D), and also a narrow U SS coex- minima are absent as in the classical GP theory [8], supersolid istence phase via Maxwell construction [dark-brown domain ground-state solutions [8,9]ofEq.(4) can exist when g2 < 0, in Fig. 1(a)]. These naming conventions are directly adopted as we see shortly. from Ref. [12]. As always, mean-field predictions should be To determine the ground-state phase diagram of U[n] with supplemented with other methods to properly characterize the < supersolid-NS transition to ensure that the wave functions are g2 0, it is√ convenient to define the characteristic wave = − / > = / 2 localized in the NS state [36]. Yet the comparison with the number q0 g2 (2g4) 0, time scale t0 m (¯hq0 ), and = 2 2/ qMC simulations in Ref. [12] suggests that our mean-field energy scale 0 h¯ q0 m. Focusing on the 2D case and −1, , results capture essential aspects of their numerical results; see adopting (q0 t0 0 ) as units henceforth, we can rewrite Eq. (4)as inset in Fig. 1(a). Using the one-mode approximation, the qualitatively dif- 1 |∇ρ|2 ferent states can be inferred as follows: The uniform U phase U[ρ] = dx + α ρ2 + ρ(1 +∇2 )2ρ , (5) ρ = ρ> φ = 2 4ρ u has constant density ¯ 0, corresponding to 0 0. The supersolid SS phase is distinguished through the exis- ρ = 2/ 2 tence of energy-minimizing one-mode solutions with nonzero where mn(g4q0 h¯ ) is the rescaled number density and amplitude φ0 that yield strictly positive periodic density pat- ρ> α = 4g0g4 − terns 0 everywhere. By contrast, in the NS state, the u 2 1(6) g2 one-mode density field can become locally negative, signaling

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(a) (b) 2.0 1.0 2 2 c

R 1.5 NS NS 0.8 1 SS 1.5 U 0.5 ¯ ρ Scaled density 0 0.6 0 20406080100 1.0 Interaction strength U (

0.4 ρ

SS − ρ Average density 0.5 )

0.2 /ρ U D

0.0 0.0 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 Interaction parameter αu

FIG. 1. (a) Phase diagram showing the ground-state structure of the potential energy functional U[ρ]inEq.(5), calculated analytically using a one-mode approximation. Inset: Phase diagram for an interacting Rydberg-dressed BEC obtained from qMC simulations, adapted with permission from Fig. 2 in Ref. [12]. (b) Examples of numerically computed minima of U[ρ] for parameters indicated by the yellow symbols in (a). The uniform superfluid state (U) is stable against perturbations below the thin dashed red line. The hexagonal supersolid (SS; ∗; Supplemental Material Movie 1 [30]) phase has a lower energy than the metastable supersolid stripe phase (SS; ). The first-order transition between the U phase and the SS phase supports a narrow coexistence region (dark brown) with the uniform subphase having a lower density (x). In the NS state, the one-mode minimization of U yields locally negative density solutions, signaling failure of the approximation (see also Fig. 2). In the droplet phase (•) no real-valued one-mode solutions exist. The single-droplet solution (D; •) is shown as an inset for scale in the other panels. a breakdown of the one-mode approximation in this regime An in-depth analysis of the NS phase, which is beyond (Fig. 2). The dotted curve in Fig. 1(a) indicates where the the scope of this paper, would require going beyond the one-mode approximation becomes invalid. The comparison truncated mean-field model considered here. Similarly, in with the qMC simulations in Ref. [12] shown in the inset the droplet D phase, no real-valued density solutions exist in Fig. 1(a) suggests that this curve can be used as an within the one-mode approximation. To test the analytical approximate predictor for the boundary of the SS phase. predictions and explore the validity of the underlying one- mode approximation in detail, we performed a numerical ground-state search (Appendix H) for various parameter pairs (¯ρ,αu ). Representative examples of numerically found states ¯ ρ 14 are shown in Figs. 1(b) and 2 and generally agree well

ρ/ the analytical predictions. The hexagonal supersolid atρ ¯ = 12 1.5,αu =−0.2 [asterisk in Fig. 1(b)] is indeed dominated ∼ 10 by a single mode, with the second-highest mode being 20 times smaller. Our analytical one-mode estimates suggest 8 that the supersolid stripe states have a higher energy than the hexagonal states, with the energy difference going to 0 6 as one approaches the U-SS phase boundary [solid line in 4 Fig. 1(a)]. This opens the possibility that systems at nonzero kinetic energy or in a strained geometry might arrange in

Relative number density 2 a stripe configuration similar to the metastable state shown atρ ¯ = 0.5, αu =−0.45 [parallel lines in Fig. 1(b)]. The 0 numerical solution for α =−0.2178,ρ ¯ = 1.0[x in Fig. 1(b)] Relative position x/a u nn confirms the analytically predicted coexistence of uniform FIG. 2. Difference between supersolid (SS) and normal solid– and supersolid phases, suggesting that the one-mode approx- like (NS) states. One-dimensional cross sections of the density fields imation places the coexistence region accurately in the phase of two numerically determined ground states with the same average diagram. For coexistence to be observable in experiments, the energy of the uniform and supersolid bulk regions has to be densityρ ¯ = 0.4butαu =−0.484 (SS; orange lines) and −0.60 (NS; blue lines), respectively. In both cases, the√ positions are rescaled significantly higher than that of the interface, which requires ρ with the nearest-neighbor distance ann = 4π/ 3q to compensate the a sufficiently large system size. At high ¯ , the coexistence gap α =− . /ρ different lattice spacings. Inset: Full 2D density field for αu =−0.6; closes, approaching the asymptote u 0 22 ¯ of the phase the dashed red line indicates the 1D cross section. transition.

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Our simulations show that the one-mode approximation (a) describes the ground-state structure accurately down to av- 1.4 0.06 erage densitiesρ ¯ 0.4 near the uniform-supersolid phase 1.2 transition. Ifρ ¯ and/or αu are reduced further, higher modes will no longer be negligible. The difference between a super- 1.0 solid one-mode ground state and a normal solid multimode ω/ω 0.06 0.8 solution at the same average densityρ ¯ = 0.4isshownin Fig. 2. The density profile for the NS state resembles a 0.6 collection of Gaussian peaks, which typically have a signifi- Frequency 0.4 cantly larger separation than the peaks of a one-mode solution at similarρ ¯ . Moreover, the very low-density values in the 0.2 regions between the maxima suggest that small quantum or 0.0 thermal fluctuations may suffice to destroy phase coherence 0.0 0.2 0.4 0.6 0.8 1.0 α < − in this regime. Finally, the domain u 1 corresponds to Wave vector k/k < a negative GP parameter g0 0, thus representing attractive (b) contact interactions. The numerically determined ground state

| -4 -2 0 2 4 atρ ¯ = 0.1, α =−1.1 [ﬁlled circle in Fig. 1(b)] realizes a v u | x

single droplet with an approximately Gaussian density proﬁle, 5

qualitatively similar to recent experimental observations of 10 39 quantum droplet formation in dilute K BECs [37]. 0.5 1.0 1.5 2.0 2.5

IV. DYNAMICS ρ

We next describe how the above framework can be used FIG. 3. (a) The analytically calculated nonlinear dispersion re- to obtain predictions for the sound-wave dynamics in a su- lation (blue line) for plane waves becomes asymptotically linear for persolid (see Appendix E for details of calculations). The small k values (dashed orange line). See text for the scaling factors ω supersolid phase breaks the continuous translational sym- and k . Inset: The analytical prediction (blue line) agrees well with metry and supports lattice vibrations. These vibrations can numerical results (symbols) obtained by solving the full nonlinear ρ = . α = be studied analytically close to the uniform-supersolid phase dynamical system described by Eqs. (2)and(3)for¯ 1 0and u transition, where the one-mode approximation for the density −0.24. (b) Representative snapshot from a numerical simulation ρ is accurate. Near the phase transition the local deviation showing the magnitude of the velocity field and the density field. of ρ from its mean valueρ ¯ is relatively small, and one can See also Movie 2 (Supplemental Material [30]). Taylor-expand the nonlinear quantum potential in ρ around ρ¯ . Adopting this approximation, we now consider a change → − , of coordinates x x u(t x), where u is a displacement one has q ≈ 1 and φ0/ρ<¯ 1/3, implying k/k|| 1. This field in the Eulerian frame x. Since we are interested in suggests that vibrations with wavelengths larger than the hydrodynamic long-wavelength sound excitations, we may lattice constant generally exhibit a linear dispersion and that assume that the displacement field varies over a length scale the speed of sound is simply given by c = ω /k [dashed significantly larger than the spacing between the hexagonal orange curve in Fig. 3(a)]. density peaks |∇u| 1. Inserting the one-mode ansatz (8) The analytical prediction derived from the linear Eq. (9) into Eq. (5), and keeping only the leading terms in u, one agrees well with the numerical dispersion results obtained by obtains an energy functional U[u] that is quadratic in the simulating the full nonlinear Eqs. (2) and (3) with the open- displacement field (Appendix E). The approximative dynam- source spectral code DEDALUS [39]; see inset in Fig. 3(a). ics of u is found from Eqs. (2) and (3) by noting [38] that, In our simulations, we analyzed a longitudinal mode in a ∂ = ∂ =−∇δ /δρ = for small displacements, t u v and t v U periodic box, for four box sizes in the direction of the −(1/ρ¯ )δU/δu hold. This gives the linear equation wave, usingρ ¯ = 1.0 and α =−0.24 (Appendix H). For u 2 2 2 2 these parameters, one finds φ0 = 0.266 and q = 0.934 by 3φ q 1 − φ0/ρ¯ + 5φ /ρ¯ ∂2u = 0 0 ∇2u minimizing the energy U of the supersolid state with the one- t ρ¯ 4¯ρ mode approximation. Representative simulation snapshots 2 showing the magnitude of the velocity field and the density + (3q2 − 2)∇2u + 2q2∇(∇·u) − ∇4u , (9) 3 are shown in Fig. 3(b); see also Movie 2 (Supplemental Material [30]). The low-k mode is visible as an enveloping · − ω which is solved by a plane wave u0 exp [i(k x t )]. Since modulation of the velocity field. The sound speed was mea- ∝∇ the field v S describes an irrotational potential flow sured in the simulations by estimating w and k from a ∇× = with v 0, only longitudinal modes are allowed. linear fit, yielding c = 0.6994 ± 0.0015, in excellent agree- Inserting the plane-wave ansatz in Eq. (9) yields the ment with the theoretically predicted value ω /k = 0.6995. ω/ω = / + / 2 nonlinear dispersion relation (k k ) 1 (k k ) , Thus, although waves can scatter from inhomogeneities of 2 = 3 { φ2/ρ2 − φ /ρ + / ρ − + 2} with k 2 [(5 0 ¯ 0 ¯ 1) 4¯] 2 5q and the density field in the fully nonlinear system, the long- ω2 = 2φ2 4/ρ 2q 0 k ¯ , which is shown as the blue line in Fig. 3(a). wavelength dynamics is accurately captured by the linearized For supersolids described by the one-mode approximation, theory.

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1.0 formation [40,41], the mean-field quantum systems discussed here differ from their classical counterparts through the pres- ence of the quantum potential term which penalizes high-k modes. VI. CONCLUSION To conclude, we have introduced and studied a higher- 0.5 order generalization of the classical GP mean-field theory that accounts for the structure of pair interactions through ( ρ the relevant Fourier coefficients of the underlying potential. − The resulting hydrodynamic equations share many concep- ρ tual similarities with classical Swift-Hohenberg-type pattern )

/ formation models [15], for which a wide range of advanced ρ mathematical analysis tools exists [18,44,45]. The generalized 0.0 GP equation, (1), allows us to transfer these techniques directly to quantum systems. By focusing on the fourth-order FIG. 4. Superfluid ground state exhibiting quasicrystalline order, case, we obtained analytic predictions for ground-state phase as predicted by Eq. (1) for an interaction potential u having relevant diagrams and low-energy excitations that agree well with terms up to eighth order in the expansion (10). System param- direct numerical simulations. With regard to experiments, our eters:ρ ¯ = 1.0, αu =−0.9, b1 = 0.09, b2 = 0, a1 = 0.982, a2 = results suggest that the coexistence phase at the first-order 2cos(π/12). Here ρmin ≈ 0.395 and ρmax ≈ 2.969. The minimum superfluid-supersolid transition may be the most promising density is well above 0, suggesting that this state is superfluid. Inset: regime for observing supersolids. We expect this mean-field Absolute value of the Fourier-transformed density, |ρˆ |, indicating a prediction to be robust, as the analytically derived uniform- 12-fold rotational symmetry with two rings in the reciprocal lattice. supersolid phase transition curve (solid line in Fig. 1) agrees See also Movie 3 (Supplemental Material [30]). well with recent qMC simulations [12] that account for beyond mean-field effects [Fig. 1(a), inset]. V. SUPERFLUID QUASICRYSTALS When the interaction parameter αu in Eq. (6) becomes The supersolid phase in Fig. 1 is caused by the pat- too small, a transition to a normal solid state is expected, 2 2 tern forming term αu(1 +∇ ) in the internal energy U. as quantum fluctuations will likely destroy phase coherence This fourth-order contribution makes plane waves with wave in the low-density domains. However, grain boundaries of number k0 =|k|=1 energetically favorable, giving rise to a polycrystalline normal solid could still show superfluid be- hexagonal pattern. More complex supersolid structures can havior [46,47] and can be studied using the theory presented be expected in systems where the Fourier-transformed inter- here. Interestingly, our numerical ground-state analysis sug- action potentialu ˆ possesses multiple local minima ki, where gests that at low average densities a quantum droplet can be uˆ(ki ) < 0. To demonstrate that this is indeed the case, let us stabilized by the kinetic quantum potential without requiring replace (1 +∇2 )2 with higher-order nonlinearities to correct for quantum fluctuations + 2 +∇2 2 + 2 +∇2 2 , [37,48]. From a conceptual perspective, the generalized GP b1 a1 b2 a2 (10) framework appears well suited for future extensions: By cal- which corresponds to keeping terms up to order j = 4in culating the quasicrystalline ground state (Fig. 4 and Supple- the generalized GP equation, (1). The resulting eighth-order mental Material Movie 3 [30]), we have already demonstrated operator in Eq. (10) is energetically bounded from below how to extend the approach to systems with multiple relevant and accounts for an asymmetry in the local potential energy length scales. The theory can be refined by including Lee- 5/2 minima ki through the parameters bi. Note that ai and ki Huang-Yang [49,50] corrections ∝ ρ that penalize sharply are in general no longer equal when bi = 0. With a larger peaked densities. Generalizations to anisotropic potentials number of physically relevant length scales at play, systems and multicomponent systems seem both feasible and exper- can attain a wider range of ground-state structures [40,41]. imentally relevant [4,37,51]. We thus hope that the above To illustrate this, we computed the ground state of Eq. (1) discussion can provide useful guidance for future efforts in for the expansion (10) with parametersρ ¯ = 1.0, αu =−0.9, these and other directions. b1 = 0.09, b2 = 0, a1 = 0.982, a2 = 2 cos (π/12). As evident from the corresponding mean-field density ρ and its absolute ACKNOWLEDGMENTS |ρ| Fourier transform ˆ in Fig. 4, the ground state in this case This work was supported by the Finnish Cultural Founda- is a quasicrystalline superfluid state with 12-fold rotational tion (V.H.) and an Edmund F. Kelly Research Award (J.D.). symmetry; see also Movie 3 (Supplemental Material [30]). In this context, we mention that earlier discussions [42,43] APPENDIX A: DERIVATION OF THE of superfluid quasicrystalline states considered spin and pseu- MEAN-FIELD ENERGY dospin interactions with only one relevant length scale. By contrast, the quasicrystalline ground state in Fig. 4 forms 1. Mean-field reduction of the many-particle due to the competition between the two length scales set by Schrödinger equation the local minima of the interatomic potentialu ˆ. While this For completeness, we present the derivation of the gen- mechanism is reminiscent of classical quasicrystal pattern eralized GP equation, (1), from a many-particle Schrödinger

063621-5 HEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019) equation with a pair potential u(ri, r j ). Our starting point is over the spatial coordinates gives a bosonic N-particle quantum system, described by a wave , ,..., ∂ = N ∗ ∂ function N (t r1 rN ) that satisfies the exchange symme- ih¯ t dr N ih¯ t N try N ∂ ,..., ,..., ,... = drN ∗ ih N ∂ r N (t ri r j ) N ¯ ∂ t ( j ) = (r j ) =+(t,...,r ,...,r ,...). (A1) j 1 j i 1 N = √ N ∗ N ∂ The dynamics of the system is governed by the N-particle dr N ih¯ t (r j ) (A8) N (r j ) Schrödinger equation j=1 N ∂ = ˆ , 1 ∗ ih¯ t N HN N (A2) = dr (r )ih¯∂ (r ) N j j t j j=1 where the N-particle Hamiltonian is defined by ∗ = dr (r)ih¯∂ (r) = E . N N t 2 1 HˆN =−α ∇ + vext(ri ) + u(ri, r j ), ∗ ri 2 Taking the functional derivative with respect to yields i i i=1 j =i δE (A3) ih∂ = , ¯ t δ∗ (A9) with α = h¯2/(2m). Equation (A2) can be written as a variation where the right-hand side is given by ˆ of the expectation value of HN , δ E 2 2 = −α∇ + vext + dr |(r )| u(r, r ) . (A10) δE δ∗ ih∂ = , ¯ t N δ∗ (A4) N The functional Gross-Pitaevskii equation in Eq. (A9)was discussed by Gross in his seminal paper on superfluid vortices where [13]. This theory corresponds to a quantum density functional theory where the exchange-correlation energy is neglected; ,∗ =ˆ = N ∗ ˆ . EN [ N N ]: HN dr N HN N (A5) for strongly correlated systems, see, e.g., Ref. [52].

The key step in the derivation of the generalized GP equation 2. Fourier expansion for isotropic potentials is the mean-field approximation The energy contribution for an isotropic pair potential u is 1 N (t, r1,...,rN ) ∗ ∗ u = dr1dr2[ (t, r2 ) (t, r1) 1 2 = ψ(t, rπ(1))ψ(t, rπ(2)) ...ψ(t, rπ(N ) ), (A6) × | − | , , , N! u( r2 r1 ) (t r1) (t r2 )] (A11) π∈SN which can be rewritten as where the sum is over the permutation group SN of indices 1 1,...,N. This approximation is equivalent to assuming that u = dr[n(t, r)(u ∗ n)(t, r)], (A12) 2 there are no entangled particles in the system. It also implies ψ∗ ψ where (u ∗ n) is the convolution that the one-particle probability distributions (ri ) (ri )are independent and identical. | − | , We can use Eq. (A6) to calculate the mean-field approxi- dr [u( r r )n(t r )] (A13) mation of the energy. The symmetric form of the mean-field ansatz allows for relabeling of indices in the evaluation of the and n is the local number density ||2. Hamiltonian H ,giving The Fourier transform of a pair potential N − = −ik·r ∗ 2 2 N(N 1) uˆ(k) dr[e u(r)] (A14) E = dr(−α ∇ +|| v ) + ext 2N2 with finite moments can be expressed as a power series in k2 ∗ ∗ × dr1dr2[ (r1)(r1)u(r1, r2 ) (r2 )(r2 )]. as ∞ (A7) 2 j uˆ(k) = g2 jk . (A15) √ j=0 Here = Nψ so ||2 becomes the number density n.For large N, one has N(N − 1)/N2 ≈ 1. Using this the convolution of the pair potential becomes To derive a dynamical equation for the one-particle wave ∞ −1 j 2 j function, we evaluate the expectation value of the time evo- (u ∗ n) = F [uˆnˆ] = (−1) g2 j∇ n. (A16) ∗ lution operator. Multiplying Eq. (A2)by N and integrating j=0

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Here F −1 denotes the inverse Fourier transform. TABLE I. Units and notation. Here d is the dimension of the Now system. ⎡ ⎛ ⎞ ⎤ ∞ 1 Symbol Description Units Variable(s) Text u = dr⎣n⎝ (−1)jg ∇2 j⎠n⎦. (A17) 2 2 j j=0 r Length m – x t Time s – t The energy E becomes Wave function m−d/2 (t, r) ∗ n Number density m−d (t, r) n E [ ,] u Interaction potential J md r =|r| u h¯2 x q−1 x = + |∇|2 + ||2 Length 0 – u dr vext (A18) τ / 2 2m Time m (¯hq0 )–t s Minimization parameter m/(¯hq2 )–– and the dynamics is given by Eq. (A9). √ 0 Wave functionh ¯/(q0 mue )(τ,x)– The interaction term [Eq. (A17)] can be effectively trun- ρ 2/ 2 τ, ρ Number densityh ¯ (mueq0 )(x) cated at low j because the kinetic energy contribution ∝ 2 2/ ρ U Effective potential energyh ¯ q0 m U |∇|2 penalizes high-order Fourier modes, forcing their am- −1 τ, u Displacement field q0 ( x) u plitudes to be small. Let us examine the lowest-order truncations of Eq. (A17): (i) Zeroth order: This case corresponds to the standard GP with the possible drawback of having too many symbols in equation. the text. In the text v denotes the velocity field both in SI units (ii) Second order: This case was studied in Ref. [29]. The and in units ofhq ¯ /m, whereas in the Appendixes the field v is coefficient g must be nonnegative to ensure that the energy 0 2 only in units ofhq ¯ /m. In the Appendixes we use the notation is bounded from below, thus leading to a penalization of 0 F for integral energies, where is the field variable. We have variations in n. As shown in Appendix F, for repulsive contact omitted the field variable for functionals of the fields ρ and v. interactions (g > 0) the only possible ground state is the 0 In the text the same uppercase characters of the form F are uniform state characterized by a constant density n(t, r) = n¯. used for energy integrals in both SI units and reduced units. In (iii) Fourth order: In Appendix D we show that this order both the text and the Appendixes the integral energies that are leads to pattern formation. functionals of reduced fields are also in energy units that are (iv) Higher orders: Interaction potentials with multiple reduced. See Table I for a partial list of symbols used in the competing length scales allow the realization of more complex text and the Appendixes. symmetries, including honeycomb pattern or quasicrystals [40,41]; see example in Sec. V. APPENDIX C: HYDRODYNAMIC FORMULATION 3. Fourth-order expansion The wave function can be rewritten using the polar Let us reparametrize the contribution due to the interaction decomposition as potential as (τ,x) = R(τ,x)exp(iS(τ,x)). (C1) ur ue 2 u = dr n2 + n ∇2 + q2 n . (A19) ρ =| |2 = 2 =∇ 2 2 0 Now R . We define v : S, allowing us to write Eq. (A20) as a set of flow equations, In terms of the wave function we have

Dv 2 2 ih¯∂t = Hˆ , (A20) =−∇[Q + v˜ + (α + (∇ + 1) )ρ], (C2a) Dt ext u where ∂τ ρ =−∇·[ρv], (C2b) h¯2 ˆ =− ∇2 + + + ∇2 + 2 2 / τ = ∂ + ·∇ H vext ur ue q0 (A21) where D D τ v is the advective time derivative, 2m α = / 4 u ur (ueq0 ), and is the system Hamiltonian. √ The parameters are deﬁned as 1 ∇2 ρ Q =− √ ρ (C3) g g2 2 2 =− 2 , = , = − 2 . q0 ue g4 ur g0 (A22) 2g4 4g4 is the quantum potential. Equation (C2a) can be expressed in 2 terms of a functional derivative as For negative values of q0, no pattern forming is to be expected. Thesignofq2 is fully determined by g since g > 0 in order Dv δU 0 2 4 =−∇ , (C4) to ensure ﬁnite energy for small-wavelength Fourier modes. Dt δρ where APPENDIX B: UNITS |∇ρ|2 α 1 In an effort to make the text more readable we use some of U ρ = dx + v ρ + u ρ2 + ρ ∇2 + 2ρ [ ] ρ ˜ext ( 1) the same symbols for both SI and reduced units, whereas in 8 2 2 the Appendixes we have tried to maintain mathematical rigor, (C5)

063621-7 HEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019) is the effective potential energy. These dynamics conserve the The time evolution can be made dissipative by making a total energy of the system modiﬁcation δE E[ρ,v] = K[ρ,v] + U[ρ], (C6) ∂ s, x =−i + μ + λ s s, x , s ( ) ( )δ ∗ ( ) ( ) (D6) where where μ is some positive dissipation rate. Here λ(t )isa 1 Lagrange multiplier ensuring the global conservation of the K[ρ,v] = dx ρ(τ,x)|v(τ,x)|2 (C7) 2 density | |2 and can be calculated as μ δE μ is the kinetic energy. λ(s) = dx ∗(s, x) = H, (D7) N δ ∗ N APPENDIX D: GROUND STATES where In this section we discuss ground states of the system N = dx[| (s, x)|2](D8) described by energy E [Eq. (C6)]. We deﬁne dissipative processes that are used to minimize E and use them to analyze is the rescaled number of particles in the system and H is the the ground-state structure of the system both analytically and Hamiltonian operator. This can be interpreted as the expected numerically. energy per particle times the dissipation rate. The dynamics can be made overdamped by dropping the 1. Dissipative dynamics in the density formalism imaginary part of the mobility, giving

The energy E is locally minimized when v = 0 and U is δE ∂ s, x =−μ + λ s s, x , minimized with respect to ρ. Here we present two types of s ( ) δ ∗ ( ) ( ) (D9) density preserving dissipative flows parametrized by s that where λ is defined by Eq. (D7). This equation has no coupling minimize U. between the real and the imaginary parts of allowing for The first one is given by √ setting Im[ (s, x)] = 0, which gives (s, x) = ρ(s, x). δU[ρ] Both Eq. (D6) and Eq. (D9) can be shown to lead to a ∂ ρ(s, x) =− + λ(s) s δρ(s, x) nonincreasing energy E with respect to parameter s. =−[Q + (α + (∇2 + 1)2 )ρ] + λ(s), (D1) u 3. Linear stability analysis where Let us analyze the stability of Eq. (D3) against small peri- δU[ρ] odic perturbations about a constant state, i.e., ρ(τ,x) = ρ¯ + λ = − dx δρ (D2) ε(τ )exp(iq · x), where ε is some positive, small, spatially independent amplitude. Inserting this ansatz into Eq. (D3) and is a Lagrange multiplier that keeps the total density ρdx abbreviating q =|q| we find constant in time. Here − denotes the integral mean defined by 4 − fdx = fdx/ dx. The process described by Eq. (D1)is q 2 2 2 2 ∂τ ε(τ ) =− + αuq + q (1 − q ) ε(t ) referred to as nonlocal dissipative dynamics. 4¯ρ Local dissipative dynamics is defined by = C(q)ε(τ ) (D10) δ ρ 2 U[ ] up to linear order in ε. Maximizing the coefficient C with ∂sρ =∇ δρ respect to q defines the most unstable perturbation. If such =∇2[Q + (α + (∇2 + 1)2 )ρ]. (D3) a maximum exists, it suffices to examine the value of C at the u given maximum. If no maximum exists, C is maximized by It can be shown that both these dynamics will lead to a q = 0, which will give a stable constant solution. nonincreasing U, i.e., ∂sU (s) 0. Solving the aforementioned problem gives the condition for stability. The uniform solution is not stable against perturbations if 2. Dissipative dynamics in the wave formalism − , ρ / , Let us define the total energy in the wave formalism: α < 1 ¯ 1 8 u − ρ / ρ2 , ρ> / . (D11) ∗ (1 16¯ ) 64¯ ¯ 1 8 E [ , ] This defines the spinodal curve for the uniform phase. The 1 2 4 2 2 2 = dx{|∇ | + αu| | + [(1 +∇ )| | ] }. (D4) stability phase diagram with the value for the most unstable q 2 is shown in Fig. 5. The generalized nonlinear Schrödinger’s equation can be written as 4. One-mode approximation δE The pattern forming part of the internal energy Eq. (C5) ∂ =−i s δ ∗ can be written in Fourier space as 1 2 2 2 2 1 2 2 2 =−i − ∇ + (αu + (∇ + 1) )| | . (D5) dk[(1 − k ) ρˆ (k) ] (D12) 2 8π 2

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2.0 The terms without an oscillating component are referred to as q

1.0 resonant terms since they resonate with the linear operation dx. 0.9 Cp

¯ Now

ρ 1.5 0.8 unstable 0.7 − dx n2 = − dx(e2iqx + e−2iqx + 2) = 2 0.6 S 1.0 Cp Cp 0.5 0.4 and stable

Average density 0.3 0.5 − ρ 2 = ρ2 + φ2. 0.2 dx[ S(x) ] ¯ 2 0 Cp 0.1

0.0 Maximally unstable wave number We also need 0.50.0 - 0.5 - 1.0 - 1.5 αu Interaction parameter 2p p 2p − dx[ρS(x)∇ ρS(x)] = − dx[(−1) ρS(x)q n(x)] FIG. 5. Stability diagram of the uniform density state as a func- Cp Cp tion of the interaction parameter α and average densityρ ¯ . Contour = − p 2pφ2. u 2( 1) q 0 lines show the value for the most unstable wave number q of the linear perturbation. The pattern forming part of the average energy becomes α by using Plancherel’s theorem. For crystalline ground states u 2 1 2 2 − dx ρS(x) + ρS(x)(∇ + 1) ρS(x) we have C 2 2 p ρ = ik·x, α + 1 (x) ake (D13) = u ρ2 + − 2 2 + α φ2. ¯ ((1 q ) u ) 0 (D16) k∈G 2 where G is the point group of the reciprocal crystal lattice. The remaining part of the average energy is the part from the The pattern forming term will penalize any modes with |k| = quantum potential 1, suggesting that including only the vectors k in the ﬁrst Brillouin zone might approximate well the ground state. Here |∇ρ (x)|2 (∂ n (x))2 − dx S = − dx x S . we analyze one-mode approximations of periodic number ρ (x) ρ¯ + n (x) densities in two dimensions. Cp S Cp S

Here we use nS(x) = ρ¯ A cos ξ, where ξ = qx: a. Stripe phase We use the ansatz (∂ n (x))2 A2q2ρ¯ 2π sin 2ξ − dx x S = dξ iq·x ρ¯ + n (x) 2π 1 + A cos ξ ρ (x) = ρ¯ + φ n (x) = ρ¯ + φ (e + c.c.) Cp S 0 S 0 S 0 = ρ + · , A2q2ρ¯ π sin 2ξ ¯ (1 A cos [q x]) (D14) = dξ . 2π −π 1 + A cos ξ where c.c. stands for the complex conjugate and A = 2φ0/ρ¯ . = , We can simplify the calculations by choosing q (q 0) and This can be solved with a substitution, p = tan (ξ/2). This · = writing q x qx, leading to gives iqx ρS(x) = ρ¯ + φ0nS(x) = ρ¯ + φ0(e + c.c.) 2p 1 − p2 2 = ρ¯ (1 + A cos[qx]). sin ξ = , cos ξ = , dξ = dp. 1 + p2 1 + p2 1 + p2 Inserting the stripe ansatz ρS in Eq. (C5) gives an average energy Now π 1 |∇ρ |2 α 1 A2q2ρ¯ sin 2ξ ¯ = − S + u ρ2 + ρ ∇2 + 2ρ , dξ U dx S S( 1) S (D15) π + ξ C 8 ρS 2 2 2 −π 1 A cos p ⎡ ⎤ 2 , π/ 2 2 ∞ 2p where Cp is the interval [0 2 q] corresponding to a period of A q ρ¯ + 2 2 ρ = ⎣ 1 p ⎦ S. The integrand is constant in the y coordinate, reducing the dp − 2 2π −∞ + A 1 p 1 + p2 energy to a 1D integral. We start by calculating the quadratic 1 1+p2 term 8A2q2ρ¯ ∞ p2 ρ2 = ρ2 + ρφ + φ2 2. = dp , S ¯ 2¯ 0nS(x) 0 nS(x) π − 2 + γ 2 + 2 2 2 (1 A) −∞ p A (1 p ) All the terms that have an oscillating component exp(iq jx) = , , ,... γ 2 = + / − ρ with j 1 2 3 make no contribution to the average where A (1 A) (1 A). Note that S 0 requires A p > γ 2 > energy. Therefore the interesting terms are nS with p 1. 1. We also assume that A 0, implying that A 0 and that

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Im x giving 8A2q2ρ¯ ∞ p2 1 dp π − 2 + γ 2 + 2 2 2 (1 A) −∞ p A (1 p ) x = q2ρ¯ (1 − 1 − A2 ). Finally, Re x ⎛ ⎞ q2ρ¯ φ 2 FIG. 6. Semicircular contour in the complex plane used for cal- U¯ = ⎝1 − 1 − 4 0 ⎠ culating an integral on the real axis. The symbol x marks a singularity 8 ρ¯ on the imaginary axis. α + 1 + u ρ2 + − 2 2 + α φ2. ¯ ((1 q ) u ) 0 (D17) γA is real. The integral 2 ∞ p2 1 b. Hexagonal phase dp 2 + γ 2 2 The crystalline hexagonal ground-state solution is approxi- −∞ p A 1 + p2 mated similarly to the stripe phase with the one-mode approx- ∞ p2 1 imation = dp 2 2 −∞ (p + iγA)(p − iγA) (p + i) (p − i) ρH(r) = ρ¯ + φ0nH(r) ∞ 3 · = ρ + φ iq j r + , =: dp f(p) ¯ 0 (e c.c.) (D18) −∞ j=1 can be solved in a standard way using residue theorem for a where the reciprocal lattice vectors q j have a hexagonal semicircular path on the complex plane shown in Fig. 6 and symmetry, i.e., taking the radius to ∞. For this we need residues at singu- q2, i = j, larities in the upper-half complex plane (Im p > 0). There are q · q = j i − 1 q2, i = j. two of these, p = iγA and p = i. The latter is of order 2: 2

iγA Res ( f , iγA) = , Equation (D18) can be rewritten as 2 γ 2 − 1 2 A √ ρ = ρ + 3qx qy + , i(1 + γ 2 ) H ¯ 1 2A cos cos A cos (qy) Res ( f , i) =− A . 2 2 γ 2 − 2 4 A 1 where, again, A = 2φ0/ρ¯ . Now Calculating the resonant terms for the polynomial part of ∞ the internal energy is pretty straightforward [53]. For the dp f(p) = 2πi[Res ( f , iγA) + Res ( f , i)], quantum potential we have −∞

√ √ 2 2 2 2 3qx 2 qy + 3qx qy + 2 1 |∇ρH| ρ¯ q A 3sin cos cos sin sin(qy) − dx d = − dx 2 √ 2 2 2 , C 8 ρH C 8 3qx qy + + p p 2A cos 2 cos 2 A cos(qy) 1 √ where Cp is the primitive lattice cell shown in Fig. 7.Let 3qx/2 = x˜ and qy = y˜. Now this integral is 2 ρ¯ q2A2 2π 2π 3sin2 (x˜) cos2 y˜ + cos (x˜) sin y˜ + sin(˜y) dxd˜ y˜ 2 2 . 32π 2 y˜ + + 0 0 2A cos (x˜) cos 2 A cos(˜y) 1 Theintegrationin˜x can be done by repeating the calculation done for the stripe phase. The integrand will have two purely imaginary singularities that can be calculated by factoring the polynomial appearing in the denominator. This gives an integral iny ˜ over a period of 2π: ρ¯ q2 π 1 −2 − 6A + 10A2 + (9A2 − 4A − 4) cos(˜y) − 2A cos(2˜y) − A2 cos(3˜y) dy˜ + (A + 2) cos(˜y) + 2A + 1 . 32π −π cos(˜y) + 1 2A2 cos(2˜y) − 6A2 − 8(A − 1)A cos(˜y) + 4

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This can be dealt with by using an identity, 1 − − + − − (2 s)a0Js 3 2 a1(3 2s)Js 2 1 + − − + − − + a2(1 s)Js 1 2 a3(1 2s)Js sa4Js 1 ann = ξ − −s|1, f ( c) 0 (D21) where C p 1 1 J [ f ] = dξ √ , (D22) s s 0 f (ξ − c) f is a third- or fourth-order polynomial with FIG. 7. One-mode density ﬁeld with a hexagonal symmetry 4 3 f = a0(x − c) + a1(x − c) showing the primitive√ lattice cell Cp and the nearest-neighbor dis- 2 tance ann = 4π/(q 3). + a2(x − c) + a3(x − c) + a4, s = 1, 2, 3,..., and c is some arbitrary constant [54]. In our case s = 1, c = 0, and f = f1,giving Noting that cos(−y˜) = cos(˜y) and making the change of =− 2, =− − , =− − 2, variables cosy ˜ = 1 − 2ξ gives a0 4A a1 4(1 3A)A a2 (1 3A) = − 2, = . q2ρ¯ 1 P (ξ ) Q (ξ ) a3 (1 A) a4 0 dξ 1√ + √ 1 , 32π ξ f ξ ξ (1 − ξ ) We have 0 1 1 (1 − A)2 where dξ √ ξ 2 3 2 2 0 f1 P1(ξ ) =−16A ξ + (24A − 8A)ξ =− − 2 2 (1 A) J1 + (4A − 4)ξ + (A − 1) , √ − 1 2 2 f1 =− A J− + A A − J− + . 8 2 4 (3 1) 1 ξ 0 Q (ξ ) = (2A + 4)ξ + A − 1, 1 The only diverging part is the last one. Combination with and the earlier-diverging term gives √ 1 f (ξ ) = (4A2ξ 2 + (4A − 8A2 )ξ + A2 − 2A + 1)(1 − ξ )ξ. −2 f 1 lim 1 − 2(1 − A) −1 − 1 =−2 f (1) = 0. → ξ 1 Let us ﬁrst calculate the part 0 1 Q ξ Let = ξ √ 1( ) I1 : d = + . 0 ξ ξ (1 − ξ ) It I1 I2 (D23) 1 4 + 4A 1 1 − A Now = dξ √ − dξ √ . ξ − ξ ξ ξ − ξ 0 (1 ) 0 (1 ) 2 It = 2(2 + A)π − 4(1 − A)J0 − 12A(1 − 3A)J−1 − 24A J−2. Both parts can be calculated with trigonometric substitution. This gives The integrals Js are standard elliptic integrals and can be − ξ transformed to Legendre normal form with a change of in- 2(1 A) f = g g g = ξ − ξ I1 = 2(2 + A)π − lim √ tegration variables. We have 1 1 2, where 2 (1 ) ξ→1 ξ − ξ ξ > ξ ∈ , (1 ) and g√1( ) 0for [0 1]. We use the change of variables ξ = / −1 2 g2 g1. For details, see Ref. [54]. The contribution of = 2(2 + A)π − lim 2(1 − A) − 1. (D19) 2 →0 the quantum potential can be written as q ρ¯ It /32π, which, after a somewhat tedious calculation, gives The second part diverges and has to be combined with other terms. Now the other part needed here is 2ρ √ = q ¯ + − − UQP A 2 2 3A 1 1 2 2 16 P1(ξ ) 16A ξ 8A(1 − 3A)ξ I := dξ √ = dξ − √ − √ − 2 ξ 12C 1(E(k ) − K(k )) + (2A + 1)C K(k ) 0 f1 0 f1 f1 × A A A A A , π − A − A 2 − 4(1√ ) + (1 √ ) . (D20) (D24) f1 ξ f1 These are elliptic integrals. The only diverging part is where 1 1 − A 2 1 ξ (1 √ ) . K(k) = dt (D25) d − 2 − 2 2 0 ξ f1 0 (1 t )(1 k t )

063621-11 HEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019) ) ¯ ρ 0.10 Average density ρ¯ 2

q Solution Eq. (D24) ( 0.0 0.5 1.0 1.5 2.0 / Fit

QP 0.08 U

u -0.2 α 0.06 -0.4 0.04 -0.6

0.02 -0.8

0.00 -1.0

0.0 0.2 0.4 0.6 0.8 1.0 Interaction parameter Average quantum potential Exact solution Relative amplitude 3φ0 /ρ¯ -1.2 Asymptotic FIG. 8. Exact solution for the one-mode approximation energy of FIG. 9. Uniform-supersolid phase transition line with an asymp- the quantum potential U plotted against a polynomial ﬁt showing QP totic expression α =−0.2234/ρ¯ . that the quartic approximation sufﬁces for analytical calculations. u

5. Finding the ground state is the complete elliptic integral of the first kind and √ The ground state of the system is found by minimizing 1 − k2t 2 = √1 Eqs. (D17) and (D31) with respect to the scaler q and the E(k) dt (D26) A = φ /ρ ρ α 0 1 − t 2 amplitude 2 0 ¯ for given system parameters ¯ and u. In both cases, the extrema can be expressed as roots of poly- is the complete elliptic integral of the second kind. The nomials. The coexistence gap between the uniform superfluid complex coefficients are defined as and the supersolid phases is calculated numerically using the analytical energies via common tangent construction. We find 3A2 + (1 − A)3(3A + 1) − 1 an analytical expression for the phase transition line between kA = i (D27) the uniform superfluid and the supersolid phases that can be −3A2 + (1 − A)3(3A + 1) + 1 expressed as an asymptotic expansion for largeρ ¯ as αu = −0.2234/ρ¯ . Figure 9 shows the asymptotic expression with and the exact solution. The one-mode analysis gives an upper bound for the SS-NS transition. Minimizing the one-mode 6A2 + 2 (1 − A)3(3A + 1) − 2 C = . (D28) energy given by Eq. (D31) predicts negative densities in the A 3 A parameter range labeled NS in Fig. 1. In a similar way, the D

The value of UQP at the maximal A = 2/3 can be expressed in phase is characterized by the absence of real solutions to the terms of known constants as minimization problem of the one-mode energy. For details on √ the ground-state phase diagram see the Mathematica notebook 1 3 phase_diagram.nb [30]. U = − q2ρ.¯ (D29) QP A=2/3 6 8π APPENDIX E: WAVE EQUATION FOR LATTICE For more details see the Mathematica notebook VIBRATIONS OF THE SUPERSOLID elliptic_integral.nb [30]. UQP will be a part of the total internal energy that still We next examine the dynamics of a small displacement has to be minimized with respect to q and A. To simplify the field about the hexagonal ground state described in Sec. D4b. minimization procedure we fit a fourth-order polynomial To this end, we introduce a deformation of the coordinates x → x − u(τ,x). The one-mode approximation in Eq. (D18) = 2 + 3 + 4 p4(x) a2A a3A a4A (D30) becomes against the quantum potential (D24). We find 3 − · τ, · ρ τ, = ρ + φ iq j u( x) iq j x + . 2 2 2 ( x) ¯ 0 [e e c.c.] (E1) U¯ = (a2 + a3A + a4A )A q ρ¯ j=1 1 2 2 2 2 + (2(αu + 1) + 3A (αu + (q − 1) ))¯ρ , (D31) 4 We write the time evolution equation for the displacement where a2 = 0.226 034, a3 =−0.288 956, and a4 = field u(τ,x) in terms of the effective potential energy U. 0.401 767. The zeroth coefficients disappears because for First, we assume that the relative amplitude φ0/ρ¯ is small and zero amplitude the quantum potential is 0. On the other expand the quantum potential up to fourth order in φ0/ρ¯ .This hand, the first coefficient is 0 because the integral over the assumption should work near the uniform-supersolid phase periodic domain of any single oscillating mode gives 0. The transition line, where the amplitude of the oscillating number fourth-order fit seems to agree well with the exact solution as density is low. Second, we assume that ∇u is small and expand shown in Fig. 8. the configuration energy U up to second order in u.Third,

063621-12 QUANTUM HYDRODYNAMICS FOR SUPERSOLID CRYSTALS … PHYSICAL REVIEW A 99, 063621 (2019) we assume that the velocity field v is small and keep only APPENDIX F: NO PATTERN FORMING FOR LOW-LEVEL the linear terms in the dynamical equations. For details of EXPANSIONS WITH REPULSIVE CONTACT the calculation see Ref. [38], where a similar analysis was INTERACTION performed. Here we show that the low-level expansions of the inter- Now, Eqs.(C2a) and (C2b)give action term u will only give uniform ground-state solutions. 1 δU[u] The effective potential energy functional for the second-order ∂τ u(τ,x) = v(τ,x),∂τ v(τ,x) =− , (E2) ρ¯ δu expansion is assuming that φ and |q |=q take the equilibrium value. This h¯2 |∇n|2 g g 0 j U [n] = dr + 0 n2 + 2 |∇n|2 , (F1) can be combined into a single equation, n 8m n 2 2 δ 2 1 U[u] where we use the subscripted Un here to emphasize that the ∂τ u(τ,x) =− . (E3) ρ¯ δu energy is in SI units. We assume here that g0, g2 0. The functional derivative is given by The configuration energy U with the aforementioned approximations reduces to δ 2 ∇2 |∇ |2 Un 2 h¯ 2 n n = g0n − g2∇ n + − + . (F2) 3φ2q2 δn 8m n n2 U[u(τ,x)] = 0 dx 2 We want to minimize Un with respect to a conservation 1 φ φ2 constraint for n. This can be achieved by minimizing × 1 − 0 + 5 0 + (3q2 − 2) ∇u 2 4¯ρ ρ¯ ρ¯ 2 Uñ[n] = Un[n] − μ dr[n(t, r) − n¯] (F3) 2 + q2((∇·u)2 + (∇u)T : ∇u) + ∇2u 2 3 with respect to n.Here¯n is a given average particle number and μ is the chemical potential. Extrema of U˜ fulfill the +U (α ,φ , ρ,¯ q). (E4) n 0 u 0 condition Taking the functional derivatives with respect to u and plug- δU n = μ, (F4) ging in Eq. (E3) leads to δn φ2 2 2 3 0 q 2 i.e., the functional derivative is a constant. Inserting a uniform ∂τ u = 2q ∇(∇·u) = = μ = ρ¯ n n¯ in Eq. (F2)givesg0n¯ , showing that n n¯ is an extremum of U˜ . We show that this extremum is a global 2 2 n 1 − φ0/ρ¯ + 5φ /ρ¯ 2 ˜ + 3q2 − 2 + 0 ∇2u − ∇4u . minimum by showing that the energy Un is globally convex. 4¯ρ 3 First, some prerequisites [55]: (E5) (i) A functional is convex iff its second variation is nonnegative for all test functions ϕ. Inserting a plane wave exp [i(k · x − ωτ )] gives a dispersion (ii) The sum of convex functionals is convex. relation The second variation is given by 2 2 ω/ω = k/k 1 + (k/k ) (E6) 2 d δϕUn[n] = lim Un[n + 1ϕ + 2ϕ]. (F5) 1,2→0 d d for longitudinal modes. Here 1 2 The variations for the parts proportional to g0 and g2 can be 2 3 2 2 2 calculated easily, giving k = 5φ /ρ¯ − φ /ρ¯ + 1 /4¯ρ − 2 + 5q , 2 0 0 2 2 ω2 = 2φ2 4/ρ. g0 dr[ϕ(t, r) ], g2 dr |∇ϕ(t, r)| , 2q 0 k ¯ (E7) At small k the dispersion relation becomes linear and one respectively. Both are clearly nonnegative, implying that the can read off the speed of sound: corresponding parts in U are convex. For the last part we have 2 2 2 2 2 3φ q 1 − φ0/ρ¯ + 5φ /ρ¯ |∇n + ( + )∇ϕ| c2 = 0 q2 − + 0 . 1 2 ρ 5 2 ρ (E8) ¯ 4¯ n + (1 + 2 )ϕ We can also calculate U of Eq. (E4), giving ∞ + kϕk 0 2 ( 1 2 ) =|∇n + (1 + 2 )∇ϕ| 2 nk+1 2 q 2 2 2 2 k=0 U = φ V − φ /ρ + φ /ρ + − q , 0 3 0 ρ 1 0 ¯ 5 0 ¯ (1 ) (E9) 4¯ 2 [n∇ϕ − ϕ∇n]2 = lot + 1 2 + hot, where V is the volume of the domain. Since φ0/ρ¯ 1/3, n3 the first part due to the quantum potential penalizes any where ‘lot’ stands for lower-order terms and ‘hot’ for higher- > system with q 0, making the system expand. The second order terms. This gives the second variation part penalizes any variance from q = 1. This part stabilizes the nonuniform pattern. For large values ofρ ¯ the pattern forming h¯2 [n∇ϕ − ϕ∇n]2 dr 0, (F6) part dominates and q ≈ 1. 4m n3

063621-13 HEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019) completing the proof. Setting g2 = 0 proves the same for the TABLE II. Parameter values for Rydberg-dressed BECs. zeroth-order expansion. Alternative Parameter expression Value unit APPENDIX G: PARAMETERS FOR √ RYDBERG-DRESSED BECS l h¯/(mωtr ) 0.972674 μm rc Rc/l 2.65 Here we calculate the interaction parameters q0, ur, and ue 3 2 4 c6 C6m ω /h¯ 2.45 for a model system for Rydberg-dressed BECs described in N –104 Ref. [27]. The calculation presented here can be repeated for −1 q0 q˜0/Rc 1.87005 μm any radially symmetric interatomic potential u whose Fourier / 4 4 2/ 4 ue u˜eC6 (Rc q0 ) 0.197182399h ¯ (mq0 ) transform’s smallest extremum is a minimum. / 4 − . 2/ ur u˜rC6 (Rc ) 0 008456958h ¯ m We use parameters deﬁned in Fig. 2 in Ref. [27]. The interaction potential is

u(r) = usw(r) + uRB(r), (G1) giving αu ≈−0.042 889. We also need the average density. 4 where u (r) = gδ(0) is the collision part due to S-wave In Ref. [27] the authors study a system of 10 Rb atoms in a sw ≈ 2 scattering and volume of approximately V 9Rc .Wehave

C6 2 u (r) = (G2) mueq m u˜e N RB 6 + 6 ρ¯ = 0 n¯ = C r Rc 2 2 2 2 6 h¯ Rc h¯ q˜0 V is the long-range interaction of two excited alkaline Rydberg 4 m u˜e N h¯ c6 atoms. C6, Rc, and g are system parameters. = 2 2 2 2 3ω2 We calculate the Fourier transform Rc h¯ q˜0 9Rc m tr ∞ 2π h¯2 1 u˜ Nc = θ ikr cos(θ ) = e 6 uˆ(k) d dr[re u(r)] 2ω2 4 2 m tr Rc q˜0 9 0 0 ∞ 4 πrC6rJ0(kr) l u˜ Nc = g + dr , (G3) = e 6 ≈ 9.4. (G8) R6 + r6 2 0 c rc q˜0 9 where J is the zeroth Bessel function of the first kind. The 0 ω / π = remaining integral can be calculated numerically or expressed Here tr 2 125 Hz is√ the trapping frequency of a har- = / ω in terms of the Meijer G function [56]as monic potential, and l h¯ m tr is the characteristic length π scale for this trapping potential. The lowercase characters are −4 4,0 6 ω uˆ(k) = C R G Rck , 1 , 2 , 2 , , 1 + g. (G4) used for units defined by tr and a tilde is used for units 6 c 0,6 6 ;0 3 3 3 0 3 3 defined by C6 and Rc. For parameters, see Table II. In order to obtain the parameters we fit the polynomial The one-mode approximation predicts the phase transition at αu ≈−0.024 539 8, implying that these parameters should f = + 2 − 2 2 uˆ (k) ur ue k q0 (G5) be well within the solid regime. The nearest-neighbor distance for the hexagonal lattice is to the solutionu ˆ. There are many ways to perform the fitting. This problem has been studied in the context of approximating 4π direct pair correlation functions in the classical density func- ann = √ , (G9) tional theory of freezing with simpler energy functionals [18]. q0 3 Here we will fitu ˆf by matching q to the smallest minimum 0 ≈ . k ofu ˆ and by ensuring that the energy of the uniform and which, for the parameters in [27], gives ann 1 505 16 and m = . μ the supersolid solutions will be correct, i.e.,u ˆf(0) = uˆ(0) and Rc 3 879 69 m. f Figure 10 shows the exact solution foru ˆ and several uˆ (q0 ) = uˆ(q0 ). We find RB different fits. We use a fitting procedure matching the values of = ≈ . , q0Rc q˜0 4 8202 the fitted function with the exact solution at k = 0 and k = q0. 4 4 We also ensure that q0 minimizesu ˆRB. Other fits shown in Rc 2Rc (ur − g) = uˆ(q0 ) = u˜r, (G6) Fig. 10 include matching the curvature and the magnitude at C6 C6 k = km = q0. This would most likely approximate the elastic 4 4 (q0Rc ) ue 2Rc response of the solid phase better since it describes the energy = (uˆ(0) − uˆ(q0 )) = u˜e. C6 C6 well for modes k close to q0. However, for this system this fitting method would give an inaccurate estimate for the en- u ≈ The approximate values for these parameters are ˜r ergy of the uniform state described byu ˆ(0). The last fit shows − . u ≈ . 0 170 23 and ˜e 3 9690. a general least-squares fitting tou ˆ . This fitting method fails g = α RB We set 0 in accordance with Ref. [27]. The u param- to predict the wavelength of the one-mode approximation, the eter can be calculated as solid energy, and the energy of the uniform state, implying that ur uˆ(q0 ) u˜r global fitting might not work in this case. For more details see α = = = , (G7) u 4 − . q0ue uˆ(0) uˆ(q0 ) u˜e the Mathematica notebook parameters nb [30].

063621-14 QUANTUM HYDRODYNAMICS FOR SUPERSOLID CRYSTALS … PHYSICAL REVIEW A 99, 063621 (2019)

4.0 The√ simulation domain is a rectangle of size RB Exact solution − −

ˆ 1 1 u (4πq / 3, 4πq ). For the periodic ground states the

c Match values size of the domain has to be minimized in order to obtain

/α 3.0 4 c the physical ground state. This is done as follows: Let some r Match curvature lowest mode of the ground state of a given domain be 2.0 Least squares fit exp(iq · x). Now q =|q| defines the size of the domain. Due to the effect of the quantum potential the q that minimizes the ground-state energy is usually a little bit less than 1. In 1.0 order to find the minimal q we define an iterative process. We make an initial guess for q and calculate a number of 0.0 points around it. This gives us a chart (qn, En ) to which we fit a parabola. We calculate analytically the minimum of this

-1.0 1.0 2.0 3.0 4.0 5.0 6.0 parabola, define a new set of qn around this minimum, and Scaled Fourier transform q(i) q r k repeat the process. Let m be the minimal calculated from Scaled wave vector c the fitted parabola at the ith iteration. We stop the iteration when |q(i) − q(i−1)| is less than 10−5. FIG. 10. Rydberg contribution of the Fourier-transformed inter- m m action potentialu ˆRB and different fitting polynomials. 2. Lattice vibration simulations APPENDIX H: NUMERICAL METHODS Lattice vibrations are studied by simulating the full dynamics in the hydrodynamic formulation using the spectral PDE 1. Ground-state determination solver DEDALUS [58]. Since the mean-field potential terms are The ground state of the system is found by time-evolving linear in the density ρ, they can be integrated implicitly in the the nonconserved dissipative dynamical equation [Eq. (D1)] hydrodynamic formulation, eliminating the stiff time-stepping except in the case of the quantum droplet phase, for which restriction that would arise from integrating the higher-order we used overdamped dissipative wave dynamics [Eq. (D9)]. potential terms explicitly. Since spectral methods have no nu- In general, the performance of the density formalism is better merical dissipation, we regularize the equations by inserting because the high-order derivatives appear linearly in the dy- a viscositylike term ∝∇2v in the velocity equation, which namical equations. However, the density formalism becomes smooths small-scale variations and slowly damps velocity numerically unfeasible whenever the one-particle density ρ perturbations. is close to 0 due to difficulties in evaluating the quantum First, the ground state of the periodic domain is calculated potential Q. after which the vibration simulations are initialized with a For the ground-state calculations we impose doubly pe- low-amplitude plane-wave velocity signal. The system was riodic boundary conditions and calculate the derivatives in then evolved for ∼103 temporal units. The total kinetic energy Fourier basis. We use a semiimplicit time-stepping algorithm of the system is observed to undergo oscillations which decay that treats the linear parts of the equation implicitly and the due to the added viscous term. The square of an exponentially nonlinear parts explicitly [57]. To ensure numerical stability decaying sinusoid is fit to the time series of the kinetic energy we require that the difference of the total energy E be- to determine the oscillation frequency of the velocity signal. tween time steps is nonpositive. This difference is also used The imposed perturbation wavelength and this fit frequency as a stopping condition: whenever E < 10−12 the iterative are used to assess the agreement between the simulations and search for the ground state is stopped. the analytical dispersion relation.

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