Space- and Time-Crystallization Effects in Multicomponent Superfluids
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PHYSICAL REVIEW B 101, 020505(R) (2020) Rapid Communications Space- and time-crystallization effects in multicomponent superfluids Nikolay Prokof’ev Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA and National Research Center “Kurchatov Institute,” 123182 Moscow, Russia Boris Svistunov Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA; National Research Center “Kurchatov Institute,” 123182 Moscow, Russia; and Wilczek Quantum Center, School of Physics and Astronomy and T. D. Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China (Received 4 September 2019; revised manuscript received 20 December 2019; published 14 January 2020) We observe that space- and time-crystallization effects in multicomponent superfluids—while having the same physical origin and mathematical description as in the single-component case—are conceptually much more straightforward. Specifically, the values of the temporal and spatial periods are absolute rather than relative, and the broken translation symmetry in space and/or time can be revealed with experiments involving only one equilibrium sample. We discuss two realistic setups—one with cold atoms and another one with bilayer superconductors—for the observation of space and time crystallization in two-component counterflow superfluids. DOI: 10.1103/PhysRevB.101.020505 The superfluid long-range order—either genuine or, in superimposing two matter waves with opposite wave vectors lower dimensions, topological/algebraic—is associated with [3] nicely visualize the fact that superflows break translation the emergence of a well-defined (modulo 2π), defect-free symmetry and thus qualify to be called space crystals. Su- field of a coarse-grained phase, (r, t ); see, e.g., Ref. [1]. In percurrent states in three-dimensional superfluids do not form what follows, we discuss the genuine long-range order only. naturally by cooling the system across the transition temper- The generalization to the case of algebraic order is readily ature [4]. Nevertheless, supercurrent states can be prepared achieved along the lines described in Ref. [2]. Also, we use the by cooling the system in a rotating vessel which is stopped classical-field (matter-wave) language, which, on one hand, once the system is in the superfluid phase. In this sense, the captures the essence of superfluid phenomena and, on the period of the space crystal in the phase field depends on the other hand, is straightforwardly generalized to the case of experimental conditions used to prepare the sample, but other- quantum bosonic fields [1]. wise we are dealing with a stable thermodynamic equilibrium Long-range order in the coarse-grained matter field ψ = described by the Gibbs distribution with an emergent quan- exp[i(r, t )] means that we are dealing with the broken tized topological constant of motion (phase winding number) global U (1) symmetry state. The very nature of this state [1]. Academically speaking, such a persistent current state is implies the existence of a space crystal when the phase is a metastable. However, its relaxation time due to rare quantum- linear function of distance (and ψ is periodic in space with tunneling or thermal-activation events is exponentially large the period 2π/k), in the inverse k, and easily exceeds the time of the universe unless the period of the space crystal is microscopically small. ψ , = ψ , ik·r. (r t ) (0 t )e (1) While the existence of plane-wave states (1) is a generic This is a state with finite superflow velocity proportional to property of any statistical model with broken U (1) symmetry, the period depends on the reference frame. The Galilean the wave vector k. In the literature on superfluidity, the term “space crystal” is almost never (if at all) used in the context transformation of the field when going to the reference of the state Eq. (1), because the matter density n(r, t ) = frame moving with the velocity v0 with respect to the origi- |ψ(r, t )|2 remains homogeneous in space. (It should not be nal one, confused with a supersolid—the superfluid state with sponta- v0 · r (r, t ) → (r, t ) − , (2) neously broken translation symmetry in the particle density.) γ However, it is now conventional to call various states of matter “solids” and/or “crystals” if there is some observable implies that the state wave vector changes to revealing broken translation invariance, and this observable k → k − v0/γ . (3) need not be the particle density. One familiar example is the valence-bond crystal state of lattice bosons at half-integer Here, γ is the system-specific parameter relating the wave filling factor. In superfluids, the phase field plays the role vector of the matter wave to the flow velocity. In the quantum of such an observable. The interference fringes produced by case, γ = h¯/m, whereh ¯ is the Planck’s constant (in what 2469-9950/2020/101(2)/020505(5) 020505-1 ©2020 American Physical Society NIKOLAY PROKOF’EV AND BORIS SVISTUNOV PHYSICAL REVIEW B 101, 020505(R) (2020) follows we set it to unity) and m is the particle mass. This the relative nature of k and μk. We observe that multicom- relativity of the period is quite unique for nonrelativistic ponent (counterflow) superfluids belong to such a class. Here, crystals. the quantity of interest, ab(r, t ), is the coarse-grained field What distinguishes superfluids from purely statistical mod- of the phase difference between the components “a” and els with broken U (1) symmetry is that broken U (1) sym- “b” (the description stays exactly the same for an arbitrary metry automatically entails breaking of the time-translation number of components, so we restrict ourselves to the two- symmetry, and links superfluidity to yet another fundamental component case for simplicity). We further limit ourselves phenomenon of time crystallization [5,6]. Indeed, the phase with counterflow superfluid states (see, e.g., Ref. [1]) where evolves in time in accordance with the universal Beliaev- the superfluid order exists exclusively in the field ab(r, t ) Josephson-Anderson relation (in the reference frame of the but not in the individual phases of the components. The long- normal component) wave equilibrium statistics of the two-component counterflow superfluid is isomorphic to that of a single-component super- ˙ =−μ. (4) fluid, rendering the system particularly simple and relevant Here, μ ≡ μk is the chemical potential that depends on the for our purposes. To exclude irrelevant long-wave degrees of wave vector of the superflow. Equation (4) readily follows freedom, we also assume that the normal component is pinned from the generalized Gibbs distribution for a superfluid [1]. by either disorder, walls, or an external periodic potential. The Its remarkable simplicity and universality is rooted in the fact Beliaev-Josephson-Anderson relation for ab (its derivation that the phase is canonically conjugated to the total amount from the Gibbs distribution is directly analogous to that in the of matter, single-component case), 2 d ˙ = μ − μ , N = |ψ| d r (5) ab b a (7) [the U (1) symmetry in question is the Noether’s symmetry has the form of the Josephson relation for the standard ac responsible for the conservation of N]. With relation (4), Josephson effect between two single-component superfluids expression (1) can be upgraded to the formula (made of the same type of matter but having different chemi- · − μ cal potentials). Similarly, the protocol of detecting the rotation ψ , = ψ , ik r i kt , (r t ) (0 0)e (6) of the phase ab can be based on simply creating a “Josephson showing that, in the long-wave limit, the superfluid order link” between components “a” and “b.” In this regard, note parameter has the form of a running plane wave. Hence, the that any protocol of revealing the time crystallization effect in the field ab has to deal with interactions explicitly violating superfluid state with a superflow is a space-time crystal, or a × time crystal in the absence of the superflow. the U (1) U (1) symmetry of the original system, which implies a process converting components “a” and “b” into It is important to keep in mind that the value of μk is relative. This is formally reminiscent of (and even partially each other. The conceptual difference between this “internal” connected to) the relative nature of the wave vector k:By Josephson effect and its conventional counterpart is that now the frequency of the phase rotation—and thus the period of changing the reference frame we change k and μk. Further- more, the chemical potential is defined only up to a global oscillations of the ac Josephson current—is independent of constant prescribed by the convention about the ground-state the choice for counting energy in a single equilibrium sample. energy per particle. In the nonrelativistic physics, the rest In the presence of disorder or external periodic potential, energy of a free particle is typically set to zero. The merely the system has a natural reference frame. The absence of conventional character of this choice results in a certain Galilean invariance in this case does not yet mean that the constraint on the protocols of measuring μ and the time- period of the space crystal (1) is not relative. One can, in prin- crystallization effect in superfluids, but does not exclude the ciple, design an experiment when this period is observed from effect itself. a moving frame, in which case Eqs. (2) and (3) still apply. However, in a counterflow superfluid with two components In their paper on the no-go theorem for equilibrium time γ crystals [7], Watanabe and Oshikawa argued that it is the having equal parameters , Galilean transformation (2)leaves above-discussed relativity of the chemical potential that rec- the phase field intact, onciles their theorem with Eq.