Holographic dissipative space-time supersolids

Driving a system out of equilibrium enriches the paradigm of spontaneous symmetry breaking, which could then take place not only in space but also in time. The interplay between temporal and spatial symmetries, as well as symmetries from other internal degrees of freedom, can give rise to novel nonequilibrium phases of matter. In this study, we investigate a driven-dissipative superfluid model using holographic methods and reveal the existence of a space-time supersolid (STS) phase which concomitantly breaks the time translation, spatial translation, and the internal U(1) symmetry. The holographic methods naturally include finite temperature effects, which enables us to explore the complex phase diagram of this model and observe a cascade of out-of-equilibrium phase transitions from the STS phase to a synchronized superfluid phase, and finally to a normal fluid phase, by increasing the temperature.

Introduction-The physics of quantum many-body systems out of equilibrium is much richer than its equilibrium counterpart, but less is known in general.As a prototypical example of a non-equilibrium quantum phase of matter, time crystals (TC) [1], which are characterized by spontaneous time translation symmetry breaking, have attracted considerable interests in various branches of modern physics [2][3][4][5][6][7][8][9][10], including trapped ions [11], nitrogen-vacancy center systems [12], quantum computation [13,14], and ultracold atoms [15][16][17][18].In realistic experimental setups, any quantum system is inevitably coupled to its surroundings, i.e., to a thermal bath which inevitably induces dissipation.Understanding dissipative quantum time crystals is not only a question of practical experimental importance, but also of fundamental significance due to its relevance to broader concepts such as the stability of non-equilibrium quantum matter against thermal fluctuations and the universality class of the associated non-equilibrium phase transitions.However, in spite of recent efforts made in classical TC systems [19][20][21], the effects of a thermal bath on a quantum time crystal are still poorly understood.
Recently, remarkable progress has been made in exploring non-equilibrium interacting quantum systems by exploiting the AdS/CFT correspondence (for a review, see Ref. [22]).This holographic duality recasts the problem of strongly coupled quantum systems in the language of classical gravitational models in an asymptotically higher-dimensional Anti-de Sitter spacetime [23,24].More importantly, in this framework, the existence of a black hole in the gravitational background is equiva-lent to the presence of a thermal bath for the boundary quantum system, hence it enables us to naturally include finite temperature effects without the need of uncontrolled phenomenological modeling.In the last decade, the holographic duality has provided new opportunities to explore the dissipative dynamics of out-of-equilibrium quantum many-body systems, including non-equilibrium steady states [25][26][27], out-of-equilibrium phase transitions [28][29][30], quantum quenches [31,32], driven systems [33][34][35][36] and quantum turbulence [37][38][39].In addition to that, AdS/CFT has been very successful in describing strongly-correlated phases of matter exhibiting spontaneous symmetry breaking of both internal (e.g., superfluids [40]) and spatial translation (e.g., solids [41], charge density waves [42], and supersolids [43,44]) symmetries, but has never been applied so far to the case of time translations.In this study, we investigate the late-time dynamics of a periodically-driven holographic superfluid, and we uncover the existence of a space-time crystal phase which simultaneously breaks the continuous spatial arXiv:2304.02534v2[hep-th] 28 Nov 2023 translation symmetry and the discrete time translation symmetry (DTTS).This space-time crystal also breaks the U(1) symmetry as a superfluid and it can thus be considered as a space-time supersolid (STS).The emergent space-time orders appear to be intertwined into a rich phase diagram which can be explored via a linear instability analysis based on Floquet theory.The holographic methods allow to further study the effect of thermal fluctuation on this quantum STS.It is observed that the system undergoes two continuous out-of-equilibrium phase transitions, each of which restores the DTTS and U(1) symmetry respectively.In particular, the system goes from the STS phase to the synchronized phase when the temperature increases, in accordance with the physical intuition that crystals melt upon heating (similar to classical TC [20]).
Methods-A holographic superfluid [40] is described by an Abelian-Higgs model in a (3+1)-dimensional AdS black hole spacetime: where Ψ is a complex scalar field and A µ is a U(1) gauge field with For simplicity, we have set the AdS radius to unit.Throughout the paper, we work in the probe limit in which the backreaction of all the matter fields on the metric is assumed to be negligible, which is a justified assumption whenever the temperature is not too low [45].Hence, the equations of motion (EOM) for the scalar and gauge fields read: The (3 + 1)-dimensional background spacetime is a Schwarzschild-AdS black hole: The holographic coordinate z spans from the AdS boundary z = 0 to the location of the horizon z = z H .The blackening factor is given by f

H
. The temperature of the dual field theory is T H = 3 4πz H .In the rest of the manuscript, we set m 2 = −2, and we further choose the axial gauge A z = 0.
Following the AdS/CFT dictionary [24], the dual field theory is a large N CFT with a global U(1) symmetry and a corresponding conserved current J µ .The gauge field A µ is dual to the current operator J µ , and the complex scalar Ψ is dual to a scalar operator O with conformal dimension ∆ = 2, charged under the U(1) global symmetry.Close to the AdS boundary (z → 0), the matter fields have the following time-dependent asymptotic expansion: where Ψ 1 is the source for the operator O, while µ and ρ are the chemical potential and the conserved charge density respectively.With Ψ 1 = 0, the condensate ⟨O⟩ = Ψ 2 ̸ = 0 forms below a certain critical temperature, signaling the onset of U(1) spontaneous symmetry breaking to the superfluid phase from the normal fluid phase.In the following, we consider a periodic chemical potential (see Fig. 1): which breaks translation invariance along the x direction, as in [46].The amplitude Θ(t) is modulated in time, with a period Periodic boundary condition in the x, y directions are chosen using a box L x × L y .More details about the numerical methods can be found in the supplementary material (SM).Without the periodic potential, the critical temperature for the onset of superfluidity is given by T c = 0.0587µ 0 .For later convenience, we define a dimensionless reduced temperature τ = T H /T c .Synchronized phase versus space-time crystal-To characterize the crystalline pattern, we focus on the charge density ρ(t, r), which can be extracted from Eq.( 6).In particular, it is sufficient for us to check the late-time dynamics of ρ(t, r 0 ) at a specific position Lx dxρ(t, r) along the y-direction in the synchronized phase with weak driving (red solid), and in the space-time crystal phase with strong driving (dashed blue).r 0 = (0, 0).Fig. 2 shows that, in the presence of a weak driving, ρ(t, r 0 ) oscillates with a period identical to that of the external driving, thus the system is in a synchronized phase.However, when the driving amplitude exceeds a critical value, the system displays a period-doubling discrete time crystal phase, which spontaneously breaks the Z 2 translational symmetry of the system, ρ(t, r 0 ) = ρ(t + 2T, r 0 ) ̸ = ρ(t + T, r 0 ).This period doubling phenomenon is reminiscent of the Faraday waves observed in Bose-Einstein condensate systems [47], which can be classically described using the Gross-Pitaevskii (GP) equation [48].
Next we study the spatial profile of these two nonequilibrium phases.The translation invariance along the x-direction is broken by the inhomogeneous chemical potential, while the system is translation invariant in the y-direction.To characterize the spontaneous symmetry breaking of translations along the y-direction, we define a normalized average density ρ x (t, y) = 1 Lx dxρ(t, r).As shown in Fig. 3, at a fixed time slice t = t * , the normalized density is homogeneous along the y-direction in the synchronized phase, thus it retains the continuous translation symmetry of the system.On the contrary, in the presence of a strong driving, the homogeneous pattern is no longer stable, and a crystalline phase with a characteristic length l c spontaneously emerges.The continuous translation symmetry in the y-direction is spontaneously broken into a discrete one, ρ x (t * , y) = ρ x (t * , y + l c ).In a summary, a strong periodical driving in the non-equilibrium holographic superfluid leads to a space-time crystal phase, which simultaneously breaks the spatial and temporal translation symmetry, a STS.
Linear instability analysis-The instability of the synchronized phase and the nature of the holographic spacetime crystal can be understood in the framework of lin- ear instability analysis.We start from a synchronized solution where both Ψ s and A s µ periodically oscillates in time with a period T , and are homogeneous along the y-direction.To study the stability of this solution, we introduce the following perturbations: where k y = 2πn Ly with n an integer.By substituting Eq.( 9) and ( 10) into the EOM, and keeping only the linear terms in δΨ and δA µ , one obtains the EOM for the perturbations ⃗ δ ky = [δΨ, δA µ ] T which can be written in vectorial form as with the matrix M ky (t) = M ky (t + T ).The periodicity of M ky (t) enables us to employ the Floquet description for the stroboscopic dynamics in Eq.( 11) and derive a time-independent Floquet matrix H ky satisfying: where T is the time-ordering operator T e T 0 dtM ky (t) = e dtM ky (T −dt) • • • e dtM ky (dt) e dtM ky (0) , and U ky is the evolution operator within one period, ⃗ δ ky (T ) = U ky ⃗ δ ky (0).
The stability of the background solution Ψ s and A s µ depends on the imaginary part of the eigenvalues of the Floquet matrix H ky .For our purpose, it is sufficient to focus on the eigenvalues with the largest and second largest imaginary parts, denoted respectively by ε 1 ky and ε 2 ky .In particular, whether the background solution is stable or unstable depends on whether the imaginary part of ε 1 ky is greater than or less than zero, where its real part ℜ[ε 1  ky ] indicates the frequency of the oscillation accompanying the exponential divergence or decay. .With the observation that the two modes colliding right at the edge of Floquet zone give rise to the degeneracy of their real parts, the above pattern can accordingly be well understood by the similar argument arising in the black hole dynamics [51], Bose-Einstein condensates [52,53], and other systems [54,55].In the presence of weak driving, ℑ[ε 1 ky ] is always negative for arbitrary k y as shown in Fig. 4 (a), indicating that the homogeneous synchronized phase is stable against perturbations.On the contrary, in the strongly driven case, ℑ[ε 1  ky ] becomes positive and reaches its maximum at k y = k c y .These results indicate that the homogeneous synchronized phase is not stable against the spatial fluctuations along the y-direction.In other words, the system will spontaneously develop a crystalline pattern with a characteristic length l c = 2π kc .Furthermore, a non vanishing ℜ[ε 1  ky ] = ω d 2 at k y = k c indicates that the spatial crystal is accompanied by a temporal oscillation with a frequency ω = ω d 2 , which explains the period doubling in the space-time crystal observed in the real-time numerical simulations in Fig. 2.
Non-equilibrium phase transitions and phase diagram-Now we consider the effects of temperature.In the presence of a strong driving, at low temperature, the system is in a STS phase, which is characterized by Bragg peaks in the Fourier spectrum for the density distribution ρ(ω, k y ) = dtdye −ikyy+iωt ρ x (t, y) located at (±k c , ± ω d 2 ).As temperature increases, the height of the peak |ρ( ω 2 , k c )| decreases and finally vanishes at a critical temperature τ 1 , which indicates a phase transition from a STS to a synchronized superfluid (SF) phase.In the intermediate temperature regime, the crystalline order is suppressed by thermal fluctuations, while the SF order survives.This synchronized SF phase is characterized by a nonzero SF order parameter ⟨O(t, r)⟩.We consider its Fourier component O(ω d ) = dte iω d t O(t, r 0 ).As shown in Fig. 5  The phase diagram in terms of the driving amplitude Θ and temperature τ is plotted in Fig. 6, which shows that the holographic STS exists in the regime with strong driving and low temperature.As the temperature increases, the system will experience two continuous non-equilibrium phase transitions, which are characterized by the restoration of the DTTS and U(1) symmetry respectively.The relationship between the DTTS and U(1) symmetry breaking was discussed in Ref. [56].In general, the Z 2 long-range temporal crystalline order is unstable against the presence of stochastic π-phase shift in time domain (for the same reason that 1D spatial Z 2 long-range order is unstable against the propagation of the kink excitations activated by thermal fluctuations).However, for a system that simultaneously breaks the DTTS and U(1) symmetry, a π-phase shift in time domain is accompanied by a phase slip ϕ → ϕ + π in the U(1) symmetry breaking order parameter, which is energetically disfavored in the U(1) symmetry breaking phase.As a consequence, the discrete time crystalline order is protected by the U(1) symmetry breaking, thus its critical temperature cannot be larger than that of the SF condensate, i.e. τ 2 > τ 1 , as observed in Fig. 6.
Conclusions-In summary, by using holographic methods, we studied the late-time out of equilibrium dynamics of a driven-dissipative quantum system, and discovered a space-time supersolid phase which simultaneously breaks the spatial and temporal translation symmetries together with the internal U(1) symmetry.As the temperature increases, we find that the thermal fluctuations first restore the DTTS, and then the U(1) symmetry, leaving an intermediate synchronized SF phase between them.This cascade of non-equilibrium phase transitions from STS to normal fluid could be checked in future experiments, for example using ultra-cold atom systems in an optical lattice.Further developments of our analysis will include the generalizations of our results to time crystal phases with different translation symmetry breaking, for instance, the continuous time crystal [18] and time quasicrystal [57], where the interplay between the spatial U(1) symmetry and continuous time translation symmetry might give rise to novel non-equilibrium phases and phase transitions.
From a physical point of view, we emphasize that the probe limit we are working with corresponds to considering the dissipative time crystal in open quantum systems (see [58,59] for a classification and discussion about the various scenarios), where the number of degrees of freedom (dof) of the thermal bath is parametrically large.This is a common limit for open quantum systems in quantum optics, atomic and molecular physics where one assumes that the backreaction of the system on the bath remains negligible.From the gravitational point of view, this argument can be made explicit in probe brane setups, where the suppression is controlled by N f /N c , with N f , N c respectively the number of colors and flavors (see for example [60]).This hierarchy allows to dissipate heat very efficiently and maintain a steady state for a long timescale without destroying the time-crystalline order, as realized experimentally in [17,18].Furthermore, as shown explicitly for a scalar toy model in the SM, even with backreaction, the heating induced by the driving can be kept parameterically small by controlling the relative number of dof between the bath and the system.This implies that the time-crystalline order survives in the backreaction limit up to a time-scale which can be made arbitrarily long.
and the constrain function is one eventually obtains the matrix valued problem defined in the main text, which can be solved by Floquet analysis.

SM2. Time crystal and synchronized phases in Fourier space
As mentioned in the main text, the space-time supersolid phase can be characterized by the Fourier modes ρ = ρ(±k c , ω d 2 ) for the normalized average density ρ x (t, y).The space-time supersolid phase will have a non-zero contribution from ρ (Fig. S1a), while the synchronized superfluid phase doesn't have any contribution from ρ (Fig. S1b).To distinguish the synchronized superfluid phase from the normal fluid phase, one can check the Fourier modes O(ω d ) of the superfluid order parameter ⟨O(t, r 0 )⟩.Similarly, for synchronized superfluid phase we have O(ω d ) ̸ = 0 (Fig. S1c), but for normal fluid phase we have O(ω d ) = 0.

Figure 1 :
Figure 1: (Color online) Sketch of the periodic and inhomogeneous chemical potential applied to the superfluid.Θ(t) is a periodic function with frequency ω d and wavelength a x = 2π/k 0 .

Figure 2 :
Figure 2: (Color online) (a) The time-dependent chemical potential driving the system out of equilibrium.The late-time dynamics of the local density ρ(t, r 0 ) at r 0 = (0, 0) in (b) the synchronized phase with weak driving, and (c) the space-time crystal phase with strong driving.Parameters are chosen as µ 0 = 5.4,Θ 0 = 4.1 and ω d = 0.16π.

Figure 4 :
Figure 4: (Color online) The k y −dependence of the imaginary (upper panel) and the real parts (lower panel) of the 1st and 2nd eigenvalues of the Floquet matrix H ky in the linear stability analysis for (a) the synchronized phase with Θ = 0.6ω d and (b) the space-time crystal phase Θ = 1.92ω d .The dashed line in (a) indicates the dispersion of the Goldstone modes for the synchronized SF ℜ[ϵ] ∼ v c k y with sound speed v c = 1.21π, as computed from hydrodynamics.

Figure 5 :
Figure 5: (Color online) The dependence of the STS order parameter (top panel) and the synchronized SF order parameter (bottom panel) as a function of the reduced temperature τ .
, |O(ω d )| vanishes at a temperature τ 2 , indicating a phase transition from a synchronized SF to a normal fluid.

Figure 6 :
Figure 6: (Color online) Phase diagram in terms of the driving amplitude Θ and the reduced temperature τ .

Figure S1 :
Figure S1: Fig.S1a is the Fourier spectrum ρ(ω, k y ) in the STS phase with τ = 0.5.The dominant mode locates at (±k c , ± ω d2 ) and signals the breaking of time translations.Fig.S1bis the Fourier spectrum ρ(ω, k y ) of the synchronized SF phase with τ = 0.7.As expected, there is only a zero mode (0, 0) corresponding to the homogeneous and static ρ x (t, y).Fig.S1cis the Fourier spectrum O(ω) of the synchronized SF phase with τ = 0.7.The dominant mode appears at (±ω d ).All of these spectra are calculated with Θ = 1.5ω d .

Figure S2 :
Figure S2: The growth of the dual field theory energy as a function of time in the scalar toy model, setting A = 0.1 and ω D = 5.0.The lines from bottom to top correspond respectively to κ = 0, 0.25, 0.5, 0.75, 1.