Turbulent relaxation after a quench in the Heisenberg model
TTurbulent relaxation after a quench in the Heisenberg model
Joaquin F. Rodriguez-Nieva
Department of Physics, Stanford University, Stanford, CA 94305, USA
We predict the emergence of turbulent scaling in the quench dynamics of the two-dimensionalHeisenberg model for a wide range of initial conditions and model parameters. In the isotropicHeisenberg model, we find that the spin-spin correlation function exhibits universal scaling consistentwith a turbulent energy cascade. When the spin rotational symmetry is broken with an easy-planeanisotropic exchange, we find a dual cascade of energy and quasiparticles. The scaling is shown tobe robust to quantum fluctuations, which tend to inhibit the turbulent relaxation when the spinnumber S is small. The universal character of the cascade, insensitive to microscopic details or theinitial condition, suggests that turbulence in spin systems can be broadly realized in cold atom andsolid-state experiments. Systems far from thermodynamic equilibrium can ex-hibit universal dynamics en route to thermalization.Examples of such phenomena arise when systems arequenched close to a critical point ( i.e. , ageing[1]) or deepin a broken symmetry phase ( i.e. , coarsening[2]). Turbu-lence is a different instance of scaling out of equilibriumthat can emerge even in the absence of a critical pointor long range order[3, 4]. In its simplest form, an exter-nal drive at short wavevectors gives rise to a steady-stateflux of conserved charges that span many lengthscalesup to some dissipative UV scale, see Fig.1(a). Withinthis broad lengthscale range—the inertial range—, thesefluxes govern the scaling of experimentally relevant cor-relations. As a result, turbulent states are specifiedby fluxes of conserved charges, in contrast to thermalstates which are specified by thermodynamic potentials.Importantly, the same scaling can also emerge in anintermediate-time but long-lived prethermal regime afterquenching an isolated system, reflecting that the turbu-lent scaling is intrinsic to the system rather than a featureof the drive.Turbulence in quantum systems has been broadlydiscussed in the context of Bose-Einstein condensates(BECs), both in theory[5–11] and experiments[12–19].The typical scenario is to drive a BEC across a dynamicinstability which generates a complex network of vor-tices, the topological defects of the broken U(1) phase[20–22]. Because vortices are energetically stable and long-lived, they play a central role in BEC turbulence[23, 24]and give rise to rich physics, from Kolmogorov scalingresembling hydrodynamic turbulence[5] to Kelvin wavecascades[25, 26] to self-similar relaxation[27–33] to con-nections with holography[34].Here we inquire about the nature and feasibility of re-alizing turbulent relaxation in closed spin systems, witha special focus on the Heisenberg model in dimension d = 2. The d = 2 case is special as the absence of aphase transition and long range order precludes scalingdue to ageing or coarsening. We consider generic initialconditions far from the ferromagnetic ground state, ana-lytically derive the scaling exponents in the limit of largespin number S , and numerically show the robustness of the scaling for finite S . Our results reveal turbulent dy-namics that is qualitatively distinct from BECs in severalimportant ways. Crucially, the key ingredient in BECturbulence—vortices—is absent. In addition, our resultsalso differ from turbulence in the spin sector of spinorBECs[35, 36]: spinor BECs host conservation laws thathave no analogue in spin models, i.e. , particle numberand momentum. As a result, spin turbulence in spinorBECs is coupled to orbital turbulence and, typically, or-bital turbulence dominates the dynamics[35] except forspecific (low energy) initial conditions[36]. Finally, quan-tum fluctuations must be taken into consideration giventhe typically small local Hilbert space in physical spinsystems, unlike BECs where quantum fluctuations havenegligible effects on turbulent dynamics. Phenomenology & regimes —We study the relax-ation of an excited state with a characteristic wavevector q and local magnetization m z = S sin θ , see Fig.1(c), FIG. 1. (a) Turbulent states are characterized by fluxes ofconserved charges ( e.g. , the energy flux Π E induced by pump-ing energy at a rate P E ) ranging from the wavevector q of thedrive to a dissipative lengthscale a . In this range, Π E gov-erns the scaling of the energy distribution E k . (b) Relaxationregimes of a spin spiral characterized by a wavevector q andamplitude θ [panel (c)]. The isotropic wave turbulence (IWT)and transverse wave turbulence (TWT) regimes describe tur-bulence of weakly-coupled waves and exhibit universal scal-ing. The SC regime indicates strongly-coupled waves withnon-universal behavior. The QF regime indicates that quan-tum fluctuations inhibit the turbulent relaxation. a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p evolved under the Heisenberg model with local exchangecoupling J , spin number S , and lattice constant a . Theinterplay between quantum fluctuations, q , and θ givesrise to several relaxation regimes schematically shownin Fig.1(b). Turbulent scaling emerges when the non-dissipative reactive couplings responsible for distributingenergy over local degrees of freedom are faster than quan-tum fluctuation rates (which give rise to noisy dynam-ics). This condition motivates defining a parameter Q quantifying the ratio between typical spin precession fre-quencies, 1 /τ pr = Ja |(cid:104) S (cid:105)×∇ (cid:104) S (cid:105)| ∼ JS (sin θqa ) , andtypical quantum fluctuation rates, 1 /τ q = J (cid:104) S ⊥ · S ⊥ (cid:105) ∼ JS [37]: Q = τ q τ pr = ( qa sin θ ) /S , (1)with (cid:104) . (cid:105) denoting statistical average and S ⊥ denotingtransverse magnetization. The parameter Q resemblesthe Reynolds number in hydrodynamics with the caveatthat Eq.(1) contains microscopic rates rather than trans-port coefficients, but Q is more suitable as a proxy ofquantum noise at short times.When Q < ∼ Q ∗ , with Q ∗ estimated below, we findthat quantum fluctuations dominate the initial relax-ation and no turbulent scaling emerges [QF in Fig.1(b)].When Q > ∼ Q ∗ and qa < ∼ .
7, we find universal scal-ing in the spin-spin correlation function consistent withwave turbulence, i.e. , weakly coupled incoherent waves.The global magnetization of the initial state determineswhether correlations are isotropic and all components ofmagnetization exhibit scaling or, instead, whether onlythe transverse spin components exhibit scaling [IWT andTWT regions in Fig.1(b), respectively]. When qa > ∼ .
7, we find turbulence consistent with strongly coupledwaves and non-universal scaling. Here we mainly focuson the short time dynamics of the universal IWT/TWTregimes, whereas the long time dynamics and large en-ergy densities will be considered in future work.
Microscopic model and conserved fluxes —Weconsider the two-dimensional Heisenberg model on asquare lattice with short range interactions:ˆ H = − (cid:88) (cid:104) ij (cid:105) J ˆ S i · ˆ S j + ∆ ˆ S zi ˆ S zj . (2)Each lattice site has a spin S degrees of freedom and weassume periodic boundary conditions. We mainly focuson the isotropic point (∆ = 0) and, after this case hasbeen discussed, extension of the results to the anisotropiccase will be straightforward.Starting from a spin spiral state, (cid:104) ˆ S ± i (cid:105) = S sin θe ± i q · r i , (cid:104) ˆ S zi (cid:105) = S cos θ, (3)we study the evolution of magnetization fluctuationsthrough the equal time spin-spin correlation function C α k ( t ) = (cid:104) ˆ S α − k ( t ) ˆ S α k ( t ) (cid:105) for α = x, y, z . The parameters q and θ control the energy and total magnetization of theinitial state. Triggering turbulence from an ordered stateresembles the typical scenario in BEC turbulence[38]: inboth cases, a dynamic instability gives rise to exponen-tial growth of classical fluctuations on timescales muchshorter than the thermalization time and, subsequently,turbulent scaling emerges.Because the energy operator has a strong overlap withˆ S α − k ˆ S α k for all values of k , the time evolution of C α k ( t )is constrained by the flow of energy in k -space. Inaddition, the system may exhibit emergent conservedquantities—quantities which are conserved statisticallyrather than microscopically—that may also constrain theevolution of C α k ( t ). This gives rise to multiple scalingexponents that characterize different regions of phasespace. Notably, the amplitude of transverse magneti-zation ˆ N = (cid:80) i ( ˆ S xi ) + ( ˆ S yi ) may become statisticallyconserved in several scenarios. One example is whenthe system is close to the ferromagnetic ground state.In this case, the probability of having two spin flips onthe same site is negligible and the picture of a weakly-interacting magnon gas with conserved particle numberemerges. Turbulence in this regime was already discussedin Ref.[36]. Another example for statistically conservedˆ N occur in the presence of strong anisotropy ∆. In thiscase, ˆ N appears to be conserved due to suppression oflongitudinal ( z ) spin fluctuations. This scenario will bediscussed below.With these considerations in mind, the central quan-tity characterizing spin turbulence is the energy flux in k space. To define this quantity explicitly, we first ex-press ˆ H in momentum space, ˆ H = (cid:80) k J k ˆ S − k · ˆ S k , withˆ S k = N (cid:80) i ˆ S i e − i k · r i and J k = J (cid:80) a (1 − e i k · a ) (herewe already assumed the isotropic case ∆ = 0 for simplic-ity and a are lattice vectors). The flux of energy in amomentum shell of radius p can then be computed asΠ E ( p, t ) = (cid:88) | k |
∼ /ξ . The exponents (7)characterize turbulence down the ferromagnetic groundstate[36], but their emergence for highly excited states isconditional on the validity of the incoherent fluctuationapproximation used above. Remarkably, we will see that ν E is legitimate even for highly excited states. However,strong coherent fluctuations at long wavelengths suppressthe inverse cascade with exponent ν N in the isotropic(but not XXZ) model, as revealed in the numerics below. Numerical Simulations —To capture the short timescales in which the turbulent cascade develops, we use
FIG. 2. Equal time spin-spin correlation function (cid:104) ˆ S α − k ( t ) ˆ S α k ( t ) (cid:105) after a quench computed via the TruncatedWigner Approximation for t/τ pr = 25 [see definition of τ pr inEq.(1)]. Shown are simulation in the (a) IWT and (b) TWTregimes. Indicated with dotted lines is the energy turbulencepower law ∼ /k ν E [Eq.(7)] and the shaded areas indicatethe intertial range 1 /ξ < ∼ | k | < ∼ /a . The energy flux Π E (seeinsets) exhibit a plateau in this range. Shown with arrows inthe x-axis is the wavevector q of the initial state. Parametersused: (a) q x a = 0 . q y = 0, and θ = π/
2; (b) q x a = 0 . q y = 0, and θ = π/
6. In both panels we used a linear length L = 400, and S = 10. the Truncated Wigner Approximation[40–42]. In thisapproximation, the semiclassical equations of motion (5)are supplemented with quantum fluctuations drawn froma Wigner function. Defining (cid:104) ˆ S ⊥ i (cid:105) as the transversemagnetization with respect to the magnetization axis inEq.(3), we sample trajectories using Gaussian fluctua-tions of ˆ S ⊥ i such that (cid:104) ˆ S ⊥ i (cid:105) = 0 and (cid:104) ˆ S ⊥ i · ˆ S ⊥ i (cid:105) = S .Once Gaussian fluctuations are included, connected cor-relations which were not included in the semiclassicalanalysis above become finite [ i.e. , (cid:104) ˆ S i ˆ S j (cid:105) (cid:54) = (cid:104) ˆ S i (cid:105)(cid:104) ˆ S j (cid:105) ].Figure 2 shows the spin-spin correlation function af-ter the spin spiral order has been destroyed, which oc-curs on a timescale t/τ pr ≈
5. In panel (a), we consideran initial state with zero net magnetization ( θ = π/ q x a = 0 .
15. In this case, although the initial stateis anisotropic, the dynamic instability restores the ro-tational symmetry and all components of magnetiza-tion exhibit the same scaling behavior. In our simu-
FIG. 3. (a) Real space snapshop of (cid:104) ˆ S zi ( t ) (cid:105) at t/τ pr =10 shown for a single semiclassical realization with energy E/N = J/ /q x of the initial state. (b) Value of Q [Eq.(1)] com-puted via TWA. The range 3 ≤ S ≤ lations, we observe the development of a single powerlaw at wavevectors | k | > ∼ | q | , see shaded area. The ob-served power law is consistent with an energy cascade( ν E ≈ . E exhibits a plateau in the inertial range(see inset). For an initial state with finite magnetiza-tion [Fig.2(b)], there is remaining anisotropy between thetransverse and longitudinal fluctuations after the spiralorder has been destroyed. In this case, only the tranversemagnetization exhibit scaling with the same characteris-tics as the ones described in panel (a).We do not observe the scaling 1 / | k | ν N in any of oursimulations for the isotropic Heisenberg model. Indeed,we observe that large coherent magnetization fluctua-tions persist for wavevectors | k | < ∼ | q | at the onset ofturbulence, as shown in Fig.3(a), therefore impeding theinverse quasiparticle cascade towards the infrared.We emphasize two important points. First, the tur-bulent scaling is insensitive to the details of the initialcondition and the same qualitative behavior occurs, forinstance, with an incoherent initial state. In this case,turbulence appears in a much shorter timescale t/τ pr ≈ Quantum fluctuations —At short timescales and forsmall S , quantum fluctuations may give rise to noisy dy-namics without turbulent scaling. We numerically com-pute Q = τ q /τ pr in Eq.(1) to determine the minimum S below which the initial build-up of classical fluctuations issubleading to quantum fluctuations (in the nearest neigh-bour Heisenberg model). To quantify τ q , we first computethe connected component (cid:104) ˆ S i ˆ S j (cid:105) c = (cid:104) ˆ S i ˆ S j (cid:105) − (cid:104) ˆ S i (cid:105)(cid:104) ˆ S j (cid:105) atsmall t = τ pr , then evaluate σ αi = (cid:80) j (cid:15) αβγ (cid:104) ˆ S βi ˆ S γj (cid:105) c , andfinally average σ = avg i,α [ σ αi ] over all sites and direc-tions. Using the definition Q = S (sin θqa ) σ , we obtain FIG. 4. Spin-spin correlation function shown for the XXZmodel. A dual cascade, indicated with two shades of gray,is observed. The scaling at large k is consistent with an en-ergy cascade ( ν E = 2), whereas the scaling at smaller k isconsistent with a particle cascade ( ν N = 4 / k -space, see insets. Parameters used: incoherent initial con-ditions with q x a = 1, q y = 0, θ = π/ t/τ pr = 15, ∆ = − J . Fig.3(b) which has the same S dependence as Eq.(1) upto a numerical prefactor. Inspection of the scaling ex-ponents for different values of S suggests that S = 3( Q ∗ = 8) marks the transition from noisy to turbulentdynamics. Indeed, already for S >
S > ∼ S = 1 / The Heisenberg XXZ model —The analysis of tur-bulence in the anisotropic Heisenberg model proceeds inthe same way as in the isotropic case. The main differ-ence in the derivation of the exponents (7) is that the in-teraction V becomes wavevector independent, β = 0[39].In this case, the wave turbulence exponents are ν E = 2and ν N = 4 /
3. The simulations exhibit a qualita-tively distinct behavior with respect to the isotropic case,shown in Fig.4. We observe that the power-law exponentassociated to the energy cascade ( ν E =2) appears onlyin a small sector of the inertial range. Indeed, a sec-ond cascade with an exponent consistent with a quasi-particle cascade ( ν N = 4 /
3) is found to dominate theinertial range. We confirm the dual nature of the cas-cade by plotting the energy and quasiparticle fluxes in k -space (see inset). Unlike the isotropic case shown inFig.2, the emergence of a conserved ˆ N can be justifiedby the smallness of the longitudinal fluctuations (cid:104) ˆ S z − k ˆ S z k (cid:105) (dashed-dotted lines) within the inertial range: because (cid:104) ˆ S z − k ˆ S z k (cid:105) (cid:28) (cid:104) ˆ S x,y − k ˆ S x,y k (cid:105) , turbulence is effectively occur-ring in the transverse magnetization sector. Connection to experiments —Our predictions canbe readily tested in recent quench experiments in coldatoms[43–45]. In this case, a spin spiral can be gener-ated by applying a pulsed gradient of magnetic field tomake individual spins precess at different speeds, the ex-change anisotropy can be tuned through Feshbach res-onances, and experiments can probe sufficiently longtimescales t ∼ J to capture the turbulent relaxationthat follows the dynamic instability. It is also feasibleto drive ferromagnetic insulators, such as YIG[46], atthe ferromagnetic resonance and measure the spectrumof quasiparticles at large energies, e.g. , through noisemagnetometry[47, 48]. In this case, however, it is nec-essary to study in more detail the role of dipolar interac-tions. Concluding remarks —The turbulent scaling in theisotropic and XXZ Heisenberg models is insensitive to mi-croscopic details and emerges for a broad range of initialconditions, either coherent or incoherent, far from theferromagnetic ground state. This suggests that signa-tures of spin turbulence may be predominant and read-ily accessible in various experimental platforms. Openproblems that remain to be addressed are studying theemergence of spatial/temporal scaling after the quenchand understanding the interplay between thermal andquantum fluctuations at long timescales[49–51]. In addi-tion, the possibility to engineer the spin-spin interactionin cold atom platforms motivates the study of turbulencein different settings, such as dipolar gases with long rangeinteractions.
ACKNOWLEDGEMENTS
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Joaquin F. Rodriguez-Nieva
Department of Physics, Stanford University, Stanford, CA 94305, USA
The Supplement provides additional numerical results of turbulence in different regimes. In Sec.S1, we discussturbulence obtained from evolving an incoherent initial state, different from the coherent spin spiral discussed in themain text. In Sec.S2, we discuss the non-universal energy cascade in the large energy density regime. In Sec.S3, westudy the sensitivity of the scaling exponents as a function of the spin number S . S1. INCOHERENT INITIAL CONDITIONS
The spin spiral state is a typical state in cold atom experiments. In solid state systems, however, it is also commonto incoherently drive the system with an oscillating magnetic field. As we show next, the scaling exponents areinsensitive to the details of the initial state.We consider an incoherent initial condition parametrized as (cid:104) ˆ S + i (cid:105) = (cid:80) k f k e i k · r i . To do a one-to-one comparisonwith the numerics of the main text, we consider that the distribution | f k | is Gaussian and peaked at the wavevector q , and the value of (cid:80) k | f k | defines the total magnetization of the initial incoherent state. The numerical resultsare shown in Fig.S1(a) for a state with finite magnetization in the z direction. Compared to Fig.2(b) of the maintext, the qualitative agreement between both results is notable: the power law cascade is formed at the wavevector q (indicated with an arrow), and the same power law scaling (associated to an energy cascade) is observed. The energyflux also exhibits a plateau in the inertial range. S2. BEYOND WAVE TURBULENCE
The wave turbulence exponents are not valid if the correlation length is small ( i.e. , at large energy density). In thiscase, the picture of wave turbulence developing within large islands of magnetization breaks down. Figure S1(b) showsa simulation in which the initial state has a large energy density, q x a = 1 and θ = π/
2. Clearly, in this case there isno power law scaling of the distribution function. Indeed, because the initial wavevector is imprinted at intermediatetimescales, the long wavelength expansion to describe wave turbulence breaks down. Nonetheless, it is still possibleto define an energy cascade at large momenta (see inset).
FIG. S1. (a) Equal time spin-spin correlation function obtained from quenching an initial incoherent state using the TruncatedWigner Approximation, see discussion in Supplementary text. The initial condition has wavevector q x a = 0 . q y = 0, andfinite magnetization in the z axis ( θ = π/ q x a = 1, q y = 0, θ = π/
2. In both panels, the shaded areaindicates the inertial range. The insets show the energy flux in k -space. FIG. S2. Spin-spin correlation function for different values of the spin number S . The initial state is the same as that in Fig.2(a)of the main text. Indicated with decreasing shades of blue is the results for decreasing S for S = 10 , , , ,
2. Deviations in thescaling exponents larger than 10% with respect with the wave turbulence exponents are obtained for spin numbers
S < S3. LIMIT OF SMALL SPIN NUMBER S At short times, quantum fluctuations give rise to noisy dynamics and compete with the turbulent relaxation ofenergy when the spin number S is small (particularly in the nearest neighbour Heisenberg model). Here we considerthe effects of quantum fluctuations on the scaling exponents as a function of S .Figure S2 reproduce the conditions studied in Fig.2(a) of the main text for different values of S . For S >
S < S = 1 //