QUANTUM QUENCH, CRITICAL POINTS AND HOLOGRAPHY. Sumit R. Das

Transcription

1 QUANTUM QUENCH, CRITICAL POINTS AND HOLOGRAPHY Sumit R. Das

2 QUANTUM QUENCH Consider a quantum theory with a Hamiltonian with a time dependent coupling e.g. Suppose we start, at early times, in the ground state. What is the time evolution?

3 We will call this QUANTUM QUENCH though this terminology is sometimes used only for very sudden changes. This question is, of course, relevant to many areas of physics. Recently, however, this question has attracted a lot of attention because this kind of process can be now experimentally studied in a clean fashion in cold atoms. Among many other things, the question is interesting for two related reasons (1) Thermalization does the system reach some kind of steady state and does the final state resemble a thermal state in some sense? (2) Critical Dynamics if the quench crosses an equilibrium critical point, adiabaticity will be inevitably lost is there any universal feature of the subsequent time evolution?

4 Kibble-Zurek Scaling Consider a coupling which starts far from the critical point and approaches the critical point in a linear fashion The rate is assumed to be slow compared to the asymptotic mass gap. Then one expects that the time evolution is adiabatic till a time when the instantaneous gap function violates the adiabaticity condition, i.e. Assuming that the time evolution after this is diabatic, and that the correlation length is the only scale in this region one can derive several scaling relations using naïve scaling arguments.

5 For example the one point function of an operator (which is not one of the conserved quantities) obeys Where There are many situations this kind of scaling in fact holds, and many other situations where it does not. However, unlike equilibrium or near-equilibrium situations, there is no theoretical framework which justifies this kind of scaling argument, or tells us when they don t apply.

6 Sudden Quench in 1+1 dim In 1+1 dimensions, Calabrese and Cardy used methods of boundary conformal field theory to obtain a set of general results for sudden quench from gapped phase to a critical point. For example the one point function again for an operator which is not conserved under time evolution relaxes as Where The ratios of relaxation times are therefore universal

7 Holographic Quench There are very few theoretical tools to analyze quantum quench in strongly coupled systems. It is then natural to try to apply holographic techniques to this problem. In this talk I will summarize some results in this direction with emphasis on quench across critical points.

8 The Setup In AdS/CFT the couplings of a field theory correspond to boundary values of bulk fields. Thus in the situation where supergravity is applicable, the problem becomes that of solving bulk equations of motion in the presence of time dependent boundary conditions. Typically one would start with an equilibrium ground state which specifies an initial condition the bulk solution then provide the boundary correlators at late times using the standard AdS/CFT dictionary. Here we are interested in the non-linear response for a driving which can be very fast.

9 If we do this in the CFT starting from the ground state i.e. AdS initial conditions this is the problem of gravitational collapse. Under suitable conditions, a black hole or black brane is formed the late time geometry approaches a stationary black hole. This is manifested as thermalization in the field theory. (Janik and Peschanski, Lin and Shuryak; Chesler and Yaffe, Bhattacharyya and Minwalla; Balasubramanian et. al.; Abajo-Arrista, Aparicio and Lopez; Albash and Johnson; Ebrahim and Headrick; Allais and Tonni; Keranen, Keski- Vakkuri and Thorlacius; Galante and Schvellinger; Caceres and Kundu; Garfinkle and Pando-Zayas; Aparicio and Lopez; Buchel Lehner and Myers; Bhaseen, Gauntlett & Weisman)

10 We are interested in quench across critical points. Many holographic critical points can be studied in the probe approximation where a subset of bulk fields can be treated separately and their back-reaction to the metric can be ignored. Examples include holographic superconductors, and transitons on probe branes. In the probe approximation a black hole formation is not visible. However, it turns out that signals of thermalization still persist at least in a large class of examples which involve probe branes in bulk AdS space-times. The dual theories are defect CFT s.

11 In these examples, a time dependent coupling of the dual CFT is described by branes rotating in e.g. in a non-uniform fashion. The induced metric on the brane worldvolume can develop apparent horizons. Fluctuations of the brane feel this induced metric and their correlators display thermal behavior e.g. brownian motion of endpoint of strings, or conductivity on higher dimensional branes. (S.R.D., T. Nishioka and T. Takayanagi) The background metric is always AdS. This kind of thermal behavior is similar to acoustic black holes in hydrodynamics. Also similar to Rindler radiation for accelerated branes. The mechanism appears to be quite general. (Izuka, Hashimoto, Oda; Ali-Akbari and Ebrahim,.)

12 There are several interesting models of holographic critical phenomena on probe branes e.g. the BKT type quantum critical point on D5 branes in the presence of a magnetic field. While the zero temperature equilibrium transition is analytically tractable, we have so far not been able to understand quench dynamics in a clean fashion. Let us therefore gain some insight by looking at some phenomenological models.

13 A Simple Holographic Critical Point To explore quench across a critical point, we will start with a simple model due to Iqbal, Liu, Mezei and Si, involving a self interacting neutral scalar field and a Maxwell field in addition to gravity in 3+1 dimensions In the limit of the scalar field may be treated as a probe whose back-reaction to the metric can be ignored. (This does not couple to the gauge field anyway). This is the limit in which we are going to work. The mass of the scalar field lies in the window

14 Consider, therefore the dynamics of this scalar field in the presence of a charged black brane. This has a Hawking temperature and charge given by The scalar field is living on this fixed background. For any given temperature, this theory has a critical phase transition for some mass, below which the field condenses the dual operator then has a vev.

15 The bulk equations of motion are in units Where Near the boundary at the solution to this equation is of the form In standard quantization need static solutions with and constant which is regular at the horizon. There is always the trivial solution. Its stability is determined by the spectrum of the linearized Schrodinger operator

16 As the becomes more negative this potential develops a bound state this signals instability. Exactly at there is a zero mode which is regular at the horizon and has the right behavior at the boundary. For it turns out that there is another regular solution of the non-linear equation which has nontrivial dependence. The behavior of this new solution near the boundary then determines the expectation value of the dual operator

17 At any non-zero temperature the transition has standard meanfield exponents While exactly at However at the transition is of BKT type

18 Quench across the critical point We now study quench across the critical point at and turning on a nontrivial which asymptotes to constant values in the far past and the far future and crosses the critical point at, at some time.

19 Adiabaticity Analysis We will examine the dynamics for small To maintain regularity at the horizon it is useful to use Eddington Finkelstein coordinates In which the equations of motion become Note that on the boundary and are the same. To incorporate the boundary condition it is useful to write Where

20 Starting from some early time with initial conditions determined by the adiabatic solution the time evolution is adiabatic so long as does not come close to zero. The adiabatic expansion may be written as Where is the solution in the presence of a static The leading correction satisfies the equation The order n solution acts as a source for the equation of the order (n+1) contribution.

21 Adiabaticity breaks down when An analysis of the equation shows that this happens when For the profile which goes linearly near the critical point this gives The condensate which is proportional to is then given by This is in fact Kibble-Zurek scaling with a dynamical critical exponent

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23 Dynamics in Critical Region The inevitable failure of adiabaticity means there is no power series expansion in derivatives. We will now show that in the critical region, a new smallexpansion becomes valid an expansion in Furthermore, the dynamics is dominated by a single mode of the field. To see this, let us rescale the fields and the time as follows as suggested by the adiabatic analysis

24 The equation of motion can be written as an expansion in powers of Where This is an inhomogeneous equation. The source term arises from the boundary value of the field. Let us now expand the field in eigenfunctions of Where Note that since the potential goes to zero exponentially at infinity the spectrum is continuous.

25 The equations for the modes become, to lowest orders This immediately suggests an expansion If this is valid the zero mode now satisfies where

26 However, the spectrum has a continuum starting from zero and we are trying to separate out a zero mode! This should not work and in fact for a general potential it does not. Assuming that the separation is valid, the nonzero modes satisfy In fact for the potential which arises for a generic the coefficients like And the non-zero modes would diverge, rendering the proposed expansion invalid.

27 However, exactly when And this separation indeed makes sense. Thus for the critical value of the zero mode dominates the dynamics for small A similar thing happens for scattering states of a square well potential

28 In this case the zero mode dynamics is exactly a Landau- Ginsburg dynamics of a real order parameter with first order time derivate, driven by a source at the critical point, Rescaling One gets an equation which is exactly like the equation for the zero mode. Reverting to original variables we therefore conclude that in our holographic model the order parameter obeys

29 Some features of the dynamics The analysis was at non-zero temperature which could be small. The dynamics of the order parameter has, and is dissipative. This happens because of the presence of the black hole horizon and arises in our treatment because of the requirement of regularity at the horizon in Eddington-Finkelstein coordinates. Here the equations have a single time derivative. The dissipation is of course the disappearance beyond the horizon.

30 Zero temperature limit At zero temperature the situation is quite different. The transition is of BKT type. Now the background is an extremal black hole which has a near the horizon. The effective potential now goes as a power law rather than an exponential. Even away from criticality, the small frequency expansion is not an adiabatic expansion but typically an expansion in fractional powers of. This also leads to a susceptibility which is finite at the critical point. We do not expect any decoupling of modes. We have not been able to come up with a practical approach to look at the nonlinear dynamics, other than brute force numerical solution which also becomes subtle.

31 Another Quantum Critical Point (P. Basu, D. Das, S.R.D. & T. Nishioka to appear) An interesting example of a zero temperature transition appears when one of the boundary coordinates is compact in the presence of a Maxwell field and a complex scalar. In the absence of the scalar the phase diagram is

32 On the zero temperature axis, the AdS soliton solution is, The BH is the extremal AdS-RN black hole Ryu, Nishioka and Takayanagi showed that in the probe approximation,, the scalar field condenses in the soliton phase, when the chemical potential is large enough

33 The proposed phase diagram is The scalar condensate is obtained by solving the coupled set of equations for the complex scalar and the Maxwell field. The order-disorder transition is critical with standard mean field exponents even at zero temperature.

34 We have shown that the same phenomenon at least at T= 0 is present for a self coupled complex scalar in the limit when the back-reaction of the scalar to both the metric and the gauge field can be ignored. We can now apply our previous analysis to the nonlinear equation satisfied by the scalar

35 The physics is now quite different not only do we have a zero temperature, but there is no horizon. The tip of the soliton is in fact a flat disc Clearly there can be no dissipation any disturbance from the boundary reflects off the tip.

36 Real part of order parameter for Real part of order parameter for As expected the oscillations of the order parameter do not die out

37 Despite this key difference in the physics, the dynamical exponents are dominated by the chemical potential term in the equation of motion that gives rise to Adiabaticity breakdown once again gives rise to The bulk dynamics is again dominated by a zero mode in the critical region : in fact this is cleaner than the earlier case since the spectrum of the corresponding Schrodinger problem is in fact discrete. The effective equation satisfied by the order parameter in this region is Which has the scaling solution

38 Even though the equation for the order parameter involves a first order time derivative, this is an oscillatory system. This is of course as it should be there is no horizon through which energy can disappear.

39 In fact, this is also the behavior for the Landau-Ginsburg dynamics for a complex order parameter Rescaling it is easy to see that the second derivative term is subdominant in the small limit. This is easy to verify numerically for the LG model. We are currently working on a direct numerical verification of the scaling behavior for the holographic AdS soliton model.

40 Plot of versus in complex LG dynamics

41 Epilogue So far we have concentrated on understanding any scaling behavior in the critical region for (initially) slow quench. Universal features are expected to appear for infinitely fast quenches as well one should be able to investigate this along similar lines. It would be interesting to explore this for the BKT transitions.

42 Finally, we have not addressed the issue of thermalization at late times i.e. after the system has passed through the equilibrium critical point. For the phenomenological models, it is likely that the probe approximation is inadequate. However it is possible that if we can make progress in understanding quench across critical points in probe brane models, it may be possible to study this in the probe approximation along the lines of our treatment of thermalization on probe branes.