Aspects of non-equilibrium dynamics in closed quantum systems

1. Aspects of non-equilibrium dynamics in closed quantum systems K. Sengupta Indian Association for the Cultivation of Science, Kolkata Collaborators: ShreyoshiMondal, AnatoliPolkovnikov, DiptimanSen, Alessandro Silva, Sei Suzuki, and MukundVengalattore.

2. Overview • Introduction: Why dynamics? • 2. Nearly adiabatic dynamics: defect production • 3. Correlation functions and entanglement generation • 4. Ergodicity and thermalization • 5. Statistics of work distribution • 6. Conclusion: Where to from here?

3. Introduction: Why dynamics Progress with experiments: ultracold atoms can be used to study dynamics of closed interacting quantum systems. Finding systematic ways of understanding dynamics of model systems and understanding its relation with dynamics of more complex models: concepts of universality out of equilibrium? Understanding similarities and differences of different ways of taking systems out of equilibrium: reservoir versus closed dynamics and protocol dependence. 4. Key questions to be answered: What is universal in the dynamics of a system following a quantum quench ? What are the characteristics of the asymptotic, steady state reached after a quench ? When is it thermal ?

4. Nearly adiabatic dynamics: Scaling laws for defect production

5. Landau-Zenner dynamics in two-level systems Consider a generic time-dependent Hamiltonian for a two level system The instantaneous energy levels have an avoided level crossing at t=0, where the diagonal terms vanish. The dynamics of the system can be exactly solved. The probability of the system to make a transition to the excited state of the final Hamiltonian starting from The ground state of the initial Hamiltonian can be exactly computed

6. Defect production and quench dynamics Kibble and Zurek: Quenching a system across a thermal phase transition: Defect production in early universe. Ideas can be carried over to T=0 quantum phase transition. The variation of a system parameter which takes the system across a quantum critical point at QCP The simplest model to demonstrate such defect production is For “adiabatic” evolution, the system would stay in the ground states of the phases on both sides of the critical point. Describes many well-known 1D and 2D models.

7. Specific Example: Ising model in transverse field Ising Model in a transverse field: Jordan Wigner transformation : Hamiltonian in term of fermion in momentum space: [J=1]

8. Defining We can write, where The eigen values are, The energy gap vanishes at g=1 and k=0: Quantum Critical point.

9. Let us now vary g as εk 2∆ g The probability of defect formation is the probability of the system to remain at the “wrong” state at the end of the quench. Defect formation occurs mostly between a finite interval near the quantum critical point.

10. While quenching, the gap vanishes at g=1 and k=0. • Even for a very slow quench, the process becomes non adiabatic at g = 1. • Thus while quenching after crossing g = 1 there is a finite probability of the system to remain in the excited state . • The portion of the system which remains in the excited state is known as defect. • Final state with defects, x

11. Scaling law for defect density in a linear quench In the Fermion representation, computation of defect probability amounts to solving a Landau-Zenner dynamics for each k. pk : probability of the system to be in the excited state Total defect : From Landau Zener problem if the Hamiltonian of the system is Then

12. For the Ising model Here pk is maximum when sin(k) = 0 Linearising about k = 0 and making a transformation of variable Scaling of defect density For a critical point with arbitrary dynamical and correlation length exponents, one can generalize - Ref: A Polkovnikov, Phys. Rev. B 72, 161201(R) (2005)

13. Generic critical points: A phase space argument The system enters the impulse region when rate of change of the gap is the same order as the square of the gap. For slow dynamics, the impulse region is a small region near the critical point where scaling works The system thus spends a time T in the impulse region which depends on the quench time In this region, the energy gap scales as Thus the scaling law for the defect density turns out to be

14. Critical surface: Kitaev Model in d=2 Jordan-Wigner transformation a and b represents Majorana Fermions living at the midpoints of the vertical bonds of the lattice. Dn is independent of a and b and hence commutes with HF: Special property of the Kitaev model Ground state corresponds to Dn=1 on all links.

15. Solution in momentum space Off-diagonal element Diagonal element z z

16. Gapless phase when J3 lies between(J1+J2) and |J1-J2|. The bands touch each other at special points in the Brillouin zone whose location depend on values of Ji s. J1 J2 J3 Quenching J3 linearly takes the system through a critical line in parameter space and hence through the line in momentum space. In general a quench of d dimensional system can take the system through a d-m dimensional gapless surface in momentum space. For Kitaev model: d=2, m=1 For quench through critical point: m=d Question: How would the defect density scale with quench rate?

17. Defect density for the Kitaev model Solve the Landau-Zenner problem corresponding to HF by taking For slow quench, contribution to nd comes from momenta near the line . For the general case where quench of d dimensional system can take the system through a d-m dimensional gapless surface with z= =1 It can be shown that if the surface has arbitrary dynamical and correlation length exponents , then the defect density scales as Generalization of Polkovnikov’s result for critical surfaces Phys. Rev. Lett. 100, 077204 (2008)

18. Non-linear power-law quench across quantum critical points For general power law quenches, the Schrodinger equation time evolution describing the time evolution can not be solved analytically. Models with z= D=1: Ising, XY, D=2 Extended Kitaev l can be a function of k Two Possibilities Quench term vanishes at the QCP. Quench term does not vanish at the QCP. Scaling for the defect density is same as in linear quench but with a non-universal effective rate Novel universal exponent for scaling of defect density as a function of quench rate

19. Quench term vanishes at the QCP Schrodinger equation Scale Defect probability must be a generic function Contribution to nd comes when Dk vanishes as |k|. For a generic critical point with exponents z and

20. Comparison with numerics of model systems (1D Kitaev) Plot of ln(n) vsln(t) for the 1D Kitaev model

21. Quench term does not vanish at the QCP Consider a slow quench of g as a power law in time such that the QCP is reached at t=t0. At the QCP, the instantaneous energy gap must vanish. If the quench is sufficiently slow, then the contribution to the defect production comes from the neighborhood of t=t0 and |k|=k0. Effective linear quench with

22. Ising model in a transverse field at d=1 There are two critical points at g=1 and -1 For both the critical points Expect n to scale as a0.5 15 10 20

23. Anisotropic critical point in the Kitaev model Linear slow dynamics which takes the system from the gapless phase to the border of the gapless phase ( take J1=J2=1 and vary J3) J1 J2 The energy gap scales with momentum in an anisotropic manner. J3 Both the analytical solution of the quench problem and numerical solution of the Kitaev model finds different scaling exponent For the defect density and the residual energy Other models: See Divakaran, Singh and Dutta EPL (2009). Different from the expected scaling n ~ 1/t

24. Phase space argument and generalization Anisotropic critical point in d dimensions for m momentum components i=1..m for the rest d-m momentum components i=m+1..d with z’ > z Need to generalize the phase space argument for defect production Scaling of the energy gap still remains the same The phase space for defect production also remains the same However now different momentum components scales differently with the gap Kitaev model scaling is obtained for z’=d=2 and Reproduces the expected scaling for isotropic critical points for z=z’

25. Deviation from and extensions of generic results: A brief survey Topological sector of integrable model may lead to different scaling laws. Example of Kitaev chain studied in Sen and Visesheswara (EPL 2010). In these models, the effective low-energy theory may lead to emergent non-linear dynamics. If disorder does not destroy QCP via Harris criterion, one can study dynamics across disordered QCP. This might lead to different scaling laws for defect density and residual energy originating from disorder averaging. Study of disordered Kitaev model by Hikichi, Suzuki and KS ( PRB 2010). Presence of external bath leads to noise and dissipation: defect production and loss due to noise and dissipation. This leads to a temperature ( that of external bath assumed to be in equilibrium) dependent contribution to defect production ( Patane et al PRL 2008, PRB 2009). Analogous results for scaling dynamics for fast quenches: Here one starts from the QCP and suddenly changes a system parameter to by a small amount. In this limit one has the scaling behavior for residual energy, entropy, and defect density (de Grandi and Polkovnikov , Springer Lecture Notes 2010)

26. Correlation functions and entanglement generation

27. Correlation functions in the Kitaevmodel The non-trivial correlation as a function of spatial distance r is given in terms of Majorana fermion operators Only non-trivial correlation function of the model For the Kitaev model Plot of the defect correlation function sans the delta function peak for J1=J and Jt =5 as a function of J2=J. Note the change in the anisotropy direction as a function of J2.

28. Entanglement generation in transverse field anisotropic XY model -1 1 Quench the magnetic field h from large negative to large positive values. h PM FM PM One can compute all correlation functions for this dynamics in this model. (Cherng and Levitov). No non-trivial correlation between the odd neighbors. Single-site entanglement: the linear entropy or the Single site concurrence What’s the bipartite entanglement generated due to the quench between spins at i and i+n?

29. Measures of bipartite entanglement in spin ½ systems Concurrence (Hill and Wootters) Negativity (Peres) Consider a wave function for two spins and its spin-flipped counterpart Consider a mixed state of two spin ½ particles and write the density matrix for the state. Take partial transpose with respect to one of the spins and check for negative eigenvalues. C is 1 for singlet and 0 for separable states Could be a measure of entanglement Use this idea to get a measure for mixed state of two spins : need to use density matrices Note: For separable density matrices, negativity is zero by construction

30. Steps: 1. Compute the two-body density matrix 2. Compute concurrence and negativity as measures of two-site entanglement from this density matrix 3. Properties of bipartite entanglement Finite only between even neighbors Requires a critical quench rate above which it is zero. c. Ratio of entanglement between even neighbors can be tuned by tuning the quench rate. The entanglement generated is entirely non-bipartite for reasonably fast quenches For the 2D Kitaev model, one can show that the entire entanglement is always non-bipartite

31. Evolution of entanglement after a quench: anisotropic XY model T=0 T=0 t=10 a=0.78 t=1 Prepare the system in thermal mixed state and change the transverse field to zero from it’s initial value denoted by a. The long-time evolution of the system shows ( for T=0) a clear separation into two regimes distinguished by finite/zero value of log negativity denoted by EN. The study at finite time shows non-monotonic variation of EN at short time scales while displaying monotonic behavior at longer time scales. For a starting finite temperature T, the variation of EN could be either monotonic or non-monotonic depending on starting transverse field. Typically non-monotonic behavior is seen for starting transverse field in the critical region. Sen-De, Sen, Lewenstein (2006)

32. Ergodicity and Thermalization

33. Ergodicity in closed quantum systems Classical systems: Well-known definition of equivalence of phase-space and time averages Basics Consider a classical system of N particles with a point X in 2dN dimensional phase space. Choose an initial condition X=X0 so that H(X0)=E where H is the system Hamiltonian Also define a) X(t) to be the phase space trajectory with the initial condition X=X0 b) rmc(E) to be the microcannonical distribution on the hyper-surface S of the phase space with constant energy E. Then the statement of ergodicity can be translated to the mathematical relation stating the equality of the phase space and the time averages Question: How do you define ergodicity for closed quantum systems?

34. Ergodicity in closed quantum systems Consider a quantum system with energy and corresponding eigenstates The microcannonical ensemble for such a system can be defined as for a set of states a between energy shell E and E + dE . The interval dE is chosen so that there are large number of states within dE; but dE is small compared to any macroscopic energy scale. If N is the total number of states within this interval, then one defines Now consider a generic state of the system constructed out of these states within the energy shell The long-time average of the density matrix for such a state is The diagonal and the microcannonical density matrices are not the same except for very special situations: closed non-integrable quantum systems are not ergodic in this strict sense. [ von Neumann 1929] Diagonal density matrix

35. Ergodicity in closed quantum systems Question: How does one define ergodicity in closed non-integrable quantum systems? The idea is to shift the focus from states to physical observables. (von Neumann 1929, Mazur 1968, Peres 1989) Consider a set of operator Mb (t) whose expectation values correspond to macroscopic physically measurable observables, the criteria for ergodicity is given by Equality between the diagonal and microcannonical ensembles for long times at the level of expectation values of operators corresponding to physical observables More precisely, at long time the mean square difference between the RHS and LHS of the equation becomes vanishingly small for large enough system where Poincare recurrence time is very large. Alternatively one can define These two definitions are identical if a steady state is reached

36. Quantum Integrable systems: Breakdown of ergodicity Integrability in classical systems implies breakdown of ergodicity since the presence of thermodynamic number of constants of motion do not allow arbitrary trajectories in phase: foliation of phase space into tori. To demonstrate this for the quantum system, let us go back to Ising model. Define quasiparticle operators which diagonalize the Hamiltonian. Consider a local observable (say Mx(t)) after a time t in terms of these operators The long time limit of the magnetization is given by the diagonal terms and hence described by occupation numbers of the states (constants of motion) for any initial condition Conjecture: The asymptotic states of integrable systems are described by genralized Gibbs ensemble (GGE).

37. Breaking integrability in quantum systems Fermi-Pasta-Ulam (FPU) problem for nonlinear oscillators For small non-linearity, the dynamics of these coupled oscillators remain non-ergodic. There are quasiperiodic resonances rather than thermalization Thermalization is absent in classical nearly-integrable dynamical systems; one requires a threshold to break away from integrability. What happens for quantum systems? The precise criteria for setting in of thermalization for a system is largely unclear. Experiments seem to indicate absence of thermalization in nearly-integrable systems. There is a hypothesis ( numerically shown for certain model systems) that when thermalization sets in, it does so eigenstate-by eigenstate. This is called the Eigenstate Thermalization Hypothesis (ETH) Expectation value of any observable on an eigenstate is a smooth function of the energy Ea and is essentially a constant within a given Microcannonical shell. (Deutsch 1991, Srednici 1994, Rigol 2008)

38. Absence of thermalization in 1D Bose gas Blue detuned laser used to create tightly bound 1D tubes of Rb atoms. The depth of the lattice potential is kept large to ensure negligible tunneling between the tubes. The bosons are imparted a small kinetic energy so as to place them initially at a superposition state with momentum p0 and –p0. The evolution of the momentum distribution of the bosons are studied by typical time of flight experiments for several evolution times. The distribution of the bosons are never gaussian within experimental time scale; clear absence of thermalization in nearly-integrable 1D Bose gas. Kinoshita et al. Nature 2006

39. Work statistics for out of equilibrium quantum systems

40. Statistics of work done: General Expression Consider a closed quantum system driven out of equilibrium by a tine dependent Hamiltonian H(t). The time evolution of such a system is described by the evolution operator We assume that the system dynamics starts at t=0 when H=H0 and ends at t=t when H=H1 The probability of work distribution for such a system is given by Where is the probability that the system starts from a state |a> at t=0 and ends at a state |b> at t=t. For a quantum system which starts from a thermal state at t=0 described by an equilibrium density matrix r0

41. Thus the work distribution can be written as Note that computation of P(w) even at zero temperature requires the knowledge of the excited states of the final Hamiltonian and the matrix elements of S. A quantity which is simpler to compute is the Fourier transform of the work distribution L is often called the Lochsmidt echo; its computation avoids use of |b>

42. Another interpretation: Laplace transform of work distribution Consider the work done WN for a system of size N when one of its Hamiltonian parameter is quenched from g0 to g Define the moment generating function of WN One can think of G(s) as the partition function of a classical (d+1)-dimensional systems in the slab geometry where the area of the slab is N and thickness s. The transfer matrix of this Classical system is exp[-H(g)] and the boundary condition of the problem is set by One can thus define a free energy in this geometry which has contributions from the bulk and the surface One defines an excess free energy and it can be shown that G(s) can be written as The inverse Laplace transform of G(s) can now be computed using standard techniques in the limit of large N (thermodynamic limit) Confirms the large deviation principle in case of sudden quench

43. Conclusion 1. We are only beginning to understand the nature of quantum dynamics in some model systems: tip of the iceberg. Many issues remain to settled: i) specific calculations a) dynamics of non-integrable systems b) correlation function and entanglement generation c) open systems: role of noise and dissipation. d) applicability of scaling results in complicated realistic systems 3. Issues to be settled: ii) concepts and ideas a) Role of the protocol used: concept of non-equilibrium universality? b) Relation between complex real-life systems and simple models c) Relationship between integrability and quantum dynamics. d) Setting in of thermalization in quantum systems.