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: DiptimanSen, ShreyoshiMondal, Christian TrefzgerAnatoliPolkovnikov, MukundVengalattore, Alsessandro Silva, Arnab Das, Bikas K Chakraborti, Sei Suzuki, Takumi Hikichi.

2. Overview • Introduction: Why dynamics? • 2. Nearly adiabatic dynamics: defect production • 3. Correlation functions and entanglement generation • 4. Non-integrable systems: a specific case study • 5. Experiments • 7. 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: 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 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. Defect Correlation in Kitaev model The defect correlation as a function of spatial distance r is given in terms of Majorana fermion operators Only non-trivial correlator 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 multipartite for reasonably fast quenches For the 2D Kitaev model, one can show that the entire entanglement is always multipartite

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. Non-integrable systems: a specific case study

33. Dynamics of the Bose-Hubbard model Bloch 2001 Transition described by the Bose-Hubbard model:

34. Mott-Superfluid transition: preliminary analysis Mott state with 1 boson per site Stable ground state for 0 < m < U Mott state is destabilized when the excitation energy touches 0. Adding a particle to the Mott state Removing a particle from the Mott state Destabilization of the Mott state via addition of particles/hole: onset of superfluidity

35. Beyond this simple picture Higher order energy calculation by Freericks and Monien: Inclusion of up to O(t3/U3) virtual processes. Mean-field theory (Fisher 89, Seshadri93) Phase diagram for n=1 and d=3 Quantum Monte Carlo studies for 2D & 3D systems: Trivedi and Krauth, B. Sansone-Capponegro O(t2/U2) theories MFT Superfluid Predicts a quantum phase transition with z=2 (except at the tip of the Mott lobe where z=1). Mott No method for studying dynamics beyond mean-field theory

36. A more accurate phase diagram: building fluctuations over MFT Distinguish between two types of hopping processes using a projection operator technique (A) (B) Define a projection operator Divide the hopping to classes A and B Design a transformation which eliminate hopping processes of class A perturbatively in J/U.

37. Equilibrium phase diagram Use the effective Hamiltonian to compute the ground state energy and hence the phase diagram Reproduction of the phase diagram with remarkable accuracy in d=3: much better than standard mean-field or strong coupling expansion in d=2 and 3. Allows for straightforward generalization for treatment of dynamics

38. We were not that lazy……

39. Non-equilibrium dynamics Consider a linear ramp of J(t)=Ji +(Jf-Ji) t/t. For dynamics, one needs to solve the Sch. Eq. Make a time dependent transformation to address the dynamics by projecting on the instantaneous low-energy sector. The method provides an accurate description of the ramp if J(t)/U <<1 and hence can treat slow and fast ramps at equal footing. Takes care of particle/hole production due to finite ramp rate

40. Absence of critical scaling: may be understood as the inability of the system to access the critical (k=0) modes. Fast quench from the Mott to the SF phase; study of superfluid dynamics. Single frequency pattern near the critical Point; more complicated deeper in the SF phase. Strong quantum fluctuations near the QCP; justification of going beyond mft.

41. Experiments

42. Experimental Systems: Spin one ultracold bosons Spin one bosons are loaded in an optical trap with mz=0 and subjected to a Zeeman magnetic field. Second order QPT at q= 2c2n between the FM (q << 2c2n) and the scalar (q>> 2c2n) phases. One can study quench dynamics by quenching the magnetic field L. Sadler et al. Nature (2006). In the experiment, quench was done from the scalar to the FM phase. Defects are absence of ferromagnetic domains. For a rapid quench, one starts with very small domain density which then develop in time. Perform a slow power law quench of the magnetic field across the QCP from the FM to the scalar phase and measure magnetization at the end of the quench.

43. Experiments with ultracold bosons on a lattice: finite rate dynamics 2D BEC confined in a trap and in the presence of an optical lattice. Single site imaging done by light-assisted collision which can reliably detect even/odd occupation of a site. In the present experiment they detect sites with n=1. Ramp from the SF side near the QCP to deep inside the Mott phase in a linear ramp with different ramp rates. The no. of sites with odd n displays plateau like behavior and approaches the adiabatic limit when the ramp time is increased asymptotically. No signature of scaling behavior. Interesting spatial patterns. W. Bakr et al. arXiv:1006.0754

44. 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-equlibrium universality? b) Relation between complex real-life systems and simple models c) Relationship between integrability and quantum dynamics.