Kondo Physics in Graphene: A STM study

Aspects of non-equilibrium dynamics in closed quantum systemsK. Sengupta

Indian Association for the Cultivation of Science, KolkataCollaborators: Shreyoshi Mondal, Anatoli Polkovnikov, Diptiman Sen, Alessandro Silva, Sei Suzuki, and Mukund Vengalattore.1Overview 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?

2Introduction: Why dynamicsProgress 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 ?3Nearly adiabatic dynamics: Scaling laws for defect production4Landau-Zenner dynamics in two-level systems

Consider a generic time-dependent Hamiltonian for a two level system

The instantaneous energy levelshave an avoided level crossing att=0, where the diagonal terms vanish.

The dynamics of the system can be exactly solved.

The probability of the system tomake a transition to the excited stateof the final Hamiltonian starting fromThe ground state of the initial Hamiltonian can be exactly computed

5Defect production and quench dynamicsKibble 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 systemacross a quantum critical pointat

QCPThe simplest model to demonstratesuch defect production is

Describes many well-known1D and 2D models.

For adiabatic evolution, the system would stay in theground states of the phases on both sides of the criticalpoint.

6Jordan Wigner transformation :

Hamiltonian in term of fermion in momentum space: [J=1]

Specific Example: Ising model in transverse fieldIsing Model in a transverse field:

7Defining

We can write,

where

The eigen values are,

The energy gap vanishes at g=1 and k=0: Quantum Critical point.8Let us now vary g as

2kg

The probability of defect formation is the probability of the systemto remain at the wrong state at the end of the quench.

Defect formation occurs mostly between a finite interval near thequantum critical point. 9While 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,

x10In 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

Scaling law for defect density in a linear quench11For the Ising model

Here pk is maximum when sin(k) = 0

Linearising about k = 0 and making a transformation of variable

For a critical point with arbitrary dynamical and correlation length exponents, one can generalize

Ref: A Polkovnikov, Phys. Rev. B 72, 161201(R) (2005)Scaling of defect density -12Generic critical points: A phase space argumentThe system enters the impulse region whenrate of change of the gap is the same orderas the square of the gap. For slow dynamics, the impulse region is a small region near the critical point wherescaling worksThe 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

13Critical 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 aand b and hence commutes with HF: Special property of the Kitaev modelGround statecorresponds toDn=1 on all links.

14

Solution in momentum spacezz

Off-diagonal elementDiagonalelement15J1J2

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=dGapless 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. Question: How would the defect density scale with quench rate?J316Defect density for the Kitaev modelSolve 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 dimensionalgapless 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 Polkovnikovs result for critical surfaces

Phys. Rev. Lett. 100, 077204 (2008)17Non-linear power-law quench across quantum critical pointsFor general power law quenches, the Schrodinger equation time evolution describing the time evolution can not be solved analytically. Models withz= D=1: Ising, XY,D=2 Extended Kitaevl can be a function of k

Two PossibilitiesQuench term vanishesat the QCP.Novel universal exponentfor scaling of defect densityas a function of quench rateQuench 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

18Comparison with numerics of model systems (1D Kitaev)

Plot of ln(n) vs ln(t) for the 1D Kitaev model

20Quench term does not vanish at the QCPConsider a slow quench ofg as a power law in timesuch that the QCP is reachedat 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

21

Ising model in a transverse field at d=1

101520There are two critical points at g=1 and -1

For both the critical pointsExpect n to scale as a0.522Anisotropic critical point in the Kitaev modelJ1J2J3Linear slow dynamics which takes the system from the gapless phase to the border of the gapless phase ( take J1=J2=1 and vary J3)The energy gap scales with momentum in an anisotropic manner.

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

Different from the expected scaling n ~ 1/tOther models: See Divakaran, Singh and Dutta EPL (2009).

23Phase space argument and generalizationAnisotropic 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 > zNeed to generalize the phase space argument for defect productionScaling of the energy gap stillremains the same

The phase space for defect production also remains the same

However now different momentum components scales differently with the gap

Reproduces the expected scaling for isotropic critical points for z=z

Kitaev model scaling is obtained for z=d=2and

24Deviation from and extensions of generic results: A brief surveyTopological 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 lossdue 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 QCPand 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 andPolkovnikov , Springer Lecture Notes 2010)

25Correlation functions and entanglement generation26Correlation functions in the Kitaev model

The non-trivial correlation as a function of spatial distance r is given in terms of Majoranafermion operators

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.Only non-trivial correlation function of the model27Entanglement generation in transverse field anisotropic XY model

Quench the magnetic field h from large negative to large positive values.PMPMFMhOne can compute all correlation functions for this dynamics in this model. (Cherng and Levitov).1-1

No non-trivial correlation between the odd neighbors.Whats the bipartite entanglement generated due to the quench between spins at i and i+n?Single-site entanglement: the linear entropy or the Single site concurrence

28Measures of bipartite entanglement in spin systemsConcurrenceNegativity(Hill and Wootters)(Peres)Consider a wave function for two spins and its spin-flipped counterpart

C is 1 for singlet and 0 for separable statesCould be a measure of entanglementUse this idea to get a measure for mixed state of two spins : need to use density matrices

Consider a mixed state of two spin particles and write the density matrix for the state.Take partial transpose with respectto one of the spins and check for negative eigenvalues.

Note: For separable density matrices,negativity is zero by construction

29Steps:1. Compute the two-body density matrix

2. Compute concurrence and negativity as measures of two-siteentanglement from this density matrix

3. Properties of bipartite entanglementFinite 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 quenchesFor the 2D Kitaev model, one can show that the entire entanglement is always non-bipartite30

Evolution of entanglement after a quench: anisotropic XY model

Prepare the system in thermal mixed state and change the transverse field to zero fromits initial value denoted by a.

The long-time evolution of the system shows ( for T=0) a clear separation into two regimesdistinguished 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 behavioris seen for starting transverse field in the critical region.

T=0t=10T=0a=0.78t=1Sen-De, Sen, Lewenstein (2006)31Ergodicity and Thermalization32Ergodicity in closed quantum systemsClassical systems: Well-known definition of equivalence of phase-space and time averages 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

Basics

Question: How do you define ergodicity for closed quantum systems? 33Ergodicity in closed quantum systemsConsider a quantum system with energy and corresponding eigenstates

The microcannonical ensemble for such a system can be defined as for a setof states a between energy shell E and E + dE . The interval dE is chosen so that thereare 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 Diagonal density matrixThe 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]34Ergodicity in closed quantum systemsQuestion: 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 ensemblesfor long times at the level of expectation values of operatorscorresponding to physical observables More precisely, at long time the mean square difference between the RHS and LHS of the equation becomes vanishinglysmall 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 reached35Quantum Integrable systems: Breakdown of ergodicityIntegrability 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 magnetizationis 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).

36Breaking integrability in quantum systemsFermi-Pasta-Ulam (FPU) problem for nonlinear oscillatorsFor small non-linearity, the dynamicsof these coupled oscillators remainnon-ergodic. There are quasiperiodic resonances rather than thermalizationThermalization 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 asmooth function of the energy Ea and is essentially a constant within a givenMicrocannonical shell. (Deutsch 1991, Srednici 1994, Rigol 2008)

37

Absence of thermalization in 1D Bose gasKinoshita et al. Nature 2006Blue detuned laser used to create tightly bound1D 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 ofthe bosons are studied by typical time of flightexperiments 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.

38Work statistics for out of equilibrium quantum systemsStatistics of work done: General ExpressionConsider 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

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>Another interpretation: Laplace transform of work distributionConsider 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 surfaceOne 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 quenchConclusion1. 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. 43