Aspects of non-equilibrium dynamics in closed quantum systems
K. Sengupta Indian Association for the Cultivation of Science,
Kolkata Collaborators: Shreyoshi Mondal, Anatoli Polkovnikov,
Diptiman Sen, Alessandro Silva, Sei Suzuki, and Mukund
Vengalattore.
Slide 2
Overview 1.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?
Slide 3
Introduction: Why dynamics 1.Progress with experiments:
ultracold atoms can be used to study dynamics of closed interacting
quantum systems. 2.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? 3.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 ?
Slide 4
Nearly adiabatic dynamics: Scaling laws for defect
production
Slide 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
Slide 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 Describes
many well-known 1D and 2D models. For adiabatic evolution, the
system would stay in the ground states of the phases on both sides
of the critical point.
Slide 7
Jordan Wigner transformation : Hamiltonian in term of fermion
in momentum space: [J=1] Specific Example: Ising model in
transverse field Ising Model in a transverse field:
Slide 8
Defining We can write, where The eigen values are, The energy
gap vanishes at g=1 and k=0: Quantum Critical point.
Slide 9
Let us now vary g as 22 kk 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.
Slide 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
Slide 11
In the Fermion representation, computation of defect
probability amounts to solving a Landau-Zenner dynamics for each k.
p k : 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 quench
Slide 12
For the Ising model Here p k 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 -
Slide 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
Slide 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. D n is independent
of a and b and hence commutes with H F : Special property of the
Kitaev model Ground state corresponds to D n =1 on all links.
Slide 15
Solution in momentum space zz Off-diagonal element Diagonal
element
Slide 16
J1J1 J2J2 Quenching J 3 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 Gapless phase when J 3 lies between(J 1 +J 2 )
and |J 1 -J 2 |. The bands touch each other at special points in
the Brillouin zone whose location depend on values of J i s.
Question: How would the defect density scale with quench rate?
J3J3
Slide 17
Defect density for the Kitaev model Solve the Landau-Zenner
problem corresponding to H F by taking For slow quench,
contribution to n d 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 Polkovnikovs result for critical surfacesPhys.
Rev. Lett. 100, 077204 (2008)
Slide 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.
Novel universal exponent for scaling of defect density as a
function of quench rate 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
Slide 19
Quench term vanishes at the QCP Schrodinger equation Scale
Defect probability must be a generic function Contribution to n d
comes when D k vanishes as |k|. For a generic critical point with
exponents z and
Slide 20
Comparison with numerics of model systems (1D Kitaev) Plot of
ln(n) vs ln(t) for the 1D Kitaev model
Slide 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=t 0.
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=t 0 and |k|=k 0.
Effective linear quench with
Slide 22
Ising model in a transverse field at d=1 10 15 20 There are two
critical points at g=1 and -1 For both the critical points Expect n
to scale as a 0.5
Slide 23
Anisotropic critical point in the Kitaev model J1J1 J2J2 J3J3
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) 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/ t Other models: See Divakaran, Singh and Dutta EPL
(2009).
Slide 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 Reproduces the expected
scaling for isotropic critical points for z=z Kitaev model scaling
is obtained for z=d=2 and
Slide 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)
Slide 26
Correlation functions and entanglement generation
Slide 27
Correlation functions in the Kitaev model The non-trivial
correlation as a function of spatial distance r is given in terms
of Majorana fermion operators For the Kitaev model Plot of the
defect correlation function sans the delta function peak for J 1 =J
and J t =5 as a function of J 2 =J. Note the change in the
anisotropy direction as a function of J 2. Only non-trivial
correlation function of the model
Slide 28
Entanglement generation in transverse field anisotropic XY
model Quench the magnetic field h from large negative to large
positive values. PM FM h One can compute all correlation functions
for this dynamics in this model. (Cherng and Levitov). 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
Slide 29
Measures of bipartite entanglement in spin systems
ConcurrenceNegativity(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 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 Consider a mixed state of two
spin particles and write the density matrix for the state. Take
partial transpose with respect to one of the spins and check for
negative eigenvalues. Note: For separable density matrices,
negativity is zero by construction
Slide 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
a.Finite only between even neighbors b.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
Slide 31
Evolution of entanglement after a quench: anisotropic XY model
Prepare the system in thermal mixed state and change the transverse
field to zero from its 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 E N. The study at finite time shows non-monotonic
variation of E N at short time scales while displaying monotonic
behavior at longer time scales. For a starting finite temperature
T, the variation of E N 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. T=0 t=10 T=0 a=0.78 t=1 Sen-De, Sen, Lewenstein (2006)
Slide 32
Ergodicity and Thermalization
Slide 33
Ergodicity in closed quantum systems Classical 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=X 0
so that H(X 0 )=E where H is the system Hamiltonian Also define a)
X(t) to be the phase space trajectory with the initial condition
X=X 0 b) r mc (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?
Slide 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 Diagonal density matrix 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]
Slide 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 M b (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
Slide 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 M x
(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).
Slide 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 E a and is
essentially a constant within a given Microcannonical shell.
(Deutsch 1991, Srednici 1994, Rigol 2008)
Slide 38
Absence of thermalization in 1D Bose gas Kinoshita et al.
Nature 2006 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 p 0 and p 0. 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.
Slide 39
Work statistics for out of equilibrium quantum systems
Slide 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=H 0 and ends at t= t when H=H 1 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 r 0
Slide 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 >
Slide 42
Another interpretation: Laplace transform of work distribution
Consider the work done W N for a system of size N when one of its
Hamiltonian parameter is quenched from g 0 to g Define the moment
generating function of W N 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
Slide 43
Conclusion 1. We are only beginning to understand the nature of
quantum dynamics in some model systems: tip of the iceberg. 2.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.