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.
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?
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 ?
Nearly adiabatic dynamics: Scaling
laws for defect production
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
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
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
Describes many well-known
1D and 2D models.
Specific Example: Ising model in transverse field
Ising Model in a transverse field:
Jordan Wigner transformation :
si  ci  ci
 1  2c
 1  2c
j i
si  ci  ci
j i
s i  1  2c j c j
Hamiltonian in term of fermion in momentum space: [J=1]
H 
 2 g  cos ka c
c k  sin  ka
c k c  k
 c k c k
We can write,
 ck
ψ k   
 c k
H 
ψkH kψk
 2 g  cos  k 
H k  
 2sin  k 
The eigen values are,
ε k  2
2sin  k 
 2 g  cos  k  
g  cos(k)
2  sin(k) 2 
The energy gap vanishes at g=1 and k=0: Quantum Critical point.
Let us now vary g as
1 k 
2 k 
1 k 
2 k 
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.
 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
 Final state with defects,
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 1 k 
Total defect :
n 
 p k    dk
From Landau Zener problem
if the Hamiltonian of the system is
Δ 
 ε 1 t 
H k  
 Δ * ε 2 t 
 2π
 exp 
 d  ε  t   ε  t  
 dt
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)
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
Critical surface: Kitaev Model in d=2
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.
Solution in momentum space
J 3  
J3  
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.
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
Quenching J3 linearly
takes the system
through a critical line in
parameter space and
hence through the line
in momentum space.
For quench through critical point: m=d
Question: How would the defect density scale with quench rate?
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)
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
Quench term vanishes at the QCP
Schrodinger equation
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
Comparison with numerics of model systems (1D Kitaev)
Plot of ln(n) vs ln(t) for the 1D Kitaev model
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
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
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)
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).
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
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)
Correlation functions and entanglement generation
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
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.
Entanglement generation in transverse field anisotropic XY model
Quench the magnetic field h from
large negative to large positive values.
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?
Measures of bipartite entanglement in spin ½ systems
Concurrence (Hill and Wootters)
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
Negativity (Peres)
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.
Note: For separable density matrices,
negativity is zero by construction
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
Evolution of entanglement after a quench: anisotropic XY model
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)
Ergodicity and Thermalization
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=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?
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]
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
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
Conjecture: The asymptotic states of
integrable systems are described by
genralized Gibbs ensemble (GGE).
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)
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
Work statistics for out of equilibrium quantum systems
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
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 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
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.

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