quantum systems Pasquale Calabrese University of Pisa Cargese, - - PowerPoint PPT Presentation

quantum systems

Pasquale Calabrese

University of Pisa Based on joint works with:

Cargese, September 2014

John Cardy, Mario Collura, Fabian Essler, Maurizio Fagotti, Marton Kormos, Spyr

• A many-body quantum system is prepared in the ground-state |Ψ
• At t=0, H0 ➔H, i.e. an Hamiltonian parameter is quenched
• Isolated: No contact with an external world
• For t>0, it evolves unitarily: |Ψ(t)⟩=e-iHt |Ψ0⟩
• How can we describe the dynamics?
• What about a “stationary state”?

von Neumann in 1929 posed these questi

It stayed a purely academic question: for condensed matter sys the coupling to the environment is unavoidable

Essentially unitary time evolution few hundreds 87Rb atoms in a 1D trap

0τ 2τ 4τ 9τ

• 2D and 3D systems relax quickly and thermalize:

0τ 2τ 4τ 9τ

• 2D and 3D systems relax quickly and thermalize:

The 1D case is special because the system is almost integra

Non-equilibrium new states of matter

When and why a steady state is thermal??

.".". t (i)"Preparation (ii)"Evolution (iii)"Readout

d c b

nodd neven

1 2 3 8 10 12 14 16 18 20 0.0 0.5 1.0 " "

t"(ms) nodd Position"(hk)

2 >2 >4 4 1 2 3 4

t"(ms)

U

0.2 0.4 0.6 1 2 3 4 5 0.2 0.4 0.6 1 2

U/J"="2.44(2) K/J$=$5·10.3 U/J"="5.16(7) K/J$=$9·10.3 K

4Jt$/"h nodd

a c

• Numerics and experiment agree perfectly
• The stationary state looks thermal

.".". t (i)"Preparation (ii)"Evolution (iii)"Readout

d c b

nodd neven

1 2 3 8 10 12 14 16 18 20 0.0 0.5 1.0 " "

t"(ms) nodd Position"(hk)

2 >2 >4 4 1 2 3 4

t"(ms)

U

0.2 0.4 0.6 1 2 3 4 5 0.2 0.4 0.6 1 2

U/J"="2.44(2) K/J$=$5·10.3 U/J"="5.16(7) K/J$=$9·10.3 K

4Jt$/"h nodd

a c

• Numerics and experiment agree perfectly
• The stationary state looks thermal

Common Belief: - Generic systems “thermalizes”

• Integrable systems are different

But the system is always in a pure state!

Reduced density matrix: ρA(t)=TrB ρ |Ψ(t)⟩ time dependent pure state ρ(t) = |Ψ(t)⟩⟨Ψ(t)| density matrix of The expectation values of all local observables in A are ⟨Ψ(t)|OA(x) |Ψ(t)⟩ = Tr[ρA(t) OA(x)]

B

A

Reduced density matrix: ρA(t)=TrB ρ |Ψ(t)⟩ time dependent pure state ρ(t) = |Ψ(t)⟩⟨Ψ(t)| density matrix of The expectation values of all local observables in A are ⟨Ψ(t)|OA(x) |Ψ(t)⟩ = Tr[ρA(t) OA(x)]

B

A

Stationary state: if for any finite subsystem A of an infinite system, it exi

lim ρA(t) = ρA(∞)

t→∞

Note: in a finite system, the stationary state is the regime |A|<< vt<<

Consider the Gibbs ensemble for the entire system AUB with Reduced density matrix for subsystem A: ρA,T=TrB ρT The system thermalizes if for any finite subsystem A

ρA,T = ρA(∞)

The infinite part B of the system “acts as an heat bath for A”

ρT= e-H/Teff /Z

⟨Ψ0| H |Ψ0⟩ = Tr[ρT H]

Teff is fixed by the energy in the initial state: no fre

Proposal by Rigol et al 2007: The GGE density matrix

ρGGE= e-∑ λm Im /Z

⟨Ψ0| Im |Ψ0⟩ = Tr[ρGGE with λm fixed by

Again no fre

Im are the integrals of motion of H, i.e. [Im ,H]=0

Proposal by Rigol et al 2007: The GGE density matrix

ρGGE= e-∑ λm Im /Z

⟨Ψ0| Im |Ψ0⟩ = Tr[ρGGE with λm fixed by Reduced density matrix for subsystem A: ρA,GGE=TrB ρGGE The system is described by GGE if for any finite subsystem A of a infinite system

ρA,GGE = ρA(∞)

[Barthel-Schollwock ’08] [Cramer, Eisert, et al ’08] + ........ [PC, Essler, Fagotti ’12]

Again no fre

Im are the integrals of motion of H, i.e. [Im ,H]=0

Any quantum system has too many integrals of motion, regardless of integrability, e.g.

Om = |Em⟩⟨Em|

Any quantum system has too many integrals of motion, regardless of integrability, e.g.

Om = |Em⟩⟨Em|

In this case B is not a standard heat bath for A: infinite information on the initial state is retained!

[PC, Essler, Fagotti ’12]

ρGGE= e-∑ λm Im /Z

where Im is a complete set of local (in space) integrals of moti

[Im ,In]=0 [Im ,H]=0 Im =∑ Om(x)

x

New proposal:

Solving Heisenberg equation of motion Not surprisingly, the harmonic oscillator oscillates

H = p2 2 + ω2 2 x2

H0 = p2 2 + ω2 2 x2

H0 ⇤

⌃x2(t)⌥ = ω2 + ω2 4ω0ω2 + ω2 ω2 4ω0ω2 cos 2ωt

• ⇥

Quenching the frequency in one harmonic oscillator

H(m) = 2

• n=0
• aπn + am ϕn + a(ϕn+1 − ϕn)
• ,

Ω2

k = m2 +

Each momentum mode is a free oscillator

⌃φr(t)φ0(t)⌥ ⌃φr(0)φ0(0)⌥ = ⇧

BZ

eikr (Ω2

0k Ω2 k)(1 cos(2Ωkt))

Ω2

kΩ0k

dk

• ⇤

⇧

BZ

eikr (Ω2

0k

Ω2

k

t ⇤ ⇧

This compatible with the GGE

ρGGE = e−P

k λknk

Z nk = a†

kak

λk = ln 1 + 4ΩkΩ0k (Ωk Ω0k)2

!

H(m) = 2

• n=0
• aπn + am ϕn + a(ϕn+1 − ϕn)
• ,

Ω2

k = m2 +

Each momentum mode is a free oscillator

⌃φr(t)φ0(t)⌥ ⌃φr(0)φ0(0)⌥ = ⇧

BZ

eikr (Ω2

0k Ω2 k)(1 cos(2Ωkt))

Ω2

kΩ0k

dk

• ⇤

⇧

BZ

eikr (Ω2

0k

Ω2

k

t ⇤ ⇧

This compatible with the GGE

ρGGE = e−P

k λknk

Z nk = a†

kak

λk = ln 1 + 4ΩkΩ0k (Ωk Ω0k)2

!

Non local... but linear combinations of loca

X

k

λknk =

X

m

γmIm

The GGE built with nk and with Im with are equivalent!

• Mass quenches in (lattice) field theories
• Luttinger model quartic term quench
• Transverse field quench in Ising/XY model
• Quench to the Tonks-Girardeau model
• Few more.....

PC-Cardy ’07, Barthel-Schollwock ’08, Cramer, Eisert, et al ’08, Sotiriadi Cazalilla ’06, Cazalilla-Iucci ’09, Mitra-Giamarchi ’10.... Barouch-McCoy ’70, Igloi-Rieger ’00-13, Sengupta et al ’04, Rossini et al. ’10, PC, Ess Foini-Gambassi-Cugliandolo’12, Bucciantini, Kormos, PC ’14........

The GGE always turned out to work

Rostunov, Gritsev, Demler ’10, Collura, Sotiriadis, PC ’13, Kormos, Collura

If we take a linear superposition of a finite number of eigenstates system will obviously oscillate forever Can we find some conditions for the initial state/Hamiltonian gua steady state and GGE/thermalization?

Sotiriadis, PC 2014

For a free theory, the steady state is described by the GGE if the state satisfy the cluster decomposition property

lim

R→∞

D Y

i

φ(xi) Y

j

φ(xj + R) E = D Y

i

φ(xi) ED Y

j

φ(xj) E .

see also Crame

A simple general condition

If we take a linear superposition of a finite number of eigenstates system will obviously oscillate forever Can we find some conditions for the initial state/Hamiltonian gua steady state and GGE/thermalization?

The calculations become immensely more complicated

Lieb-Liniger gas

HLL = 1 2

N

X

j=1

∂2 ∂x2

j

+ c X

i6=j

δ(xi xj),

XXZ Spin chain

H = J

L

X

i=1

⇥ Sx

i Sx i+1 + Sy i Sy i+1

We developed a method to calculate expectation values in the G

2 4 6 8

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 k = 1 k = 2 k = 3

|Neel,θ⟩ ⟶ Δ = 2 θ = 0° ⟨σxjσxj+k⟩t t

2 4 6

• 1
• 0.5

0.5 1

|Neel,θ⟩ ⟶ θ = 0° ⟨σzjσzj+k⟩t t

Fagotti, Collura, Es

Analytics vs Numerics for the Neel →Δ quench: Similar agreement with other initial states and final H

A new method to compute the exact time evolution developed

Ess

Particularly effective to compute the long-time limit

Brockmann, Wouters, Fioretto, De Nardis

Applied to XXZ chain for the Neel quench:

hσz

1σz 2isp

= 1 + 2 ∆2 7 2∆4 + 77 16∆6

hσz

1σz 2iGGE = 1 + 2

∆2 7 2∆4 + 43 8∆6

1 2 3

t

• 0.62
• 0.60
• 0.58
• 0.56

(a)

iTEBD GGE

• TBA

1 0.10 0.20 0.30

The difference is more evident starting from the dimer state

Pozsgay, Mestyan, Werner, Kormos, Zarand, Takacs ’14

A new method to compute the exact time evolution developed

Ess

Particularly effective to compute the long-time limit

Brockmann, Wouters, Fioretto, De Nardis

Applied to XXZ chain for the Neel quench:

hσz

1σz 2isp

= 1 + 2 ∆2 7 2∆4 + 77 16∆6

hσz

1σz 2iGGE = 1 + 2

∆2 7 2∆4 + 43 8∆6

1 2 3

t

• 0.62
• 0.60
• 0.58
• 0.56

(a)

iTEBD GGE

• TBA

1 0.10 0.20 0.30

The difference is more evident starting from the dimer state

Pozsgay, Mestyan, Werner, Kormos, Zarand, Takacs ’14

The GGE does not work?? :( more work to be done! But what about the time evolution?

‹O(t,x)› ∝ e-π xot/2τo

• 1. One-point function of a primary operator with ‹ψ0|O(x) |ψ0› ≠ 0:

Exponential relaxa

• 2. Two-point function of a primary operator with ‹ψ0|O(x) |ψ0› ≠ 0:

Connected correlations vanish for t<r/

τ0 related to the init

‹O(t,r)O(t,0)› ∝ e-π xor/2τo for t>r/2 e-π xot/τo for t<r/2

{

If ‹ψ0|O(x) |ψ0› ≠ 0, for t<r/2 ⇒ ‹O(t,r)O(t,0)›= ‹O

Asymptotic r-decay is expone Thermalization of CF

t r/2

ln ‹O(t,r)O(t,0)›

Horizon

Correlations saturate to t-independent values for Finite T Sharp horizon and thermalization are consequences

• f perfectly linear dispersion relation and specific initial state

Not true in general CFT calculations have been generalized to different situations such as correlations, systems with boundaries, different initial states, et

〈O〉≠0

• |ψ0› acts as a source of quasi-particle at t=0
• particles emitted from regions of size ~τ0 are entangled
• For t>0 quasi-particles move at fixed velocity ±v (linear dispersion)
• Horizon: points at separation r become correlated when left- and ri

particles originating from the same spatial region ~τ0 first reach the

• If all particles move at speed v, correlations are then frozen for t >

• |ψ0› acts as a source of quasi-particle at t=0
• particles emitted from regions of size ~τ0 are entangled
• For t>0 quasi-particles move at fixed velocity ±v (linear dispersion)
• Horizon: points at separation r become correlated when left- and ri

particles originating from the same spatial region ~τ0 first reach the

• If all particles move at speed v, correlations are then frozen for t >

General lattice dispersion:

vk = dΩk

dk

vmax exists correlations form at t = Slower particles change correlations after t = large t is driven by slowest particles

1 2 3 4 5 time t 1 2 3 4 5 time t 10 20 1 2 3 4 |i−j| time t 10 1 2 3 4 time t (b) V = 2 (a) V = 0

Manmana et al ’08 Fermi-Hubbard

0.001 0.01 0.1 1

<b0br>(t)

0.01 0.1 <b0br>(t) low pass

r=2 r=3 r=4 r=5 r=6 + +

Kollath-Lauechli ’08 Bose-Hubbard

t r/2 ln ‹O(t,r)O(t,0)›

CFT: linear dispersion non-linear dispersion

• Lattice
• non-zero momentum vmax

Numerical checks

t ~τ0

~ ~ position time b d = v t a

quench

• FIG. 1. Spreading of correlations in a quenched atomic

Mott insulator. a, A 1d ultracold gas of bosonic atoms (black balls) in an optical lattice is initially prepared deep in the Mott-insulating phase with unity filling. The lattice depth is then abruptly lowered, bringing the system out of

• equilibrium. b, Following the quench, entangled quasiparticle

pairs emerge at all sites. Each of these pairs consists of a doublon (red ball) and a holon (blue ball) on top of the unity- filling background, which propagate ballistically in opposite

• directions. It follows that a correlation in the parity of the

site occupancy builds up at time t between any pair of sites separated by a distance d = vt, where v is the relative velocity

• f the doublons and holons.

Thank you for your attention

Quantum quenches represent a theoretical and experimental challenge raising many fundamental questions in many-body quantum mechanics