Ground State and out of equilibrium fidelities:
from quantum phase transitions
to equilibration dynamics
P. Zanardi
USC
KITPC April 2011
(Quantum) Phase Transition: dramatic change of the
(ground) state properties of a quantum system with respect
A smooth change of some control parameter
e.g., temperature, external field, coupling constant,…
Characterization of PTs:
“Traditional”:
Local Order parameter (OP), symmetry breaking (SB)
(correlation functions/length) Landau-Ginzburg framework
Smoothness of the (GS) Free-energy (I, II,.. order QPTs)
Quantum Information views:
Quantum Entanglement (concurrence, block entanglement,..)
Geometrical Phases
Question: How to map out the phase diagram of a system
with no a priori knowledge of its symmetries and ops?
Answer(?): distinguishability =distance in the information space
PZ, N. Paunkovic, PRE 74 031123 (2006)
| Y0 (l)ñ
Ground State of
H (l )
F (l, l' ) =| á Y0 (l) | Y0 (l' )ñ |
F is a universal (Hilbert space) geometrical quantity:
No a priori understanding of the SB pattern or OP is needed
d 2 (l, l' ) :=|| Y0 (l) - Y0 (l' ) ||2 = 2 - 2 Reá Y0 (l) | Y0 (l' )ñ » l®l'
æ ¶2F ö 2
çç 2 ÷÷d l
è ¶l ø
THE KEY (naive) IDEA
At the quantum critical points (QCPs) F should have a sharp drop
i.e., the induced metric d should have a sharp increase. Weak
topology vs norm topology: a METRIC APPROACH
THE XY Model (I)
H ( l, g ) = å
L
i =1
1+ g x x 1- g y y
(
si si +1 +
si si +1 + lsiz )
2
2
g=anysotropy parameter, l =external magnetic field
g=0
XX line II-order QPT
QCPs:
Ferro/para-magnetic II order QPT
l = ±1
Jordan-Wigner mapping H
EXACTLY SOLVABLE!
L k = (cos
Free-Fermion system:
2pk
2pk
- l) 2 + g 2 sin 2
L
L
Quasi-particle spectrum: zeroes in the TDL in all the
QCPs
Gaplessness of the many-body spectrum
THE XY Model (II)
Ground State
2pk
cos
-l
-1
L
qk = cos
Lk
ck c- k | 00ñ k , - k = 0
| Y0 (l, g )ñ = Äk >0 cos
qk
2
| 00ñ k , - k - i sin
L/2
F =| á Y0 | Y0 ñ |= Õ | cos
'
qk - q k '
2
k =1
F (l, l + dl) = exp(-dl2 S l 2 (l) + o(dl4 ))
qk
2
| 11ñ k , - k
|
M
S 2 (l, g ) = å L- 4 k g 2 sin 2
L
k =1
Fidelity
2pk
L
Strong L-dependent peak at l c = ±1universality features in the TDL
O( L2 ) (l = l c )
®¥
S2L (l, g) ¾L¾
¾®
O(L) (l ¹ l c )
¥
S2 (l, g ) µ| l - l c |-a ( g ) µ
a( g) » const » 1
Orthogonalization in the TDL: enhanced @ QCPs!
PZ, N. Paunkovic, Phys. Rev E (2006)
XY Model (III): Overlap Functions
F (l + dl, g + dg) » exp(-dl2 S2l + dg 2 S2g )
S2g (l, g )
S 2l (l, g )
Differential Geometry of QPTs:QGT
H (l ) Î L ( H )
Control parameters
Hamiltonians
lÎM
| Y (l)ñ Î H
Hermitean metric over the projective space
Ground
states
Q(dY, dY) := á dY, (1- | Yñá Y |)dYñ = åmn Qmn dlm d ln
Qmn = ᶠm Y | ¶n Yñ - ᶠm Y | Yñá Y | ¶n Yñ
g mn := Re Qmn
Riemannian metric
F ( l , l + dl ) » 1 - 1 / 2åmn g mn dlm dln » e
- ds 2
Fmn := Im Qmn = Imᶠm Y | ¶n Yñ µ ¶ m An - ¶n Am
Am := á Y | ¶n Yñ
2-form (=Berry curvature!)
Metric scaling behaviour: the XY model
æ ¶K ¶K ö
÷dln dlm
ds = å g mn dlm dln µ å Tr ç
ç
÷
¶
l
¶
l
mn
mn
n
m
è
ø
2
æ 1
L
1 ö
g=
diag çç
,
÷
2
2 ÷
16 | g |
è1- l 1+ g ø
Ricci Scalar
1
(1+ | g |)
R(| l |£ 1) =
16 L | g |
ö
1 æç
|l |
÷
R(| l |> 1) = 1+
16 L çè
h 2 + g 2 - 1 ÷ø
QGT: Spectral representations
Qmn = å
á Y0 (l ) | ¶ m H | Yn (l )ñá Yn (l ) | ¶n H | Y0 (l )ñ
n
( E0 (l ) - En (l ))2
1
1
*
2
| Qmn |£|| Q ||= á L | Q | L ñ =£ (á H L H L ñ- | á H L ñ | ):= X
D
D
H L := åm L m ¶ m H
Local operator (trans inv)
D := min n ( En - E0 )
limL®¥ D( L) > 0
Gapped system
Spectral gap
limL®¥ L- d X ( L) < ¥
(Hastings 06)
limL®¥ L- d | Qmn |< ¥
For gapped systems the QGT entries scales at most extensively
Superextensive scaling implies gaplessness
QGT Critical scaling
¶ m H ® ò d d xVm (x )
Continuum limit
x ® ax;t ® a z t
Scaling transformations
-n
x =| l - lc |
Vm ® a
nDQ
q Sing mn := Qmn / Ld ®| l - lc |
q
Sing
mn
- DQ
:= Qmn / L ® L
d
-D m
Vm
Proximity of the critical point
At the critical points
DQ := D m + Dn - 2z - d
Scal dim of QGT: the smaller the faster the orthogonalization rate
Super-extensivity
D £ (z + d ) - d / 2
Criticality it is not sufficient, one needs enough relevance….
Dynamical Fidelity:Loschmidt Echo
= å pn pm exp[-it(E n - E m )]
n,m
Spectral resolution
Probability distribution(s)
Different Time-Scales & Characteristic quantities
•Relaxation Time (to get to a small value by dephasing and oscillate around it)
•Revivals Time (signal strikes back due to re-phasing)
Q1: how all these depend on H,
, and system size?
Q2: how the global statistical features of L(t) depend on H,
and system size N?
Typical Time Pattern of L(t)
Transverse Ising (N=100)
For a given initial state L-echo is a RV over the time line [0,∞) with Prob Mea
cA
Characteristic function of
A Ì [0,¥)
Probability distribution of L-echo
Goal: study P(y) to extract global information about the
Equilibration process
1 Moments of P(y)
Each moment is a RV over the unit sphere (Haar measure) of initial states
Mean:
Long time average of L(t) is the purity of the time-average
density matrix (or 1 -Linear Entropy)
Remark
D1 Is a projection on the algebra of the fixed points
Of the (Heisenberg) time-evolution generated by H
Remark dephased state min purity given the constraints I.e., constant of motion
Question: How about the other moments e.g., variance and
initial state dependence? Are there “typical” values?
Ising in transverse field:
g =1
H.T Quan et al, Phys. Rev. Lett. 96, 140604 (2006)
L(t) = Õ(1- sin2 (2a k )sin 2 (L2k t))
cosqn ( l) =
cos
2pk
-l
L
Ln
a k = qk (l1) - qk (l2 )
k
Large size limit (TDL)==> spec(H) quasi-continuous ==> Large t limits exist
(R-L Lemma) =time averages
L(t) = e-Ls(t )
(m=band min, M=band max
s(¥) = -
1
p
ò ln[(1- | cos(a ) | /2)]dk
k
Inverting limits I.e., 1st t-average, 2nd TDL
g=-
1
2p
ò ln[(1- sin (a ) /2)]dk
2
k
•g and s are qualitatively the same but when we consider different phases
First Two Moments
a k := sin2 (
k1 = P k [1- a k /2],
N = 100
3
1
var(L) = P k [1- a k + a 2 k ] - P k [1- a k + a 2 k ]
8
4
N = 100
qk (h1 ) - qk (h1 )
2
)
P(L=y) Different Regimes
•
dh
Large ==>L for (moderately) large (quasi) exponential
• dh Small and off critical a)
b) Otherwise Gaussian
L >>| dh |-2 Exponential
• dh Small and close to criticality a) L >>| dh |-1 Exponential
b) Quasi critical I.e., L <<| h(i) -1 |µ x universal “Batman Hood”
L=18, h(1)=0.3, h(2)=1.4
L=20, h(1)=0.1, h(2)=0.11
L(t) = k1 + å c(L k )cos(tL k )
k
+ n-body spectrum contributions
L=40, h(1)=0.99, h(2)=1.1
Different regimes depend on how many
frequencies have a non-negligible weight
a
c(w ) := k |w = L k
2
Approaching exp for large sizes
L=10,20,30,40
h(1)=0.9, h(2)=1.2
L=20,30,40,60,120
h(1)=0.2, h(2)=0.6
Joint work with
Nikola Paunkovic
Marco Cozzini
Paolo Giorda
Radu Ionicioiu
L. Campos-Venuti
and now @USC
Damian Abasto
Toby Jacoboson
Silvano Garnerone
Stephan Haas
THANKS!
Summary Part I
•We analyze the phase diagram of a systems in terms
of the induced geometric tensor Q on the parameter space
•Boundaries bewteen different phases can be detected
by singularities of Q (real & immaginary parts)
•Q and related functions show critical scaling behaviour and
Universality (response function representation)
•Extensions to finite temperature and to classical systems
Quantum Tensor(s) approach to (Q)PTs:
Geometrical, QI-theoretic, Universal
Summary Part II
•Unitary equilibrations: measure convergence/concentration
•Universal content of the short-time behavior
of L(t) and criticality
•Moments of Probability distribution of LE (large time)
•Transverse Ising chain: mean, variance, regimes for P(L)
Phys. Rev. A 81,022113 (2010)
Phys. Rev. A 81, 032113 (2010)
Short-time behavior
Square of a characteristic function --> cumulant expansion
H sum of N local operator in the TDL N-> ∞ one expects CLT to hold I.e.,
TR = O(1)
Off critical (or large quench)
TR = O(Lz ) µ x z
Critical (& small quench)
•0 for
Gaussian
Bn /B2n / 2 for
Non Gaussian (universal)
Non Gaussian & non universal)
Relaxation time
Quantum Fidelity and Thermodynamics
Uhlmann fidelity between Gibbs states of the same Hamiltonian
at different temperatures Z (b) = tr exp(-bH ) = exp(-bA(b))
=partition function
F (b0 , b1 ) =
Z (b0 / 2 + b1 / 2)
Z (b0 ) Z (b1 )
F (b0 , b1 ) = exp(b0 A(b0 ) / 2 + b1 A(b1 ) / 2 - (b0 + b1 ) A(b0 / 2 + b1 / 2) / 2)
cV (T )
F (T , T + dT ) » exp( -dT
)
2
8T
2
Singularities of the specific heat cV (T ) shows up at the fidelity level:
Divergencies e.g., lambda points, results in a singular drop of fidelity
Temperature driven phase transitions can be detected by
mixed state fidelity in both classical and quantum systems!
The
“Hey dude! If you have the GS you have EVERYTHING…”
Slide
Ok cool. Let’s suppose you’re somehow given the GS e.g.,a MPS
| Y ( g )ñ =
d
å Tr ( A
i1, … ,i N =1
Aik ( g ) Î M D (C ),
i1
AiN ) | i1 ,
, iN ñ
g = ( g1 ,
, gQ ) Î RQ
Please, now tell me where the QCPs are….
You may answer:
“Great! Now I compute the OP and its correlations! ….”
Well… you have infinitely many candidates: Which one?!?
and perhaps none of them is gonna work out (e.g., TOPorder)…
“Ok, fair enough, then I compute the energy!”
I see, so you want also the H? You didn’t ask for that so far…
Anyways, here we go: identically vanishing E(g)=0….
Distinguishability of Exps
Distance in Prob space
Qualitative: the difference between the means should be
(much) bigger than the finite size variances….
d(P,Q)= number of (asymptotically) distinguishable
preparations between P and Q: ds=dX/Variance(X)
Quantitative: Fisher Information metric
E
( pi - qi ) 2 E
d F ( P, Q = P + dP) = å
=å pi (d log pi ) 2
pi
i =1
i =1
Problem: Quantumly each Q-preparation | Pñ defines infinitely
many prob distributions, one for each observable: pi =| ái | Pñ |2
one has to maximize over all possible experiments!
Surprise!
max Exps d F ( P, Q) = Projective Hilbert space distance!
DFS ( P, Q) º cos -1
| áQ | Pñ |
º cos -1 ( F )
|| Q || × || P ||
Q-fidelity!
(Wootters 1981)
Statistical distance and geometrical one collapse:
Hilbert space geometry is (quantum) information
geometry…..
Question: What about non pure preparations
rP
?!?
d (r0 , r1 ) = cos -1 ( F )
Answer: Bures metric & Uhlmann fidelity!
(Braunstein Caves 1994)
F (r0 , r1 ) = tr r11/ 2r0r11/ 2
Remark: In the classical case one has commuting objects, simplification ….