frustration in spin systems

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Nov 15, 2013 (3 years and 8 months ago)

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1

Ground state factorization versus
frustration in spin systems

Gerardo Adesso


School of Mathematical Sciences

University of Nottingham


joint work with

S. M. Giampaolo

and

F. Illuminati

(University of Salerno)

"Hamiltonian & Gaps", 7/9/2010

2

Outline


Spin systems and
frustration


What we want to do and why


Theory of ground state factorization


Factorized solutions to frustration
-
free models


Frustration vs factorization and order


Summary
and outlook




"Hamiltonian & Gaps", 7/9/2010

3

"Hamiltonian & Gaps", 7/9/2010

Quantum spin systems


N

spin
-
1/2 particles
on regular lattices



anisotropic
interactions of
arbitrary range


arbitrary spatial
dimension


translationally
invariant & PBC


external field along
z

H

=

1

2

X

i

;

l

J

r

x

S

x

i

S

x

l

+

J

r

y

S

y

i

S

y

l

+

J

r

z

S

z

i

S

z

l

¡

h

X

i

S

z

i

(
)
r i l
= -
4

"Hamiltonian & Gaps", 7/9/2010

Ground states


No known exact analytical solution
in general,
except
for
a few simple
subcases
(Ising, XY,...) and now a
wider class
of models with nearest
-
neighbor interactions (see
JE
)


Difficult
to be determined
even numerically
, especially
for high
-
dimensional
lattices (2D, 3D, ...)


Rich phenomenology
: different magnetic
orderings,
critical points and quantum phase transitions


Typically
exhibit highly correlated quantum fluctuations,
i.e.,
they are
typically entangled

5

"Hamiltonian & Gaps", 7/9/2010

Frustration

AF

?

AF

AF


Occurs when the
ground state
of the
system cannot satisfy all the couplings


Even
richer phase
diagram
(high
degeneracy),
hence
even harder
to
find ground states


In frustrated systems a magnetic order
does not freeze,
which typically results
in
even more
correlations


At the root of statistically fascinating phenomena and exotic
phases such as
spin liquids
and
glasses


Frustrated systems may play a crucial role to
model
high
-
Tc

superconductivity
and certain biological processes

6

Relevant questions


How to define natural signatures and measures of
(classical and/or quantum)
frustration
?


More generally, is it possible to tune an external field
so that a many
-
body model admits as exact ground
state a completely
factorized

(“classical
-
like”) state?


This would be an instance of mean field becoming exact


If yes, under which conditions? Does this possibility
depend on the presence of frustration? In turn, does
the
fulfillment

or not of this condition define a regime
of weak versus strong frustration?

"Hamiltonian & Gaps", 7/9/2010

7

"Hamiltonian & Gaps", 7/9/2010

Ground state factorization


Answer to the 2
°

question:
YES
!

There
can exist special
points in the phase diagram of a spin system such that the
ground state is exactly a completely uncorrelated
tensor
product of single
-
spin states:
factorized ground state


The “factorization point” is obtained for specific, finite values
of the external magnetic field
(dubbed
factorizing field
) which
depend on the Hamiltonian parameters


First devised by Kurmann, Thomas and Muller (1982) for 1
-
d
Heisenberg chains with nearest neighbor antiferromagnetic
interactions


ground
=
⊗ ⊗

⊗ ⊗ …


8

"Hamiltonian & Gaps", 7/9/2010

Motivations


Many
-
body condensed matter perspective


To find
exact particular solutions
to non
-
exactly solvable models


To devise ansatz for
perturbative analyses
, DMRG, …


Quantum information and technology perspective


For several applications (e.g.
quantum state transfer
, dense coding,
resource engineering for one
-
way quantum computation), both in
the case of protocols relying on “natural” ground state
entanglement for quantum communication (in which case
factorization points should be avoided!), and for tasks which
instead require a qubit register initialized in a product state


Statistical perspective


To investigate the occurrence of
“phase transitions in
entanglement”

with no classical counterpart


For
frustrated

systems: to characterize the
frustration
-
driven
transition

between order (signaled by a factorized ground state)
and disorder (landmarked by correlations in the ground state
), thus
achieving a quantitative handle on the frustration degree

9

"Hamiltonian & Gaps", 7/9/2010

History


Direct method

(product
-
state ansatz)


Analitic brute
-
force method, guess a product state and verify that it is the ground state
via the Schrödinger equation, becomes nontrivial for more complicate models …



Kurmann et al. (1982):

1d Heisenberg, nearest neighbors


Hoeger et al. (1985); Rossignoli et al. (2008):

1d Heisenberg, arbitrary
interaction range


Dusuel & Vidal (2005):

Fully connected Lipkin
-
Meshkov
-
Glick model


Giorgi (2009):

Dimerized XY chains




Numerical method

(Monte Carlo simulations)

Nightmarish for spatial dimensions bigger than two (never attempted !)



Roscilde et al. (2004, 2005):

1d & 2d
Heisenberg, nearest neighbors


10

"Hamiltonian & Gaps", 7/9/2010

Our method


Quantum informational approach



Inspired by tools
of
entanglement
theory


Fully analytic
method


Requires
no ansatz
: the magnetic order, energy, and specific
form of the factorized ground state are obtained as a result of
the method


Encompasses previous findings and enables the identification of
novel factorization points


Provides self
-
contained
necessary and sufficient conditions
for
ground state factorization (in absence of frustration) in terms of
the Hamiltonian parameters


Straightforwardly applied to cases with
arbitrary range of the
interactions and arbitrary spatial dimension
(e.g. cubic
Heisenberg lattices), and to systems with spatial anisotropy, etc
.

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

100
, 197201 (2008);
Phys. Rev. B

79
, 224434 (2009)

11

"Hamiltonian & Gaps", 7/9/2010

The ingredients /1


Under translational invariance,
the ground state is completely
factorized iff the entanglement between any spin and the
block of all the remaining ones vanishes
, i.e., if the marginal
(linear) entropy of a generic spin, say on site
k
, is zero



We have:


so this factorization condition would depend on the
magnetizations, which are indeed the objects one cannot
compute in general models

½

k

S

L

(

½

k

)

=

4

D

e

t

½

k

=

1

¡

4

£

h

S

x

k

i

2

+

h

S

y

k

i

2

+

h

S

z

k

i

2

¤

12

"Hamiltonian & Gaps", 7/9/2010

The ingredients /2


A generic
N
-
qubit state is factorized iff for any qubit
k

there exist a unique
Hermitian, traceless, unitary operator
U
k

(which takes in general the form
of a linear combination of the three Pauli matrices), whose action on qubit
k

leaves the global state unchanged
(Giampaolo & Illuminati, 2007)


We can define in general the
“entanglement excitation energy”

(
EXE
)
associated to spin
k

as the increase in energy after perturbing the system,
in its ground state, via this special local unitary
U
k

(Giampaolo et al., 2008)


In formula:



One can prove that, under translational invariance and under the
hypothesis [
H
,
S
a
]≠0 (
a
=
x,y,z
),
the ground state is completely factorized iff
the entanglement excitation energy vanishes for any generic spin
k


½

k

U

k

¢

E

(

U

k

)

=

h

ª

j

U

y

k

H

U

k

j

ª

i

¡

h

ª

j

H

j

ª

i

13

"Hamiltonian & Gaps", 7/9/2010

The ingredients /3

1.
A factorized ground state must have vanishing local entropy

2.
A factorized ground state must have vanishing EXE

3.
The ground state must minimize the energy

4.
The Hamiltonian model
H

does not give rise to frustration

general theory of ground state factorization

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

100
, 197201 (2008);
Phys. Rev. B

79
, 224434 (2009)

14

"Hamiltonian & Gaps", 7/9/2010

Net interactions


All

the results (form of the state, factorizing field, conditions
for ground state factorization) are
only functions of the
Hamiltonian coupling parameters
and of lattice geometry
factors, or more compactly, of the
“net interactions”:



Z
r

is the coordination number, i.e.
the number of spins at a distance
r

from a given site



the magnetic order is determined by

¹

=

m

i

n

f

J

F

x

;

J

A

x

;

J

F

y

;

J

A

y

g

¹

=

8

>

>

<

>

>

:

J

F

x

)

F

e

r

r

o

m

.

o

r

d

e

r

a

l

o

n

g

x

;

J

F

y

)

F

e

r

r

o

m

.

o

r

d

e

r

a

l

o

n

g

y

;

J

A

x

)

A

n

t

i

f

e

r

r

o

m

.

o

r

d

e

r

a

l

o

n

g

x

;

J

A

y

)

A

n

t

i

f

e

r

r

o

m

.

o

r

d

e

r

a

l

o

n

g

y

.

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. B

79
, 224434 (2009)

,,
1
,,
1
,
,
1
( 1),
,
.
A r r
x y r x y
r
F r
x y r x y
r
A r
x y r
r
z
F
Z J
Z J
Z J
¥
=
¥
=
¥
=
= -
=
=
å
å
å
J
J
J
15

"Hamiltonian & Gaps", 7/9/2010

Results: Frustration
-
free

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. B

79
, 224434 (2009)

16

"Hamiltonian & Gaps", 7/9/2010

Heisenberg lattices


The method is versatile
and the result is
totally
general
: the complexity
is the same
for any
spatial dimension
, one
only needs to put the
correct coordination
numbers in the
definition of the net
interactions


(e.g. for nearest
neighbor models: Z=2 for
chains, Z=4 for planes,
Z=6 for cubic lattices)

F
(x)

F
(y)

AF
(x)

AF
(y)

Kurmann‘82

1D nearest neighbor

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

100
, 197201 (2008);
Phys. Rev. B

79
, 224434 (2009)

17

"Hamiltonian & Gaps", 7/9/2010

Other applications


Long
-
range and infinite
-
range models


Models with spatial anisotropy









a)

b)

c)

d)

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. B

79
, 224434 (2009)

18

"Hamiltonian & Gaps", 7/9/2010

Frustrated systems


We consider a subclass of the original Hamiltonian, comprising models with
anisotropic antiferromagnetic
(along
x
)
interactions up to a maximum range
r
max


Frustration arises from the
interplay between the couplings
at different ranges


We focus on
1d systems
(chains) of infinite length


For simplicity, we consider the interaction anisotropies independent on the
distance, but
overall the couplings are rescaled by a range
-
dependent factor
f
r


If all the

f
r
’s beyond
r
=1 vanish, the system is not frustrated. Vice versa,
if the
f
r
’s are all equal, the system is
fully frustrated
.

AF

AF

AF



H

=

X

i

;

r

·

r

m

a

x

f

r

(

J

x

S

x

i

S

x

i

+

r

+

J

y

S

y

i

S

y

i

+

r

+

J

z

S

z

i

S

z

i

+

r

)

¡

h

X

i

S

z

i

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

104
, 207202
(
2010)

19

"Hamiltonian & Gaps", 7/9/2010

Short
-
range systems


Simplest case:

r
max
= 2

(nearest and next
-
nearest neighbors)



We set
f
1
= 1,
f
2

f


The parameter
f


[0,1]

plays the role of a
“frustration degree”


(a more general definition of frustration degree was given by
Sen(De) et al., PRL 2008
)




Magnetic order of the ground state

f
<
½

standard antiferromagnet

f

½

dimerized antiferromagnet

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

104
, 207202
(
2010)

20

"Hamiltonian & Gaps", 7/9/2010

Factorized ground states


We can determine in general the form of the candidate factorized state
and the factorizing field






The nontrivial part is now in the
verification steps


We find that for the candidate factorized state to be an eigenstate, a
necessary condition is


J
z
= 0

(other possibilities lead to saturation instead of proper factorization)



By decomposing the Hamiltonian into triplet terms, we can derive a
sufficient condition for the candidate state to be the ground state, by
testing whether its projection on three spins is the
ground state of
the triplet
Hamiltonian


For frustration
-
free, the factorized state was always the ground state

h

f

=

1

2

p

J

x

J

y

=

(

1

¡

f

)

p

J

x

J

y

f

<

1

=

2

f

p

J

x

J

y

f

¸

1

=

2

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

104
, 207202
(
2010)

21

"Hamiltonian & Gaps", 7/9/2010

ground state factorization

Factorization vs frustration


From the
triplet
decomposition we find, analytically, that if the frustration
is weaker than a
“critical” value
, ,


then the ground state is factorized


The actual
“compatibility
threshold”

(i.e. the maximum
frustration degree that allows
ground state factorization) can
be determined numerically by
considering decompositions
into blocks of more than three
spins.


Above this boundary the system
admits a factorized eigenstate
at
h
=
h
f
, but this does not
minimize energy and instead
the ground state is entangled

1
2
x x y y
c
x y
J J J J
f f
J J
- +
£ º
+
S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

104
, 207202
(
2010)

22

"Hamiltonian & Gaps", 7/9/2010

Remarks


Frustration naturally induces correlations which tend to suppress ground
state factorization
: for strong enough frustration it is not energetically
favourable for the system to arrange in a factorized state (although a
factorized state can exist in the higher
-
energy spectrum)


At the factorizing field, we witness a
first order quantum
phase
transition

(level crossing) from a factorized to an entangled ground state when the
frustration crosses the compatibility threshold


Qualitative agreement with the results on the scaling of correlations
(Sen(De)
et al., PRL 2008
)

and on tensor network representability
(Eisert
et al., PRL
2010)


Factorized antiferromagnetic
ground state

Factorized antiferromagnetic excited eigenstate

Factorized dimerized excited eigenstate

FRUSTRATION

(
h

=
h
f

)

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

104
, 207202
(
2010)

23

"Hamiltonian & Gaps", 7/9/2010

Remarks


Reversing the perspective
, we can define the regime of
weak frustration
as
the one
compatible with ground state factorization
, and the regime of
strong frustration

as the one where
no factorization points are allowed
.


Ground state factorization implies a definite magnetic order
, thus it is a
precursor to a
quantum phase transition
, with critical field
h
c

h
f


The regime of strong frustration is thus characterized by the fact
that a
magnetic order does not freeze

even at zero temperature (in layman’s
words, the ground state remains always entangled), in accordance with
other criteria to assess the frustration degree
(Ramirez, Balents, ...)

Factorized antiferromagnetic
ground state

Factorized antiferromagnetic excited eigenstate

Factorized dimerized excited eigenstate

FRUSTRATION

(
h

=
h
f

)

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

104
, 207202
(
2010)

24

"Hamiltonian & Gaps", 7/9/2010

Longer
-
range models


The same general features emerge by investigating frustrated
systems with interactions beyond next
-
nearest neighbors


Factorized eigenstates are only allowed for
J
z
=0
(this limitation could be relaxed in more general non
-
translationally
-
invariant models
where the anisotropies depend individually on the distance)


There is a
compatibility threshold

dividing the phase diagram into a
region of
weak frustration/order/ground state factorization

and a
region of
strong frustration/disorder/ground state entanglement

C

o

m

p

a

t

i

b

i

l

i

t

y

t

h

r

e

s

h

o

l

d

s

:

M

a

x

i

m

u

m

v

a

l

u

e

o

f

t

h

e

f

r

u

s

t

r

a

t

i

o

n

f

a

s

a

f

u

n

c

t

i

o

n

o

f

t

h

e

r

a

t

i

o

J

y

=

J

x

f

o

r

w

h

i

c

h

g

r

o

u

n

d

s

t

a

t

e

f

a

c

t

o

r

i

z

a

t

i

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n

p

o

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n

f

r

u

s

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t

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a

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e

d

a

n

t

i

f

e

r

r

o

m

a

g

n

e

t

s

w

i

t

h

r

m

a

x

=

4

.

T

h

e

b

l

a

c

k

l

i

n

e

s

t

a

n

d

s

f

o

r

s

y

s

t

e

m

s

w

i

t

h

f

2

=

f

,

f

3

=

f

=

2

a

n

d

f

4

=

f

=

3

,

t

h

e

r

e

d

l

i

n

e

f

o

r

f

2

=

f

,

f

3

=

f

=

2

a

n

d

f

4

=

f

=

4

,

a

n

d

t

h

e

b

l

u

e

l

i

n

e

f

o

r

f

2

=

f

,

f

3

=

f

=

p

2

a

n

d

f

4

=

f

=

6

.

r
max
= 4

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

104
, 207202
(
2010)

25

"Hamiltonian & Gaps", 7/9/2010

Infinite
-
range models


To verify ground state factorization in fully connected models (
r
max
=
∞)
,
one should decompose the Hamiltonian in terms involving
n
→∞ spins,
i.e., basically solve the Hamiltonian itself!


A workaround is possible if the frustration coefficients
f
r

follow a
decreasing functional law with
r

and vanish in the limit
r
→∞


In this case one can impose a cutoff and deal with decompositions into
blocks of a finite number
n

of neighboring spins which are most effectively
coupled



Then one takes the limit
n
→∞. Numerically, this means that
ground state
factorization occurs if, for
n

large enough, the difference
D

between the
minimum eigenvalue
m

潦⁴桥o
n
-
spin Hamiltonian component and the
energy associated to the candidate factorized state, vanishes
asymptotically
. If
r
’ is the cutoff range (such that
n
=2
r
’+1), then

P

D

(

r

0

)

=

¹

(

r

0

)

+

1

4

(

J

x

+

J

y

)

r

l

=

1

(

¡

1

)

l

f

l

f
r

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

104
, 207202
(
2010)

26

"Hamiltonian & Gaps", 7/9/2010


Case
f
r
=1/
r
2

(weak frustration)


The difference
D
(
r
’)=0 for any cutoff
r




Exact factorized ground state

at



Case
f
r
=1/
r


(medium frustration)


The difference
D
(
r
’) seems to converge



to 0
(more numerics needed)



Conjectured factorized



ground state

at




Case
f
r
=1/√
r

(strong frustration)


The difference
D
(
r
’) converges


to a finite value



No factorized ground state !



Results for infinite range

h

f

=

(

¼

2

=

1

2

)

p

J

x

J

y

h

f

=

l

n

(

2

)

p

J

x

J

y

D
D
_

S. M. Giampaolo, GA, F. Illuminati,
Phys. Rev. Lett.

104
, 207202
(
2010)

27

"Hamiltonian & Gaps", 7/9/2010

Summary


We approached the problem of finding
exact factorized ground state
solutions to general cooperative spin models


We devised
a method to identify and fully characterize such solutions
thanks to some tools borrowed from quantum information theory


In
frustration
-
free

systems, necessary and sufficient conditions are derived
and several
novel factorized exact solutions
are straightforwardly
obtained for various translationally invariant models, with interactions of
arbitrary range, and arbitrary lattice spatial dimension


In
frustrated

systems, a universal behaviour emerged in which
frustration
and ground state factorization are competing phenomena
, the former
inducing correlations and disorder, and the latter relying on ordered,
uncorrelated magnetic arrangements. Notably short
-
range as well as
infinite
-
range (weakly) frustrated antiferromagnetic models have been
shown to admit exact factorized solutions


Ground state factorization is an effective tool to probe quantitatively
frustrated quantum systems.
The possibility vs impossibility of having a
classical
-
like ground state

at a given value of the magnetic field
defines
the regimes of weak vs strong frustration

28

"Hamiltonian & Gaps", 7/9/2010

Discussion and outlook


In this talk we only considered spin
-
1/2 systems, however due to a
theorem by Kurmann et al. (1982), any spin
-
S (S> ½) Hamiltonian which is
of the same form as a spin
-
1/2 Hamiltonian that admits a factorized
ground state at
h
=
h
f
, will also admit a factorized ground state at the same
value of the field: greater scope of our results


For a generic model (frustrated or not), the factorizing field (when it
exists) is a precursor to the critical field associated to a quantum phase
transition (where the external field is the order parameter), i.e.
h
f

h
c


A fascinating perspective is the investigation of the ground state
entanglement structure near a factorization point: it is conjectured that
entanglement undergoes a global reshuffling and can change its typology
(demonstrated in the XY and XXZ models,
Amico et al. 2007
): an

entanglement phase transition
” with no classical counterpart, and
signaled by a diverging range of pairwise entanglement


More perspectives


Generalize the method: relax translational invariance, identify exactly
dimerized solutions, etc.


Area laws for frustrated systems?


Define a measure of frustration, able to distinguish quantum from classical


...

29

"Hamiltonian & Gaps", 7/9/2010

Thank you

?