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
o
n
p
o
i
n
t
s
e
x
i
s
t
i
n
f
r
u
s

t
r
a
t
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
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