Exact Ground States of Finite Ising Spin Glasses Obtained by

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Exact Ground States of
Finite Ising Spin Glasses
Obtained by ”Branch-and-Bound”
S. Kobe
.
[email protected]
http://www.physik.tu-dresden.de/itp/members/kobe.html
Technische Univ. Dresden, Inst. für Theor. Physik, D-01062 Dresden, Germany
S. Kobe, Goettingen/Leipzig 2004 – p.1/??
Outline
•
•
•
I SING
model: A short history
Exact ground states of finite I SING spin glasses
Results:
– Lattice models:
Energy landscape
Relaxation
– Mean field models:
Energy exponents and correction to scaling
Ferromagnetic – spin-glass transition
Historical remarks
Letter of WOLFGANG PAULI to H. B. G. C ASIMIR:
Princeton, 11. Oktober 1945
” ... A few weeks will be sufficient for you and others to learn everything of
scientific interest which happened during these ‘lost years‘.
I am sending you today a package with reprints, please divide
them among persons who are interested.
There is a paper of Onsager included (...) of which I think
that it is a masterpiece of mathematical analysis.
It contains the rigorous solution of the Kramers - Wannier order-disorder problem
for the two dimensional model (unfortunately the method cannot be generalized
for three dimensional crystals). ... "
Ising Model: Phase transition, scaling behaviour
Zero field I SING model:
H=−
X
1≤i<j≤N
Jij Si Sj
Si = 1 ∨ −1
– Jij > 0 between nearest neighbours of a lattice; J ij = 0 else
– Ground state is trivial
– Phase transition: (ferromagnetic) order - disorder
Ernst Ising 1925
Peoria 1996
Ising Spin Glass Models
Zero field I SING model:
H=−
X
1≤i<j≤N
Jij Si Sj
Si = 1 ∨ −1
– Jij arbitrary
– Ground state is not known!!!
Spin Glass Models (Lattice vs. Mean-Field Models)
•
Lattice models:
Jij = ±1
or
Gij ;
i.e. the couplings between spins beeing nearest neighbours in a (cubic) lattice are
randomly distributed
•
Mean-field models:
e.g S HERRINGTON -K IRKPATRICK (SK) model:
Gij
Jij = √
N
Gij → independent identically distributed Gaussian random numbers
with zero means and variance one.
A Related Mean-Field Model
SK model with non-symmetric distribution of Gauss couplings:
|Gij |
|Gij |
P (Jij ) = xδ(Jij + √ ) + (1 − x)δ(Jij − √ )
N
N
i.e. the sign of interactions are inversed according the probability x
limiting cases:
x=0
x = 0.5
x=1
→ ferromagnetic system
→ SK model
→ antiferromagnetic system
No shift of the Gauss distribution !!!
How much states ?
N
1
2
3
4
5
...
10
20
...
24
...
47
...
58
...
64
...
90
...
1010
2N
2
4
8
16
32
...
1024
1048576
...
16 777 216
...
≈ 1.4 x 1014
...
≈ 3 x 1017
...
≈ 2 x 1019
...
≈ 2 x 1027
...
unimaginable!
notes
≈ (Maximal-)number of ancestors in 20th generation (14th century)
≈ combinations in German Lotto
≈ age of mankind in seconds (4 Mio. a)
≈ time since "big bang" in seconds (13 Mrd. a)
"chessboard"-bet: fields vs. grains
≈ time since "big bang" in ns
Exact Ground States: Optimization
Complexity: NP-complete
Method: "branch-and-bound":
exact nonlinear discrete optimization
S. K., A. H ARTWIG , Comp. Phys. Commun. 16 (1978) 1
example: N = 8
0
B
B
B
B
B
B
J=B
B
B
B
B
B
@
0
−5
0
−2
−10
0
−5
−4
0
0
−6
0
0
−3
0
−1
−2
−3
−5
−4
0
0
−1
0
−7
−5
0
0
0
0
−1
−4
−8
−1
0
0
1
C
C
C
C
C
C
C
C
C
C
C
C
A
Branch-and-Bound Tree
Energy of the branching level El (with El=N = Estates ):
El = El−1 + 2
l−1
X
k(kl)
|Jkl |
El ≥ El−1
example: N = 8; E1 = Eid = −77
Branch-and-Bound Tree: Part
N = 8; Eid = −77; Ebound = −47 (heuristic solution: − − −)
N = 8; Eid = −77; Ebound = −47 (heuristic solution: − − −)
N = 8; Eid = −77; Ebound = −47 (heuristic solution: − − −);
exact solution: E0 = −51.
Misfit parameter: A measure for frustration
•
•
•
Misfit parameter → a useful rescaling of the ground-state energy per spin
`
´
Definition: µ0 = 12 1 − e0 /eid
0
with eid
0 → reference energy of a related non-frustrated system:
Jij = |Jij | (lattice model)
Jij =
•
|Gij |
√
N
1/2 (SK and related models)
→ eid
0 = (N − 1)/(2πN )
Properties:
– µ0 is the fraction of each bond of the system, which is on average not
satisfied.
– Example: Antiferromagnetic triangular lattice → µ 0 = 13 ,
because one of three bonds of equal strength cannot be satisfied.
– Maximum value: µ0 = 12 for highly frustrated systems
(e.g. high-dimensional hypercubic and fcc fully frustrated ±J systems).
– SK and related models belongs also to the class of systems with maximum
occurring frustration.
Results I
Lattice Models
Misfit parameter: Lattice models (±J spin glass)
lattice type
D
µ0
e0 from
honeycomb
square
triangular
simple cubic
hypercubic
hypercubic
2
2
2
3
4
5
0.09
0.1495
0.22
0.202
0.24
0.26
W. L EBRECHT, E.E. VOGEL (1994)
I.A. C AMPBELL , A.K. H ARTMANN , H.G. K ATZGRABER (2004)
W. L EBRECHT, E.E. VOGEL (1994)
various authors
S. B ÖTTCHER , A. G. P ERCUS (2001)
S. B ÖTTCHER, private communication
(S. KOBE , J. K RAWCZYK in: Computational Complexity and Statistical Physics, in press)
•
•
•
~
>?
&'
./
67
<=
FG
NO
VW
^_
fg
no
vw
|}
\]
TU
45
$%
,-
23
:;
DE
LM
RS
Z[
de
lm
tu
z{
rs
jk
bc
JK
BC
*+
"#
!
()
01
89
@A
HI
PQ
XY
`a
hi
pq
xy
Ground state clusters: Configuration vs. real space
two (fixed) spin clusters in the real space: "red" and "yellow" spin domains
Reversal of one spin domain ⇒ another cluster in the configuration space
"free" spins between the spin domains ⇒ degeneracy of ground state clusters
Exact Landscape
420 clusters,13628 states altogether
E
54 clusters,
2185 states
224 clusters,
8922 states
1503690
states
E3
E2
E1
E0
System 4
•
•
•
4
4 the first excitation
configuration space
18 states
12 states
- 50 states, the second excitation
- 300 states
1.635.796 states up to E3 ,
clusters of states form valleys (#1, #2),
valleys are connected via saddle clusters
Number of configurations
Landscape: Inner profile of the saddle cluster
Hamming distance
•
•
•
Restriction of "transition" between clusters via saddle cluster
Hamming distance of all pairs of states in the saddle cluster (right)
"Bottleneck" has to be passed: "entropic barrier"
L = 12: Landscape and dynamics
1
0.98
E
E1
95744 199168 54016 25344
28160 52864 20998 14080
saddel cluster
>500000 states
Ec0
276544
Ec2
59008
saddel cluster
55168
Ec0’
Ec2’
14560
3840
0.96
10260 37600
5600 10848
20400 17280 10240
5600 6080 4480
q(t)
0.94
0.92
0.9
c0
E0
5632 states
c2
1280 states
0.88
configuration space
0.86
0.01
0.1
1
10
100
1000
10000
100000
WTM TIME
•
•
More complex saddle cluster structure (left)
Dynamics: q vs. WTM time for 20 runs at T = 0.37 starting from one ground state
(right) qpl = 0.915 ± 0.02 ⇒ hd = 73 ± 2 ⇒ "width of the valley"
Ground states from A.K. H ARTMANN: Genetic Cluster-Exact Approximation
Results II
Mean – Field Models
Energy exponents (zero temperature)
Notation of
J.-P. B OUCHAUD, F. K RZAKLA , O.C. M ARTIN , Phys. Rev. B 68 (2003) 224404
Lattice models: N = Ld
EJ (L) = e0 Ld + e1 LΘs + . . . .
eJ (L) = EJ (L)/Ld = e0 + e1 L−ω + . . . .
Scaling of the energy fluctuations:
2
[EJ2 (L) − EJ (L) ]1/2 = σ0 LΘf + . . . .
Scaling exponents:
shift exponent Θs → ω = d − Θs
fluctuation exponent Θf
Energy exponents ctd.
L → N 1/d
0
eJ (N ) = e0 + e1 N −ω + . . . .
2
σJ (N ) = [e2J (N ) − eJ (N ) ]1/2 = σ0 N −ρ + . . . .
with
ω 0 = ω/d = 1 −
ρ=1−
Θs
d
Θf
d
System sizes and numbers of realizations
# of samples
N
x = 0.5 (SK)
x = 1.0
19
257909
438242
25
229086
207149
32
123220
39
74827
16519
42
50797
13828
49
7933
1486
50
7724
1181
56
6274
282
59
2082
136
64
1779
70
634
75
236
81
126
85
112
90
42
Results: Ground-state energy (SK model)
0.76
e0(N)
0.74
0.72
0.70
0.68
0.66
0.64
0.00
0.05
0.10
N
0.15
-1/2
0.20
Results: Ground-state energy (x = 1: AFM model)
No analytical solution known!
0.475
e0(N)
0.470
0.465
0.460
0.00
0.05
0.10
N
0.15
-1/2
0.20
Results: Fluctuation exponent
sigma
0.06
0.04
0.02
0.00
0
0.00
0.02
0.04
N
upper curve: x = 0.5 (SK);
0.06
-0.75
lower curve: x = 1 (afm)
0.08
0.10
Results: Misfit parameter
0.5
>
<µ(N)
0
0.45
0.4
0.35
0.3
0.05
0.1
0.15
N
0.2
0.25
-1/2
upper curve: x = 1 (afm); lower curve: x = 0.5 (SK)
blue stars: M. PALASSINI , cond-mat/0307713: hybrid genetic algorithm
Results of fitting procedures
#
x = 0.5 (SK)
e0
ω0
e0 (N ) 3-p
-0.7615(25)
item 2-p
x = 1.0
e0
ω0
0.698(23)
-0.4755(5)
1.66(9)
*)
0.684(2)
-
-
µ0 (N −1/2 ) 3-p
-0.7655(38)
0.652(31)
-0.4756(4)
1.63(8)
item 2-p
*)
0.671(3)
-
-
σJ (N ) 2-p
ρ
0.710(5)
ρ
0.736(9)
*) The analytical value eRSB
= −0.76321(3) is used for the 2-parameter fit
0
(M. A. C RISANTI , T. ROSSI , Phys. Rev. E 65, 046137 (2002))
Phase transition
0.5
0.4
µ0
0.3
0.2
0.1
0
0.2
0.4
0.6
0.8
1
x
Summary
•
•
•
Optimization algorithms → exact results for finite spin glass models
Semi-quantitative understanding of dynamics (relaxation) in lattice models
For mean-field models: SK ground states for small N are consistent with RSB
solution and other numerical results.
•
•
Related models are introduced: AFM model is ”higher” frustrated for finite N .
•
Predictions from energy scaling:
Ground-state energy e0,afm is estimated.
– SK and AFM model have the same fluctuation exponent: Θ f /d ' 1/4.
•
– The shift exponent is different: Θs /d ' 1/3 (SK) and -2/3 (AFM).
Outlook:
– Phase transition between ferromagnetic and spin-glass ground state near
x = 1/2?
– Another related model: A fully connected ±J model
Acknowledgement
K LAUS H ANDRICH
A NDREAS H ARTWIG
F RANK DASKE
T HOMAS K LOTZ
G ÜNTER M ILDE
PAWEL P OLASZEK
M ARIUSZ N OGALA
J AROSŁAW K ŁOS
J AROSŁAW K RAWCZYK
E UGENIO VOGEL , J AIME C ARTES , PATRICIO VARGAS , D ORA A LTBIR

Exact Ground States of Finite Ising Spin Glasses Obtained by

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