arXiv:1302.5843v3 [cond-mat.stat-mech]
Ising formulations of many NP problems
arXiv:1302.5843v3 [cond-mat.stat-mech]
Ising formulations of many NP problems
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Ising model?
Quantum Version?
How is related to TSP?
the ising model
Ludmila Augusta Soares Botelho
History time: around 100 years ago
Is there is a very simple model to describe ferromagnetism?
Is there a phase transition in the magnetism of a linear sequence of little magnetic moments where only neighbors are energetically coupled?
(Ferro)magnetism
Wilhelm Lenz
{
{
{
z
History time: around 100 years ago
(Ferro)magnetism
Wilhelm Lenz
Ernst Ising
?
!
Wilhelm Lenz
Ernst Ising
?
Model
It does not have phase transition!
... what about some quantum mechanics?
... more dimensions?
History time: around 100 years ago
( the best time)
Why do we care about Statistical Mechanics?
Physics time
( the best time)
Physics time
Why do we care about Statistical Mechanics?
It relates the
microscopic
with the
MACROSCOPIC
How is that possible?
(x,y,z)
m
m
p
(x,y,z,px,py,pz)
(x,y,z)
v
Statistical mechanics
m
(x,y,z,px,py,pz)
- Microstate
Statistical mechanics
m
j=(x,y,z,px,py,pz)
- Microstate
j=(x1,...,x3np1,...,p3N)
- For N particles:
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
Statistical mechanics
m
- Microstate
j=(x1,...,x3np1,...,p3N)
- For N particles:
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
Statistical mechanics
m
- Microstate
j=(x1,...,x3np1,...,p3N)
- For N particles:
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
Statistical mechanics
- Microstate
j=(x1,...,x3np1,...,p3N)
- For N particles:
thermodynamic Quantities
- Macrostate
- Equilibrium observable state
microstate
thermodynamic Quantities
- Macrostate
Condensed matter physics
The bridge
Ω(E)=Number of Microstates
S=−kbi∑pi lnpi
- Entropy
S=kblnW
W=∏iNiN!
S=kblnΩ
- Gibbs Entropy
- Postulate: All microstates are equally probable
The bridge
- Entropy
S=kblnΩ
- State Function/Fundamental Equation
S=S(U,V,N)
T1=∂U∂S
Tp=∂V∂S
T−μ=∂N∂S
temperature
pressure
chemical potential
The bridge
- Helmholtz Free Energy
F=U−TS
U=−μH
m=−(∂H∂F)T,N
z
H
- Legendre Transformation for internal energy U=Q+W
χ(T,H)=−(∂H∂F)T,N
U=−μ0H
→TCH
M=N⟨m⟩
dF=MdH−SdT
Statistical Mechanics
Why do we care about Statistical Mechanics?
It relates the
microscope
with the
MACROSCOPE
- Few assumptions
Canonical Ensemble
S
R
~
~
~
- Probability of particular microscopic state
Pj=Ze−βEj
Z=σ∑e−βEσ
ustandssumme
- Partition function
(T, V, N)
- Few assumptions
Canonical Ensemble
S
- Probability of particular microscopic state
Pj=Ze−βEj
Z=σ∑e−βEσ
ustandssumme
- Partition function
(T, V, N)
- Free Energy
- Systems Total Energy
- Heat Capacity
Connection with Thermodynamics
- Entropy
- Magnetic Susceptibility
- Magnetization
R
Statistical Mechanics
quiz: why the "-" signal?
Systems energy
H=−Ji∑sisi+1−hi∑Nsi
- Hamiltonian
{
interaction
{
external field
Z=?
z
h
⋯
⋯
{
Free Energy
Magnetization
Partition Function
1D
quantum mechanics
H=−Ji∑sisi+1−hi∑Nsi
H=−Ji∑σizσi+1z
Pauli Matrices
Reduces to classical version
H=−Ji∑σizσi+1z−hi∑Nσiz
σx=(0110)
\sigma^x= \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
σy=(0i−i0)
\sigma^y= \begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}
σz=(100−1)
\sigma^z= \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
E=−Ji∑σiσi+1
{
\{
transverse field
energy levels σi=±1
diagonal!
{
\{
- It has phase transition!
-Ferromagnetic vs Paramagnet
σiα=I⊗i−1⊗σαI⊗N−i
H=−Ji∑σizσi+1z−hi∑Nσix
quantum mechanics
H=−Ji∑σizσi+1z−hi∑Nσix
H=−21i∑N[Jxσixσi+1x+Jyσiyσi+1y+ Jzσizσi+1z−hσiz]
H=−21i∑N[Jxσixσi+1x+Jyσiyσi+1y+ −hσiz]
Jx=(21+γ)
Jy=(21−γ)
- Tensor Networks
- Quantum Many-Body Problems
- Heisenberg Model
- XY Model
- 1D Quantum Ising Model
H=−Ji∑σizσi+1z−hi∑Nσix
J>0
J<0
J=?
H=−Ji∑σizσi+1z
Ground state
Spin Glass Model
H=−i,j∑Ji,jσizσjz
≤
}
min
?
magnet
Thank you!
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arXiv:1302.5843v3 [cond-mat.stat-mech] Ising formulations of many NP problems
Ising Model
By ludmilaasb
Ising Model
- 222
