Off Diagonal Long Range Order in Superconductor
C.N.Yang Rev.Mod.Phys. 34 (1962) 694
Suzuki-Kusakabe Laboratory
Fumiya Kanetake
Contents
1.
2.
3.
4.
5.
6.
Introduction
- Macroscopic quantum phenomenon
- Order parameter of superconductivity
- Quantization of magnetic flux
- Off Diagonal Long Range Order
Density matrix
Diagonalization
Diagonalization of 2-particle density matrix
Logic flow
Summary
2
Macroscopic quantum phenomenon
Macroscopic quantum phenomenon:巨視的量子現象


Superconductivity and superfluidity are macroscopic
quantum phenomena.
They are achieved by a macroscopic number of
particles occupying the same state. (Bose-Einstein
condensation)
boson
fermion
3
Bose-Einstein condensation
Order parameter of superconductivity
Order parameter:秩序パラメータ
Order parameter (G-L wave function)
(macroscopic wave function of Cooper-pairs)
Wave function
Density of electron
Density of Cooper-pairs
The Schrödinger equation
The Ginzburg-Landau equation
Superconducting current
Current
Free energy in superconductivity
4
Quantization of magnetic flux
Meissner effect
Stokes’ theorem
Order parameter has to be single valued. 秩序パラメータの一価性
Quantization of magnetic flux:磁束の量子化
Quantization of Magnetic Flux needs that
“order parameter is coherent over the entire ring”
5
coherent:可干渉性の,位相がそろった
Off Diagonal Long Range Order
ODLRO:非対角長距離秩序
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
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We need an index of “coherence of order parameter”.
We adopt “Off Diagonal Long Range Order”
(ODLRO) as its index.
If some off-diagonal elements of the density matrix
have finite values for infinite separation,
ODLRO is defined to exist.
↓2-particle density matrix means 2-particle correlation.
⇒The system has ODLRO in ρ2 .
6
Off Diagonal Long Range Order


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Off-diagonal elements of the 2-particle density matrix
means that “2-particle can travel from (y1,y2) to (x1,x2)
without disturbing the state of the system”
∞ means that the distance has a macroscopic scale.
(~mm, cm)
The existence of ODLRO confirms superconductivity
regardless of the origin (electron-phonon coupling,
spin fluctuation, charge fluctuation, …) of the
appearance of superconductivity.
7
Density matrix
Density Operator:密度演算子
Partition function:分配関数
Tr: 行列の対角和
⇒ρ has all Information of the system
1-particle density matrix
2-particle density matrix
Field operator :場の演算子
8
Diagonalization of matrix
diagonalization: 対角化
Hermitian matrix:エルミート行列
Unitary matrix:ユニタリー行列
Diagonal matrix:対角行列
9
Diagonalization of 2-particle density matrix
If ρ2 has large eigen value.
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ρ2 has a large eigen value. ⇒ The system has ODLRO in ρ2 .
This condition implies existence of Cooper-pair.
This result does NOT relate detail of Hamiltonian.
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Logic flow
ρ2 has a large eigen value.
⇒ The system has ODLRO in ρ2.
⇒ Quantization of magnetic flux appears.
⇒ The system shows superconductivity.
When does a big eigen value appear?
⇒It seems that it is time when the system condense in
Bose-Einstein condensation.
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Summary




Existence of ODLRO means “2-particle can travel
everywhere without disturbing the state of the
system”.
Existence of ODLRO indicates “coherence of order
parameter”.
ODLRO is a good index characterizing
superconductivity. (or superfluidity)
The Cooper-pair system has ODLRO in ρ2.
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Appendix
1.
2.
3.
Reduced density matrix
Density matrix of free boson system
Density matrix of free fermion system
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Reduced density Matrix
1-particle Density Matrix (M×M)
2-particle Density Matrix (M2×M2)
means 1-particle correlation from x to y
means 2-particle correlation from (x1,x2) to (y1,y2)
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Density Matrix of free boson system
p is diagonal for ρ1
Complete relation
Riemann-Lebesgue’s
theorem
T>Tc → 0
T<Tc → ≠0
ODRLO in ρ1 at T<Tc 15
Density Matrix of free fermion system
No ODRLO in ρ1
16

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