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Critical behaviors near Lambda transition of 4He
quantized vortexquantized circulation (k)
(n = 0, 1, 2, …)
:superfluid velocity
macroscopic coherence, i.e., const. q
𝒗" =ℏ𝑚𝛻𝜃
q
3D-XY model
𝜌"�Order parameter (O.P.) in superfluid state:
rS: superfluid densityq : phaseΨ = 𝜌"� 𝑒,-.
𝜅 = 0 𝒗1 2 𝑑𝒍5
=ℎ𝑚 𝑛
𝜉
macroscopic quantum state
Landau theory for 2nd order phase transitions
Free energy F can be expanded with O.P. within mean field theory for second-order transition
A0, B > 0 (B depends weakly on T)
When T < Tc , minimum of F is given by:
rS: superfluid densityq : phase
Order parameter (O.P.) in superfluid state:
Higgs modeNambu-Goldstonemode
O.P. (real part)
free energyT > Tc
T = Tc
T < Tc
0
Ψ = 𝜌"� 𝑒,-.
Mean field theory neglecting OP fluctuations
Off diagonal long-range order in BEC
Single-particle density matrix (two-point correlation function) r1(r, r') is :
: particle number operator
Fourier transforms of annihilation and creation operators (in momentum space). Introduce bosonic field operators , :
which follow ordinary Boson commutation rules:
𝜓: 𝜓:;
, , . 𝜓: 𝒓 ,𝜓:; 𝒓> = 𝛿 𝒓− 𝒓> 𝜓: 𝒓 ,𝜓: 𝒓> = 0 𝜓:; 𝒓 ,𝜓:; 𝒓> = 0
𝑁C 𝒓 = 𝜓:; 𝒓 𝜓: 𝒓
.𝜌D 𝒓, 𝒓> = 𝜓:; 𝒓> 𝜓: 𝒓
C(r, r') → 0 (|r - r'| → ∞) : short rangecorrelation
In Bose condensed systems,
.
(|r - r'| → ∞): off diagonal long-range order (ODLRO)
𝜌D 𝒓,𝒓> = 𝜓:; 𝒓> 𝜓: 𝒓 + 𝐶 𝒓, 𝒓>
Order parameter is or .Ψ 𝒓 = 𝜓: 𝒓 Ψ∗ 𝒓 = 𝜓:; 𝒓
𝜓: 𝒓 =1𝑉�J𝑒-𝒌2𝒓𝑎M
�
M
𝜓: ; 𝒓 =1𝑉�J𝑒-𝒌2𝒓𝑎M
N�
M,
Critical phenomena and scaling laws for phase transitions
Critical exponents
specific heat:
𝑇 > 𝑇Q𝐶 ∝ 𝑡,TU𝑇 < 𝑇Q
𝑀 ∝ 𝑡X
𝜒 ∝ 𝛾,[\
𝑡 ≡ 𝑇 − 𝑇Q 𝑇Q⁄
𝑀 ∝ ℎD _⁄
𝑇 = 𝑇Q
𝜉 ∝ 𝑡,`\𝜉 ∝ 𝑡,`U
magnetization:
magnetic susceptibility:
magnetization curve:
reduced temperature:critical temperature: 𝑇Q
correlation length:
𝑔 𝑟 ∝ 𝑟, c,d;ecorrelation function:
spatial dimension: 𝑑
anomalous dimension: 𝜂ℎ ≡ 𝐻 𝑘i𝑇Q⁄reduced magnetic field:
𝜒 ∝ 𝑡,[U
𝐶 ∝ 𝑡,T\
𝑔 𝑟 ∝ 𝑟, c,D d⁄ exp −𝑟𝜉𝑟 ≫ 𝜉
∞ self-similarity
𝛼 + 2𝛽 + 𝛾 = 2𝛿 = 1 + 𝛾 𝛽⁄
Scaling relations
𝛾 = 𝜈 2 − 𝜂
2− 𝛼 = 𝜈𝑑
𝛿 =𝑑 + 2 − 𝜂𝑑 − 2 + 𝜂
,,,
,
Scaling hypothesis𝐹"tuvwxyz ∝ 𝑡{𝑓 ℎ 𝑡}⁄ : singular part of free
energy density: arbitrary function of zf (z)
x, y : related to scaling indices
Universality classMaterials which belong to the same universality class show the same critical behaviors with the same critical exponents regardless of system details.
• SO(2) symmetryXY spins, ...
• U(1) symmetry (imaginary OPs)superfluid (lambda) transition, superconducting transition, ...
Quantum effects are not important at finite-T phase transitions.
• Z2 symmetryI-sing spins, gas-liquid critical point, critical solution of mixture liquids, ...
• SO(3) symmetryHeisenberg spins, ...
C
𝜶 = −0.0145(5)
TcC 𝜶 = −0.141(7)
C 𝜶 = 0.1098(2)𝐶 ∝ 𝑡,T
Universality class of two dimensional systemMaterials which belong to the same universality class show the same critical behaviors with the same critical exponents regardless of system details.
• SO(2) symmetryXY spins, ...
• U(1) symmetry (imaginary OPs)superfluid (lambda) transition, superconducting transition, ...
Quantum effects are not important at finite-T phase transitions.
• Z2 symmetryI-sing spins, gas-liquid critical point, critical solution of mixture liquids, ...
• SO(3) symmetryHeisenberg spins, ...
C
𝜶 = 0𝐶 ∝ 𝑡,T
No finite-T transition
C
Tc
TKT0BKT transition
No scaling
Renormalization group theory
Universal parameters:
Renormalization group theory for 3D O(2) universality class
series expansion(not based on divergence at Tc)
cf. From RG theory, 𝛼+ = 𝛼−.
determined from 2nd sound experiment
Subtract shorter scale fluctuations from a system, by coarse-graining with appropriate scale transformations.
At T = Tc, there are fluctuations of any length scales (self-similarity). ⟺ power-law of 𝜉
𝑆� → 2�𝑆� d⁄When , we should transform as for physical quantity S.𝑟 → 𝑟> = 𝑟 2⁄
𝑟,T → 2𝑟 ,T = 2 ,T 𝑟 ,T 𝑒,� �⁄ → 𝑒,d� �⁄ = 𝑒,� d�⁄When , or .𝑟 → 2𝑟
Experimental determinations of critical exponents for lambda transition
From the following relation, when a ≈ 0, logarithmic and power-law behaviors become indistinguishable.
Since experimental a value (= −0.01264) for lambda transition is small but finite and negative, the specific heat does not diverge but has a cusp with a finite value at T = Tl.
A+, A- and B are in J/mol K
(T < Tl)
(T > Tl)
universal parameters:Exp. data are fitted to:
M. Barmatz, I. Hahn, J.A. Lipa and R.V. Duncan, Rev. Mod. Phys. 79, 1 (2007)
𝑐dd =𝜌1𝑇𝑆d
𝜌�𝐶�+ 𝒪
𝑐d𝑐D
ddetermined from 2nd sound experiment
F. J. Wegner, Phys. Rev. B 5, 4529 (1972)
RG theory for 3D XY −0.0135(21)
𝜈 𝛼 + 𝜈d
0.67117(72)
0.67049(32) 1.9988(6)
2.0000(1)
High precision specific heat measurements for lambda transition under micro-gravity
Tl = 2.1768 [email protected].
a = − 0.01264± 0.00024
A+/A- = 1.05251± 0.0011
flight
ground
ground
T < TlT > Tl
flight
flight
-0.03
-0.02
-0.01
0
1970 1980 1990 2000 2010
α
Year
J.A. Lipa et al., Phys. Rev. Lett. 76, 944 (1996); Phys. Rev. B 68, 174518 (2003)M. Barmatz, I. Hahn, J.A. Lipa and R.V. Duncan, Rev. Mod. Phys. 79, 1 (2007)
-0.03
-0.02
-0.01
0
1970 1980 1990 2000 2010
αYear
RG theory
experimental result
Tl = 2.1768 [email protected].
A lnt + B
High precision specific heat measurements for lambda transition under micro-gravity
J.A. Lipa et al., Cryogenics 34, 341 (1994); Phys. Rev. B 68, 174518 (2003)
3He不純物0.45 ppb
DT ≈ 3×10-11 Knoise ≈ 10-4 F0/√Hzband width = 0.16 Hz
high-sensitivity magnetic thermometer: Cu(NH4)2Br4•2H2O
flight
ground
thermal noises due to cosmic-ray
thermal overshooting
T < Tl
T > Tl
4-step radiation shield stages
+ 3He exchange gas
flight experimentresidual gravity ≤ 2×10-6 g
Lambda transitions in magnetic systems
���(K)���������������������������� �����������������������
�
�
�
��
��
�
��
c P(k
J/kg
· K)
����
Tl Tc
P = 0.1013 MPa
0.2275 MPa
0. 5 MPa
liquid 4He
T (K)
CP
(J/g
K)
T𝜆 : 3D-XY type (lambda cusp of C)
Tc : 3D-Ising type (divergence of C at Tc)
CuK2Cl4・2H2O
10−3 ≤ t ≤ 10−1
3D-XY type
(Heisenberg? ferromagnet with bcc structure)A. R. Miedema et al., Physica 31, 1585 (1965)
P = 2.87 MPa
Superfluid transitions in 4He and 3He
Tl = 2.17 [email protected]. x(0) ≈ 0.1 nm Tc = 0.93 mK @s.v.p. x(0) ≈ 80 nm
T.A. Alvesalo, T. Haavasoja and M.T. Manninen,J. Low Temp. Phys. 45, 373 (1981)
BCS type with extremely narrowtG ≈ 1×10-5
planar phase at t ≤ 1×10-6 ?
Lambda type with widetG ≈ 10-1
M.J. Buckingham and W.M. Fairbank (1961)
Ginzuburg conditionfor critical T region (tG): (3D-XYモデル),
liq. 4He liq. 3He
BCS↔BEC crossover in superconducting transitions (?)
YBa2Cu3O6.92Bi2Sr2CaCu2Ox(optimally doped high-Tc superconductor)
A. Junod, A. Erb and C. Renner, Physica C 317-318, 333 (1999)
Tc = 5.3 K
vanadium
el.ex.:bosonic dimmer?
3D-XY type(lambda type)
el. ex.: fictitious molecule ?(well isolated Cooper pairs)
el.ex.: fermionicquasi particles
BCS typeBEC type
(low-Tc superconductor)(high-Tc superconductor)
G.S.: deeply over-lapped Cooper pairs
G.S.: BEC of molecules?
na03 >> 1na0
3 << 1