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 K@s.v.p.

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 K@s.v.p.

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 K@s.v.p. 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