( e-e interaction energy is V(r) ~1/rg )
Electrons (g=1) form Wigner crystals at T=0 and small n when rs >> 1 and Epot>>Ekin In the opposite limit the system is in the Fermi liquid state.
3He and 4He (g>2 ) are crystals at large n At T=0 the crystal melts by quantum fluctuations at rs =rc >>1 Most quantum liquids in nature are strongly correlated with rS >1.
Dg
s
Dgpot
Dkin
nr
nEnE2
//2
−
∝
∝∝
Interparticle interaction can be characterized by a parameter rs =Epot /Ekin
Hierarchy of characteristic energies in weakly interacting liquids rs<<1
Bose liquid
Ed =EF
Weakly interacting Fermi liquid
Ed =Tc
Weakly interacting classical gas
Weakly interacting superfluid liquid
Θ is Debye (or plasma) frequency, Tc is the Bose condensation critical temperature EF is the Fermi energy
Epot
Fermi liquid
Weakly interacting classical gas
Epot Θ
Θ
Hierarchy of characteristic energies in strongly interacting quantum liquids rc>>rs>>1
Epot
strongly correlated degenerate Fermi or Bose liquid
Ed Θ
Semi-Quantum liquid classical liquid weakly interacting
classical gas
The subject of the talk: If Ed << T << θ << Epot the liquid is not degenerate. It is still not a gas ! It is also not a classical liquid !
,2/1 pots
pot
s
potd E
rE
rE
E <<=Θ<<=
A digression: features of classical liquids: Classical liquids exist because crystal melting temperature is about 100 times less than Epot They exhibit short range order and, in this sense, they are close to crystals. (Frenkel 1946, Shockley 1949) a model: Particles quickly oscillate in a cage made by other particles with the frequency Θ. Lifetime in the cage: τ>>1/Θ.
viscositysliquid'theis
???exp1
τη
τ
∝
⎟⎟⎠
⎞⎜⎜⎝
⎛−Θ≈TEpot
Viscosity of classical liquids decreases exponentially with temperature. Viscosity of gases increases with temperature.
gas classical liquid
T
η
What is the viscosity of semi-quantum liquids? What is the heat capacity of semi-quantum liquids? What is the resistivity of conductors in semi- quantum regime?
Features of strongly correlated quantum degenerate liquids: The effective mass and the spin susceptibility of Fermi liquids are significantly enhanced. Bose-condensed fraction in Bose liquids is small.
A model for strongly correlated degenerate liquids: particles quickly oscillate in a cage made by other particles with frequency Θ. The life time in the cage τ >>1/EF
distancecleinterpartianis
; 2*
aa
mEdτ
τ!!
≈≈
T <<θ
Epot
snapshot of the potential:
θ
Snapshot of the single particle potential in semi-quantum liquids. On the time scale t<τ the system looks like a glass!
The low energy (ε<Θ) density of states of glasses ν is temperature independent. (Halperin, Anderson, Varma) A conclusion: the heat capacity of semi-quantum liquids at T<θ is proportional to T, independently of the particle’s statistics (Bose or Fermi)
What about experiment?
Temperature dependence of the heat capacity of He3 and He4 in the temperature interval 1-20K is not known. WOW!
KK HeHe 204034≈Θ≈Θ
TC ν≈
Steps to calculate the viscosity of semi-quantum liquids : 1. Assume: there is no characteristic frequencies in the system smaller than Ed ~1/τ 2. Apply shear deformation e(ω) . Calculate the energy dissipation 3. Put ω=1/τ
( ) ( ) ( )
( ) ( )
( ) 2
22
222
22
22
||||0
||||||||2
1;21
0
TVSVn
T
neVT
neVT
thdtdE
ETedtdE
f
d
!!
!
≻≻≻≻
=∝=
≈=
==
==
νωη
νπω
νωωπ
τωωηωω
ωτωηωηT
ω
He4
Temperature dependence of the viscosity of He3 and He4 in the temperature interval 1-20K is not known. WOW!
T
TED
1∝
Θ
η
≺≺
H
( ) 2
22 ||||0
TVSVn
T×=∝= !
!νωη
S is the entropy density
Sl
lSvl
!
!
≈⇒=
∝∝
ηλλ
ρη
:resulttransportBoltzman“Minimal” viscosity:
T
η
Θ EF Epot
2
1T
∝T1
∝⎟⎠
⎞⎜⎝
⎛∝T?exp
Does the semi-quantum regime exist in the electron system in semiconducting quantum wells? In semi-quantum liquids hydrodynamics works starting with the spatial scale of order inter-electron spacing
( )
T
Dau
uF
1
2/ln1
∝∝
=∝
ηρ
ηη
The case of rare strong impurities Stokes law:
The case of smooth potential: Andreev, Kivelson, Spivak
Tba 11
∝+∝κ
ηρ
Steps to calculate of conductivity of semi-quantum electronic liquids : 1. Assume: no characteristic frequencies in the system smaller than Ed ~1/τ 2. Calculate energy dissipation in the presence of electric field E(ω) 3. Put ω=1/τ
( ) ( )
( ) ( ) ( )
( ) ( ) !;|~|0
|||~||||~|2
0
1;
22
2
222
22
2
TnVeaT
naeVT
naeVT
thdtdE
f
ETdtdE
d
∝∝=
Ε≈Ε=
==
=Ε=
ρντ
ωσ
νωπ
νωωπ
ωτωσωσ
τωωσω
!
!!
≻≻≻≻
1
2ω
f
ω
Sr2RuO4
Tyler & Mackenzie, Physica C (97)
Sr2RuO4
Linear in T resistivity of strongly correlated metals at high temperatures
How small could the conductivity be in the framework of the semi-quantum liquid theory ?
( ) MFF
Msq VEET σνσσ <<Θ
=Θ= 2**
|~|)(
Yoffe –Regel limit
tyconductiviminimalsMott'theis
2
M
MDF aeal
σ
σσλ!
≈≈⇒≈≈
3:tyconductiviDrude
2 νσ
lve FD =
A problem:
1
2ω
σ
ω
Drude conductivity based on conventional Fermi liquid theory decreases with frequency
( )( )21 tr
D
ωτσ
ωσ+
=
In the framework of the semi-quantum liquid the conductivity increases with the frequency
La1.9 Sr0.1CuO4 at specified temperatures. The inset shows the in-plane resistivity data up to 1000 K. From Hussey et al., 2004.
A suspect for semi-quantum regime
Conclusion:
Semi-quantum interval of temperature in strongly correlated liquids, probably, exists.
Almost nothing is known about it either experimentally or theoretically
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