B. Spivak University of Washington

Semi-quantum liquids

A.F. Andreev Pisma JETP 28, 603, 1978 JETP 77, 2518, 1979

( 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

T

rs

Quantum liquid (Fermi or Bose)

crystal

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

a suspect for semi-quantum region

Frequency dependence of resistivity is unavailable

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

The Pomeranchuk effect.

The temperature dependence of the heat capacity of He3. The semi-quantum regime. The Fermi liquid regime.

He3 phase diagram:

The liquid He3 is also strongly correlated liquid: rs; m*/m >>1.