Universal phase diagram
of interacting quantum liquids
near the unitarity limit
Predrag Nikolić and Subir Sachdev
Harvard University
Overview
Introduction and motivation
Universality in quantum liquids
Review of scattering theory
Interacting atoms: one-channel model
Interacting atoms and molecules: two-channel model
Large-N expansion
Critical point at the Feshbach resonance
Zero-temperature phase diagram
Some universal ratios
Unitarity in optical lattices (work in progress)
Conclusions
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Introduction and motivation
Fundamental issues
Attractive interactions between particles => pairing correlations
Weak pairing => BCS superconductor
Strong pairing => Bose-Einstein condensate (BEC) of molecules
The effects of temperature, magnetic field, disorder...
Unitarity limit – the strongest pairing correlations
Infinite scattering length => universality
Crossover between BEC and BCS superfluidity
Exotic phases
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)
Breached pair superfluid (Sarma)
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Exotic phases in Zeeman fields
Breakdown of superconductivity
Pauli-Clogston limit: μ↑-μ↓~Δ
First-order phase transition: superconductor→ normal...
Or: superconductor → ??? → normal
FFLO: non-uniform
magnetized superfluid
Q=kF↑-kF↓
Breached pair (Sarma)
phase
Deformed Fermi sea
phase
Q=0
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Recent experiments at Rice
G.B.Partridge, W.Li, R.I.Kamar, Y.A.Liao, R.G.Hulet
Science 311, 503 (2006)
G.B.Partridge, W.Li, Y.A.Liao, R.G.Hulet, M.Haque, H.T.C.Stoof
Phys.Rev.Lett. 97, 190407 (2006)
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Recent experiments at MIT
M.Zwierlen, A.Schirotzek, C.H.Schunck,
W.Ketterle; Science 311, 492 (2006)
Y.Shin, M.W.Zwierlein, C.H.Schunck,
A.Schirotzek, W.Ketterle;
cond-mat/0606432
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Universality in quantum liquids
Scattering theory in Quantum Mechanics
Hard sphere repulsive potential
,
Spherical potential well
,
,
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Unitarity limit
Unitarity limit
Universality
Irrelevant microscopic details of the scattering potential
Resonant scattering
Energy close to a bound or metastable state
=>
Unitarity in one-dimension
when
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Field theory at unitarity
Field theories
One-channel approach
Two-channel approach
Sp(N) generalization
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Feshbach resonance
Fixed point
Predrag Nikolić
One channel model
Critical field theory of interacting atoms
Relevant perturbations
Exact renormalization group
Perturbing about the vacuum (T=0, μ=h=0)
=0
=0
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
≠0
Predrag Nikolić
RG in one channel model
Renormalization group flow
d<2, repulsive interactions
stable fixed point: universal (dilute) quantum liquid
d>2, attractive interactions
unstable fixed point: Feshbach resonance
Detuning from the Feshbach resonance:
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Two channel model
Critical field theory of interacting atoms and molecules (s-wave)
Relevant perturbations
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Feshbach resonance
Exact renormalization group
=0
≠0
Renormalization group flow
The critical exponents of the relevant operators are the same
as in the one-channel model
Both models describe the same fixed point => Feshbach resonance
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
ε-Expansions
Thermodynamic properties are universal functions at the resonance
Expressed as expansions in powers of ε
One-channel: ε=d-2; two-channel: ε=4-d
Fundamental limitations
Perturbation theory in the interaction strength (u0, g0)
Critical coupling must be small (u0, g0~ ε)
Tractable effects of pairing fluctuations are small
One-channel => negligible pairing amplitude
Two-channel => deeply in the superfluid, or 1 Fermi sea
Large-N expansion allows strong coupling at the resonance...
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Large-N expansion
Sp(2N) generalization of the two-channel model
N-pairs of spin-up and spin-down fermions, ψσ (σ=1..2N)
Fermions coupled to a single molecule field,Φ
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Relevant perturbations
μ ... chemical potential
h ... Zeeman field
ν ... detuning from the Feshbach resonance
Renormalization group flow:
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Universal mean-field phase diagram
Free energy of molecules
Static uniform superfluid (mean-field Φ=const. at N=∞)
Integrate-out fermions (Gaussian)
Obtain |Φ| by minimizing F(|Φ|)
Substitute |Φ| in the fermion action to check the Fermi seas
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
T=0 phases and transitions
hc(0)=0.807μ+O(1/N)
Predictions
1st order transition between the superfluid and normal phases
Smooth BEC-BCS crossover
Uniform magnetized BEC superfluid phase for μ<0
Normal phases with one (1N) or two (2N) Fermi seas
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Other views of the T=0 phase diagram
The number of Fermi seas is trivially decided by h
in the absence of superfluid
Magnetized superfluid seems fragile and occurs only
when pairing fluctuations in vaccum can nucleate stable molecules
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Equation of state: Normal phase at T=0
Pairing fluctuations
increase pressure
Longer-lived pairs
=> larger pressure
No pressure increase
without two Fermi seas
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Finite temperatures
2nd order superluid-normal phase transition at T=Tc
Monte-Carlo:
N=1
MC
4.28948
3.247
7.33124
6.579
0.53648
0.776
E.Burovski, N.Prokof'ev, B.Svistunov, M.Troyer
Phys.Rev.Lett. 96, 160402 (2006)
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
To do...
Effects of pairing fluctuations on fermions
Single particle Green's function, 1/N expansion
Density profiles and correlations in the normal phase
1/N corrections to the phase diagram
Translation to the canonical ensemble
Numbers of particles are fixed in experiments
rather than chemical potentials
Non-uniform superfluid phases (FFLO)
Unitarity in optical lattices (work in progress)
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Unitarity in optical lattices
2nd order superluid-insulator phase transition at T=0, h=0
Band insulator at N=∞
Mott physics .... 1/N corrections
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić
Conclusions
Theoretical treatment of unitarity in interacting quantum liquids
Interacting atoms: one-channel model (ε=d-2)
Interacting atoms and molecules: two-channel model (ε=4-d)
Sp(N) generalization
Universal phase diagram
Feshbach resonance is an RG fixed point
Uniform superfluid and normal phases
1st order superfluid-normal transition by population imbalance
Exotic magnetized uniform superfluid at T=0
Universal ratios of observables at T>0 phase transition
Unitarity in optical lattices (work in progress)
Universal Phase Diagram of interacting quantum liquids near the unitarity limit
Predrag Nikolić