Universal phase diagramof fermionic quantum liquids

near the unitarity limit

Predrag Nikolić and Subir Sachdev

Harvard University

Motivation from recent experiments

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

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.StoofPhys.Rev.Lett. 97, 190407 (2006)

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

Field theory at unitarity

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

Field theories One-channel approach Two-channel approach Sp(N) generalization

Feshbach resonanceUnitarity = universalityRG Fixed point

One channel model

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

Critical field theory of interacting atoms

Relevant perturbations

Exact renormalization group Perturbing about the vacuum (T=0, μ=h=0)

= 0 = 0 ≠ 0

RG in one channel model

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

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:

Two channel model

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

Critical field theory of interacting atoms and molecules (s-wave)

Relevant perturbations

Feshbach resonance

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

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

ε-Expansions

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

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, g

0)

Critical coupling must be small ( u0, g

0~ 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 overcomes these limitations

Large-N expansion

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

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

Relevant perturbations

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

μ ... chemical potential h ... Zeeman field ν ... detuning from the Feshbach resonance

Renormalization group flow:

Universal mean-field phase diagram

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

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

Uniform T=0 phases and transitions

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

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

hc(0)=0.807μ+O(1/N)

Other views of the T=0 phase diagram

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

The number of Fermi seas is trivially decided by h in the absence of superfluid (mean-field)

Universal phase diagram in proximity of the RG fixed point

Equation of state: Normal phase at T=0

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

Pairing fluctuations increase pressure

Longer-lived pairs => larger pressure

No pressure increase without two Fermi seas

Finite temperatures

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

2nd order superluid-normal phase transition at T=Tc

4.28948 3.247

7.33124 6.579

0.53648 0.776

N=1 MC

Monte-Carlo: E.Burovski, N.Prokof'ev, B.Svistunov, M.TroyerPhys.Rev.Lett. 96, 160402 (2006)

Conclusions

Universal Phase Diagram of interacting quantum liquids near the unitarity limit Predrag Nikolić

Theory of unitarity in fermionic 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 BEC-BCS crossover 1st order superfluid-normal transition by population imbalance Exotic magnetized uniform superfluid at T=0 1/ N corrections to the equation of state in the T=0 normal phase Universal ratios of observables at T>0 phase transition