universal phase diagram of fermionic quantum liquids near...
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Universal phase diagramof fermionic quantum liquids
near the unitarity limit
Predrag Nikolić and Subir Sachdev
Harvard University
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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
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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
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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
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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:
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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
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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
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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
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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
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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:
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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
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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)
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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
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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
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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)
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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