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Quantum Phase Transition in the BCS-to-BEC evolution of p-wave Fermi gases
Sergio Botelho and Carlos Sa de Melo
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Recent experiments in cold fermionic gases have shown that s-wave magnetic field induced Feshbach resonance can be used to study the BCS-to-BEC evolution from large Cooper pairs on the higher magnetic field side of the resonance (BCS regime) to small diatomic molecules on the lower magnetic field side of the resonance (BEC regime) [1,2,3]. These studies led to the first experimental realization of the theoretically proposed BCS-to-BEC crossover in three-dimensional continuum s-wave superfluids [4]. Although a great deal of both theoretical and experimental activity has taken place in the area, most of the efforts described only the BCS-to-BEC crossover in s-wave systems.
Our research aims at investigating the BCS-to-BEC evolution in p-wave fully spin-polarized Fermi gases, where p-wave Feshbach resonances have already been observed experimentally [5]. Using a functional integral formalism, we showed that a quantum phase transition takes place when the chemical potential crosses a critical value, instead of the usual smooth BCS-to-BEC crossover that occurs in s-wave superfluids. We study the case of two-dimensional systems, which can be prepared experimentally through the formation of a one-dimensional optical lattice, where atom transfer between latice sites is suppressed by a large trapping potential, as shown in Fig. 1 below.
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Fig. 1: One-dimensional optical lattice that can be used as the atom trap.
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We consider the case of a homogeneous (trap potential not included in the Hamiltonian) and uniform continuum model of spin-polarized (all atoms in the same hyperfine state) fermionic atoms in the presence of an external magnetic field. The direction of the magnetic field, which is chosen to define the spin quantization axis, need not coincide with the spatial direction of the laser beam.
The inter-atomic potential energy is obtained by expanding the k-space interaction in plane waves, and isolating only the contribution from the lth angular momentum channel to the scattering process that is responsible for the interaction between the fermionic atoms. This results in a separable potential, that is, a V(k,k') which is a product of a function of k only and a function of k' only. This greatly simplifies the analytical treatment of the problem, and allows one to proceed much further with the investigation of the BCS-to-BEC evolution. See Ref. [6] for more details on the Hamiltonian of the system and on the derivation of a separable expression for the interaction potential.
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The saddle-point solution of the functional integral that determines the partition function of the system yields the BCS order parameter and number equations. These equations can be self-consistently solved for the chemical potential and order parameter amplitude as functions of the binding energy. This allows the BCS-to-BEC evolution region to be studied as a function of the interatomic coupling strength at fixed particle density. Our numerical results for these quantities are shown below in Fig. 2.
Fig. 2: Chemical potential and order parameter amplitudes as functions of the binding energy at T=0 for p-wave symmetry. |
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We also calculated the momentum distribution n(k) of the fermionic atoms in the case of p-wave interaction. The results are shown in Fig. 3 for different values of the chemical potential. Notice that n(k) becomes discontinuous when the chemical potential crosses the bottom of the band,
which coincides with the collapse of the Dirac points to a single point k=0 and the appearence of a full gap to the addition of quasiparticles.
Fig. 3: Momentum distribution n(k) for three different values of the chemical potential across the BCS-to-BEC transition region. Notice the discontinuity of n(k) at k=0 when mu=0. |
This major rearrangement of the momentum distribution has a dramatic effect in the atomic compressibility of the system. In fact, we found that the compressibility develops a cusp when expressed as a function of the binding energy (or chemical potential), as shown in Fig. 4. This non-analytic behavior can be explained in terms of the collapse of the Dirac points toward k=0, together with the vanishing of the excitation energy when the chemical potential crosses zero. This strongly suggests the existence of a quantum critical point at mu=0.
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Fig. 4: Atomic compressibility as a function of the binding energy. Inset: First derivative of the compressibility with respect to the binding energy. |
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The superfluid density tensor is associated with phase twists of the superconductor order parameter. We also analyzed the behavior of this quantity as a function of temperature (for small T), and observed that it undergoes a sudden change of behavior as the critical binding energy is crossed, as shown in Fig. 5. In fact, it goes from a power-law behavior on the BCS side (consistent with the nodal structure of the p-wave excitation spectrum) to an exponential behavior on the BEC side (reflecting the appearence of a full gap to the addition of quasiparticles as the chemical potential becomes negative). The figure also shows (inset) the zero-temperature slope of the superfluid density over T^2 as a function of the binding energy, which is clearly discontinuous at the critical point.
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Fig. 5: Superfluid density divided by T^2 as a function of temperature for various values of the binding energy. Inset: Zero-temperature slope of the superfluid density over T^2. |
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We are proposing the existence of a quantum phase transition in the BCS-to-BEC evolution of p-wave fully spin-polarized Fermi gases as a function of the two-body bound state energy. We have shown that the momentum distribution undergoes a major rearrangement in k-space at a critical value of the binding energy, which leads to a non-analytic behavior of the atomic compressibility of the gas. Also, the low-temperature superfluid density of the system was shown to change dramatically as the critical point is crossed, with a zero-temperature slope that is discontinuous at the critical binding energy.
We suggest that this phase transition may be observable in traps of Li and K gases which exhibit p-wave Feshbach resonances, and may be investigated through the direct measurement of the atomic compressibility, spin susceptibility or superfluid density as functions of binding energy or magnetic field.
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[1] C.A. Regal, M. Greiner, and D.S. Jin, Phys. Rev. Lett. 92, 040403 (2004).
[2] T. Bourdel et al, Phys. Rev. Lett. 93, 050401 (2004).
[3] K.E. Strecker, G.B. Partridge, and R.G. Hulet, Phys. Rev. Lett. 91, 080406 (2003).
[4] C.A.R. Sa de Melo, M. Randeria, and J.R. Engelbrecht, Phys. Rev. Lett. 71, 3202 (1993).
[5] C.A. Regal, C. Ticknor, J.L. Bohn, and D.S. Jin, Phys. Rev. Lett. 90, 053201 (2003).
[6] S.S. Botelho and C.A.R. Sa de Melo, cond-mat/0504263 (2005).
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