PRL 113, 019602 (2014)
Feiguin and Fisher Reply: In Ref. [1], we studied a generalized version of the attractive U Hubbard model, with spin-dependent hoppings. Because of the resulting mismatch in the Fermi surfaces, exotic paired states could potentially be realized. A BCS mean-field treatment of the Hamiltonian revealed a phase with a gapless superfluid, in which a fraction of fermions pair with zero total momentum, while another fraction remains unpaired in a normal state. This phase—dubbed “gapless superfluid” or GSF—is reminescent of the Sarma phase in conventional superconductivity, where phase separation takes place in momentum space, but translational invariace is not broken. In their Comment [2], Chiesa and Batrouni (CB) argue that a treatment of the attractive U Hubbard model considering Hartree terms in the mean-field decoupling of the Hamiltonian moves the phase boundaries in such a way that the gapless superfluid phase is completely replaced by phase separation. Considering different decouplings of the interactions points to revealing the competition between different phases. One would typically search for certain instabilities, based on physical intuition. We have considered only superfluidity in our work. In this case, it is reasonable to use a decoupling of the BCS form. One could also consider other decoupling channels. In the positive U Hubbard model, one would normally seek antiferromagnetism, while in the case that concerns us, one should consider a charge density wave. A proper treatment of this term should consider a charge density wave order parameter defined as X † ni ¼ hci;σ ci;σ i ¼ n þ δ cos ðQ · ri Þ; σ
where n ¼ N=L2 is the total density, which is fixed in the canonical ensemble, and the order is characterized by a wave vector Q. Conventionally, one would solve separately for both decouplings and compare energies. When adding both terms at the same time, self-consistency presumably would pick one order parameter over the other, but it might lead to a phase where both the superfluid order parameter and δ acquire a finite value. This kind of coexistence is unlikely and the system might prefer to phase separate. CB studied a uniform or homogeneous phase. In this case the Hartree-type correction above is often ignored in BCS theory, since with δ ¼ 0, ni ¼ n does not represent an order at all, but the lack thereof, i.e., the normal state. It is not determined self-consistently but introduced by hand: it is constant and just equal to the density. This translates into a correction to the chemical potential, as CB describe. The self-consistent equations remain unchanged, but now
0031-9007=14=113(1)=019602(1)
week ending 4 JULY 2014
PHYSICAL REVIEW LETTERS
the quasiparticle dispersion has a nontrivial density dependence on n. This illustrates that within mean-field theory there is not one “correct” decoupling scheme, so that the decoupling must be chosen with a physical picture in mind. Moreover, while the CB decoupling is certainly reasonable, it would presumably overestimate tendencies towards phase separation. Although the results with the Hartree term cast doubt on the stability of the GSF phase at the mean-field level, it does not rule out the possibility nor does it change the physics at a conceptual level. As mentioned in our work, the choice of parameters for the model is not unique, and the particular example used in the calculation—with Fermi surfaces rotated 90° with respect to one another—is a proof of concept scenario. Any mismatch in the Fermi surfaces could favor unconventional pairing and other possibilities should be explored, such as anisotropic hoppings for only one spin species. Leaving these arguments aside, it is still clear that a meanfield treatment is quite limited, and would not reveal the presence of other competing phases, such as the “d-wave paired phase.’. Indeed, CB do not cite a density matrix renormalization group study of this model on 2-leg ladders, presented by Feiguin and Fisher in Ref. [3]. In this work, numerically exact simulations reveal a quite different picture to the one described by the mean-field solution. In fact, phase separation and the GSF phase are no longer present, and are replaced by a quasisuperfluid with finite center of mass momentum and a d-wave paired state at large anisotropy. This shows that the exotic paired phases can be realized and are particularly robust in low dimensions. But they also point at the fact that a mean-field has a quite limited predictive power when it comes to low dimensions and phases that have a nontrivial order parameter. Adrian E. Feiguin1 and Matthew P. A. Fisher2 1
Department of Physics Northeastern University Boston, Massachusetts 02115, USA 2 Department of Physics University of California Santa Barbara, California 93106, USA Received 25 March 2014; published 2 July 2014 DOI: 10.1103/PhysRevLett.113.019602 PACS numbers: 67.85.-d, 03.75.Lm, 74.20.-z, 74.25.Dw [1] A. E. Feiguin and M. P. A. Fisher, Phys. Rev. Lett. 103, 025303 (2009). [2] S. Chiesa and G. Batrouni, preceding Comment, Phys. Rev. Lett. 113, 019601 (2014). [3] A. E. Feiguin and M. P. A. Fisher, Phys. Rev. B 83, 115104 (2011).
019602-1
© 2014 American Physical Society
Feiguin and Fisher Reply: In Ref. [1], we studied a generalized version of the attractive U Hubbard model, with spin-dependent hoppings. Because of the resulting mismatch in the Fermi surfaces, exotic paired states could potentially be realized. A BCS mean-field treatment of the Hamiltonian revealed a phase with a gapless superfluid, in which a fraction of fermions pair with zero total momentum, while another fraction remains unpaired in a normal state. This phase—dubbed “gapless superfluid” or GSF—is reminescent of the Sarma phase in conventional superconductivity, where phase separation takes place in momentum space, but translational invariace is not broken. In their Comment [2], Chiesa and Batrouni (CB) argue that a treatment of the attractive U Hubbard model considering Hartree terms in the mean-field decoupling of the Hamiltonian moves the phase boundaries in such a way that the gapless superfluid phase is completely replaced by phase separation. Considering different decouplings of the interactions points to revealing the competition between different phases. One would typically search for certain instabilities, based on physical intuition. We have considered only superfluidity in our work. In this case, it is reasonable to use a decoupling of the BCS form. One could also consider other decoupling channels. In the positive U Hubbard model, one would normally seek antiferromagnetism, while in the case that concerns us, one should consider a charge density wave. A proper treatment of this term should consider a charge density wave order parameter defined as X † ni ¼ hci;σ ci;σ i ¼ n þ δ cos ðQ · ri Þ; σ
where n ¼ N=L2 is the total density, which is fixed in the canonical ensemble, and the order is characterized by a wave vector Q. Conventionally, one would solve separately for both decouplings and compare energies. When adding both terms at the same time, self-consistency presumably would pick one order parameter over the other, but it might lead to a phase where both the superfluid order parameter and δ acquire a finite value. This kind of coexistence is unlikely and the system might prefer to phase separate. CB studied a uniform or homogeneous phase. In this case the Hartree-type correction above is often ignored in BCS theory, since with δ ¼ 0, ni ¼ n does not represent an order at all, but the lack thereof, i.e., the normal state. It is not determined self-consistently but introduced by hand: it is constant and just equal to the density. This translates into a correction to the chemical potential, as CB describe. The self-consistent equations remain unchanged, but now
0031-9007=14=113(1)=019602(1)
week ending 4 JULY 2014
PHYSICAL REVIEW LETTERS
the quasiparticle dispersion has a nontrivial density dependence on n. This illustrates that within mean-field theory there is not one “correct” decoupling scheme, so that the decoupling must be chosen with a physical picture in mind. Moreover, while the CB decoupling is certainly reasonable, it would presumably overestimate tendencies towards phase separation. Although the results with the Hartree term cast doubt on the stability of the GSF phase at the mean-field level, it does not rule out the possibility nor does it change the physics at a conceptual level. As mentioned in our work, the choice of parameters for the model is not unique, and the particular example used in the calculation—with Fermi surfaces rotated 90° with respect to one another—is a proof of concept scenario. Any mismatch in the Fermi surfaces could favor unconventional pairing and other possibilities should be explored, such as anisotropic hoppings for only one spin species. Leaving these arguments aside, it is still clear that a meanfield treatment is quite limited, and would not reveal the presence of other competing phases, such as the “d-wave paired phase.’. Indeed, CB do not cite a density matrix renormalization group study of this model on 2-leg ladders, presented by Feiguin and Fisher in Ref. [3]. In this work, numerically exact simulations reveal a quite different picture to the one described by the mean-field solution. In fact, phase separation and the GSF phase are no longer present, and are replaced by a quasisuperfluid with finite center of mass momentum and a d-wave paired state at large anisotropy. This shows that the exotic paired phases can be realized and are particularly robust in low dimensions. But they also point at the fact that a mean-field has a quite limited predictive power when it comes to low dimensions and phases that have a nontrivial order parameter. Adrian E. Feiguin1 and Matthew P. A. Fisher2 1
Department of Physics Northeastern University Boston, Massachusetts 02115, USA 2 Department of Physics University of California Santa Barbara, California 93106, USA Received 25 March 2014; published 2 July 2014 DOI: 10.1103/PhysRevLett.113.019602 PACS numbers: 67.85.-d, 03.75.Lm, 74.20.-z, 74.25.Dw [1] A. E. Feiguin and M. P. A. Fisher, Phys. Rev. Lett. 103, 025303 (2009). [2] S. Chiesa and G. Batrouni, preceding Comment, Phys. Rev. Lett. 113, 019601 (2014). [3] A. E. Feiguin and M. P. A. Fisher, Phys. Rev. B 83, 115104 (2011).
019602-1
© 2014 American Physical Society
