week ending 4 JULY 2014
PHYSICAL REVIEW LETTERS
Comment on “Exotic Paired States with Anisotropic Spin-Dependent Fermi Surfaces”
-1.70
2 mu2 (left axis) mu1 (right axis)
-1.80
0.4 1
-1.90
In Ref. [1] the authors investigated equal populations of spin-half fermions characterized by mismatched Fermi surfaces. The mismatch is caused by having up (down) particles moving preferentially along the x (y) direction on a square lattice. One of the claims of the authors is that, in the presence of on-site attraction, such a model supports an exotic gapless superfluid phase when treated at the meanfield level. Even more interesting, but outside the realm of mean-field solutions, is the possibility that the system will enter a Bose metal phase whose basic constituents are uncondensed d-wave fermionic pairs. Both phases are supposed to exist in the regime of large x-y anisotropy. The appeal of this model, compared to others supporting some of the same physics, resides in the fact that it invokes only the use of an experimentally realizable local attractive interaction. Here we show, however, that the conclusions that the authors draw from mean-field theory are modified when the Hartree contribution to the energy is taken into account. In drawing the phase diagram the authors employed Bogoliubov–de–Gennes theory, which corresponds to replacing the interaction Uni↑ ni↓ with H1MF ¼
X Uhc†i↑ c†i↓ ici↓ ci↑ þ H:c:
ð1Þ
i
and determining hc†i↑ c†i↓ i self-consistently. In a more complete Hartree–Bogoliubov–de Gennes mean-field treatment the potential energy is instead decoupled as H2MF ¼ H 1MF þ
X Uhc†iσ ciσ ic†i−σ ci−σ :
ð2Þ
iσ
As long as one is concerned with the determination of the character of a homogeneous phase, the distinctions brought by employing either Eq. (1) or Eq. (2) are largely irrelevant: using one or the other decoupling simply corresponds to a redefinition of the chemical potential. In particular, when targeting a particular density n, the chemical potentials obtained using the decoupling in Eqs. (1) and (2) are related by μ1 ¼ μ2 − Un:
ð3Þ
Once the dependence of μ on n is established, the issue of phase separation can be easily settled by looking for regions where dμ=dn is negative and locating the phase boundaries via Maxwell construction. This is where the choice of μ1 rather than μ2 alters the results. In particular, 0031-9007=14=113(1)=019601(2)
0.5
1.5
0.5
BCS
-2.00
0.3 ta/tb
PRL 113, 019601 (2014)
0 -2.10
0.2
-0.5 -2.20
-1
-2.30 -2.40
N
-1.5
0.2
0.4
0.6 Density
0.8
1
-2
0.1
GS 0
0.2
0.4
0.6
0.8
1
0
Density
FIG. 1 (color online). Left: μ1 and μ2 as functions of density for tx =ty ¼ 0.1. μ2 displays a large region of phase separation. The black line corresponds to the Maxwell construction. Right: Phase diagram. N is the normal state while GS is the gapless superfluid. The dash-dotted lines contain the region where the system phase separates according to μ1 and that was previously reported in Ref. [1] . The dotted lines enclose the region where the system phase separates according to μ2 .
Eq. (3) shows that large and negative U values will tend to mask the onset of phase separation when using μ1 . The left-hand panel of Fig. 1 illustrates this point for the case with tx =ty ¼ 0.1 and U ¼ −3.5. Unfortunately, using μ2 leads to the mean-field scenario reported in the right-hand panel of Fig. 1 where, due to a larger regime of phase separation, the gapless superfluid phase does not appear. We could have alternatively proceeded by first determining the many-body wave function jΨi using the same decoupling of Ref. [1]. jΨi defines the variational energy E ¼ hΨjHjΨi hΨjΨi and, from it, the chemical potential μ3 ¼ dE=dn. Wick’s theorem would then imply that μ3 ¼ μ2 . To obtain μ1 , one needs to omit the Hartree contribution from E. In light of the fact that this contribution is larger than the pairing one, this omission, and the use of μ1 rather than μ2 , seems hard to justify. We have also tried to go beyond the possibility of finitemomentum single-q paired states (shown not to be the correct ground state in [1]) by including multiple q vectors. We chose a parameter regime where the Fermi surfaces have elliptical shapes and, therefore, possess almost parallel pieces. In this regime, we tried to stabilize patterns where the order parameters had nodes in correspondence to the zeros of cos qxi and cos qxi þ cos qyi but found no value of q for which this led to a solution lower in energy than the BCS state. In conclusion, we believe that the model discussed in Ref. [1] does not support any exotic paired state (neither the gapless state proposed by the authors nor a Larkin-Ovchinnikov-type solution) at the mean-field level. Of course, we cannot draw any conclusion on whether a possible Bose-metal phase is stable against phase separation. However, the fact that such a phase is supposed to occur around the parameter regime where the
019601-1
© 2014 American Physical Society
PRL 113, 019601 (2014)
PHYSICAL REVIEW LETTERS
system shows a tendency toward phase separation indicates that this last possibility will have to be seriously taken into account in studies of this and related models. S. Chiesa1 and G. Batrouni2 1
Department of Physics College of William & Mary Williamsburg, Virginia 23188, USA 2 INLN, Université de Nice-Sophia Antipolis CNRS; 1361 route des Lucioles
week ending 4 JULY 2014
06560 Valbonne, France and Institut Universitaire de France 103, Boulevard Saint-Michel 75005 Paris, France Received 25 September 2013; published 2 July 2014 DOI: 10.1103/PhysRevLett.113.019601 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).
019601-2
PHYSICAL REVIEW LETTERS
Comment on “Exotic Paired States with Anisotropic Spin-Dependent Fermi Surfaces”
-1.70
2 mu2 (left axis) mu1 (right axis)
-1.80
0.4 1
-1.90
In Ref. [1] the authors investigated equal populations of spin-half fermions characterized by mismatched Fermi surfaces. The mismatch is caused by having up (down) particles moving preferentially along the x (y) direction on a square lattice. One of the claims of the authors is that, in the presence of on-site attraction, such a model supports an exotic gapless superfluid phase when treated at the meanfield level. Even more interesting, but outside the realm of mean-field solutions, is the possibility that the system will enter a Bose metal phase whose basic constituents are uncondensed d-wave fermionic pairs. Both phases are supposed to exist in the regime of large x-y anisotropy. The appeal of this model, compared to others supporting some of the same physics, resides in the fact that it invokes only the use of an experimentally realizable local attractive interaction. Here we show, however, that the conclusions that the authors draw from mean-field theory are modified when the Hartree contribution to the energy is taken into account. In drawing the phase diagram the authors employed Bogoliubov–de–Gennes theory, which corresponds to replacing the interaction Uni↑ ni↓ with H1MF ¼
X Uhc†i↑ c†i↓ ici↓ ci↑ þ H:c:
ð1Þ
i
and determining hc†i↑ c†i↓ i self-consistently. In a more complete Hartree–Bogoliubov–de Gennes mean-field treatment the potential energy is instead decoupled as H2MF ¼ H 1MF þ
X Uhc†iσ ciσ ic†i−σ ci−σ :
ð2Þ
iσ
As long as one is concerned with the determination of the character of a homogeneous phase, the distinctions brought by employing either Eq. (1) or Eq. (2) are largely irrelevant: using one or the other decoupling simply corresponds to a redefinition of the chemical potential. In particular, when targeting a particular density n, the chemical potentials obtained using the decoupling in Eqs. (1) and (2) are related by μ1 ¼ μ2 − Un:
ð3Þ
Once the dependence of μ on n is established, the issue of phase separation can be easily settled by looking for regions where dμ=dn is negative and locating the phase boundaries via Maxwell construction. This is where the choice of μ1 rather than μ2 alters the results. In particular, 0031-9007=14=113(1)=019601(2)
0.5
1.5
0.5
BCS
-2.00
0.3 ta/tb
PRL 113, 019601 (2014)
0 -2.10
0.2
-0.5 -2.20
-1
-2.30 -2.40
N
-1.5
0.2
0.4
0.6 Density
0.8
1
-2
0.1
GS 0
0.2
0.4
0.6
0.8
1
0
Density
FIG. 1 (color online). Left: μ1 and μ2 as functions of density for tx =ty ¼ 0.1. μ2 displays a large region of phase separation. The black line corresponds to the Maxwell construction. Right: Phase diagram. N is the normal state while GS is the gapless superfluid. The dash-dotted lines contain the region where the system phase separates according to μ1 and that was previously reported in Ref. [1] . The dotted lines enclose the region where the system phase separates according to μ2 .
Eq. (3) shows that large and negative U values will tend to mask the onset of phase separation when using μ1 . The left-hand panel of Fig. 1 illustrates this point for the case with tx =ty ¼ 0.1 and U ¼ −3.5. Unfortunately, using μ2 leads to the mean-field scenario reported in the right-hand panel of Fig. 1 where, due to a larger regime of phase separation, the gapless superfluid phase does not appear. We could have alternatively proceeded by first determining the many-body wave function jΨi using the same decoupling of Ref. [1]. jΨi defines the variational energy E ¼ hΨjHjΨi hΨjΨi and, from it, the chemical potential μ3 ¼ dE=dn. Wick’s theorem would then imply that μ3 ¼ μ2 . To obtain μ1 , one needs to omit the Hartree contribution from E. In light of the fact that this contribution is larger than the pairing one, this omission, and the use of μ1 rather than μ2 , seems hard to justify. We have also tried to go beyond the possibility of finitemomentum single-q paired states (shown not to be the correct ground state in [1]) by including multiple q vectors. We chose a parameter regime where the Fermi surfaces have elliptical shapes and, therefore, possess almost parallel pieces. In this regime, we tried to stabilize patterns where the order parameters had nodes in correspondence to the zeros of cos qxi and cos qxi þ cos qyi but found no value of q for which this led to a solution lower in energy than the BCS state. In conclusion, we believe that the model discussed in Ref. [1] does not support any exotic paired state (neither the gapless state proposed by the authors nor a Larkin-Ovchinnikov-type solution) at the mean-field level. Of course, we cannot draw any conclusion on whether a possible Bose-metal phase is stable against phase separation. However, the fact that such a phase is supposed to occur around the parameter regime where the
019601-1
© 2014 American Physical Society
PRL 113, 019601 (2014)
PHYSICAL REVIEW LETTERS
system shows a tendency toward phase separation indicates that this last possibility will have to be seriously taken into account in studies of this and related models. S. Chiesa1 and G. Batrouni2 1
Department of Physics College of William & Mary Williamsburg, Virginia 23188, USA 2 INLN, Université de Nice-Sophia Antipolis CNRS; 1361 route des Lucioles
week ending 4 JULY 2014
06560 Valbonne, France and Institut Universitaire de France 103, Boulevard Saint-Michel 75005 Paris, France Received 25 September 2013; published 2 July 2014 DOI: 10.1103/PhysRevLett.113.019601 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).
019601-2
