PHYSICAL REVIEW B 87, 219904(E) (2013)

Erratum: Length scales, collective modes, and type-1.5 regimes in three-band superconductors[Phys. Rev. B 84, 134518 (2011)]

Johan Carlstrom, Julien Garaud, and Egor Babaev(Received 18 April 2013; published 18 June 2013)

DOI: 10.1103/PhysRevB.87.219904 PACS number(s): 74.70.Xa, 74.20.Mn, 74.20.Rp, 99.10.Cd

The example shown in Fig. 2 does not correspond to the model parameters listed in the caption. Here, Fig. 1 shows thequantities corresponding to that parameter set. Compared to the original plot, the differences are only quantitative and do notchange any conclusions in the paper.

We also take this opportunity to correct a few minor typos:(i) In Eq. (11), for the contribution associated with the Proca field, the equation should read

EProca ≡ 1

2

(a′

r

)2

+ e2

2r2

∑i

u2i a

2.

(ii) There is a typo in the definition of the component of the mass matrix operator in Eq. (15). For the submatrix Mππ , itshould read

Mππ =

⎛⎜⎝

u2η12+u3η13

−u1η12 η13

η12u1η12+u3η23

−u2η23

η13 η23u1η13+u2η23

−u3

⎞⎟⎠ .

(iii) The sentence in the paragraph after Eq. (15) should read as follows: “Its eigenspectrum is the set of six squared massesM2

i

2 , whose corresponding lengths �i = 1/√

2Mi are the physical length scales of a three band superconductor.”(iv) In Eq. (2), the absolute values of the fields in the prefactors of the third, fourth, and fifth terms in Eq. (2) should be

squared: |ψl|2|ψk|2/(2∑

i |ψi |2) instead of |ψi ||ψj |/(2∑

i |ψi |2).(v) Following Eq. (1), the definition of φij should be φij = φi − φj .

We thank Daniel Weston for pointing out the discrepancy between the parameters in the caption and the curves in Fig. 2, andfor finding typos. This work was supported by US National Science Foundation CAREER Award No. DMR-0955902, by theKnut and Alice Wallenberg Foundation through the Royal Swedish Academy of Sciences, and by the Swedish Research Council.The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at theNational Supercomputer Center at Linkoping, Sweden.

219904-11098-0121/2013/87(21)/219904(2) ©2013 American Physical Society

ERRATA PHYSICAL REVIEW B 87, 219904(E) (2013)

3.75 3.7 3.65 3.61.15

1.2

1.25

1.3

1.35

1.4

1.45

1,3 &

2,3

Ground state amplitudes

123

3.75 3.7 3.65 3.6

0.4

0.2

0

0.2

0.4

Ground state phases

1,3 &

2,3

1

2

3

3.75 3.7 3.65 3.60

1

2

3

4

Mass spectrum

1,3 &

2,3

Mas

s

123456

3.75 3.7 3.65 3.60.4

0.2

0

0.2

0.4

0.6

EigenVector 1

1,3 &

2,3

123123

3.75 3.7 3.65 3.6

0.6

0.4

0.2

0

0.2

0.4

0.6

EigenVector 2

1,3 &

2,3

123123

3.75 3.7 3.65 3.6

0.6

0.4

0.2

0

0.2

0.4

EigenVector 3

1,3 &

2,3

123123

3.75 3.7 3.65 3.6

0

0.2

0.4

0.6

0.8

EigenVector 4

1,3 &

2,3

123123

3.75 3.7 3.65 3.6

0.6

0.4

0.2

0

0.2

0.4

0.6

EigenVector 5

1,3 &

2,3

123123

3.75 3.7 3.65 3.60.6

0.5

0.4

0.3

0.2

0.1

0

EigenVector 6

1,3 &

2,3

123123

( )

FIG. 1. (Color online) Ground-state field values, eigenspectrum, and eigenvectors of the mass matrix. The x axis gives the two parametersη13 = η23 while the other parameters are α1 = −3, β1 = 3, α2 = −3, β2 = 3, α3 = 2, β3 = 0.5, η12 = 2.25. The eigenvectors are sortedaccording to the corresponding eigenvalue, starting with the largest. The smallest eigenvalue is the zero mode associated with the spontaneouslybroken U (1) symmetry (no coupling to the gauge field A is implied here, which otherwise renders this mode massive via the Anderson-Higgsmechanism). At η13 = η23 ≈ −3.69, there is a transition from a U (1) to a U (1) × Z2 state. The eigenvalue (5) becomes zero at the transitionpoint, so there appears a divergent length scale at this point which corresponds to the eigenvector (5), i.e., the phase-difference mode becomesa scaleless collective excitation. Observe that in the U (1) region, the eigenvectors exhibit no mixing between densities and phases, while in theU (1) × Z2 region there is in fact not a single eigenvector that does not exhibit a mixing of phases and densities. Therefore, the perturbationof densities is generically associated with perturbations of the phase differences in this regime. The arrows on the panels in the bottom rowillustrate the variation of the fields associated with the corresponding collective modes. The lengths of the arrows correspond to modulusfluctuation and the directions of the arrows correspond to phase fluctuation. The modulus field of component 1 is in red, that of component 2is in green, and that of component 3 is in blue.

219904-2