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The mobility edges which separate the localized energy eigenstates from the extended ones exist normally only in three dimensional systems. For one-dimensional systems with random on-site potentials, one never encounters mobility edges, where all the eigenstates are localized. However, there are two kinds of 1D systems such as correlated disordered models, and the systems of exponentially decaying hopping kinetics, features of mobility edges at some specific values become possible. We study in this paper the properties of the mobility edges in a one-dimensional p-wave superfluid on an incommensurate lattice with exponentially decaying hopping kinetics. Without the p-wave superluid, the system displays a single mobility edge, which separates the extended regime from the localized one at a certain energy. Without the exponentially decaying hopping term, the system displays a phase transition from a topological superconductor to an Anderson localization at a certain disorder strength, where no mobility edge exists. We are interested in the influence of the p-wave superfluid on the mobility edge. By solving the Bogoliubov-de Gennes equation, the eigenvalues and the eigenfunctions are obtained. In order to identify the extending or the localized properties of the eigenvectors, we define an inverse participation ratio IPR. For an extended state, IPRn～1/L which goes to zero at a large L, and for a localized one, IPRn being constant. Therefore, the IPR can be taken as a criterion to distinguish the extended state from the localized one, while the mobility edge is defined as the boundary between two different states. We find that, with a p-wave superfluid, the system changes from a single mobility edge to a multiple one, and the number of mobility edges increases with the increased superfluid pairing order parameter. To further obtain the energy or the location of the mobility edge, we investigate the scaling behavior of wave functions by using a multifractal analysis, which is calculated through the scaling index . The minimum value of the index, with the values min= 1, 0min1, and min= 0, mean the extended, critical, and localized states, respectively. For the two consecutive states, the minima of the scaling index min when extrapolating to the large size limit between 0 and 1 signal the mobility edge. By exploring the corresponding Bogoliubov quasi-particle wave functions for the system under open boundary conditions together with the multifractal analysis for the system under periodic boundary conditions, we identify two mobility edges for the system of the p-wave superfluid pairing. Furthermore, we will investigate how the existence of the mobility edges influences the p-wave superfluid, and identify the phase diagram at the given parameters. We will in the future try to understand the relationship between the topological superfluid and the mobility edges.
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Keywords:
- p-wave superfluid /
- mobility edges /
- multifractal analysis
[1] Anderson P W 1958 Phys. Rev. 109 1492
[2] Billy J, Josse V, Zuo Z, Bernard A, Hambrecht B, Lugan P, Clment D, Sanchez-Palencia L, Bouyer P, Aspect A 2008 Nature 453 891
[3] Roati G, D' Errico C, Fallani L, Fattori M, Fort C, Zaccanti M, Modugno G, Modugno M, Inguscio M 2008 Nature 453 895
[4] Aubry S, Andr G {1980 Ann. Isr. Phys. Soc. 3 18
[5] Mott N 1987 J. Phys. C: Solid Stat. Phys. 20 3075
[6] Hiramoto H, Kohmoto M 1989 Phys. Rev. B 40 8225
[7] Zhou P Q, Fu X J, Guo Z Z, Liu Y Y 1995 Solid State Commun. 96 373
[8] Ganeshan S, Pixley J H, Sarma S D 2015 Phys. Rev. Lett. 114 146601
[9] Biddle J, Sarma S D 2010 Phys. Rev. Lett. 104 070601
[10] Ivanov D A 2001 Phys. Rev. Lett. 86 268
[11] Beenakker C W J 2013 Annu. Rev. Condens. Matter Phys. 4 113
[12] Kitaev A Y 2001 Phys. Usp. 44 131
[13] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
[14] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
[15] Das A, Ronen Y, Most Y, Oreg Y, Heiblum M, Shtrikman H 2012 Nat. Phys. 8 887
[16] Motrunich O, Damle K, Huse D A 2001 Phys. Rev. B 63 224204
[17] Brouwer P W, Furusaki A, Gruzberg I A, Mudry C 2000 Phys. Rev. Lett. 85 1064
[18] Gruzberg I A, Read N, Vishveshwara S 2005 Phys. Rev. B 71 245124
[19] Lobos A M, Lutchyn R M, Sarma S D 2012 Phys. Rev. Lett. 109 146403
[20] Cai X M, Lang L J, Chen S, Wang Y P 2013 Phys. Rev. Lett. 110 176403
[21] Ingold G L, Wobst A, Aulbach C, Hnggi P 2002 Eur. Phys. J. B 30 175
[22] Thouless D J 1974 Phys. Rep. 13 93
[23] Wang J, Liu X J, Gao X L, Hu H {2015 Phys. Rev. B 93 104504
[24] Hiramoto H, Kohmoto M 1992 Int. J. Mod. Phys. B 06 281
[25] Kohmoto M, Kadanoff L P, Tang C 1983 Phys. Rev. Lett. 50 1870
[26] Ostlund S, Pandit R, Rand D, Schellnhuber H J, Siggia E D 1983 Phys. Rev. Lett. 50 1873
[27] Kohmoto M 1983 Phys. Rev. Lett. 51 1198
[28] Thouless D J 1983 Phys. Rev. B 28 4272
[29] Kohmoto M, Tobe D 2008 Phys. Rev. B 77 134204
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[1] Anderson P W 1958 Phys. Rev. 109 1492
[2] Billy J, Josse V, Zuo Z, Bernard A, Hambrecht B, Lugan P, Clment D, Sanchez-Palencia L, Bouyer P, Aspect A 2008 Nature 453 891
[3] Roati G, D' Errico C, Fallani L, Fattori M, Fort C, Zaccanti M, Modugno G, Modugno M, Inguscio M 2008 Nature 453 895
[4] Aubry S, Andr G {1980 Ann. Isr. Phys. Soc. 3 18
[5] Mott N 1987 J. Phys. C: Solid Stat. Phys. 20 3075
[6] Hiramoto H, Kohmoto M 1989 Phys. Rev. B 40 8225
[7] Zhou P Q, Fu X J, Guo Z Z, Liu Y Y 1995 Solid State Commun. 96 373
[8] Ganeshan S, Pixley J H, Sarma S D 2015 Phys. Rev. Lett. 114 146601
[9] Biddle J, Sarma S D 2010 Phys. Rev. Lett. 104 070601
[10] Ivanov D A 2001 Phys. Rev. Lett. 86 268
[11] Beenakker C W J 2013 Annu. Rev. Condens. Matter Phys. 4 113
[12] Kitaev A Y 2001 Phys. Usp. 44 131
[13] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
[14] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
[15] Das A, Ronen Y, Most Y, Oreg Y, Heiblum M, Shtrikman H 2012 Nat. Phys. 8 887
[16] Motrunich O, Damle K, Huse D A 2001 Phys. Rev. B 63 224204
[17] Brouwer P W, Furusaki A, Gruzberg I A, Mudry C 2000 Phys. Rev. Lett. 85 1064
[18] Gruzberg I A, Read N, Vishveshwara S 2005 Phys. Rev. B 71 245124
[19] Lobos A M, Lutchyn R M, Sarma S D 2012 Phys. Rev. Lett. 109 146403
[20] Cai X M, Lang L J, Chen S, Wang Y P 2013 Phys. Rev. Lett. 110 176403
[21] Ingold G L, Wobst A, Aulbach C, Hnggi P 2002 Eur. Phys. J. B 30 175
[22] Thouless D J 1974 Phys. Rep. 13 93
[23] Wang J, Liu X J, Gao X L, Hu H {2015 Phys. Rev. B 93 104504
[24] Hiramoto H, Kohmoto M 1992 Int. J. Mod. Phys. B 06 281
[25] Kohmoto M, Kadanoff L P, Tang C 1983 Phys. Rev. Lett. 50 1870
[26] Ostlund S, Pandit R, Rand D, Schellnhuber H J, Siggia E D 1983 Phys. Rev. Lett. 50 1873
[27] Kohmoto M 1983 Phys. Rev. Lett. 51 1198
[28] Thouless D J 1983 Phys. Rev. B 28 4272
[29] Kohmoto M, Tobe D 2008 Phys. Rev. B 77 134204
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