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Quantum phase transitions and phase diagram for a one-dimensional p-wave superconductor with an incommensurate potential
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 155701 (http://iopscience.iop.org/0953-8984/26/15/155701) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 141.70.80.5 This content was downloaded on 23/04/2014 at 10:24
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 155701 (9pp)
doi:10.1088/0953-8984/26/15/155701
Quantum phase transitions and phase diagram for a one-dimensional p-wave superconductor with an incommensurate potential X Cai State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China E-mail: [email protected] Received 12 November 2013, revised 10 February 2014 Accepted for publication 11 February 2014 Published 27 March 2014 Abstract
The effect of the incommensurate potential is studied for the one-dimensional p-wave superconductor. It is determined by analyzing various properties, such as the superconducting gap, the long-range order of the correlation function, the inverse participation ratio and the Z2 topological invariant, etc. In particular, two important aspects of the effect are investigated: (1) as disorder, the incommensurate potential destroys the superconductivity and drives the system into the Anderson localized phase; (2) as a quasi-periodic potential, the incommensurate potential causes band splitting and turns the system with certain chemical potential into the band insulator phase. A full phase diagram is also presented in the chemical potential– incommensurate potential strength plane. Keywords: 1D p-wave superconductor, Anderson localization, topological phase transition (Some figures may appear in colour only in the online journal)
1. Introduction
merits much theoretical effort. Under the large Zeeman field approximation [20] the quantum wire can be modeled by the spinless p-wave SC model originally studied by Kitaev [3]. It is a prototype model unveiling the topological features of the 1D TSC. The 1D p-wave SC model was proposed some time ago as a toy model [3]. In the system, the chemical potential can drive topological phase transitions. The system in the topological superconducting phase can host one Majorana fermion at each end of a chain under open boundary conditions. As the topological superconducting phase is protected by particle– hole symmetry, the system should be immune to perturbations of weak disorder [21]. However, for strong enough disorder the TSC phase will be destroyed and the system will turn into an Anderson insulator [22]. Due to the existence of a finite SC gap, the interplay between superconductivity and disorder naturally gives rise to topological phase transitions. The Anderson localization in a 1D disordered SC system has long been an active research area [23–26].
The zero-energy Majorana fermion [1–4], which satisfies the non-Abelian statistics [2, 5], has received a great deal of interest in recent studies [6–10]. Among various proposals [11–15] for the practical realization of the Majorana fermion, the one-dimensional (1D) topological superconductor (TSC) in nanowires [14, 15] provides a promising opportunity to study Majorana fermions. The 1D TSC is a quantum wire in a hybrid superconductor–semiconductor heterostructure with strong spin–orbital coupling, proximity-induced superconductivity and the presence of a Zeeman field. It is theoretically predicted that in a suitable parameter region the 1D TSC can host one Majorana fermion at each end of the superconducting (SC) wire. The smoking gun evidence of the presence of the Majorana fermion is still lacking in spite of the recent experimental observation of a charge current zero bias peak in the 1D TSC [16–19]. Therefore, a comprehensive understanding of the Majorana fermion in 1D still 0953-8984/14/155701+9$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
X Cai
J. Phys.: Condens. Matter 26 (2014) 155701
In this scenario, most theoretical works focus on the system with completely random disorder [24–28]. The system with disorder produced by the incommensurate potential has been considered only very recently [29–31]. In [29], the topological phase transitions caused by the chemical potential have been studied in a 1D TSC quantum wire with incommensurate modulation. However, the phase nature and the phase transitions caused by the incommensurate potential have still not been comprehensively studied. Our previous paper [31] focus on the topological phase transition from the TSC to the Anderson localized phase, which is purely induced by the incommensurate potential, for the 1D p-wave SC model with zero chemical potential. The topological phase transition point was numerically determined and agreed well with the theoretical prediction for a system in the thermodynamical limit. The incommensurate potential can now be engineered with ultra-cold atoms loaded in 1D bichromatic optical lattices [32], opening up an experimental way to study the localization properties of quasi-periodic systems. Extending the previous study in [31], this paper is aimed at investigating the properties of a 1D p-wave superconductor with the incommensurate potential. In particular, we focus on the subtle effect resulting from the chemical potential and the incommensurate potential. The paper is organized as follows. In section 2, we introduce the model of a 1D p-wave SC with the incommensurate potential and the Bogoliubov–de Gennes (BDG) method, which will be used to diagonalize the Hamiltonian. In section 3, we will study the properties of a system from different aspects, such as the SC gap, the particle number, the longrange order of the correlation function, the inverse participation ratio and the Z2 topological invariant. A summary is given in the final section.
Following the method proposed by Lieb [35], one can rewrite the Hamiltonian equation (1) in quadratic form ⎡ ⎞⎤ 1⎛ H = ∑ ⎢ ci†Aij cj + ⎜⎜ ci†Bijc†j + h.c. ⎟⎟ ⎥ . (3) 2⎝ ⎣ ⎠ ⎥⎦ i, j ⎢ with A being a Hermitian matrix and B being an antisymmetric matrix. The Hamiltonian in quadratic form can be diagonalized by using the BDG transformation:
=
{
L
∑
⎡ ⎤ ⎢φniγiA + iψniγiB ⎥ , ⎢⎣ ⎥⎦
}
the form H = ∑ Λn (ηn†ηn − ½), where Λn is the spectrum. n=1 The coefficients φni and ψni in BDG transformation equation (4) are determined by the diagonalization condition, i.e., [ηn, H] = Λnηn. Then they satisfy the following coupled equations: L
(6) (A + B )ϕn = Λnψn, (A − B )ψn = Λnϕn . Either φn or ψn can be eliminated from the above coupled equations giving either or
(A − B )(A + B )ϕn = Λ2nϕn
(7)
(A + B )(A − B )ψn = Λ2nψn .
(8) For a given solution (φn, ψn) to the last two equations with eigenvalue Λn, one can prove by equations (6) that (φn, −ψn) is also a solution to these two equations with eigenvalue −Λn. It implies that ηn(Λn) = ηn†(−Λn) . After diagonalizing the Hamiltonian equation (1), one can obtain the ground state of system, in which all the negative energy levels are filled up. If the quasi-particle energies are arranged in ascending order, i.e., Λn ⩽ Λn+1, for Λn > 0, the gap of the system is just given by Δg = 2Λ1.
We consider a typical lattice model of a 1D p-wave superconductor with an incommensurate potential, which is described by the following Hamiltonian: ⎡⎛ ⎤ ⎞ † H = ∑⎢⎜⎜−tci†ci + 1 + Δ cici + 1 + H. c.⎟⎟ + Vc i i ci ⎥ . ⎢ ⎥⎦ ⎠ i ⎣⎝
1 2
(4)
(5) i=1 where L is the number of lattice sites and n = 1, …, L. γiA ≡ ci† + ci and γiB ≡ i ( ci − ci† ) are the operators of two Majorana fermions corresponding to one site [3]. They fulfil relations ( γiα )† = γiα and γiα, γ jβ = 2δijδαβ with α and β taking A or B. In terms of operators ηn and ηn†, the Hamiltonian equation (1) is diagonal with
2. A 1D p-wave superconductor with an incommensurate potential
L ⎡ φ + ψni † φni − ψni ⎤ ηn† = ∑ ⎢ ni ci + ci ⎥ 2 2 ⎦ i=1 ⎣
(1)
where ci† (ci) is the creation (annihilation) operator of spinless fermions and t is the nearest-neighbor hopping amplitude. For convenience, we set it to the unit of energy (t = 1) throughout this paper. Δ is the p-wave superconducting pairing amplitude and can be chosen to be a positive real number [3]. Vi is the potential at site i with the form
3. Novel properties of a 1D p-wave superconductor with an incommensurate potential In this section, we study the properties of a 1D p-wave superconductor with an incommensurate potential. When Δ = 0, the Hamiltonian equation (1) reduces to the Aubry–André model [33, 34]. Without superconductivity, the system has U (1) symmetry and the total particle number is conserved. As the strength of the incommensurate potential V increases, the system undergoes a delocalization to localization transition. In the thermodynamic limit the phase transition point is at V = 2t, which can be mapped out by a self-duality mapping.
Vi = −μ + V cos(2πβi ) . (2) Here μ is the chemical potential, controlling the total particle number of system. The second term in the above equation is the incommensurate potential with strength V; here β is an irrational number. When Δ = 0 this model reduces to the Aubry–André model [33, 34], while when V = 0 this Hamiltonian (equation (1)) describes Kitaev’s 1D p-wave superconducting chain [3]. 2
X Cai
J. Phys.: Condens. Matter 26 (2014) 155701
For a system with V 2t they are Anderson localized. When V = 0, the Hamiltonian equation (1) describes the famous Kitaev 1D p-wave SC chain [3]. With a finite Δ, as the absolute value of chemical potential μ increases, the uniform system undergoes a topological phase transition at |μ| = 2t. The system with |μ| 2t is in the normal SC phase. The model for μ = 0 has been studied in [31]. As the strength of incommensurate potential V increases, the gap of the system decreases and is closed around Vc = 2(t + Δ) (see also figure 1(c)); the long-range order of the correlation function also decreases and becomes zero around Vc = 2(t + Δ); the mean inverse participation ratio increases and experiences a sudden increase around Vc = 2(t + Δ) (see also figures 4(b) and (d). When V 2(t + Δ), ν = 1 and the system is in the topological trivial phase. Under open boundary conditions, when V 2(t + Δ) there is no zero energy state or Majorana fermion under open boundary conditions and all the quasi-particle states are Anderson localized. So as V increases, there is a topological phase transition from the TSC phase to the Anderson localized phase with the transition point at Vc = 2(t + Δ) in the thermodynamic limit. Next we study the subtle physical properties of the system, the effect of μ on the topological phase transition and other many-body phenomena. In numerical calculations, the value of irrational number β is the inverse golden ratio ( 5 − 1) / 2.
the system with |μ| 0 (μ 2(t + Δ), the system is in the Anderson localized phase with all the single-particle states being localized. In figure 4(f) we also show the IPR of all the single-particle states for systems with Δ = 0.5, μ = 1, and different V. One finds that the band insulator phase cannot effect the Anderson localization of a system which still happens around Vc = 2(t + Δ). Therefore, for a system with possible partial single-particle states being localized, the sudden increase in the MIPR is not a good criterion for the Anderson localization transition. Regardless of the chemical potential, the Anderson localization transition happens
3.4 Inverse participation ratio
The inverse participation ratio provides information about the spatial extension of a single-particle state. For a normalized s ingle-particle state |φ〉 = ∑ici|i〉 expressed in terms of the Wannier states |i〉, the inverse participation ratio (IPR) [39, 40] is defined as P = ∑i|ci|4. For an extended state, P → 1/L and as the system size L increases the IPR tends to zero, whereas the IPR tends to a finite number for a localized state. Therefore, the IPR can be taken as a criterion to distinguish the extended states from the localized states. In this paper the IPR is defined as: Pn =
∑ |ϕn(i )|4 ,
(14)
with condition ∑i|φn(i)|2 = 1. Similarly one can define an IPR for the wave function ψn(i). ψn(i) and φn(i) are related by equation (6), and should have similar IPR properties, which is confirmed by numerical data. Here we only consider the IPR properties of φ. In order to characterize the localization of a many-body ground state we define the mean inverse participation ratio (MIPR) P = ∑ Pn / L. n In figures 4(a) and (c), we show 3D plots of the mean inverse participation ratio for two systems with different Δ. They have similar behaviors. Firstly, when μ = 0 the MIPR increases monotonically as V increases and experiences a sudden increase around Vc = 2(t + Δ), which characterizes the localization transition (see also figures 4(b) and (d). These agree with the results obtained in [31]. When μ = 0 all the single-particle states are extended for a system with V
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Quantum phase transitions and phase diagram for a one-dimensional p-wave superconductor with an incommensurate potential
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 155701 (http://iopscience.iop.org/0953-8984/26/15/155701) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 141.70.80.5 This content was downloaded on 23/04/2014 at 10:24
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 155701 (9pp)
doi:10.1088/0953-8984/26/15/155701
Quantum phase transitions and phase diagram for a one-dimensional p-wave superconductor with an incommensurate potential X Cai State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China E-mail: [email protected] Received 12 November 2013, revised 10 February 2014 Accepted for publication 11 February 2014 Published 27 March 2014 Abstract
The effect of the incommensurate potential is studied for the one-dimensional p-wave superconductor. It is determined by analyzing various properties, such as the superconducting gap, the long-range order of the correlation function, the inverse participation ratio and the Z2 topological invariant, etc. In particular, two important aspects of the effect are investigated: (1) as disorder, the incommensurate potential destroys the superconductivity and drives the system into the Anderson localized phase; (2) as a quasi-periodic potential, the incommensurate potential causes band splitting and turns the system with certain chemical potential into the band insulator phase. A full phase diagram is also presented in the chemical potential– incommensurate potential strength plane. Keywords: 1D p-wave superconductor, Anderson localization, topological phase transition (Some figures may appear in colour only in the online journal)
1. Introduction
merits much theoretical effort. Under the large Zeeman field approximation [20] the quantum wire can be modeled by the spinless p-wave SC model originally studied by Kitaev [3]. It is a prototype model unveiling the topological features of the 1D TSC. The 1D p-wave SC model was proposed some time ago as a toy model [3]. In the system, the chemical potential can drive topological phase transitions. The system in the topological superconducting phase can host one Majorana fermion at each end of a chain under open boundary conditions. As the topological superconducting phase is protected by particle– hole symmetry, the system should be immune to perturbations of weak disorder [21]. However, for strong enough disorder the TSC phase will be destroyed and the system will turn into an Anderson insulator [22]. Due to the existence of a finite SC gap, the interplay between superconductivity and disorder naturally gives rise to topological phase transitions. The Anderson localization in a 1D disordered SC system has long been an active research area [23–26].
The zero-energy Majorana fermion [1–4], which satisfies the non-Abelian statistics [2, 5], has received a great deal of interest in recent studies [6–10]. Among various proposals [11–15] for the practical realization of the Majorana fermion, the one-dimensional (1D) topological superconductor (TSC) in nanowires [14, 15] provides a promising opportunity to study Majorana fermions. The 1D TSC is a quantum wire in a hybrid superconductor–semiconductor heterostructure with strong spin–orbital coupling, proximity-induced superconductivity and the presence of a Zeeman field. It is theoretically predicted that in a suitable parameter region the 1D TSC can host one Majorana fermion at each end of the superconducting (SC) wire. The smoking gun evidence of the presence of the Majorana fermion is still lacking in spite of the recent experimental observation of a charge current zero bias peak in the 1D TSC [16–19]. Therefore, a comprehensive understanding of the Majorana fermion in 1D still 0953-8984/14/155701+9$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
X Cai
J. Phys.: Condens. Matter 26 (2014) 155701
In this scenario, most theoretical works focus on the system with completely random disorder [24–28]. The system with disorder produced by the incommensurate potential has been considered only very recently [29–31]. In [29], the topological phase transitions caused by the chemical potential have been studied in a 1D TSC quantum wire with incommensurate modulation. However, the phase nature and the phase transitions caused by the incommensurate potential have still not been comprehensively studied. Our previous paper [31] focus on the topological phase transition from the TSC to the Anderson localized phase, which is purely induced by the incommensurate potential, for the 1D p-wave SC model with zero chemical potential. The topological phase transition point was numerically determined and agreed well with the theoretical prediction for a system in the thermodynamical limit. The incommensurate potential can now be engineered with ultra-cold atoms loaded in 1D bichromatic optical lattices [32], opening up an experimental way to study the localization properties of quasi-periodic systems. Extending the previous study in [31], this paper is aimed at investigating the properties of a 1D p-wave superconductor with the incommensurate potential. In particular, we focus on the subtle effect resulting from the chemical potential and the incommensurate potential. The paper is organized as follows. In section 2, we introduce the model of a 1D p-wave SC with the incommensurate potential and the Bogoliubov–de Gennes (BDG) method, which will be used to diagonalize the Hamiltonian. In section 3, we will study the properties of a system from different aspects, such as the SC gap, the particle number, the longrange order of the correlation function, the inverse participation ratio and the Z2 topological invariant. A summary is given in the final section.
Following the method proposed by Lieb [35], one can rewrite the Hamiltonian equation (1) in quadratic form ⎡ ⎞⎤ 1⎛ H = ∑ ⎢ ci†Aij cj + ⎜⎜ ci†Bijc†j + h.c. ⎟⎟ ⎥ . (3) 2⎝ ⎣ ⎠ ⎥⎦ i, j ⎢ with A being a Hermitian matrix and B being an antisymmetric matrix. The Hamiltonian in quadratic form can be diagonalized by using the BDG transformation:
=
{
L
∑
⎡ ⎤ ⎢φniγiA + iψniγiB ⎥ , ⎢⎣ ⎥⎦
}
the form H = ∑ Λn (ηn†ηn − ½), where Λn is the spectrum. n=1 The coefficients φni and ψni in BDG transformation equation (4) are determined by the diagonalization condition, i.e., [ηn, H] = Λnηn. Then they satisfy the following coupled equations: L
(6) (A + B )ϕn = Λnψn, (A − B )ψn = Λnϕn . Either φn or ψn can be eliminated from the above coupled equations giving either or
(A − B )(A + B )ϕn = Λ2nϕn
(7)
(A + B )(A − B )ψn = Λ2nψn .
(8) For a given solution (φn, ψn) to the last two equations with eigenvalue Λn, one can prove by equations (6) that (φn, −ψn) is also a solution to these two equations with eigenvalue −Λn. It implies that ηn(Λn) = ηn†(−Λn) . After diagonalizing the Hamiltonian equation (1), one can obtain the ground state of system, in which all the negative energy levels are filled up. If the quasi-particle energies are arranged in ascending order, i.e., Λn ⩽ Λn+1, for Λn > 0, the gap of the system is just given by Δg = 2Λ1.
We consider a typical lattice model of a 1D p-wave superconductor with an incommensurate potential, which is described by the following Hamiltonian: ⎡⎛ ⎤ ⎞ † H = ∑⎢⎜⎜−tci†ci + 1 + Δ cici + 1 + H. c.⎟⎟ + Vc i i ci ⎥ . ⎢ ⎥⎦ ⎠ i ⎣⎝
1 2
(4)
(5) i=1 where L is the number of lattice sites and n = 1, …, L. γiA ≡ ci† + ci and γiB ≡ i ( ci − ci† ) are the operators of two Majorana fermions corresponding to one site [3]. They fulfil relations ( γiα )† = γiα and γiα, γ jβ = 2δijδαβ with α and β taking A or B. In terms of operators ηn and ηn†, the Hamiltonian equation (1) is diagonal with
2. A 1D p-wave superconductor with an incommensurate potential
L ⎡ φ + ψni † φni − ψni ⎤ ηn† = ∑ ⎢ ni ci + ci ⎥ 2 2 ⎦ i=1 ⎣
(1)
where ci† (ci) is the creation (annihilation) operator of spinless fermions and t is the nearest-neighbor hopping amplitude. For convenience, we set it to the unit of energy (t = 1) throughout this paper. Δ is the p-wave superconducting pairing amplitude and can be chosen to be a positive real number [3]. Vi is the potential at site i with the form
3. Novel properties of a 1D p-wave superconductor with an incommensurate potential In this section, we study the properties of a 1D p-wave superconductor with an incommensurate potential. When Δ = 0, the Hamiltonian equation (1) reduces to the Aubry–André model [33, 34]. Without superconductivity, the system has U (1) symmetry and the total particle number is conserved. As the strength of the incommensurate potential V increases, the system undergoes a delocalization to localization transition. In the thermodynamic limit the phase transition point is at V = 2t, which can be mapped out by a self-duality mapping.
Vi = −μ + V cos(2πβi ) . (2) Here μ is the chemical potential, controlling the total particle number of system. The second term in the above equation is the incommensurate potential with strength V; here β is an irrational number. When Δ = 0 this model reduces to the Aubry–André model [33, 34], while when V = 0 this Hamiltonian (equation (1)) describes Kitaev’s 1D p-wave superconducting chain [3]. 2
X Cai
J. Phys.: Condens. Matter 26 (2014) 155701
For a system with V 2t they are Anderson localized. When V = 0, the Hamiltonian equation (1) describes the famous Kitaev 1D p-wave SC chain [3]. With a finite Δ, as the absolute value of chemical potential μ increases, the uniform system undergoes a topological phase transition at |μ| = 2t. The system with |μ| 2t is in the normal SC phase. The model for μ = 0 has been studied in [31]. As the strength of incommensurate potential V increases, the gap of the system decreases and is closed around Vc = 2(t + Δ) (see also figure 1(c)); the long-range order of the correlation function also decreases and becomes zero around Vc = 2(t + Δ); the mean inverse participation ratio increases and experiences a sudden increase around Vc = 2(t + Δ) (see also figures 4(b) and (d). When V 2(t + Δ), ν = 1 and the system is in the topological trivial phase. Under open boundary conditions, when V 2(t + Δ) there is no zero energy state or Majorana fermion under open boundary conditions and all the quasi-particle states are Anderson localized. So as V increases, there is a topological phase transition from the TSC phase to the Anderson localized phase with the transition point at Vc = 2(t + Δ) in the thermodynamic limit. Next we study the subtle physical properties of the system, the effect of μ on the topological phase transition and other many-body phenomena. In numerical calculations, the value of irrational number β is the inverse golden ratio ( 5 − 1) / 2.
the system with |μ| 0 (μ 2(t + Δ), the system is in the Anderson localized phase with all the single-particle states being localized. In figure 4(f) we also show the IPR of all the single-particle states for systems with Δ = 0.5, μ = 1, and different V. One finds that the band insulator phase cannot effect the Anderson localization of a system which still happens around Vc = 2(t + Δ). Therefore, for a system with possible partial single-particle states being localized, the sudden increase in the MIPR is not a good criterion for the Anderson localization transition. Regardless of the chemical potential, the Anderson localization transition happens
3.4 Inverse participation ratio
The inverse participation ratio provides information about the spatial extension of a single-particle state. For a normalized s ingle-particle state |φ〉 = ∑ici|i〉 expressed in terms of the Wannier states |i〉, the inverse participation ratio (IPR) [39, 40] is defined as P = ∑i|ci|4. For an extended state, P → 1/L and as the system size L increases the IPR tends to zero, whereas the IPR tends to a finite number for a localized state. Therefore, the IPR can be taken as a criterion to distinguish the extended states from the localized states. In this paper the IPR is defined as: Pn =
∑ |ϕn(i )|4 ,
(14)
with condition ∑i|φn(i)|2 = 1. Similarly one can define an IPR for the wave function ψn(i). ψn(i) and φn(i) are related by equation (6), and should have similar IPR properties, which is confirmed by numerical data. Here we only consider the IPR properties of φ. In order to characterize the localization of a many-body ground state we define the mean inverse participation ratio (MIPR) P = ∑ Pn / L. n In figures 4(a) and (c), we show 3D plots of the mean inverse participation ratio for two systems with different Δ. They have similar behaviors. Firstly, when μ = 0 the MIPR increases monotonically as V increases and experiences a sudden increase around Vc = 2(t + Δ), which characterizes the localization transition (see also figures 4(b) and (d). These agree with the results obtained in [31]. When μ = 0 all the single-particle states are extended for a system with V
