What is a particle-conserving Topological Superfluid?

1.2 MB·

by  Gerardo Ortiz

What is a particle-conserving Topological Superfluid?
6
1 The fate of Majorana modes beyond mean-field theory
0
2
Gerardo Ortiz1 and Emilio Cobanera2
n
a 1 Department of Physics, Indiana University, Bloomington, IN 47405, USA
J 2 Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory,

8 Hanover, NH 03755, USA
2
E-mail: [email protected]

]
n
o Abstract. We establish a criterion for characterizing superfluidity in interacting, particle-
c number conserving systems of fermions as topologically trivial or non-trivial. Because our
- criterion is based on the concept of many-body fermionic parity switches, it is directly
r
p associated to the observation of the fractional Josephson effect and indicates the emergence
u of zero-energy modes that anticommute with fermionic parity. We tested these ideas on the
s Richardson-Gaudin-Kitaev chain, a particle-number conserving system that is solvable by way
.
t of the algebraic Bethe ansatz, and reduces to a long-range Kitaev chain in the mean-field
a
approximation. Guided by its closed-form solution, we introduce a procedure for constructing
m
many-bodyMajoranazero-energymodesofgappedtopologicalsuperfluidsintermsofcoherent
- superpositionsofstateswithdifferentnumberoffermions. Wediscusstheirsignificanceandthe
d
physical conditions required to enable quantum control in the light of superselection rules.
n
o
c
[

1
1. Introduction
v
4 Most investigations of fermionic condensed matter are based on a type of mean-field
6 approximation popularized by Bogoliubov [1]. One recent and conspicuous example is the ten-
7
foldway,atopologicalclassificationoffermionicsystemsbasedonthreediscrete—timereversal,
7
0 charge conjugation, and chiral — symmetries and K-homology [2, 3]. The main reasons for the
. prevalence of the mean-field approximation are clear. First, it leads to a very intuitive and
1
0 natural picture of fermionic systems thanks to Landau’s theory of Fermi liquids and their quasi-
6 particles. Second, it leads to a mathematically simple Lie-algebraic formalism by which the
1
problem of diagonalizing the Hamiltonian in Fock space becomes of polynomial complexity in
:
v the total number of degrees of freedom. And last but not least, topological invariants such as
i
X (full or partial) Chern numbers, Berry phases [4], Bott and Hopf indexes, and others are easily
r associated to mean-field theories and evaluated for concrete instances.
a
The thermodynamic state of a fermionic superfluid (or superconductor) is characterized
by the spontaneous breaking of the global continuous U(1) symmetry related to particle-
number conservation. The positive features of the mean-field approximation for fermions
come at a surprising cost in the context of superfluidity (electrically neutral fermions) or
superconductivity (charged fermions): the explicit breaking of the symmetry of particle-number
conservation. However, electronic matter is composed of interacting electrons whose number is
locally conserved. Whether this mismatch between models and systems being modeled matters
or not is bound to depend on the physical quantities to be computed. Many calculations of
thermodynamic and transport properties of fermionic superfluids have firmly established the
phenomenological success of particle-number non-conserving mean-field theories. But these
successes do not imply that every experimentally accessible feature of the fermionic superfluid
state is well described by breaking particle-number conservation, see for example Ref.[5].
Inthispaperwewillsetupthegroundforasystematicinvestigationoftheinterplaybetween
the mean-field topological classification of superfluid systems of fermions and more realistic
models where, for closed systems, the number of fermions is conserved. Our discussion will be
organized around three fundamental and interrelated questions.
What makes a particle-number conserving fermionic superfluid/superconductor topologi-
•
cally trivial or non-trivial?
What are the experimental signatures of fermionic topological superfluidity ?
•
What is the fate of Majorana zero-energy modes beyond the mean-field approximation?
•
And even more basically, what is their very meaning?
Concern with the role of particle (non-)conservation in the mean-field theory of
superconductivity is as old as the theory itself [6]. We feel prompted to revisit this issue by
the recent experimental efforts to detect and control Majorana zero-energy modes. The presence
of Majorana modes has been typically considered a key mean-field manifestation of topological
fermion superfluidity since the work of Ref.[7]. These quasi-particles emerge from the interplay
betweentheexistenceofatopologicallynon-trivialvacuumanda,typically,symmetry-protected
physical boundary (or defect). In recent literature, this connection goes under the name of
bulk-boundary correspondence. Because of the expected resilience against decoherence and
non-Abelian braiding properties, Majorana modes, or simply Majorana fermions for short, are
key components of many blueprints of quantum-information processing devices. Given that
either electrons or fermionic atoms are in fact locally conserved, it is imperative to investigate
the conditions for the emergence of Majorana fermions and procedures for their experimental
detection beyond mean field.
Our work here suggests the idea that a Majorana fermion may be quantum controlled is
so deeply rooted in the mean-field picture that it might not have a natural counterpart in
more realistic particle-number conserving frameworks. Briefly stated, if a zero-energy mode
of a superfluid system creates a superposition of states that differ in particle number/electric
charge, it may not be possible to manipulate this mode without exchanging particles with an
environment, as opposed to exchanging, say, energy only. Then the question becomes whether
it is possible in practice to exchange coherently (charged) particles with a reservoir big enough
to grant the mean-field picture of the (sub)system of interest. This claim is ultimately rooted
in the venerable [8], but still much investigated and debated [9], subject of superselection rules.
Our last statement requires some clarification, since it takes for granted the existence of
Majorana modes for closed, hence necessarily particle-number conserving, systems. Here we will
investigate whether this assumption is reasonable based on our work published in Ref. [10]. By
exploitingthealgebraicBetheansatz,wesucceededinestablishingandcharacterizingtopological
superconductivity for prototypical particle number-conserving, and thus necessarily interacting,
superconducting chains beyond mean-field theory. Before this work, it was not known how to
test for topological fermion superfluidity in number-conserving systems. The fact that a witness
based on fermionic parity switches works in spite of the conservation of particle number can be
firmly established thanks to the realization of an exactly-solvable topological fermion superfluid,
the Richardson-Gaudin-Kitaev (RGK) wire. Moreover, the RGK wire allows derivation of an
exact topological invariant. Consequently, it follows that the fractional Josephson effect remains
a signature of topological fermion superfluidity (see Fig. 1).
To summarize, in this paper we intend to convey a certain amount of caution as to what
is physically possible in terms of manipulation and control of Majorana modes, and hope to
Fermion parity Switches Fractional Josephson effect
⌘
E [�]
J
S S
✓ ✓ �
1 2 !Φ0 !Φ20 Φ20 Φ0(�=2⇡)
�= ✓ ✓
1 2
�
Despite particle-conservation the Fractional Josephson effect remains
a physical experimental signature of Topological Superfluidity
Figure 1. (Left panel) Schematics of an SNS Josephson junction. (Right panel) Standard (2π-
periodic) and fractional (4π-periodic) Josephson effects depicting the Josephson energy E [Φ]
J
φ
as a function of the magnetic flux Φ = Φ , where Φ is the superconducting flux quantum.
2π 0 0
shed some light on this exciting field [11]. The outline is as follows. We begin in Section 2 by
presenting an exactly solvable model of a particle-conserving superfluid that, as will be shown
explicitly, displays topologically trivial and non-trivial phases [10]. In Section 3 we consider
the problem of characterizing a particle-conserving fermionic superfluid as topologically trivial
or non-trivial. This problem has been shown to have a definite answer only very recently [10].
As explained in this paper, a topological fermion superfluid (or superconductor), in addition to
the global U(1) symmetry of particle-number conservation, spontaneously breaks the discrete Z
2
symmetry of fermionic parity. In Section 4 we focus on the meaning and nature of Majorana
zero-energy modes in gapped, interacting many-electron systems. (One may compare to recent
work based on a quasi-exactly solvable, particle-number conserving, two-leg ladder model of
spinless fermions with open boundary conditions and a gapless excitation spectrum [12, 13].
The concocted Majorana-like modes are not related to total fermionic parity.) And in Section
5 we discuss the physical picture that emerges, from the standpoint of superselection rules, as
far as the quantum manipulation and control of Majorana modes is concerned. We conclude in
Section 6 with an outlook.
2. Particle-number Conserving Fermionic Superfluids: The RGK chain
We now present a model, dubbed the RGK chain, introduced and solved in Ref. [10]. The
RGK chain is the first example of an interacting, particle-conserving, fermionic superfluid in one
spatial dimension shown to display a topologically non-trivial superfluid phase. It was designed
to benchmark possible criteria of topological superfluidity in number conserving systems.
2.1. The Hamiltonian in position and momentum representations
The Hamiltonian of the RGK chain, in the momentum representation, is given by
† † †
HRGK = εkcˆkcˆk −8G ηkηk(cid:48)cˆkcˆ−kcˆ−k(cid:48)cˆk(cid:48), (1)
k(cid:88)∈Skφ k,k(cid:88)(cid:48)∈Skφ+
†
in terms of spinless (or fully spin-polarized) fermion creation operators cˆ , with momentum
k
k-dependent single-particle spectrum
ε = 2t cosk 2t cos2k, (2)
k 1 2
− −
⌘(m) Long-range
1.0
t =1,t =0
0.8 1 2
0.6 ( 1)m+1
0.4 �
0.2 ⇠ m
10 20 30 40 50
m
-0.2
-0.4
Figure 2. Functional form of the pairing interaction η(m) for L = 100, t = 1 and t = 0.
1 2
where t ,t are the nearest and next-nearest neighbor hopping amplitudes, and G > 0 the
1 2
attractive interaction strength. The interaction strength is modulated by the potential
η = sin(k/2) t +4t cos2(k/2) (3)
k 1 2
odd in k, η = η , as is characteristic of(cid:112)p-wave superconductivity.
k −k
−
The pair potential and the single-particle spectrum are connected by the simple relation
4η2 = ε +2t (t = t +t ). (4)
k k + + 1 2
This property of the model is the key for achieving exact solvability. Nonetheless, their
functional forms have been chosen so that they also realize a new exactly solvable model that is
physically sound in position, as well as momentum, space. In position representation, we define
c = L−1/2 eijkcˆ for a chain of length L, measured in units of the lattice constant. We
j k∈Sφ k
k
take φ-dependent boundary conditions c = eiφ/2c . In a ring geometry, periodic boundary
(cid:80) j+L j
conditions (φ = 0) correspond to enclosed flux Φ = 0 and antiperiodic boundary conditions
φ
(φ = 2π) correspond to Φ = Φ = h/2e. The resulting sets of allowed momenta are
0 Sk
0 = 0 0 0, π and 2π = 2π 2π, (5)
k k+ k− k k+ k−
S S ⊕S ⊕{ − } S S ⊕S
with
0 = L−1 2π, 4π, , (πL 2π) and 2π = L−1 π, 3π, , (πL π) . (6)
k± k±
S {± ± ··· ± − } S {± ± ··· ± − }
The number of momenta per sector is Card[ φ ] = L δ , so that it totals to Card[ φ] = L.
Sk± 2 − φ,0 Sk
After Fourier transformation, the RGK Hamiltonian in the position representation is given by
L 2
† †
H = t c c +H.c. 2GI I , where (7)
RGK − r i i+r − φ φ
(cid:88)i=1(cid:88)r=1(cid:16) (cid:17)
L
I 2i η cˆ cˆ = η(i j)c c . (8)
φ k k −k i j
≡ −
k∈(cid:88)Skφ+ (cid:88)i>j
There are at least two cases in which the pairing function η(m) can be determined in closed
form by Fourier transformation of Eq. (3). For t = 0 and t = 0, η(m) = √t δ , and so we
1 2 2 m1
(cid:54)
obtain nearest-neighbor pairing only. For t = 0 and t = 0 instead we obtain
1 2
(cid:54)
( 1)m8√t m
1
η(m) = − , for L , (9)
π 1 4m2 → ∞
−
describing a long-range pairing interaction with a slow 1/m decay with distance m = i j, see
−
Fig. 2. In general η(m) is a monotonically decaying function of m with η(0) = 0 = η(L/2),
η(m) > 0 or < 0 for m odd or even, respectively.
This long-ranged pairing interaction, a main difference with the original Kitaev chain [14],
allows for an exact solution beyond the mean-field approximation and a gapped spectrum in
spite of the one-dimensional character of the model. As we shall see in a moment, the long-
range coupling allows as well for a topologically non-trivial phase. It may also be physically
relevantforchainsofmagneticnanoparticlesonasuperconductingsubstrate[15,16],whichhave
recently been shown to support topologically protected Majorana zero-modes in the presence of
a long-range coupling [17].
2.2. Mean-field approximation
Before we present the exact solution of the RGK chain, we would like to establish whether
it displays a non-trivial topological phase in the mean-field approximation. For simplicity we
consider the pairing function (9), that is, t = 0. The mean-field approximation is obtained
2
from the substitution
2GI†I ∆∗I +∆I†, (10)
φ φ → φ φ
with gap function
∆ = 2G I = eiθ ∆ . (11)
φ
(cid:104) (cid:105) | |
Let us define Majorana fermion operators a = e−iθ/2c +eiθ/2c†, ib = e−iθ/2c eiθ/2c†. In
i i i i i − i
terms of these degrees of freedom the mean-field Hamiltonian is
L−1 L
it i
1
H = (b a a b ) ∆ (b a +a b ), (12)
mf i i+1 i i+1 i−j i j i j
2 − − 2
i=1 i>j
(cid:88) (cid:88)
with∆ = ∆ η(m),andcanbeshowntodisplayatopologicalphasecharacterizedbypower-law
m
| |
Majorana edge modes and the associated 4π-periodic Josephson effect. Due to the long-range
nature of the interaction, the wavefunction of the Majorana edge modes decays algebraically
rather than exponentially in the bulk, and their energy approaches zero as a power law in 1/L.
2.3. Exact solution
In Ref. [10] we showed that the RGK chain is exactly solvable. To this end we re-wrote it in the
algebraic form
H = 8H +δ (ε cˆ†cˆ +ε cˆ† cˆ ) 4t Sz +C , with (13)
RGK φ φ,0 0 0 0 −π −π −π − + φ
Hφ = ηk2 Skz −G ηkηk(cid:48) Sk+Sk−(cid:48), (14)
k∈(cid:88)Skφ+ k,k(cid:88)(cid:48)∈Skφ+
C = 2t δ , Sz = 1(cˆ†cˆ +cˆ† cˆ 1), and S+ = cˆ†cˆ† for each pair (k, k) of pairing-active
φ 2 φ,0 k 2 k k −k −k− k k −k −
momenta. The operators Sz,S± satisfy the algebra of SU(2). It follows that
k k
Sz = Sz (15)
k
k∈(cid:88)Skφ+
represents a conserved quantity, [H ,Sz] = 0 = [H ,Sz]. The relation to the total fermion
φ RGK
number operator Nˆ is
Nˆ (cˆ†cˆ +cˆ† cˆ ) L 1 if φ = 0
2Sz = − 0 0 −π −π − 2 − . (16)
(cid:40) Nˆ L (cid:16) (cid:17) if φ = 2π
− 2
N = 2M +N +N ( L orbitals)
⌫ inactive
Paired Unpaired Inactive levels
k k k k k k
� � �
seniority: ⌫ = 0 ⌫ = +1 ⌫ = 1
k k k
�
M
⌘
k
| Ni = ↵Y=1✓k2XSk�+ ⌘k2 �E↵cˆ†kcˆ†�k◆|⌫i⌦|n0n�⇡i
N = ⌫ N = n +n (only PBC)
⌫ k inactive 0 ⇡
| | �
k
X
Figure 3. Theeigenstates Ψ ofH areconstructedoutofthreedifferentkindsofelectrons
N RGK
| (cid:105)
1−|ν |
(s = k ): paired, unpaired, and inactive (only for periodic boundary conditions (PBC)).
k 2
Written in this form, one immediately recognizes H as an exactly solvable pairing
φ
HamiltonianbelongingtothehyperbolicfamilyofRichardson-Gaudinintegrablemodels[18,19].
Eigenstates with exactly 2M +N fermions are given by
ν
M
η
k † †
Φ = cˆ cˆ ν , (17)
| M,ν(cid:105) η2 E k −k | (cid:105)
α(cid:89)=1(cid:18)k∈(cid:88)Skφ+ k − α (cid:19)
where M is the number of fermion pairs. The state ν with N unpaired fermions satisfies
ν
S− ν = 0 for all k. Moreover, Sz ν = s ν , |w(cid:105)ith s = 0 if the level k is singly-
k | (cid:105) k| (cid:105) − k| (cid:105) k
occupied or s = 1/2 if it is empty (see Fig. 3). The corresponding energy levels are
k
M
E = ν H ν + E , with spectral parameters E determined by the Richardson-
M,ν (cid:104) | φ| (cid:105) α=1 α α
Gaudin (Bethe) equations
(cid:80)
s 1 Q
k φ
= , (18)
η2 E − E E E
k∈(cid:88)Skφ+ k − α β(cid:88)((cid:54)=α) β − α α
where Q = 1/2G s +M 1.
φ − k∈Skφ+ k −
For periodic boundary conditions (φ = 0), the two momenta k = 0, π are not affected by
(cid:80)
−
the interactions and must be included separately. Then, eigenvectors of H are given by
RGK
Ψ = Φ n n , (19)
N M,ν 0 −π
| (cid:105) | (cid:105)⊗| (cid:105)
where N = 2M +N +n +n is the total number of fermions and n ,n 0,1 . See Fig.
ν 0 −π 0 −π
∈ { }
3.
Topological E = 0 ⇢ = 1 1/g
) �
⇢
µ = 0 ⇢ = 1/2 1/2g
) �
Topologically trivial
g
Figure 4. Quantum phase diagram of the RGK wire in the (ρ,g)-plane. Dashed and full lines
representtheMoore-Read(g−1 = 1 ρ)andRead-Green(g−1 = 1 2ρ)boundaries,respectively.
− −
Asshowninthetext, theweak-pairingphaseistopologicallynon-trivialwhilethestrongpairing
phase is topologically trivial. The horizontal dashed arrow corresponds to the density ρ = 1/4.
2.4. Quantum phase diagram
The quantum phase diagram of the RGK chain is determined from the analytical dependence
of its ground energy (ρ,g) on the density ρ = N/L and scaled coupling strength g = GL/2.
0
E
Depending on the boundary condition φ and fermion-number parity, one should consider either
N = 0 or 1. For periodic boundary conditions, since the levels k = 0, π decouple from the
ν
−
rest, N = 0 for both even and odd N. If N is odd, then the unpaired particle occupies the
ν
k = 0 level without blocking an active level. For antiperiodic boundary conditions the ground
state has N = 0 for N even, while for N odd it has N = 1 with blocked level k . The resulting
ν ν 0
ground state energy is given by
M
φ(N) = 8 E 4t M +J +δ (4η2 2t ), (20)
E0 α− + φ,0 Nν,1 k0 − +
α=1
(cid:88)
where J = δ (ε δ +ε δ ). In the thermodynamic limit (N,L , such that ρ is
φ,0 φ,0 0 n0,1 −π n−π,1 → ∞
kept constant) the energy density becomes
4 4 π
e lim φ/L = 2t ρ ∆2+ η2v2dk, (21)
0 ≡ L→∞E0 − + − g π 0 k k
(cid:90)
where the chemical potential µ and gap function ∆ are related to the original parameters of the
model by
2π π η2 1 π
= k dk, ρ = v2dk, (22)
g E π k
0 k 0
(cid:90) (cid:90)
η2 2 1 η2 2µ
E = k µ +η2∆2, v2 = k − . (23)
k (cid:115) 2 − k k 2 − 4Ek
(cid:18) (cid:19)
The resulting phase diagram is shown in Fig. 4. The RGK chain is gapped for all g > 0,
except for the Read-Green coupling g = g = 1/(1 2ρ) where it becomes critical in the
c
−
thermodynamic limit, independently of the choice of boundary conditions φ. This critical line
defines the phase boundary separating weak from strong pairing phases, and thus is a line of
non-analyticities. At g a cusp develops in the second derivative of e , that leads to singular
c 0
discontinuous behavior of the third-order derivative. Hence the transition from a weakly-paired
to a strongly-paired fermionic superfluid is of third order, just like for the two-dimensional
chiral p-wave superconductor [20, 21]. Which one of these two superfluid phases, if any, may be
properly characterized as topologically non-trivial?
3. Fermionic Parity Switches and the Fractional Josephson Effect
Reference [10] introduced a quantitative criterion for establishing the emergence of topological
superfluidity in particle-number conserving, many-fermion systems. The criterion exploits the
behavior of the ground state energy of a system of N, and N 1 particles, for both periodic
±
and antiperiodic boundary conditions. The emergence of topological order in a superconducting
wire, closedinaringanddescribedinmean-field, isassociatedwithswitchesintheground-state
fermionparity (φ)uponincreasingtheenclosedfluxΦ = (φ/2π) Φ [14,22,23,24,25,26,27].
0
P ×
Any spin-active superconductor, topologically trivial or not, may experience a crossing of the
ground state energies for even and odd number of electrons [28, 29, 30, 31]. Regardless of spin,
what matters is the number of crossings N between Φ = 0 = φ and Φ = Φ , φ = 2π. The
X 0
superconductor is topologically non-trivial if N is odd, otherwise it is trivial.
X
In the many-body, number conserving, case we need to identify the relevant parity switches
signaling the emergence of a topological fermion superfluid phase. Our exact solution gives us
access to (φ) only at φ = 0 and φ = 2π, but this is sufficient to determine whether N is even
X
P
or odd. Notice that odd N means that the flux Φ should be advanced by 2Φ , rather than Φ ,
X 0 0
in order to return to the initial ground state. This is the essence of the 4π-periodic Josephson
effect [14, 32].
Topological Topologically Trivial
(�=� )=+1
PN 0
0.3
0.2
n
ve 0.1
e0
�E0.0 Antiperiodic
Periodic
d
d
o0E-0.1 PN(�=0)=PN(�=�0)=+1
-0.2 ⇢=1
4
-0.3
1.4 1.6 1.8 2.0 2.2 2.4 2.6
gc g
(�=0)= 1
PN �
Figure 5. Ground state energy differences (in units of t 1, for t = 0) for even (N = 2M)
1 2
≡
and odd (N = 2M 1) number of fermions, and with periodic (φ = 0) or antiperiodic (φ = 2π)
±
boundary conditions. The odd-even difference is shown as a function of the interaction strength
g for a finite system (data points, for N = 512, L = 2048) and in the thermodynamic limit
(continuous lines). The topologically non-trivial state is entered for g < g = 2. Also indicated
c
are the values of the fermion parity switches (Φ) across the transition.
N
P
φ
Toidentifythefermionparityswitcheswecalculatethegroundstateenergy (N)foragiven
E0
number N of fermions in the chain of length L, with periodic (φ = 0) or antiperiodic (φ = 2π)
boundary conditions, and compare odd(φ) = 1 φ(N +1)+ 1 φ(N 1) and even(φ) = φ(N),
E0 2E0 2E0 − E0 E0
where we assumed N even. The difference (inverse compressibility) χ(φ) = odd(φ) even(φ)
0 0
E −E
determines (φ) = signχ(φ), so it has the opposite sign at φ = 0 and φ = 2π in the
N
P
Thermodynamic Limit
E0even Topological E0eovdedn Trivial -1581.5 E0odd
odd E0 g=1.6 g=2.5
E0 ⇢=1 -1582.0
4 even ⇢=1 even
E0 4 E 0
-1120.0
odd Special flux �
-1120.2 g=1.6 EE00even g=2.5 EE00oevdedn E0odd Topological T⇤rivial
�=0 �=2⇡ �=0 �=2⇡
�=0 �=2⇡ �=0 �=2⇡
Figure 6. Groundstateenergiesforevenandoddnumberoffermions, forthefinitesystem(left
panel) and in the thermodynamic limit (right panel), illustrating the fermion parity switches
(φ). Notice the special role played by the flux φ∗ at the crossing (degeneracy) point.
N
P
topologically non-trivial phase. We also find that (φ) = (φ) in the topologically
N∈even N∈odd
P −P
non-trivial phase. The results, shown in Figs. 5 and 6, unambiguously demonstrate the
topologically non-trivial nature of the superfluid for g < g — both in a finite system and
c
in the thermodynamic limit, and without relying on any mean-field approximation.
The ground state of the odd (2M 1) system strongly depends on the boundary conditions.
±
For periodic boundary conditions the unpaired particle always occupies the k = 0 level, while
0
for the antiperiodic case it starts blocking the Fermi momentum k = k at g = 0, continuously
F 0
decreasingitsmoduluswithincreasingg,uptok = π/Latg 1.1936(ρ = 1/4),corresponding
0 0
∼
to µ = ∆2 in the thermodynamic limit. In that limit χ(φ) has a particularly simple form:
χ(0) = 8µ, and χ(2π) = 8 µ for g > g .
0
− | |
4. Many-body Majorana Zero-energy Modes
The subject of this section is conceptually rather challenging. It is convenient to begin by
recalling how Majorana modes emerge within the standard mean-field framework before we
tackle this problem in a many-body setting.
4.1. Majorana modes in the mean-field approximation
Inthemean-fielddescription,i.e.,theBogoliubov-deGennes(BdG)approach,asecond-quantized
Hamiltonian describing a generic fermionic superfluid is [1, 33]
L
1 1
H = K c†c + ∆ c†c† + ∆∗ c c , (24)
mf ij i j 2 ij i j 2 ij j i
i(cid:88),j=1(cid:104) (cid:105)
† †
intermsofcreation/annihilationoperatorsc /c ofafermion( c ,c = δ )inthesingle-particle
j j { i j} ij
orbital φ . The labels i,j subsume arbitrary quantum numbers like position or momentum,
j
band index, spin and orbital angular momentum, and so on. The total number of single-particle
orbitals is L < in the presence of infrarred (volume) and ultraviolet (lattice spacing) cutoffs.
∞
In Eq. (24), the one-body kinetic energy and mean-field pairing interaction matrices
K† = K, ∆T = ∆, (25)
−
encode the relevant effective interactions of the superfluid system. Here † is the adjoint, T the
transpose, and ∗ complex conjugation of a matrix.
In Nambu form, the Hamiltonian H can be rewritten as
mf
1 1
H = φˆ†H φˆ+ TrK, (26)
mf BdG
2 2
where the Nambu column vector of fermion operators is given by
cˆ
φˆ= , with φˆ = c , φˆ = c† (j = 1, ,L). (27)
cˆ† j j L+j j ···
(cid:18) (cid:19)
Then the BdG single-particle Hamiltonian is the 2L 2L matrix
×
K ∆
H = = i1 (K)+iτx (∆)+iτy (∆)+τz (K), (28)
BdG ∆∗ K∗ ⊗(cid:61) ⊗(cid:61) ⊗(cid:60) ⊗(cid:60)
(cid:18)− − (cid:19)
where τν, ν = x,y,z, are Pauli matrices, and ( )( ( )) denotes the real (imaginary) part of the
(cid:60) · (cid:61) ·
matrix .
·
No matter what the specific form of the matrices K and ∆ may be, the single-particle
Hamiltonian H always anticommutes with the antiunitary operator
BdG
= τx 1, 2 = 1, (29)
C K ⊗ C
of particle-hole or charge-conjugation symmetry, H , = 0. Here denotes complex
BdG
{ C} K
conjugation. It follows that the single-particle energy spectrum
X(n)
H ϕ = (cid:15) ϕ , ϕ = , n = L, L+1, , 1,1, ,L 1,L, (30)
BdG n n n n Y(n) − − ··· − ··· −
(cid:18) (cid:19)
is symmetrically distributed around zero and so it is automatically particle-hole symmetric,
(cid:15) = (cid:15) . In turn, the single-particle spectrum can be used to write down H
n −n mf
−
1
H = (cid:15) f†f + TrK (cid:15) (31)
mf n n n n
2 −
n(cid:88)>0 (cid:16) n(cid:88)>0 (cid:17)
in terms of quasi-particle operators
L
f† = (X(n)c† +Y(n)c ). (32)
n j j j j
j=1
(cid:88)
Assume now the existence of an exact zero energy mode, then it must be, at least, two-fold
degenerate
(cid:15) = 0 = (cid:15) , (33)
1 −1
and the associated single-particle states, ϕ , ϕ , yield the quasi-particle operators
1 −1
L L
f† = X(1)(c† +c ) , X(1) R , f† = X(−1)(c† c ) , iX(−1) R. (34)
1 j j j j ∈ −1 j j − j j ∈
j=1 j=1
(cid:88) (cid:88)
These zero-energy quasi-particles are necessarily Majorana fermions because
† †
f = f ,f = f . (35)
1 1 −1 −1
In other words, in the BdG formalism, zero-energy modes are Majorana fermions by default.
Typically, in a finite-size system, Majorana quasi-particles are not exact zero-energy modes but,
strictly speaking, emerge as such only in the thermodynamic (L ) limit.
→ ∞