sites, denoted by |0Ni for clarity in this section, is defined to be the product state of zero fermion occupancy states in the position representation,
|0Ni ≡ ON
x=1
|0, xi ≡ |00. . .0i. (4.10) This state is destroyed by any cx operator. We define a state with some filled sites to have the order of increasing indices on fermion creation operators. For instance, the fully filled state, |11· · ·1i, is defined as
|11. . .1i ≡c†1c†2· · ·c†N|0Ni. (4.11) Acting with a fermion operator on some general state may therefore result in picking up a sign, depending on the fermion parity of the preceding part of the state. For instance,
cj|α1α2· · ·αj−11αj+1· · ·αNi= (−1)Pj−1i=1αi|α1α2· · ·αj−10αj+1· · ·αNi, (4.12) with αi ∈ {0,1}. We know that the ground state subspace is two-dimensional since the fermionic operatord†0 creates an excitation of zero energy. Two orthogonal states spanning the ground state, {|g1i,|g2i}, are therefore defined to satisfy
d0|g1i=d†0|g2i= 0. (4.13) The states are related by
d†0|g1i=|g2i. (4.14) We start by constructing a ground state candidate, |ai, naively by applying all dx
for x∈ {1, . . . , N −1} on the vacuum state,
|ai ≡
NY−1 x=1
dx|0Ni. (4.15)
Normalization is taken care of later. The state |ai has by construction zero en-ergy.2 An immediate question should be raised. The ground state subspace is two-dimensional, and we have a set of three possible states at our disposal,
G ≡n
|ai, d0|ai, d†0|aio
. (4.16)
2Again, the constant energy term in (4.7) is understood to be neglected. Therefore, the Hamil-tonian witht= ∆ andµ= 0 is simplyH=tPN−1
x=1 d†xdx.
Section 4.2 The Ground State Subspace 55
By some means, one of the states in G must be eliminated as candidates for |g1i and |g2i. In order to determine which, the state|ai should be calculated explicitly.
We will do this by observing a pattern occurring with just a few sites, propose a hypothesis, and prove it by induction on the system size, N. The starting point is to express the d operators in terms of ordinary fermion operators. Using equations (4.2), (4.5) and (4.8) we find, assuming that φ= 0,
dx = i 2
cx+c†x−cx+1+c†x+1
and d0 = i 2
cN +c†N −c1+c†1
. (4.17) To see the structure of |ai we look at the emerging pattern with N ∈ {2,3,4}. Applying d operators successively yields
d1|02i= i 2
h|10i+|01ii
, (4.18)
d2d1|03i= i
2 2h
|000i − |101i − |110i − |011ii
, (4.19)
d3d2d1|04i= i
2 3h
|0i ⊗
|10i+|01i
⊗ |0i − |1i ⊗
|10i+|01i
⊗ |1i +|1i ⊗
|00i − |11i
⊗ |0i+|0i ⊗
|00i − |11i
⊗ |1ii . (4.20) In (4.20) the states belonging to the end points are explicitly extracted to reveal a pattern. It motivates our induction hypothesis, denoted by IN, on the general form of |ai,
IN : |ai ≡
N−1Y
x=1
dx|0Ni= i
2
N−1h
|0b0i − |1b1i+|0c1i+|1c0ii
. (4.21)
Above,|bicontains all 2N−3 combinations of odd (even) fermion parity states ifN is even (odd). All the states in |bihave the same relative weights of either +1 or −1.
Similarly, |ci contains all 2N−3 combinations of even (odd) fermion parity states if N is even (odd). The relative weights of the states in |ci are also either +1 or −1.
We proceed by proving that IN ⇒ IN+1 when the number of sites is increased by one in the chain. This is, by induction, enough to prove that IN holds for all N. Proof. Assume first that N is odd. Thus, IN tells us that |ai takes the form in (4.21) with |bi being an even parity state and |ci as an odd parity state. We must show that|a0i ≡dN|ai ⊗ |0iis in accordance with the description imposed by IN+1. In other words, |a0i must have |b0ias an odd parity state and |c0i as an even parity state since N + 1 is even. We find that
|a0i ≡dN |ai ⊗ |0i
= i
2 Nh
|0i ⊗
|b1i − |c0i
⊗ |0i − |1i ⊗
|b1i − |c0i
⊗ |1i +|1i ⊗
|b0i+|c1i
⊗ |0i+|0i ⊗
|b0i+|c1i
⊗ |1ii
= i
2 Nh
|0b00i − |1b0 1i+|0c01i+|1c0 0ii ,
(4.22)
with
|b0i ≡ |b1i − |c0i and |c0i ≡ |b0i+|c1i. (4.23) Hence, |b0iis an odd parity state that contains all 2N−2 combinations, which follows from the induction hypothesis. In addition, |c0i must be an even parity state con-taining all 2N−2 combinations. This is exactly the properties required by IN+1. In principle, we must also show that the same implication holds withN even. However, this is not necessary since the only change would be an overall sign in the above calculation, stemming from equation (4.12). We conclude that IN ⇒ IN+1 and IN
is true for all N.
The basic properties of the state|ai are now established by equation (4.21). At this point it is convenient to split the discussion into even and odd N.
4.2.1 Odd N Ground States
According to (4.21), the state |bi contains combinations of even parity states with relative weights of either +1 or −1. Analogously, |ci contains combinations of odd parity states. Using once more the rule in equation (4.12) we find that
d†0|ai= 0. (4.24)
In other words, the stated†0|aiis trivial and should be excluded fromG. Calculating d0|ai and normalizing the resulting states gives the proper ground states, |g1i and
|g2i, and corresponding fermion parities:
d0|ai 7→ |g1i= 1
√2N−1
h− |0c0i+|1c1i+|0b1i+|1b0ii
with P =−1,
|ai 7→ |g2i= 1
√2N−1
h|0b0i − |1b1i+|0c1i+|1c0ii
with P = +1.
(4.25) Notice how all the 2N available combinations of parity states are involved in either
|g1ior |g2i.
Section 4.2 The Ground State Subspace 57
4.2.2 Even N Ground States
Still, |ai is as described in (4.21), and it has |bi containing combinations of odd parity states and |ci containing combinations of even parity states. Again, by using the rule in equation (4.12) we find that
d0|ai= 0. (4.26)
Hence, the state d0|ai should now be excluded from G. Normalizing |ai and d†0|ai results in the ground states, which this time are denoted by |g˜1i and |g˜2ito distin-guish them from the odd N ground states,
|ai 7→ |g˜1i= 1
√2N−1
h|0b0i − |1b1i+|0c1i+|1c0ii
with P =−1, d†0|ai 7→ |g˜2i= 1
√2N−1
h|0c0i − |1c1i − |0b1i − |1b0ii
with P = +1.
(4.27)
4.2.3 Relation to the Closed Chain
The deduction above was restricted to the special case of ∆ =tand µ= 0. We have just seen that the degenerate ground states involve all fermion occupancy states with weights of equal magnitude. The basic parity properties found above are in satisfactory accordance with Kitaev’s discussion [10]. However, our results give a more detailed picture of the ground state subspace structure that is not transparent a priori. Furthermore, we have seen that choosing the correct states from the set of candidates inG depends non-trivially on whether the number of sites are even or odd. A natural follow-up question is to ask what the relation between the degener-ate ground stdegener-ates of the open chain and the unique ground stdegener-ate in the closed chain is. Recall from Figure 3.8 that the state|Ωiin the periodic chain hadP =−1 when
|µ|< t.
Imagine connecting the two ends of the open chain by adding the term λit2γA,1γB,N
in equation (4.4) and gradually increasingλfrom 0 to 1. Since the state|g1i(|g˜1i) is of odd parity whenN is odd (even), it should correspond to the unique ground state
|Ωifor odd (even)N. Recall further the defining property of the ground state in the periodic chain: it should be annihilated by any quasiparticle annihilation operator.
This is in perfect agreement with the deduction above since both|g1iand|g˜1iare an-nihilated byd0 (and by construction all the otherdxoperators),d0|˜g1i=d0|g1i= 0.
Moreover, the state |Ωi was of even fermion parity with anti-periodic boundary conditions. In this case, |g2i (|g˜2i) would be the correct candidates for |Ωi for odd (even) N. However, both |g2i and |g˜2i are annihilated by d†0 and not d0. Formally, this is equivalent to introducing the term λi2tγA,1γB,N in the Hamiltonian but let λ change from 0 to −1 instead. Anti-periodic boundary conditions may therefore
be interpreted as a negative hopping between the ends of the open chain. To close the focus on the degenerate ground states, we calculate the entanglement entropy between the ends and the interior of the chain.
4.2.4 Entanglement Entropy
We seek further insight of the ground states with ∆ =t and µ= 0 by the dividing the system into two parts. Let E (edge) be the subsystem consisting of the end points, x = 1 and x = N, and I (internal) the internal part of the system, x ∈ {2, . . . , N−1}. To keep our discussion concrete, we stick to odd N and make use of (4.25). However, our results can be checked to remain equally valid for both ground states with even N. We want to quantitatively establish the entanglement entropy of the composite systemI+E when being in one of the states|g1ior|g2i. Generally, if the system is in some state |ψi, the density operator of the full system is defined to be
ρ≡ |ψi hψ|. (4.28)
The density operator of the subsystem E is found by tracing over the states from I in ρ,
ρE = TrI (ρ) =X
i∈I
hi|ρ|ii. (4.29)
The von Neumann entropy of ρE is then taken as a quantitative measure of the entanglement entropy of the composite system [23],
SE =−Tr(ρElogρE) = −X
k
λ(kE)logλ(kE), (4.30)
with λ(kE) the kth eigenvalue of ρE. This framework is applied to |ψi = |g1i and ρ=|g1i hg1|. To trace out the internal system one may split up the sum in odd and even parity states in I,
ρE =X
i∈I
hi|ρ|ii
= X
i∈I i:P=−1
hi|ρ|ii+ X
j∈I j:P=+1
hj|ρ|ji. (4.31)
The reason for this is that hi|ρ|ii only contributes for terms containing |ci hc|. Fur-thermore, we know from earlier that |ci contains 2N−3 states of odd parity with weights equal to ±1. Therefore, hi|ci hc|ii = 1 in the odd parity trace. Similarly, the sum over even parity states picks out contributions with |bi hb|in ρ. Hence,