Tamer S. Boz
Quantum Fields on Star Graphs with Bound States at the Vertex
Physics
Master-level thesis
Date/Term: XX-XX-XX
Supervisor: Professor Jürgen Fuchs Examiner: Professor Claes Uggla Serial Number: X-XX XX XX
A star graph consists of an arbitrary number of segments that are joined at a point which is called the vertex. In this work it is investigated from a pure theoretical point of view, in the framework of quantum field theory. As a concrete physical application, the electric conductance tensor is obtained. In particular it is shown that this conductance behaves differently according to whether the scattering matrix associated with the vertex of the graph has bound-state poles or not.
Sammanfattning
En stjärngraf består ov ett godtyckligt antal segment som möts i en punkt, vilken kallas nod. I denna uppsats undersöks stjärngrafer från ett teoretiskt perspektiv inom ramen för kvantfältteori. Som en konkret tillämpning erhålls den elektriska konduktanstensorn.
Speciellt visas att konduktansen uppför sig olika beroende på om spridningsmatrisen som hör till noden tillåter bundna tillstånd eller inte.
Abstract . . . 1
Sammanfattning . . . 1
1 INTRODUCTION 3 1.1 Background . . . 3
1.2 Overview . . . 4
2 MASSLESS BOSONS WITH GENERAL BOUNDARY CONDITIONS ON A STAR GRAPH 6 2.1 Preliminaries . . . 6
2.2 The Scattering Component ϕ(s) . . . 8
2.3 The Vertex Bound State (VBS) Component ϕ(b) . . . 10
2.4 Dual field, Chiral Fields and Vertex Operators . . . 12
3 CORRELATION FUNCTIONS 15 3.1 The Half-Line . . . 15
3.2 Generic Star Graph . . . 17
4 LUTTINGER LIQUID WITH GENERAL BOUNDARY CONDITIONS 19 4.1 Bosonization . . . 20
4.2 Conductance . . . 22
5 TOPOLOGICAL DEFECTS 24 6 RESULTS AND OUTLOOK 27 7 APPENDIX 28 A: Conditions for the Operator −∂x2 to Be Self-Adjoint . . . 28
B: Unitarity of the Scattering Matrix . . . 31
C: Expression of the Scattering Matrix in Diagonalized Form . . . 32
D: Consistency of the Conditions (2.17) . . . 34
E: Derivation of the Commutation Relation (2.19) . . . 34
F: Derivation of the Expression (2.27) . . . 35
G: Scattering and Bound-state Contributions to the Dual Field . . . 36
H: Derivation of the Correlation Function (3.3) . . . 38
I: Associativity of the Star Product . . . 40
J: Trivialness of the Topological Scattering Matrix . . . 43
1.1 Background
Apart from being of theoretical interest on its own, quantum wires have practical ap- plications. These are mainly carbon nanotubes, which are essentially molecules with a diameter of as little as 1 nm [1, 2]. In this work, which is based on the article [3], quantum fields on a star graph, in particular with boundary conditions that allow for bound states at its vertex, are investigated. This is a theoretical model for quantum wires joined, in general, at an arbitrary number of junctions. Such a graph, for the special case of four edges joined at a single junction, is shown in Fig. 1.
Fig. 1 A star graph with four edges
In a practical construction, the vertex corresponds to a defect or impurity which can be represented by a scattering matrix S. Depending on the boundary conditions at the vertex, this matrix may or may not allow bound states. As a result, two different regimes emerge in the theory: without bound states, the time evolution of the system is unitary and the energy is conserved, whereas the presence of the bound states results in energy flow through the boundary, i.e., the vertex. Though the underlying Hamiltonian is Her- mitian, it is not self adjoint, which is responsible for this non-unitary time evolution.
The bound states of S will be referred as the vertex boundary states because they decay exponentially in the bulk of the graph, i.e., away from the vertex.
We begin by investigating a massless scalar field on a star graph with n edges, which satisfies the Klein-Gordon equation
∂2t − ∂x2 ϕ (t, x, i) = 0, i = 1, ..., n.
Our investigation, at this stage, proceeds as considering a quantum field theory problem for a bosonic field in one space dimension. We impose the relevant commutation relations and require that microcausality is satisfied, i.e., that local commutation relations vanish for space-like separations. To have a complete set of solutions we require the operator
−∂x2 to be self-adjoint. This requirement restricts the choice of the boundary condition and allows one to determine all possibilities for S.
As a result of the scattering, the scalar field can be split into scattering and bound parts ϕ(s) and ϕ(b), respectively. The canonical commutation relation for ϕ(s) turns out to give two different expressions according to the analytical properties of S. The case with poles in the upper complex half plane gives rise to a non-trivial expression for ϕ(b), while absence of poles in the complex upper half plane corresponds to vanishing of it, i.e., bound states do not occur.
We investigate the former case for which the Hamiltonian can be split into contributions resulting from scattering and bound states as well. The scattering part resembles the Hamiltonian of a harmonic oscillator, whereas the bound part looks similar to that of a damped oscillator, which is responsible for the non-unitary time evolution. As an example to a physical observable, the vacuum energy of the system is given, which is time dependent.
To deal with electrons in a wire requires consideration of a Luttinger liquid [5]. This is a model that describes electrons in one dimension for which the Fermi liquid model breaks down. It is exactly solvable by means of the method of bosonization which involves the notion of vertex operators. The reason for considering a bosonic field on a star graph previously is that this method allows one to treat fermionic excitations as bosons.
We are particularly interested in calculating the electric conductance tensor of the system in question. To include electromagnetic interactions we do minimal substitution in the Lagrangian. As expected, the electric conductance tensor turns out to exhibit different behaviours for the two regimes of the theory, which may open a possibility for comparison with experimental results.
1.2 Overview
As pointed out above, this work closely follows the article [3], and may be regarded, in some sense, as a reproduction of large parts of it. However, much effort is put in the calculations and derivations in the appendices, which are not fully given, or not given at all, in the articles used through the work, especially in the main article [3].
In Section 2, the problem of a scalar field on a star graph is investigated. The results are required for applying the bosonization method that is used later on. The equal-time commutation relations and the scattering and bound state contributions to the total field are introduced here as well as dual fields, chiral fields and the vertex operator, which
is an important tool for bosonization. Section 3 is concerned with the correlators and some physical observations for the bosonic field propagating on the graph. This is first done on a special case of the graph -the half line, and then the results are generalized to the generic star graph. In Section 4, the main goal of obtaining the conductance tensor of a quantum wire is carried out via bosonization, which allows us to treat fermionic excitations in the wire as bosons. Section 5 discusses an aspect which, to the best of our knowledge, is not treated in the literature. We investigate the question if one can have a quantum wire with defects having topological properties. Section 6 collects the main results and mentions some problems that can be investigated. Finally we present an appendix where calculations and derivations are given that can not be fully found in the sources that are referred in the Bibliography.
GENERAL BOUNDARY
CONDITIONS ON A STAR GRAPH
2.1 Preliminaries
We begin by investigating a massless scalar field. This is a preparation to investigate the Luttinger liquid on a star graph, which is solvable by the method of bosonization. The equation of motion for a scalar field is the Klein-Gordon equation which, for a massless field reduces to:
∂t2− ∂x2 ϕ (t, x, i) = 0. (2.1) ϕ (t, x, i) denotes the value of the field on the ith edge of the graph, at time t, at a posi- tion x, which is the distance to the vertex. We next impose the equal-time commutation relations:
[ϕ (0, x, i) , ϕ (0, x, j)] = 0, [∂tϕ (0, x, i) , ∂tϕ (0, x, j)] = 0, (2.2)
∂tϕ (0, x, i) , ϕ 0, x0, j = −iδijδ x − x0 . (2.3) To proceed further, we require the operator −∂x2 to be self-adjoint. As shown in Appendix A, this is the case provided that we choose those functions which satisfy the boundary condition:
n
X
j=1
[Aijϕ (t, 0, j) − Bij(∂xϕ) (t, 0, j)] = 0, ∀ t ∈ R, i = 1, ..., n (2.4) as the domain of the operator. Moreover, the matrices A and B must satisfy AB†= BA† and the composite matrix bABc, as defined in Lemma 2 of Appendix A, must have rank n. The matrices A and B can be parametrized as [3] : A = λ (1 − U ) , B = i (1 + U ), where U is any unitary matrix and λ > 0 is a parameter with dimension of mass.
The scattering matrix S is given by [4]:
S (k) = − [λ (1 − U ) + k (1 + U )]−1[λ (1 − U ) − k (1 + U )] . (2.5)
With this form, S (k) is unitary:
S (k)†= S (k)−1, (2.6)
as shown in Appendix B, and satisfies:
S (k)†= S (−k) , (2.7)
which is referred as Hermitian analyticity.
From (2.6) and (2.7) one obtains:
S (k) S (−k) = 1. (2.8)
One also has:
S (λ) = − [λ (1 − U ) + λ (1 + U )]−1[λ (1 − U ) − λ (1 + U )] ,
[λ (1 − U ) + λ (1 + U )] S (λ) = − [λ (1 − U ) − λ (1 + U )] , which implies
S (λ) = U. (2.9)
For diagonal elements we have: Sii =hϕ (t, x, i)| S |ϕ (t, x, i)i, which is the transition amplitude for a scattered state S |ϕ (t, x, i)i to be found in |ϕ (t, x, i)i, i.e. reflection at the vertex. Analogously, an off-diagonal element,
Sij = hϕ (t, x, i)| S |ϕ (t, x, j)i ,
can be interpreted as the transition amplitude for transmission from the jth edge to the ith.
In particular, a 2 × 2 matrix which corresponds to a star graph with two edges has the form:
S =
S11 S12 S21 S22
=
R T
T L
, (2.10)
where R (L) stands for the reflection coefficient from the right (left) and T stands for the transmission coefficient. One should be warned about the convention used here in that it differs from that in quantum mechanics, where the corresponding matrix is:
Sstandard =
T R
L T
. The two conventions are related as follows ([4], p. 601):
Sstandard = S
0 1 1 0
.
Neverthless, one can check that the properties given through (2.6)-(2.8) follow for both conventions. The reason for following another convention from the standard one reveals itself when considering an n × n matrix for large n. Namely, there are n2−n transmission coefficients Sij with i 6= j, while there are just n reflection coefficients Sii, which are more natural to be placed in the diagonal.
As shown in Appendix C, diagonalization of S by means of the appropriate unitary matrix leads to the following expression:
Sd(k) = diag k + iη1
k − iη1,k + iη2
k − iη2, ...,k + iηn
k − iηn
, (2.11)
where ηi’s are parameters given by
ηi = λ tan (αi) , (2.12)
αi’s being in the range −π2 ≤ αi ≤ π2.
This structure, with poles on the imaginary axis, is characteristic of some kind of bound states in quantum field theory [9]. The poles on the upper half-plane are known as bound states while those on the lower half-plane are called antibound states. The case when the S-matrix is constant is referred as the critical or conformal case. It follows directly from (2.11) that in this case the S-matrix has the eigenvalues ±1. These correspond to poles at η = 0 and η 7→ ±∞.
The bound and antibound states are formed due to the scattering at the vertex. We can split the field into contributions coming from the elastic scattering and from the bound states:
ϕ (t, x, i) = ϕ(s)(t, x, i) + ϕ(b)(t, x, i) . (2.13)
2.2 The Scattering Component ϕ
(s)The solution to the equation of motion with the given boundary condition is known to be [6]:
ϕ(s)(t, x, i) = ˆ ∞
−∞
dk 2πp2 |k|
h
a†i(k) ei(|k|t−kx)+ ai(k) e−i(|k|t−kx)i
, (2.14)
where a†i (k) and ai(k) are creation and annihilation operators that are the generators of an associative algebra A [7, 8], which satisfy the following commutation relations:
[ai1(k1) , ai2(k2)] = 0, h
a†i1(k1) , a†i2(k2) i
= 0, (2.15)
h
ai1(k1) , a†i
2(k2)i
= 2π [δi1i2δ (k1− k2) + Si1i2(k1) δ (k1+ k2)] 1, (2.16) and the constraints:
ai(k) =
n
X
j=1
Sij(k) aj(−k) , a†i(k) =
n
X
j=1
a†j(−k) Sji(−k) . (2.17) In Appendix D, it is shown that (2.8) guarantees the consistency of these conditions.
These operators are neither even nor odd under time reversal that is implemented in the algebra A by:
T a (k) T−1 = a (−k) , T a†(k) T−1= a†(−k) , (2.18) where T is antiunitary.
The expression for ϕ(s)(t, x, i), inserted in the canonical commutation relation (2.3), gives:
h
∂tϕ(s)
(0, x1, i1) , ϕ(s)(0, x2, i2) i
= −iδi1i2δ (x1− x2) − i 2π
ˆ ∞
−∞
dk eik ˜x12Si1i2(k) , (2.19) as shown in Appendix E, where ˜x12≡ x1+ x2. The integral on the right hand side can be computed by the method of residues:
ˆ +∞
−∞
dk eik(x1+x2)Si1i2(k) = 2πiX
j
Reszj n
eiz(x1+x2)Si1i2(z) o
.
We choose the contour in the upper complex half plane:
X
j
Reszj n
eiz(x1+x2)Si1i2(z) o
= X
iη∈P+
lim
k→iη
(k − iη) eik(x1+x2)Si1i2(k)
= X
iη∈P+
e−η(x1+x2) lim
k→iη
(k − iη) Si1i2(k) ,
which implies ˆ ∞
−∞
dk eik ˜x12Si1i2(k) = 2πi X iη∈P+
e−η ˜x12 lim
k→iη
(k − iη) Si1i2(k) , (2.20) where
P+= {iη : η > 0} (2.21) denotes the set of poles of Sd(k) in the upper half-plane and similarly, P−= {iη : η < 0}
will denote the set of poles in the lower half-plane. Defining a new matrix:
R(η)i
1i2 ≡ 1 iη lim
k→iη
(k − iη) Si1i2(k) , iη ∈ P± (2.22) we have:
h
∂tϕ(s)
(0, x1, i1) , ϕ(s)(0, x2, i2) i
= −iδi1i2δ (x1− x2) + i X iη∈P+
ηe−η ˜x12R(η)i1i2. (2.23)
Two different regimes show up for the problem depending on whether P+= Ø or not:
• If P+= Ø (no poles in the upper half-plane) the summation on the right hand side of (2.23) vanishes and the scattering component satisfies the canonical commutation relation (2.3) which the total field ϕ = ϕ(s)+ ϕ(b) is supposed to satisfy. Therefore we set ϕ(b) = 0 in this case, which means that the theory does not allow bound states.
• If P+ 6= Ø, ϕ(b) should be chosen such that the total field satisfies the canoni- cal commutation relation (2.3). We say that the scattering matrix admits bound states. The problem of constructing ϕ(b)in this manner will be solved by the phys- ical requirement of microcausality, i.e., by requiring that the local commutation relations vanish for space-like separations (local commutativity).
Which of the situations is present depends on the particular self-adjoint extension chosen, i.e., on the choice of the boundary condition. To decide which of the two regimes is actually realized for a given boundary condition is a difficult problem, which we do not analyze in this work.
2.3 The Vertex Bound State (VBS) Component ϕ
(b)We now assume that P+ 6= Ø, i.e., we allow for the bound states and introduce the set ι+ = {i : ηi> 0}. The wave functions of the bound states of K = −∂x2 are then {e−ηix|i ∈ ι+} and generate the set of real solutionse−ηi(x±t)|i ∈ ι+ to the equation of motion (2.1). We associate with each index i ∈ ι+ a quantum oscillator. The field ϕ(b) in terms of these solutions is defined to be:
ϕ(b)(t, x, i) = 1
√ 2
X
j∈ι+
Uijh
b†j+ bj
e−ηj(x+t−tm)+ i
b†j− bj
e−ηj(x−t+tm) i
, (2.24)
where U , introduced in Appendix C, is the matrix that diagonalizes the unitary matrix U and biand b†i are generators of an algebra B that commute with
n
ai(k) , a†i(k) o
intro- duced in the previous section and satisfy the following canonical commutation relations:
[bi1, bi2] = 0, h
b†i
1, b†i
2
i
= 0, (2.25)
h bi1, b†i
2
i
= δi1i2. (2.26)
The expression for ϕ(b) satisfies the equation of motion (2.1) and the boundary con- dition (2.4) by construction. The functions e−ηi(x±t) can be interpreted as the wave functions associated with the bound states. Note that the solution depends on a free parameter tm∈ R which has a physical meaning that will be explained in Section 3.1.
As mentioned above, we require the total field ϕ to satisfy microcausality. We now claim that the expression for ϕ(b) as given above is such that this requirement is met. To demonstrate this explicitly we first write the result obtained in Appendix F:
h
ϕ(b)(t1, x1, i1) , ϕ(b)(t2, x2, i2) i
= −i X iη∈P+
e−η ˜x12sinh (ηt12) R(η)i1i2. (2.27)
Here R(η)i
1i2 = i1ηlimk→iη(k − iη) Si1i2(k), as defined in Section 2.2.
Likewise, the expression forϕ(s)(t1, x1, i1) , ϕ(s)(t2, x2, i2) is computed to be:
h
ϕ(s)(t1, x1, i1) , ϕ(s)(t2, x2, i2)i
= −i
4[sgn (t12+ x12) + sgn (t12− x12)] δi1i2−i
4[sgn (t12+ ˜x12) + sgn (t12− ˜x12)] Si1i2(0)
−i 2
X iη∈P+
h
θ (t12+ ˜x12) e−η(t12+˜x12)− θ (−t12+ ˜x12) eη(t12−˜x12)i R(η)i
1i2
−i 2
X iη∈P−
h
θ (t12− ˜x12) eη(t12−˜x12)− θ (−t12− ˜x12) e−η(t12+˜x12) i
R(η)i1i2, (2.28)
where θ is the Heaviside step function. It can be checked thatϕ(s), ϕ(b) = 0. Combining this with (2.27) and (2.28) we obtain:
[ϕ (t1, x1, i1) , ϕ (t2, x2, i2)] = −i
4[ε (t12+ x12) + ε (t12− x12)] δi1i2
−i
4[ε (t12+ ˜x12) + ε (t12− ˜x12)] Si1i2(0) − i
2θ (t12− ˜x12) X
iη∈P
eη(t12−˜x12)R(η)i1i2
+i
2θ (−t12− ˜x12) X
iη∈P
e−η(t12+˜x12)R(η)i1i2, (2.29)
where P = P+∪ P−.
Now, by definition, x1 > 0 and x2 > 0, which implies for spacelike separations that
|t12| < |x12| < |˜x12|. This, in turn, implies the vanishing of (2.29) at spacelike distances which shows that microcausality is indeed satisfied.
To investigate the time evolution of the system, we construct the explicit form of the Hamiltonian which can also be split into two parts corresponding to the scattering and bound state contributions:
H = H(s)+ H(b). (2.30)
The scattering contribution has the form of a quantum harmonic oscillator in the number representation:
H(s)= 1 2
n
X
i=1
ˆ ∞
−∞
dk
2π|k| a†i (k) ai(k) . (2.31) For the VBS part, the identity
[H, ϕ (t, x, i)] = −i (∂tϕ) (t, x, i) (2.32) implies
h
H(b), bj+ b†ji
= iηj
bj+ b†j
, h
H(b), bj− b†ji
= −iηj
bj− b†j
, j ∈ ι+ (2.33) which leads to:
H(b) = i 2
X
j∈ι+
ηj
b2j− b†2j
. (2.34)
This form has resemblance to the Hamiltonian of a kind of quantum damped oscillator, in agreement with the idea that there occurs energy flow through the vertex in the presence of bound states.
2.4 Dual field, Chiral Fields and Vertex Operators
Let us define a field ˜ϕ such that:
∂tϕ (t, x, i) = −∂˜ xϕ (t, x, i) , ∂xϕ (t, x, i) = −∂˜ tϕ (t, x, i) (2.35) which satisfies the equation of motion too:
∂t2− ∂x2 ˜ϕ (t, x, i) = 0. (2.36) The field ˜ϕ is called the dual field. We can split it as:
˜
ϕ (t, x, i) = ˜ϕ(s)(t, x, i) + ˜ϕ(b)(t, x, i) . (2.37) Using the definitions (2.35), ˜ϕ(s) and ˜ϕ(b) are calculated to be:
˜
ϕ(s)(t, x, i) = ˆ ∞
−∞
dk ε (k) 2πp2 |k|
h
a†i(k) ei(|k|t−kx)+ ai(k) e−i(|k|t−kx)i
(2.38) and
˜
ϕ(b)(t, x, i) = 1
√2 X
j∈ι+
Uijh
−
b†j+ bj
e−ηj(x+t−tm)+ i
b†j− bj
e−ηj(x−t+tm)i
, (2.39)
as shown in Appendix G.
We now introduce the following linear combinations:
ϕi,R(t, x) = ϕ (t, x, i) + ˜ϕ (t, x, i) , ϕi,L(t, x) = ϕ (t, x, i) − ˜ϕ (t, x, i) . (2.40) These are called chiral fields.
The VBS contributions to them, using (2.24), are:
ϕ(b)i,R(t − x) = i√
2X
j∈ι+
Uij
b†j − bj
e−ηj(x−t+tm), (2.41)
ϕ(b)i,L(t + x) =√
2 X
j∈ι+
Uij
b†j+ bj
e−ηj(x+t−tm). (2.42) These vanish or diverge in the limit t → ±∞. This is a first sign that the VBS gives origin to a kind of instability in the theory, producing a complementary damp- ing/enhancement of ϕ(b)i,R and ϕ(b)i,L in time.
With the definitions (2.40), it is straightforward to calculate ϕi,R(t − x) and ϕi,L(t + x), the results being:
ϕi,R(t − x) = ˆ ∞
−∞
dk
2πp2 |k|[1 + sgn (k)]h
a†i(k) ei(|k|t−kx)+ ai(k) e−i(|k|t−kx)i
+i√
2X
j∈ι+
Uij
b†j − bj
e−ηj(x−t+tm) (2.43)
and
ϕi,L(t + x) = ˆ ∞
−∞
dk
2πp2 |k|[1 − sgn (k)]h
a†i (k) ei(|k|t−kx)+ ai(k) e−i(|k|t−kx)i
+√
2X
j∈ι+
Uij
b†j+ bj
e−ηj(x+t−tm). (2.44)
We will need vertex operators for bosonization, which are defined in terms of the chiral fields, by:
V (t, x, i; σ, τ ) = ziκiq (i, σ, τ ) : expi√
π [σϕi,R(t − x) + τ ϕi,L(t + x)] :, (2.45) where zi∈ R are normalization constants, κi are the Klein factors and : ... : denotes the normal products in the algebras A and B. The family of vertex operators are parametrized by the continuous real parameters σ and τ . The factor q (i, σ, τ ) is given by:
q (i, σ, τ ) = expi√
π (σQi,R− τ Qi,L) , (2.46) with Qi,R(L) being the chiral charges given by:
Qi,R(L)= 1 4
ˆ ∞
−∞
dξ∂ξϕi,R(L)(ξ) . (2.47)
Correlation functions are the expectation values of products of operators, usually with respect to the ground state, that are useful objects in calculation of various physical quantities. A correlation function that includes the product of two operators is also referred as the two-point function, and similarly for other numbers of products.
As we saw above, the theory has two different regimes for P+= Ø and P+6= Ø.
The Hamiltonian corresponding to the VBS contribution is Hermitian but not self adjoint in the domain where the commutation relations (2.33) hold. As a result, the time evolution is not unitary.
3.1 The Half-Line
A star graph with one edge is the same as the half line R+. For such a graph, all indices that have been appearing have only a single value and the expressions (2.11), (2.12) reduce to:
S (k) = k + iη
k − iη, η = λ tan (α) . (3.1)
The boundary condition is a mixed boundary condition:
(∂xϕ) (t, 0) + ηϕ (t, 0) = 0, (3.2) to which (2.4) reduces for one edge. The two extreme cases as η → 0 and η → ∞ correspond to the Neumann and Dirichlet boundary conditions respectively.
There is one pole in the upper half-plane for η > 0 and the associated oscillator is generated by b and b†. The following results for the two-point correlation functions are taken from [3]. The derivation of the first one is given in Appendix H.
hϕR(ξ1) ϕR(ξ2)i0 = u (µξ12) + 2θ (η) eη(ξ˜12−2tm), (3.3)
hϕL(ξ1) ϕL(ξ2)i0 = u (µξ12) + 2θ (η) e−η(ξ˜12−2tm), (3.4)
hϕR(ξ1) ϕL(ξ2)i0 = 2θ (η) h
v+(−ηξ12) − ieηξ12 i
+ 2θ (−η) v−(−ηξ12) − u (µξ12) , (3.5)
hϕL(ξ1) ϕR(ξ2)i0 = 2θ (η)h
v−(ηξ12) + ie−ηξ12i
+ 2θ (−η) v+(ηξ12) − u (µξ12) , (3.6) where
ξ12= (t1− x1) − (t2− x2) , ξe12= (t1− x1) + (t2− x2) and
u (ξ) = −1
π ln (iξ + ) = −1
π ln (|ξ|) − i
2ε (ξ) , (3.7)
v±(ξ) = −1
πe−ξEi (ξ ± i) . (3.8)
In the last two expressions > 0 and Ei is the exponential integral function. In all the expressions µ is a mass parameter for infrared regularization, as introduced in Appendix H. Due to the step funtion θ (η) in the correlation functions, the theory gives different results for η < 0 and η > 0.
For η < 0, the S-matrix (3.1) has one antibound state. The energy is conserved due to the fact that the correlation functions are invariant under time translations in this case, which leads to ϕ with unitary time evolution.
For η > 0, there is a bound state. The step functions that multiply the eξ12-dependent exponential functions do not vanish in this case, and these exponential terms collect the contribution of this VBS on R+, having two features:
• The exponential factors in (3.3) and (3.4) depend on t1 + t2, which breaks the invariance under time translations. This is due to the VBS contribution to the total Hamiltonian, given by (2.34), which does not annihilate the vacuum:
HΩ = −i
2ηb†2Ω 6= 0. (3.9)
The energy is not conserved, as expected. This reveals a non-trivial energy flow through the boundary. It may be worth to emphasize that this is a purely boundary effect and the energy momentum tensor satisfies the continuity equation at all interior points of the wire.
• The correlation functions (3.3)-(3.6) are not invariant under time reversal trans- formation t → −t for tm 6= 0, whereas they are invariant under the transformation t → −t + 2tm. This symmetry is implemented by:
T ϕi,R(t − x) T−1= ϕi,L(−t + 2tm+ x) , (3.10)
T ϕi,L(t + x) T−1= ϕi,R(−t + 2tm− x) , (3.11)
where T is an antiunitary operator that leaves the vacuum invariant.
As an example, consider the vacuum energy density of the system. The vacuum energy density of the right and the left movers are defined, via point splitting, as:
θL,R(ξ) = 1 2 lim
ξ1,2→ξ(∂ϕL,R(ξ1)) (∂ϕL,R(ξ2)) − h(∂ϕL,R(ξ1)) (∂ϕL,R(ξ2))iline , (3.12) where the subtracted term is the contribution on the full line. Making use of the corre- lation functions (3.3) and (3.4) one has:
εL(t + x) = hθL(t + x)i0 = η2e−2η(t+x−tm) (3.13) and
εR(t − x) = hθR(t − x)i0 = η2e2η(t−x−tm). (3.14) Upon integration of these densities and taking the sum one gets the total vacuum energy which is time dependent:
E (t) = ˆ ∞
0
dx [εL(t + x) + εR(t − x)] = η cosh [2η (t − tm)] . (3.15) We see that the vacuum energy decreases in the interval (−∞, tm) while it increases in the interval (tm, ∞). This suggests energy flow out of and into the vertex. The minimum of the graph E (t) versus t is at t = tm. We now have a physical interpretation of the parameter tm, namely that it is the instant at which the energy flow through the vertex changes its direction. The time reversal transformation t → −t + 2tm inverts the two regimes.
3.2 Generic Star Graph
Using the diagonalized form, (2.11), of the S-matrix and the results for the half-line, it is possible to obtain the two point correlation functions for a general star graph. These are:
hϕi1,R(ξ1) ϕi2,R(ξ2)i0 = X iη∈P+
eη(ξ˜12−2tm)R(η)i
1i2 + δi1i2u (µξ12) , (3.16) hϕi1,L(ξ1) ϕi2,L(ξ2)i0 = X
iη∈P+
e−η(ξ˜12−2tm)R(η)i
1i2 + δi1i2u (µξ12) , (3.17)
hϕi1,R(ξ1) ϕi2,L(ξ2)i0 = X iη∈P+
h
v+(−ηξ12) − ieηξ12 i
R(η)i1i2
+ X iη∈P−
v−(−ηξ12) Ri(η)1i2 − δi1i2u (µξ12) , (3.18)
hϕi1,L(ξ1) ϕi2,R(ξ2)i0= X iη∈P+
h
v−(ηξ12) − ie−ηξ12i R(η)i
1i2
+ X
iη∈P−
v+(ηξ12) R(η)i1i2 − δi1i2u (µξ12) . (3.19)
Because of the presence of VBS, in (3.16) and (3.17), we again see breakdown of time- translation invariance that is due to the t1+t2-dependent term ˜ξ12. Analogous to the case of the half-line, the vacuum energy density of the right and left movers can be calculated using these correlators, which yields:
εi,L(t + x) = X iη∈P+
η2e−2η(t+x−tm)R(η)ii , εi,R(t − x) = X iη∈P+
η2e2η(t−x−tm)R(η)ii . (3.20)
Thus, the behaviour is the same as in the case of the half-line. This is also valid for the total vacuum energy that is calculated to be:
E (t) = X iη∈P+
ηR(η)ii cosh [2η (t − tm)] . (3.21) This resemblance for the particular examples above actually generalizes to all properties of the fields ϕR,L, i.e., they have the same qualitative features on a generic star graph, as they do on the half-line.
GENERAL BOUNDARY CONDITIONS
Interacting electrons in one space dimension, for which the Fermi liquid theory breaks down, is described by the Luttinger liquid model [5]. The scale-invariant case of the model will be referred to be at criticality.
The Luttinger liquid model is governed by the Lagrangian density:
L = iψ1∗(∂t+ vF∂x) ψ1+ iψ2∗(∂t− vF∂x) ψ2− g+(ψ1∗ψ1+ ψ2∗ψ2)2− g−(ψ1∗ψ1− ψ∗2ψ2)2. (4.1) Here ψα(t, x, i) with α = 1, 2 are complex fermion fields and vF is the Fermi velocity. In 1 + 1 dimensional space-time, the Lagrangian L has a mass dimension of 2. A straightfor- ward analysis shows that ψα has mass dimension 1/2. Therefore, the coupling constants g± ∈ R in the interaction terms are dimensionless. The Lagrangian is invariant under the phase transformations:
ψα 7→ eisψα, ψα∗ 7→ e−isψ∗α, s ∈ R (4.2) and
ψα7→ e−i˜s(−1)αψα, ψα∗ 7→ eis(−1)˜ αψ∗α, ˜s ∈ R. (4.3) Thus the model has U (1) ⊗ ˜U (1) global symmetry.
The Luttinger liquid model is exactly solvable on the real line, by the method of bosonization [10]. This method allows us to treat fermionic excitations as bosons. Ba- sically, this is due to the fact that in one dimension we lose the notion of spin as we have it in three dimensions. In three dimensions spin is associated with the group SU (2) which is homomorphic to the group SO (3) that describes rotations in three dimensions, whereas in one dimension the relevant group turns out to be U (1), which is abelian.
The Luttinger liquid model on a star graph is studied at criticality in [11, 12], where the electric conductance tensor is obtained to be
Gij = Gline (δij − Sij) . (4.4)
We will derive below the corresponding expression away from criticality. It is enough to consider the Thirring model which is the special case of the Luttinger liquid model with
g+= −g−≡ gπ > 0, vF = 1. (4.5) In this case the equations of motion for the fermion read:
i (γt∂t− γx∂x) ψ (t, x, i) = 2πg [γtJt(t, x, i) − γxJx(t, x, i)] ψ (t, x, i) , (4.6) where
ψ (t, x, i) =
ψ1(t, x, i) ψ2(t, x, i)
, γt=
0 1 1 0
, γx =
0 1
−1 0
, (4.7) and
Jν(t, x, i) = ¯ψ (t, x, i) γνψ (t, x, i) , ψ ≡ ψ¯ †γt, (4.8) is the conserved U (1)-current. The conserved ˜U (1)-current is:
J˜ν(t, x, i) = ¯ψ (t, x, i) γνγ5ψ (t, x, i) , γ5 ≡ −γtγx. (4.9)
4.1 Bosonization
In order to solve the Thirring model, we make use of the vertex operators defined by (2.45). The fermionic fields, in terms of the vertex operators, are given by:
ψ1(t, x, i) = 1
√2πV (t, x, i; σ, τ ) , ψ2(t, x, i) = 1
√2πV (t, x, i; τ, σ) . (4.10) Recall that the parameters σ and τ are real. By computing the contraction of the two- point function of the vertex operators, one gets the restriction:
σ2− τ2= 1. (4.11)
The quantum current Jν is constructed, with point-splitting, as
Jν(t, x, i) = 1 2 lim
→0+Z ()ψ (t, x, i) γ¯ νψ (t, x + , i) + ¯ψ (t, x + , i) γνψ (t, x, i) . (4.12) The renormalization constant Z () can be fixed such that
Jν(t, x, i) = − 1 (σ + τ )√
π∂νϕ (t, x, i) . (4.13) This generates the U (1)-phase transformations (4.2),
[Jt(t, x, i) , ψ (t, y, j)] = −δ (x − y) δijψ (t, x, j) . (4.14)
With (4.13), the equation of motion takes the form i (γt∂t− γx∂x) ψ (t, x, i) = − 2g√
π
(σ + τ ) : (γt∂tϕ − γx∂xϕ) ψ : (t, x, i) . (4.15) Using (4.10), it can be checked that (4.15) is satisfied provided that:
τ (σ + τ ) = g. (4.16)
This equation, together with (4.11) determines σ and τ in terms of the coupling constant:
σ = 1 + g
√1 + 2g > 0, τ = g
√1 + 2g. (4.17)
This fixes the solution completely.
The quantum current ˜Jν is treated in an anologous way. The equations corresponding to (4.12) and (4.13) are:
J˜ν(t, x, i) = 1 2 lim
→0+
Z ()˜ ψ (t, x, i) γ¯ νγ5ψ (t, x + , i) + ¯ψ (t, x + , i) γνγ5ψ (t, x, i) , (4.18) with γ5 = −γtγx, and
J˜ν(t, x, i) = − 1 (σ − τ )√
π∂νϕ (t, x, i) ,˜ (4.19) respectively. The latter is conserved, generating the ˜U (1)-phase transformations (4.3) according to:
h ˜Jt(t, x, i) , ψα(t, y, j)i
=
−δ (x − y) δijψ1(t, y, j) for α = 1
δ (x − y) δijψ2(t, y, j) for α = 2. (4.20) Next we consider symmetries. The boundary conditions (2.4) we set at the beginning of the problem naturally affect the symmetries. The relation between symmetries and boundary conditions is implemented by the Kirchhoff’s rule. In order to generate a time- independent charge, Kirchhoff’s rule must be imposed on a conserved current at the vertex of the star graph. Different conserved currents can generate different Kirchhoff’s rules. This may give rise to a contradiction for generic boundary conditions. Namely, it turns out that on the star graph we can preserve only one of the symmetries U (1) and U (1) rather than both of them.˜
First consider the U (1) symmetry. Just as the total field can be split into scattering and VBS parts, the associated current (4.13) can be split into scattering and VBS parts too:
Jν(t, x, i) = Jν(s)(t, x, i) + Jν(b)(t, x, i) . (4.21)
One has:
n
X
i=1
Jx(s)(t, 0, i) = 0 ⇔
n
X
i=1
Sij(k) = 1 ⇔
n
X
i=1
Uij = 1, ∀j = 1, ..., n. (4.22) From (2.24), one gets
n
X
i=1
Jx(b)(t, 0, i) ⇔
n
X
i=1
Uij = 0, ∀j ∈ ι+, (4.23) for the VBS contribution. It turns out that the U (1)-Kirchhoff’s rule holds if and only if the matrix U satisfies (4.22). Analogously, for ˜U (1) one gets:
n
X
i=1
J˜x(t, x, i) = 0 ⇔
n
X
i=1
Uij = −1, (4.24)
giving rise to the contradiction mentioned above. Obviously one must choose between (4.22) and (4.24) or correspondingly between the U (1) symmetry and the ˜U (1) symmetry.
Note that this was not the case for the half-line. Upon minimal coupling to an external field Aν(t, x, i), the corresponding systems have different physical features: (4.22) implies electric charge conservation wheras (4.24) does not. However, the latter equation implies by duality (2.35) that the charge density satisfies:
n
X
i=1
Jt(t, 0, i) = 0. (4.25)
This is the characteristic feature of a superconducting junction. We see that both (4.22) and (4.24) have a physical interpretation.
4.2 Conductance
We shall apply minimal coupling to the equation of motion of the system in order to extract information about the electric conductance. The substitution
∂ν 7→ ∂ν + iAν(t, x, i) (4.26)
in (4.6) couples the system to a classical external field and gives rise to a time dependent Hamiltonian. The conductance can be read off from the linear term in the expansion of hJx(t, x, i)iA
ν in terms of Aν. This term is computed to be [11]:
hJx(t, x, i)iAν = hJx(t, x, i)i + i ˆ t
−∞
dτHint (τ) , Jx(t, x, i)
= 1
π (1 + 2g)
Ax(t, x, i) + i
n
X
j=1
ˆ t
−∞
dτ ˆ ∞
0
dyAy(τ, y, j) h[∂yϕ (τ, y, j) , ∂xϕ (t, x, i)]i
. (4.27)
Consider now a uniform electric field E (t, i) = ∂tAx(t, i) that is switched on at t = t0. Making use of the commutation relation (2.29), the above expectation value is computed to be:
hJx(t, 0, i)iAν = Gline
n
X
j=1
ˆ ∞
−∞
dω
2πAˆx(ω, j) e−iωt
δij − Sji(ω) − X iη∈P
R(η)ji η
η + iωe(t−t0)(η+iω)
. (4.28)
Here Gline = 2π(1+2g)1 and ˆAx(ω, j) is the Fourier-transformed vector potential in the frequency space. From (4.28), the electric conductance tensor reads:
Gij(ω, t − t0) = Gline
δij − Sij(ω) − ei(t−t0)ω X iη∈P
ηe(t−t0)η η + iω R(η)ij
, t > t0. (4.29)
This is the generalization of (4.4) away from criticality and, in general, complex, giving rise to a non-trivial impedance.
Note that the factor ei(t−t0)ω in (4.29) produces oscillations in t − t0, whose amplitude depends on the summation factor. The existence of two regimes of the theory mentioned occasionally up to this point reveals itself on the electric conductance tensor via this summation. Namely, we can say:
• For P+ = Ø, the sum on the right hand side of (4.29) runs over the antibound states only, i.e., η < 0 for all terms in the summation. This damps the oscillations via the term e(t−t0)η. Thus,
t→∞lim Gij(ω, t − t0) = lim
t0→−∞Gij(ω, t − t0) = Gline [δij − Sij(ω)] , (4.30) which is the conductance that will be observed in this regime, long after switching the external field on.
• For P+ 6= Ø, the sum on the right hand side of (4.29) involves terms with η >
0 corresponding to bound states that give rise to oscillations with exponentially growing amplitude in the variable t − t0.
Let us consider a star graph with two vertices and three edges, as shown in Fig. 2.
Fig. 2 A star graph with two vertices and three edges
Here SA and SB are the S-matrices characterizing scattering at the first (defect A) and the second (defect B) vertices, viewed from left to right, respectively. We are interested in the overall scattering effect resulting from these two defects. To this end we must in some way fuse these two defects into one. This is in fact possible, and the resulting scattering is characterized by the S-matrix given by ([4], Sec. 4):
S =
S11A + S12ASB11S21A
1 − S22AS11Be2ia|k|−1
S12AS12B
1 − S22AS11Be2ia|k|−1
S21BSA21
1 − S22AS11Be2ia|k|−1
S22B + S22AS21BS12B
1 − S22ASB11e2ia|k|−1
. (5.1) This prescription is called the star product of the two scattering matrices SAand SB:
S = SA∗ SB. (5.2)
This is an associative product with the unit I =
0 1 1 0
, (5.3)
as shown in Appendix I. Therefore the space of 2 × 2 S-matrices, with star product, form a unital associative algebra. In fact, it is more accurate to talk about a family of star products, depending on the parameter a.
Note that in equation (5.1), for the case S22A = S11B = 1, the limit a → 0 is not allowed mathematically. This has the following physical explanation. Upon taking the star product, we can think of the result, corresponding to the overall scattering effect to be caused by a single defect. S22A = S11B = 1 means full reflection in edge 1 and in edge 3.
In this case, what we are sure is that there is no probability for a fermion approaching defect A from the left (edge 1) to be found in edge 3 as depicted in Fig. 3.
Fig. 3 A wave function penetrating the region labelled by 2 can not reach edge 3 Yet, if we let a → 0, we can no longer think of the effect of the two defects as caused by a single one: If there were indeed a single defect only, there would always be some probability for the fermion to be found in edge 3, due to quantum tunneling. A similar argument goes for a fermion approaching defect B from the right (edge 3). (Of course we are not considering infinite potentials, since these all corespond to some practical constructions.)
If the position of a defect can be changed without affecting any physical result then one calls it topological. This means that the distance a between the individual defects must not have any effect on the physical results, which in turn implies the S-matrix to be independent of a. Clearly, this is possible in either of the cases: i) S22A = 0 and ii) S11B = 0. But as shown in Appendix J each of these conditions, together with the general condition (2.8) on a scattering matrix, leads to the identitiy matrix with respect to the star product, i.e. :
Stopological =
0 1 1 0
.
Physically, this means that the “defect” does not have any effect on a current whichever direction it flows. Hence the fact that the position of a “defect” can be changed without affecting physical results corresponds to the fact that there is no defect at all. In other words, there cannot exist topological defects in a quantum wire corresponding to a star graph with two vertices.
Note that the result here means that such defects are trivial as far as the scattering is concerned only. But there can still be defects with trivial scattering which are non-trivial in other aspects. In particular, in the scale-invariant case it is known [13] that there are
non-trivial topological defects which are not specified in the way as done here. These can be analyzed in the framework of two dimensional conformal field theory. Such an analysis is much beyond the scope of our study.
In this thesis we investigated a star graph from a pure theoretical point of view. First it is found out that the scattering matrix, associated with the vertex of the star graph is a function that is analytic on the complex plane, except for some isolated points which are located on the imaginary axis and constitute the (anti)bound states. A remarkable result was that the scattering component of the total field is local only in the absence of the VBS’s. From this, we attempted to construct the bound state component such that the total field is both canonical and local. We saw that the VBS contribution is fixed up to a parameter tm, by the physical requirement of causality. After introducing the correlation functions, we were able to investigate more concretely the difference between the presence and the absence of the bound states and how this affects the time evolution of a system. This was examplified by the vacuum energy, which reveals the time dependence and clarifies the meaning of the parameter tm. As a last step, we investigated the Luttinger liquid on a star graph, by the method of bosonization, where we used some of the previous results. In particular we obtained the electric conductance tensor away from criticality and saw that its keeps track of the two regimes of the theory which we observed occasionally depending on whether there are bound states or not. All our investigations dealt with a star graph with a pointlike boundary. It may be of interest to investigate a graph with boundaries that have higher dimensions. One could also study a general star graph with more than one junction. One can expect this problem to be difficult because the most simplified case of it, which we considered in Section 5, was already quite complicated.
