1 Introduction↩︎

Differential operators on metric graphs provide an excellent opportunity to study connections between topology and spectral theory using differential operators having eigenfunctions that often can be calculated explicitly despite rather complicated topological structure of the system. In these systems the edges determine only the geometric properties, while topological structure is given by the vertex conditions. Standard vertex conditions (continuity of the function and Kirchhoff condition on its first derivatives) used in many examples are uniquely determined by the metric graph and give no possibility to change the topology of the system. Using more sophisticated vertex conditions, still ensuring that the operator is self-adjoint, allows one to consider families of systems exhibiting different topologies while using the same set of edges, (i.e. the same metric structure) (see Section 2, where we describe these conditions in detail).

Our attention was drawn to the subject by Shapere2012?, where relations between vertex conditions connecting end points in two intervals and possible topology of the corresponding graph are thoroughly discussed (following pioneering work [1]). In particular, the authors were interested in Berry’s phase appearing in adiabatic evolution of quantum systems (see [2], SchWi1?). In particular the authors raised the question (see Shapere2012?):

whether a non-trivial geometric (Berry) phase can be observed, despite bases of real eigenfunctions can be used throughout?

In the current note we answer this question positively by considering a periodic family of metric graphs (similar to one studied in Shapere2012?) exhibiting a non-trivial geometric phase equal to \(\pi\) even though all eigenfunctions can be chosen real-valued: the normalised eigenfunctions of the system are multiplied by \(-1= e^{i \pi}\) after passing one period when the system returns back to the original state. Our model can be seen as a figure eight graph where the central vertex connects four end points. The chosen family of vertex conditions depends on just one parameter \(\theta\) and is periodic with period \(2 \pi\). For certain special values of the parameter the vertex conditions do not properly connect all four end points and the graph turns into either two independent cycles or just one cycle. The suggested family of vertex conditions is also real, implying that the eigenfunctions can be chosen real-valued. This property puts a rather strong restriction on the system making the existence of a non-trivial geometric phase even more striking: once eigenfunctions are fixed for one value of the parameter, the normalised eigenfunctions for all other values are uniquely determined by continuity. Note that, without real-valued eigenfunctions, Berry’s phase was observed for an even simpler system of just two half-lines coupled together at one degree two vertex ExGr?.

The article is organised as follows. In Section 2 we revise the notion of metric graph and describe relations between vertex conditions and topology of the graph. Our explicit model is presented in Section 3. Spatial symmetry of the model described in Section 4 allowing to treat even and odd eigenfunctions separately. The spectrum is determined in Section [SecSpectrum]; it is independent of the parameter in the special case of equal edge lengths. The eigenfunctions are calculated explicitly depending continuously on the parameter. It appears that after one period the eigenfunctions are multiplied by \(-1\) generating a geometric phase equal to \(\pi: e^{i \pi} = -1\).

More details can be found in Master’s theses Tibbling2025?, Shubin?.

2 Topology of metric graphs and vertex conditions↩︎

In our construction we are going to follow the approach to spectral theory of metric graphs described in detail in Book?, see also BeKu?, Mugnolo?. Let \(\Gamma\) be a metric graph formed by a finite number of edges \(E_n = [x_{2n-1}, x_{2n}], \; n = 1,2, \dots, N,\) each identified with a compact interval on an individual copy of the real line \(\mathbb{R}\). Then the vertices in \(\Gamma\) can be seen as a partition of the set of end points \(\mathbf{V} = \{ x_j \}_{j=1}^{2N}\) into equivalence classes \(V_m, \; m =1,2, \dots, M\) (\(V_i \cap V_j = \emptyset, i \neq j, \bigcup_{m=1}^M V_m = \mathbf{V}\)), called vertices. The differential operator, say the Laplacian \(- \frac{d^2}{dx^2}\), is well-defined acting on the functions determined on the edges. The functions on different edges are independent and therefore the differential operator is not symmetric. To make the operator self-adjoint one introduces vertex conditions connecting limiting values of the functions \[u(x_j) := \lim_{x \rightarrow x_j} u(x)\] and their normal derivatives \[\partial u(x_j) = (-1)^{j+1} \lim_{x \rightarrow x_j} u'(x_j),\] where the limits are taken from inside the intervals. The derivatives are taken in the direction pointing inside the edges, hence the extra sign in the formula. For each vertex \(V_m\) one introduces the vectors \[\vec{U}_m := \begin{pmatrix} u(x_{i_1}) \\ u(x_{i_2}) \\ \vdots \\ u(x_{i_{d_m}}) \end{pmatrix}_{x_{i_j} \in V_m}, \quad \partial \vec{U}_m := \begin{pmatrix} \partial u(x_{i_1}) \\ \partial u(x_{i_2}) \\ \vdots \\ \partial u(x_{i_{d_m}}) \end{pmatrix}_{x_{i_j} \in V_m} .\] The dimension of the vectors coincides with the degree \(d_m\) of the vertex \(V_m\) – the number of end points joined at \(V_m\). Then the most general vertex conditions at \(V_m\) can be written as (see (3.21) in Book?) \[i (\mathbf{S}_m - \mathbf{I}) \vec{U}_m = (\mathbf{S}_m + \mathbf{I}) \partial \vec{U}_m,\] where \(\mathbf{S}_m\) is any unitary \(d \times d\) matrix. This parametrisation of all possible vertex conditions grew up from Kostrykin-Schrader parametrisation using pairs of matrices KoSch? and first appeared in Ha1?, Ha2?, KuNo?.

If the matrix \(\mathbf{S}_m\) has block-diagonal form, then the vertex \(V_m\) can be divided into two (or more) smaller vertices so that the vertex conditions are introduced only joining limiting values belonging to the new vertices. Therefore to have vertex conditions that correctly reflect the topology of the graph the matrices \(\mathbf{S}_m\) should be chosen not only unitary, but also being not block-diagonal.

To get metric graph models with changing topology one may simply consider families of unitary matrices \(\mathbf{S}_m = \mathbf{S}_m (\theta)\) depending on a parameter \(\theta\) and having block-diagonal structure for certain values of the parameter. This would correspond to splitting of the vertex for these values of \(\theta\), while preserving original topological structure for all other values of the parameter.

The dependence between vertex conditions and topology of the metric graph was first discussed in KuSt? and is well-described in Section 3.3.3 of Book?. Note that vertex conditions can be used not only to reflect different topologies but also to model physical properties of the system since the parameter \(\mathbf{S}_m\) can be seen as the vertex scattering matrix at unit energy [3], AsKuUs?, ExTa?, ExTa2?, KuMa?, KuEn?, BaEx?.

3 The model↩︎

A differential operator on metric graphs can be seen as a triple consisting of a metric graph, a differential operator acting on the edges and vertex conditions. The role of vertex conditions is two-fold: they are needed to make the differential operator self-adjoint and they determine how different edges are attached to each other, i.e. the topology of the metric graph. In what follows we consider metric graphs on two edges of lengths \(\ell_1\) and \(\ell_2\) \[E_1 = [x_1, x_2] \equiv [-\ell_1/2,\ell_1/2] \quad and \quad E_2 = [x_3, x_4] \equiv [-\ell_2/2,\ell_2/2].\] The differential operator acting in the Hilbert space \(L_2 (\Gamma) = L_2 (E_1) \oplus L_2 (E_2)\) will always be the Laplacian \[\label{tau} \tau = - \frac{d^2}{dx^2}.\tag{1}\] The vertex conditions will be given by the following one-parameter family of unitary and Hermitian matrices: \[\mathbf{S}_\theta = \begin{pmatrix} 0&\sin\theta&0&\cos\theta\\ \sin\theta&0&\cos\theta&0\\ 0&\cos\theta&0&-\sin\theta\\ \cos\theta&0&-\sin\theta&0 \end{pmatrix}, \quad \quad \theta \in [0,2 \pi],\] via the formula \[\label{eq95bc95ss} i(\mathbf{S}_\theta -I)\vec{u} = (\mathbf{S}_\theta +I)\partial {\vec{u}},\tag{2}\] where the boundary values of the functions involve their values at the end points of the intervals as well as their normal derivatives: \[\label{eq:limit46values} \vec{u} = \begin{pmatrix}u (x_1)\\u (x_2)\\u (x_3)\\u (x_4)\end{pmatrix},\quad \partial {\vec{u}} = \begin{pmatrix}\partial {u}(x_1)\\\partial {u}(x_2)\\\partial {u}(x_3)\\\partial {u}(x_4)\end{pmatrix} := \begin{pmatrix}u' (x_1)\\-u' (x_2)\\u' (x_3)\\- u' (x_4)\end{pmatrix} .\tag{3}\] Since the vertex conditions are scaling-invariant, i.e. the matrix \(\mathbf{S}_\theta\) is not only unitary, but also Hermitian, the ranges of the matrices \(\mathbf{S}_\theta - \mathbf{I}\) and \(\mathbf{S}_\theta + \mathbf{I}\) are orthogonal to each other (see formula (3.33) in Book?) and therefore the vertex conditions can be explicitly written as follows, separating limiting values of the functions from the normal derivatives: \[\label{vc} \begin{array}{l} \displaystyle \begin{pmatrix} -1 & \sin \theta & 0 & \cos \theta \\ \sin \theta & -1 & \cos \theta & 0 \\ 0 & \cos \theta & -1 & - \sin \theta \\ \cos \theta & 0 & - \sin \theta & -1 \end{pmatrix} \begin{pmatrix}u (x_1)\\u (x_2)\\u (x_3)\\u (x_4)\end{pmatrix} = \vec{0}, \\[10mm] \begin{pmatrix} 1 & \sin \theta & 0 & \cos \theta \\ \sin \theta & 1 & \cos \theta & 0 \\ 0 & \cos \theta & 1 & - \sin \theta \\ \cos \theta & 0 & - \sin \theta & 1 \end{pmatrix} \begin{pmatrix}u' (x_1)\\-u' (x_2)\\u' (x_3)\\- u' (x_4)\end{pmatrix} = \vec{0}. \end{array}\tag{4}\] Note that the matrices above have rank \(2\) for any value of \(\theta\); hence we shall always have two conditions on function values and two conditions on the derivatives.

The corresponding operator acting on the functions from the Sobolev space \(W_2^2 (E_1) \oplus W_2^2 (E_2)\) satisfying vertex conditions 4 will be denoted by \(L^\theta\) and it is self-adjoint for any value of \(\theta \in [0,2 \pi].\)

Note that since the matrix \(\mathbf{S}_\theta\) is not only unitary, but also Hermitian, the corresponding vertex scattering matrix \(S_{\mathbf{v}}\) does not depend on the energy parameter \(k, \; k^2 = \lambda\), and is given by \[\mathbf{S}_{\mathbf{v}} (k) \equiv \mathbf{S}_\theta.\]

This form of vertex conditions was first used in KuEn? to model ballistic scattering of electrons on a wire coupled to a ring with magnetic flux. Similar vertex conditions determined by the matrix \[\begin{pmatrix} 0 & 0 & \sin \theta & \cos \theta \\ 0 & 0 & - \cos \theta & \sin \theta \\ \sin \theta & - \cos \theta & 0 & 0 \\ \cos \theta & \sin \theta & 0 & 0 \end{pmatrix}\] instead of \(\mathbf{S}_\theta\) was used in [3], where model with anomalous dependence of the spectrum on the magnetic fluxes was considered.

Depending on \(\theta\), metric graphs with different topological structure correspond to the operator \(L^\theta\). Thus, for all \(\theta \neq 0, \frac{1}{2} \pi, \pi, \frac{3}{2} \pi, 2 \piNone\) all the end points \(x_1, x_2, x_3,\) and \(x_4\) are connected together by the vertex conditions and the corresponding metric graph is the figure eight graph depicted in Fig. 1 – the matrix \(\mathbf{S}_\theta\) is not block-diagonal.

For other values of \(\theta\) the matrix \(\mathbf{S}_\theta\) is block-diagonal and the end points can be divided into pairs so that the corresponding metric graph is either formed by two loops of lengths \(\ell_1\) and \(\ell_2\) \[\begin{array}{lcccc} \displaystyle \theta = \frac{1}{2} \pi & \Rightarrow & \displaystyle \left\{ \begin{array}{l} \sin \theta = 1 \\ \cos \theta = 0 \end{array} \right.& \Rightarrow &\begin{array}{l} \left\{ \begin{array}{ccc} u(x_1) & = & u(x_2) \\ u'(x_1) & = & u'(x_2) \end{array} \right. \\[3mm] \left\{ \begin{array}{ccc} u(x_3) & = & - u(x_4) \\ u'(x_3) & = & - u'(x_4) \end{array} \right. \end{array}; \\[10mm] \displaystyle \theta= \frac{3}{2} \pi & \Rightarrow & \left\{ \begin{array}{l} \sin \theta = -1 \\ \cos \theta = 0 \end{array} \right. & \Rightarrow & \begin{array}{l} \left\{ \begin{array}{ccc} u(x_1) & = & - u(x_2) \\ u'(x_1) & = & - u'(x_2) \end{array} \right. \\[3mm] \left\{ \begin{array}{ccc} u(x_3) & = & u(x_4) \\ u'(x_3) & = & u'(x_4) \end{array} \right. \end{array}; \end{array}\] or by single loop of length \(\ell_1 + \ell_2\) with reflectionless conditions at two internal points: \[\begin{array}{lcccc} \displaystyle \theta = 0 & \Rightarrow & \displaystyle \left\{ \begin{array}{l} \sin \theta = 0 \\ \cos \theta = 1 \end{array} \right.& \Rightarrow &\begin{array}{l} \left\{ \begin{array}{ccc} u(x_1) & = & u(x_4) \\ u'(x_1) & = & u'(x_4) \end{array} \right. \\[3mm] \left\{ \begin{array}{ccc} u(x_2) & = & u(x_3) \\ u'(x_2) & = & u'(x_3) \end{array} \right. \end{array}; \\[10mm] \displaystyle \theta= \pi & \Rightarrow & \left\{ \begin{array}{l} \sin \theta = 0 \\ \cos \theta = -1 \end{array} \right. & \Rightarrow & \begin{array}{l} \left\{ \begin{array}{ccc} u(x_1) & = & - u(x_4) \\ u'(x_1) & = & - u'(x_4) \end{array} \right. \\[3mm] \left\{ \begin{array}{ccc} u(x_2) & = & - u(x_3) \\ u'(x_2) & = & - u'(x_3) \end{array} \right. \end{array}. \end{array}\]

This dependence of the topology of the metric graph \(\Gamma\) on the parameter \(\theta\) is best illustrated by Fig. 2. In this figure we use the following notations for the vertices:

• filled large circles: degree four vertex with non-separable conditions involving parameter \(\theta \neq 0, \frac{1}{2} \pi, \pi, \frac{3}{2} \pi, 2 \pi\);

• empty small circles: degree two vertices with reflectionless conditions involving multiplication of the function and its first derivative by \(-1\);

• no circle: degree two vertices with standard conditions (continuity of the function and its first derivative).

We see that the topology of the metric graph associated with the operator \(L^\theta\) changes with \(\theta\) and the corresponding graphs are not even connected for \(\theta = \frac{1}{2} \pi, \frac{3}{2} \pi\). The number of independent cycles also changes and is equal to \(1\) for \(\theta = 0, \pi\).

4 Graph’s symmetry↩︎

The constructed operator possesses the following symmetry, which will help us to calculate its eigenfunctions. Let us denote by \(J\) the unitary operator corresponding to the horizontal symmetry of the figure eight graph (see Fig. 3): \[\begin{array}{cccc} \displaystyle J: & \displaystyle L_2(E_1) \oplus L_2 (E_2) & \rightarrow & \displaystyle L_2(E_1) \oplus L_2 (E_2); \\[3mm] & \displaystyle (u_{\rm 1} (x), u_{\rm 2} (x) ) & \mapsto & \displaystyle (u_{\rm 1} (-x), u_{\rm 2} (-x) ). \end{array}\]

The symmetry transformation \(J\) acts on the vectors \(\vec{u}\) and \(\partial \vec{u}\) of limiting values introduced in 3 as multiplication by the matrix \[\mathbf{J} := \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}.\] We have obviously \[\mathbf{J} \mathbf{S}_\theta = \mathbf{S}_\theta \mathbf{J},\] which together with \[J \tau = \tau J ,\] (where \(\tau\) was defined in 1 ) implies that the operator \(L^\theta\) commutes with the symmetry operator \(J\) \[J L^\theta = L^\theta J.\]

It follows that the eigenfunctions may be divided into two classes to be studied separately:

• even eigenfunctions satisfying \(J \psi= \psi\);

• odd eigenfunctions satisfying \(J \psi = - \psi\).

The defined operator is real in the sense that if \(u\) belongs to the domain of the operator, then the complex conjugate \(\overline{u}\) also belongs to the domain and \(\tau \overline{u} = \overline{\tau u}.\) Therefore if \(\psi_\lambda\) is an eigenfunction corresponding to the eigenvalue \(\lambda\), then \(\overline{\psi}_\lambda\) is also an eigenfunction corresponding to the same eigenvalue (which is real since the operator is self-adjoint). Hence both \(\Re \psi := \frac{\psi + \overline{\psi}}{2}\) and \(\Im \psi := \frac{\psi - \overline{\psi}}{2i}\) are eigenfunctions implying that without loss of generality the eigenfunctions may always be chosen not only even/odd, but also real-valued.

5 The secular equation↩︎

Non-zero eigenvalues↩︎

To get the secular equation describing the non-zero spectrum of the operator we shall use secular polynomials well described in Section 6.1 of Book?. We introduce the secular polynomial \(P(z_1, z_2)\) determined by \[\begin{array}{ccl} \displaystyle P(z_1, z_2) & = & \displaystyle \det \Big(\mathbf{S}_{\rm e} - \underbrace{\mathbf{S}_{\rm v}}_{\displaystyle = \mathbf{S}_\theta} \Big) \\ & = & \displaystyle \det \left(\begin{pmatrix} 0 & z_1 & 0 & 0 \\ z_1 & 0 & 0 & \\ 0 & 0 & 0 & z_2 \\ 0 & 0 & z_2 &0 \end{pmatrix} - \begin{pmatrix} 0&\sin\theta&0&\cos\theta\\ \sin\theta&0&\cos\theta&0\\ 0&\cos\theta&0&-\sin\theta\\ \cos\theta&0&-\sin\theta&0 \end{pmatrix} \right) \\[10mm] & = & \displaystyle \left( (z_1 - \sin \theta) (z_2 + \sin \theta) - \cos^2 \theta \right)^2. \end{array}\] Then the spectrum of the operator is given by zeroes of the trigonometric polynomial \[\begin{array}{ccl} p(k) & := & P(e^{ik \ell_1}, e^{i k \ell_2}) \\ & = & \displaystyle - 4 e^{i k (\ell_1+\ell_2)/2} \left( \sin \left(k \frac{\ell_1 + \ell_2}{2}\right) + \sin \theta \cdot \sin \left( k \frac{\ell_1-\ell_2}{2} \right) \right)^2. \end{array}\] Ignoring the non-vanishing factor we conclude that all non-zero eigenvalues are double degenerate and are given by the solutions of the equation \[\label{secular} \sin k \frac{\ell_1 + \ell_2}{2} + \sin \theta \cdot \sin k \frac{\ell_1-\ell_2}{2} = 0.\tag{5}\] It will be proven in the next section that for each zero one eigenfunction can be chosen even and one odd. The secular equation 5 can also be derived directly by noting that any even eigenfunction on \(\Gamma\) is given by \[\psi_n (x) = a_j \cos n \pi x, \quad x \in E_j, j= 1,2\] and every odd by \[\psi_n (x) = b_j \sin n \pi x, \quad x \in E_j, j= 1,2.\]

In the special case \(\ell_1 = \ell_2 = \ell\) the spectrum is independent of \(\theta\) and is given by \[\sin k \ell = 0 \quad \Rightarrow \quad k_n = \frac{ \pi}{\ell} n , \quad n = 1,2,3, \dots\]

The eigenvalue \(\lambda = 0\)↩︎

To determine the multiplicity of the zero eigenvalue one needs to repeat our analysis taking into account that the solution of the eigenfunction equation \(\psi \psi (x) = 0\) on the edges is given by a linear function instead of the exponentials. It is easier to consider possible even and odd eigenfunctions separately.

Even eigenfunctions↩︎

Every even eigenfunction \(\psi\) corresponding to \(\lambda = 0\) is equal to a constant function on each of the edges: \[\psi (x) = a_j \quad x \in E_j, j= 1,2.\] Its boundary values are \[\vec{\psi} = \begin{pmatrix} a_1 \\ a_1 \\ a_2 \\ a_2 \end{pmatrix}, \quad \partial \vec{\psi} = \vec{0}.\] Substitution into the vertex conditions 4 leads to just two linear equations for \(a_1\) and \(a_2\) \[\left\{ \begin{array}{rcrcl} (-1 + \sin \theta) \; a_1 & + & \cos \theta \; a_2 & = & 0, \\ \cos \theta \; a_1 & - & (1+ \sin \theta) \; a_2 & = & 0. \end{array} \right.\] The determinant of the linear system is always equal to zero: \(1 - \sin^2 \theta - \cos^2 \theta = 0\), and all four entries do not vanish simultaneously. Hence there is precisely one even eigenfunction corresponding to \(\lambda = 0\) for any value of \(\theta,\) independently of whether the corresponding metric graph is connected or not.

Odd eigenfunctions↩︎

Every odd eigenfunction corresponding to \(\lambda = 0\) is equal to a linear function on each of the edges: \[\psi (x) = b_j x, \quad x \in E_j, j= 1,2.\] We calculate its limit values at the end points: \[\vec{\psi} = \begin{pmatrix} - b_1 \ell_1/2 \\ b_1 \ell_1/2 \\ - b_2 \ell_2/2 \\ b_2 \ell_2/2 \end{pmatrix}, \quad \partial \vec{\psi} = \begin{pmatrix} b_1 \\ - b_1 \\ b_2 \\ -b_2 \end{pmatrix}.\] Then vertex conditions 4 imply the following linear system after excluding identical equations \[\begin{pmatrix} \frac{\ell_1}{2} (1+ \sin \theta) & \frac{\ell_2}{2} \cos \theta \\ \frac{\ell_1}{2} \cos \theta & \frac{\ell_2}{2} (1-\sin \theta)\\ 1- \sin \theta & - \cos \theta \\ - \cos \theta & 1+ \sin \theta \end{pmatrix} \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \vec{0}.\] The matrix has rank \(2\) for any value of \(\theta\), hence the system has only the trivial solution implying that no odd eigenfunctions are present.

We conclude that \(\lambda _0= 0\) is a simple eigenvalue and the corresponding eigenfunction is even.

By combining all the results of this section, we obtain the following theorem.

The spectrum can be explicitly calculated not only for \(\ell_1 = \ell_2\), but for the special values of \(\theta\):

• \(\theta = 0, \pi\), then the spectrum is: \[\begin{array}{ll} 0, & simple eigenvalue, \\ n^2 \left(\frac{2\pi}{\ell_1+\ell_2} \right)^2 , \; n = 1,2,3, \dots, & double eigenvalues. \end{array}\]

• \(\theta = \pi/2\), then the spectrum is: \[\begin{array}{ll} 0, & simple eigenvalue, \\ (2n)^2 \left(\frac{\pi}{\ell_1} \right)^2 , \; n = 1,2, 3,\dots, & double eigenvalues, \\ (2n+1)^2 \left(\frac{\pi}{\ell_2} \right)^2 , \; n = 0,1,2, \dots, & double eigenvalues. \end{array}\]

• \(\theta = 3 \pi/2\), then the spectrum is: \[\begin{array}{ll} 0, & simple eigenvalue, \\ (2n+1)^2 \left(\frac{\pi}{\ell_1} \right)^2 , \; n = 0, 1,2, \dots, & double eigenvalues, \\ (2n)^2 \left(\frac{\pi}{\ell_2} \right)^2 , \; n = 1, 2, 3, \dots, & double eigenvalues. \end{array}\]

6 The eigenfunctions↩︎

In this section we are going to calculate the eigenfunctions denoted by \(\psi\). We have to bear in mind that

• the operator is real, i.e. invariant under complex conjugation, and therefore the eigenfunctions can be chosen real-valued,

• the operator is symmetric \(J L^\theta = L^\theta J\) and therefore one may look separately for even and odd eigenfunctions.

To make our presentation even more transparent we consider the case of equal unit lengths \(\ell_1 = \ell_2 = 1\), in which case all operators \(L^\theta\) are isospectral to each other.

Even eigenfunctions↩︎

Every even eigenfunction is given by \[\psi_n (x) = a_j \cos n \pi x, \quad x \in E_j, j= 1,2.\] We just need to determine the real constants \(a_1\) and \(a_2\) so that \(\psi_n\) is normalised and depends continuously on \(\theta\).

The first condition comes from the normalisation of the eigenfunction \[1 = \parallel \psi_n \parallel^2 = (a_1^2 + a_2^2 ) \int_{-1/2}^{1/2} \cos^2 n \pi x \; dx = \frac{1}{2} (a_1^2 + a_2^2) \Rightarrow\] \[\label{eq2} a_1^2 + a_2^2 = 2.\tag{6}\]

Checking that the function satisfies the vertex condition we shall need to separate the cases when \(n\) is even and odd.

Case \(n = 2 m +1 , \quad m = 0,1,2, \dots\)↩︎

\(None\)

Substituting the limiting values of the function and its first derivative into the vertex conditions 4 we get: \[\vec{\psi}_n = \vec{0}, \quad \partial \vec{\psi}_n = (-1)^m (2m +1) \pi \begin{pmatrix} a_1 \\ a_1 \\ a_2 \\ a_2 \end{pmatrix} \Rightarrow\] \[\label{eq1} (1+ \sin \theta) \; a_1 + \cos \theta \; a_2 = 0.\tag{7}\]

Our task now is to determine continuous functions \(a_1 (\theta)\) and \(a_2 (\theta)\) satisfying equations 6 and 7 . Let us fix a normalisation compatible with 6 and 7 by assuming that \[\label{eq3} a_1(0) = 1, \quad a_2 (0) = - 1.\tag{8}\] Excluding \(a_2\) from equation 7 using equation 6 we get \[\label{eq11} a_1 (\theta) = \pm \frac{\cos \theta}{\sqrt{1 + \sin \theta}}.\tag{9}\] The corresponding \(a_2 (\theta )\) is then given by \[\label{eq12} a_2 (\theta) = \mp \sqrt{1 + \sin \theta}.\tag{10}\] To ensure continuous dependence of \(a_j (\theta)\) on \(\theta\) we need to introduce the sign function \[\sigma (\theta) = \left\{ \begin{array}{ll} 1, & 0 \leq \theta < 3 \pi/2, \\ -1 & 3 \pi/2 \leq \theta \leq 2 \pi. \end{array} \right.\] Then the amplitudes given by the formulas below depend continuously on \(\theta\), the function satisfies the vertex conditions and is normalised \[a_1 (\theta) = \sigma (\theta) \frac{\cos \theta}{\sqrt{1 +\sin \theta}}, \quad a_2 (\theta) = - \sigma(\theta) \sqrt{1 + \sin \theta}.\] These formulas can be simplified using trigonometric identities \[\begin{array}{l} \displaystyle \sqrt{1 + \sin \theta} = \sqrt{\sin^2 \theta/2 + \cos^2 \theta/2 + 2 \sin \theta/2 \cos \theta/2} = \vert \sin \theta/2 + \cos \theta/2 \vert; \\[3mm] \displaystyle \sigma_1 (\theta) = {\rm sgn}\; \Big( \sin \theta/2 + \cos \theta/2 \Big), \quad \quad \theta \in [0,2 \pi]; \end{array}\] as follows \[\label{eq5} a_1 (\theta) = \cos \frac{\theta}{2} - \sin \frac{\theta}{2}, \quad \quad a_2 (\theta) = - \cos \frac{\theta}{2} - \sin \frac{\theta}{2}.\tag{11}\] It would have been possible to derive formulas 11 directly from 9 and 10 ¹ and choosing \[a_2 (\theta) = - \cos \frac{\theta}{2} - \sin \frac{\theta}{2},\] which in turn implies \[\begin{array}{ccl} a_1 (\theta) & = & \displaystyle - \frac{\cos \theta}{1+ \sin \theta} a_2 (\theta) = \frac{\cos \theta}{1 + \sin \theta} \Big( \cos \theta/2 + \sin \theta/2 \Big) \\[3mm] & = & \displaystyle \frac{\cos^2 \theta/2 - \sin^2 \theta/2}{(\cos \theta/2 + \sin \theta/2)^2} \Big( \cos \theta/2 + \sin \theta/2 \Big) \\[5mm] & = & \displaystyle \cos \theta/2 - \sin \theta/2. \end{array}\] The role of the sign function is hidden in this approach.

Case \(n = 2 m , \quad m = 1,2, 3, \dots\)↩︎

\(None\)

The limiting values are given by \[\vec{\psi}_n = (-1)^m \begin{pmatrix} a_1 \\ a_1 \\ a_2 \\ a_2 \end{pmatrix} \quad \partial \vec{\psi}_n = \vec{0},\] and lead to the equation \[\label{eq13} (-1+ \sin \theta) \; a_1 + \cos \theta \; a_2 = 0,\tag{12}\] which should be solved together with the normalisation condition 6 .

This time the normalisation, compatible with 6 and 12 , can be chosen to be given by \[\label{ypnxujer} a_1(0) = 1, \quad a_2 (0) = 1.\tag{13}\] Excluding \(a_1\) using 12 and substituting into 6 we get \[a_2 (\theta) = \pm \sqrt{1 - \sin \theta} = \pm \Big( \cos \theta/2 - \sin \theta/2 \Big).\] Then normalisation 8 requires \[\label{eq41} a_2 (\theta) = \cos \theta/2 - \sin \theta/2, \qquad 0 \leq \theta \leq \pi/2.\tag{14}\] We calculate also the first amplitude \[\label{eq42} \begin{array}{ccl} a_1 (\theta) & = & \displaystyle \frac{\cos \theta}{1 - \sin \theta} \Big( \cos \theta/2 - \sin \theta/2 \Big) = \frac{\cos^2 \theta/2 - \sin^2 \theta/2}{\Big( \cos \theta/2 - \sin \theta/2 \Big)^2} \Big( \cos \theta/2 - \sin \theta/2 \Big) \\[5mm] & = & \displaystyle \cos \theta/2 + \sin \theta/2. \end{array}\tag{15}\] Formulas 14 and 15 determine the amplitudes for \(0 \leq \theta \leq \pi/2\) and there is just a unique way to extend them for \(\pi/2 \leq \theta \leq 2 \pi\) keeping the functions continuous: \[\label{eq6} a_1 (\theta) = \cos \frac{\theta}{2} + \sin \frac{\theta}{2}, \qquad a_2 (\theta) = \cos \frac{\theta}{2} - \sin \frac{\theta}{2}.\tag{16}\]

The amplitudes depending continuously on \(\theta\) are plotted in Fig. 4. Formulas 11 and 16 determine the amplitudes as \(4 \pi\)-periodic functions of \(\theta\), while period \(2 \pi\) may be expected since the family \(L^\theta\) has period \(2 \pi\).

One clearly observes that \[\label{Berry} \psi_n \vert_{\theta = 2 \pi} = \underbrace{-}_{= e^{i \pi}} \psi_n \vert_{\theta = 0}.\tag{17}\]

Odd eigenfunctions↩︎

The analysis is entirely analogous to the even case with the only difference that the amplitudes for even and odd values of \(n\) are exchanged. The eigenfunctions demonstrate a topological phase \(\pi\) after one period \(\theta: 0 \rightarrow 2 \pi\).

The ground state \(\lambda = 0\)↩︎

The analysis is very similar to the case of even eigenfunctions \(\psi_{2m}\) with the only difference that the normalisation condition leads to \[\label{eq22} a_1^2 + a_2^2 = 1,\tag{18}\] instead of 6 . Hence the corresponidng amplitudes are given by \[\label{eq66} a_1 (\theta) = \frac{1}{\sqrt{2}} \Big(\cos \frac{\theta}{2} + \sin \frac{\theta}{2} \Big) , \qquad a_2 (\theta) = \frac{1}{\sqrt{2}} \Big(\cos \frac{\theta}{2} - \sin \frac{\theta}{2} \Big).\tag{19}\] The eigenfunction generates the topological phase \(\pi\) after one period.

The theorem has been proven in the special case of equal edge lengths, but the spectrum and the eigenfunctions depend continuously on the edge lengths. Hence the geometric phase depends continuously on the edge lengths, but it attains just two possible values of \(0\) and \(\pi\), hence the conclusion of the theorem holds even for not necessarily equal edge lengths.

7 Conclusions and discussions↩︎

Thus we have proven that the topological Berry phase for the considered family of graphs is non-trivial and equals \(\pi.\) The main reason for non-triviality of the geometric phase is connected with the fact that topology of the system changes with \(\theta\): for almost all values of \(\theta\) the graph should be seen as the figure eight graph with vertex conditions depending on the parameter, but for certain special values of the parameter the graph reduces either to one or two cycles. In other words, not only does the number of cycles change with \(\theta\), but also the connectivity of the graph. The eigenfunctions are essentially described by two amplitudes depending continuously on \(\theta\), and to gain a non-trivial topological phase these amplitudes have to change sign In other words each of the amplitudes has to vanish. The corresponding eigenfunction vanishes identically on one of the edges. In our model this appears precisely when the graph loses connectivity (\(\theta = \pi/2, 3 \pi/2\)) and reduces to two independent circles.

Identical vanishing of eigenfunctions on some edges is a typical feature of operators on metric graphs and may happen even without losing connectivity. Therefore, it might be interesting to find an example of a family of metric graphs with a non-trivial geometric phase whose topology does not change.

Data availability statement↩︎

No new data were created or analysed in this study.

Funding↩︎

The research of P.K. was been supported by The Swedish Research Council, grant number 2024-04650.

Author contributions↩︎

The authors contributed to deriving, analysing the formulas and writing the manuscript. The authors read and approved the final manuscript.

Conflict of interest↩︎

The authors declare that there are no competing interests.

References↩︎

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1. We follow this path calculating below the amplitudes for even values of \(n\).↩︎