1 Introduction↩︎

In 1946, S.S.Chern introduced the topological invariants of complex vector bundles on smooth manifolds now known as Chern classes [1]. Chern classes are derived from the coefficients of the characteristic polynomial of the curvature form associated with a vector bundle. In particular, the coefficient of the linear term in this polynomial is called the first Chern class [2], and the integral of this class over a two-dimensional base manifold is an integer known as the Chern number.

For parameter-dependent families of Hamiltonians, each nondegenerate energy band can be naturally associated with a line bundle. Specifically, let \(H(\mathbf{P})\) be a Hamiltonian at a point \(\mathbf{P}\) in the parameter space manifold \(\mathcal{M}\) with energy bands \(E_n(\mathbf{P})\). If the system is prepared in an eigenstate \(|\psi_n(\mathbf{P})\rangle\) of \(n\)th energy band , and the parameters are changed slowly enough along a closed one-dimensional path \(\mathcal{C}\subset\mathcal{M}\), the final state, \(e^{i\Delta\Phi}| \psi_n(\mathbf{P})\rangle\), acquires the geometric phase \(\Delta\Phi =\int_{\mathcal{C}} A_n\), where \(A_n=i\langle\psi_n|d \psi_n\rangle\) is the adiabatic (Berry) connection on the corresponding line bundle [3]. The adiabatic (Berry) curvature is \(\Omega_n=dA_n\), and the first Chern class of the line bundle is the cohomology class \(-\Omega_n/(2\pi)\); it follows that when \(\mathcal{M}\) is two-dimensional, the Chern number \(C_n\) of the \(n\)th energy band is \(-\frac{1}{2\pi}\int_{ \mathcal{M}}\Omega_n\).

The topological invariance of the Chern numbers implies that \(C_n\) remains constant when the Hamiltonian family is smoothly deformed as long as the gaps between the \(n\)th energy band and its neighbors remain open. In systems without time reversal symmetry, level crossings are generically codimension-three, and generic crossings are three-dimensional cones [4]. I”It follows that, for generic non-time-reversal-invariant Hamiltonians depending on two parameters \((\phi_1,\phi_2)\) and additional parameters \(\mathbf{p}\), conical intersections between bands occur on codimension-one submanifolds of the \(\mathbf{p}\) parameter space. When \(\mathbf{p}\) crosses such a submanifold transversely, the Chern number of one of the intersecting bands changes by 1, and the Chern number of the other intersecting changes by \(-1\) [5].

The first major application of Chern invariants to physics was made in the theory of the quantum Hall effect [6]. Here the base manifold is the two-dimensional torus of Bloch momenta, and the Hall conductance of a band is equal to its Chern number multiplied by the quantum of conductance [7]–[9]. Although nontrivial Chern classes can only arise when time reversal symmetry is broken, this can happen without external magnetic fields, in which case the Chern number is proportional to the anomalous quantum Hall conductance [10]. A related physical phenomenon is quantized transport in multiply connected systems, where the electromotive force integrated over the threading of a unit magnetic flux through a hole generates transport of a Chern number of charge units [11]–[13]. Later on, following the discovery of the topological aspects of the quantum spin Hall effect [14], [15], it was realized that systems with quantized Hall conductance are one class of topological states of matter that are characterized by topological invariants classified by symmetries and the dimension of the base space [16], [17].

The topological properties of complex systems with a large number of energy bands can be naturally studied by examining the distribution of topological invariants. The underlying supposition is that, under certain conditions, the distribution of topological invariants in such systems becomes universal and can be analyzed using synthetic random matrix models, where complexity is effectively replaced by randomness.

Originally, random matrix models were successfully employed to derive universal level statistics [18], [19] and transport properties [20] of complex quantum systems. The statistics of adiabatic curvature and Chern numbers, as functions of parameters, were later computed within random-matrix frameworks [21]–[23]. Similarly, the statistics of winding numbers in one-dimensional random matrix models with chiral symmetry have been studied in [24], [25], [26], and [27] for the chiral unitary, symplectic, and orthogonal classes, respectively.

Many foundational models of topological states of matter are based on tight-binding Hamiltonians defined on periodic lattices [7], [28], [29]. In the present work, we investigate the Chern numbers of energy bands in a quantum graph exhibiting a discrete two-dimensional square lattice structure, with a general four-port scatterer at each node. Quantum graphs have proven highly effective in modeling complex physical systems, from Pauling’s early work on free-electron models for organic molecules [30] to studies of the Anderson transition in disordered wires [31], mesoscopic quantum systems [32], and quantum chaos [33]–[35]. Unlike tight-binding models, quantum graphs possess an infinite number of energy levels, making them well-suited for exploring universality. In particular, even relatively small quantum graphs exhibit universal spectral statistics that are well-described by appropriate ensembles of random matrices [33], [36]. Moreover, quantum graph models enable direct access to edge states and quantized transport—key features of topological matter that are not naturally captured within conventional random matrix models.

The Chern numbers of a quantum graph lattice with an external magnetic field were computed by Goldman and Gaspard [37], who considered a special class of node scatterers that preserve time-reversal symmetry. In their work, nontrivial Chern numbers arise only in the presence of a nonzero magnetic field, leading to a Hofstadter-butterfly-like spectrum and a quantum Hall conductance governed by a Diophantine equation of the type introduced in [7]. In contrast, we consider quantum graphs in which the node scatterers break time-reversal symmetry, allowing nontrivial Chern numbers even in the absence of a net external magnetic field. In this sense, the quantum graph can be viewed as a model of the anomalous Hall effect [29], [38]. In the present work, we focus on the topological properties of the system, leaving a detailed analysis of transport phenomena for future studies.

The paper is organized as follows. Owing to translational symmetry, Bloch’s theorem is applicable, with a single plaquette per unit cell. Using this observation, we show in Section 2 that the associated fiber Hamiltonian of the lattice graph is equivalent to that of a “figure-eight” graph. This graph consists of two equal-length looped quantum wires connected to the ports of the node scatterer, with the loops pierced by magnetic fluxes proportional to the two components of the Bloch momentum. These magnetic fluxes parametrize the torus base space of the energy band line bundles.

The calculation of the energy bands of the graph is reduced, in the standard manner, to solving a generalized eigenvalue problem for a \(4 \times 4\) unitary matrix. Since the wire lengths are assumed equal, the bands are periodic, and the Chern numbers need only be computed for a fundamental set of four bands. In Section 3, we show that the only possible Chern numbers of a given band are 0—in which case all bands are topologically trivial—or \(\pm 1\), in which case all bands have Chern numbers of unit magnitude that alternate in sign.

It follows that these three types of Chern number patterns are separated by hypersurfaces in the space of the node scatterer parameters, corresponding to intersections of energy bands. The main result of this paper, presented in Section 4, is an explicit trigonometric equation describing these intersection surfaces, which form codimension-one submanifolds of the group of \(4 \times 4\) unitary matrices. Each energy-band intersection surface is characterized by an orientation that determines the change in the Chern number when the surface is crossed in a given direction. These results are validated through comparison with numerical calculations of the Chern numbers obtained via direct integration. In Section 5, we provide explicit expressions for the energy-band intersection surfaces in the special case where all matrix elements of the scatterer are real. In a sense clarified below, this case corresponds to the most extended region of topological nontriviality.

2 The quantum-graph models↩︎

2.1 The square lattice quantum graph↩︎

We study a quantum graph with a square-lattice structure. A quantum particle propagates freely along edges of equal length \(L\) connecting nearest-neighbor vertices and is scattered by identical scatterers at the vertices, as shown in the left panel of figure 1. We label the vertices of the graph by a pair of integers \(n_1,n_2\), and the edges by a vertex label and an index \(j=1,2\), representing the lattice direction.

The stationary states of the quantum graph problem are given by a set of wave functions \(\psi^{(j)}_{n_1,n_2}\) satisfying the Schrödinger equation \[\label{Sro32Eq} -\frac{d^2\psi^{(j)}_{n_1,n_2}}{d{x}^2}={k}^2\psi^{(j)}_{n_1,n_2} , \qquad x\in [0,L]\tag{1}\] on each edge of the graph, where \(k= \sqrt{2mE/\hbar^2}\) is the wave number, and \(m\) and \(E\) are the mass and the energy of the particle, respectively [39]. The solutions of these equations are given by combinations of plane waves: \[\label{124Sol32Eq} \psi^{(j)}_{n_1,n_2}=C^{(j)}_{n_1,n_2}e^{ikx}+\bar{C}^{(j)}_{n_1,n_2}e^{-ikx}.\tag{2}\]

The vertex scatterers are specified by a 4-by-4 unitary scattering matrix \(S\) that connects the incoming amplitudes \(\chi_{n_1,n_2}\) at vertex \(n_1,n_2\) with the outgoing \(\chi’_{n_1,n_2}\), where \[\label{Vect32Eq} \chi_{n_1,n_2}= \begin{pmatrix} C^{(1)}_{ { n_1-1,n_2}}e^{ikL}\\ \bar{C}^{(1)}_{n_1,n_2} \\ {C}^{(2)}_{n_1,n_2-1}e^{ikL}\\ \bar{C}^{(2)}_{n_1,n_2} \end{pmatrix}, \qquad \chi'_{n_1,n_2} = \begin{pmatrix} \bar{C}^{(1)}_{ {n_1-1,n_2}}e^{-ikL}\\ {C}^{(1)}_{n_1,n_2}\\ \bar{C}^{(2)}_{n_1,n_2-1}e^{-ikL}\\ {C}^{(2)}_{n_1,n_2} \end{pmatrix}.\tag{3}\] The scattering relations \(S\chi_{n_1,n_2}=\chi'_{n_1,n_2}\) can be viewed as a system of equations for the edge-wave amplitudes. The discrete translation symmetry of the graph implies that we can use Bloch’s theorem to seek solutions of the system of the form \(C_{n_1,n_2}^{(j)}=c^{(j)}e^{i({n_1\phi_1+n_2\phi_2})}\), \(\bar C_{n_1,n_2}^{(j)}=\bar c^{(j)}e^{i({n_1\phi_1+n_2\phi_2})}\). For each set of values of the quasimomenta \(\phi_1,\phi_2\) in the two spatial dimensions, the scattering equation becomes a generalized eigenvalue problem for \(k\) with a discrete set of solutions, each of which delineates an energy band, see [40], [41].

Since the quasimomenta are \(2\pi\)-periodic, the bands of the lattice quantum graph that are separated by gaps from the rest of the spectrum define a smooth line bundle over the two-torus, in the same manner that bands of tight-binding lattice Hamiltonians do. In the following, we study the topology of these line bundles.

Quantum graph models: Particles propagate freely on edges and are scattered at vertices by an identical scattering matrix \(S\). Left: A periodic square lattice graph with nearest-neighbor edges and no magnetic fields. The vertices are labeled by pairs of integers \((n_1,n_2)\), and the two lattice directions are labeled by \(j=1,2\). Right: A figure-eight graph with a single vertex and two edge loops pierced by Aharonov-Bohm magnetic fluxes \(\phi_1,\phi_2\). The directions \(j=1,2\) are mapped to the fluxes \(\phi_1,\phi_2\), respectively. Thus, the two quasimomenta of the lattice problem are represented as the two magnetic fluxes of the figure-eight graph.

2.2 The two-loop quantum graph↩︎

The lattice graph spectral problem for fixed values of \(\phi_1,\phi_2\) is equivalent to that of a figure-eight graph, whose two length-\(L\) loops are pierced by magnetic fluxes with corresponding phases \(\phi_1\) and \(\phi_2\), as shown in figure 1. This spectral problem can be formulated in the standard way, see, e.g., [36], as \[\label{eq:4spectral} \Lambda S\Psi=e^{-ikL}\Psi,\tag{4}\] where \(\Lambda\) is the magnetic phase-shift matrix that is block-diagonal in a basis where the first (last) two components refer to the 1st (2nd) loop, respectively: \[\label{eq:Lambda} \Lambda=\begin{pmatrix} \Lambda_{1} & 0 \\ 0 & \Lambda_{2} \end{pmatrix}, \qquad \Lambda_{j}=\begin{pmatrix} 0& e^{i\phi_j} \\ e^{-i\phi_j}& 0 \end{pmatrix}.\tag{5}\] The four components of \(\Psi\) are the forward- and backward-propagating amplitudes on the two loops of the figure-eight graph. They are linear combinations of the Bloch amplitudes \(c^{(j)},\bar c^{(j)}\), and the energy bands of the lattice graph are reproduced when the fluxes are swept across the two-torus. We note that the reduction of the nonmagnetic lattice graph to the magnetic figure-eight fiber graph also works when the lattice edge lengths in the two directions are not equal, but in this work we focus on graphs with equal-length edges, and choose \(L=1\) without loss of generality.

The energy levels of the figure-eight graph are obtained from the secular equation \[\det( \Lambda S-e^{-ik} I)=0\;,\label{SeqEq}\tag{6}\] where \(I\) is the identity matrix. It follows that a solution \(e^{-ik}\) of 6 is equal to one of the four eigenvalues \(\sigma_j\), \(j=1,\ldots 4\), of the unitary matrix \(\Lambda S\), implying that the bands repeat periodically: \[k_{j,n}(\phi_1,\phi_2)=-\log\sigma_j(\phi_1,\phi_2)+2\pi n, \qquad n\in \mathbb{Z}\;. \label{EnergyBands}\tag{7}\] The corresponding eigenvectors, and therefore the Chern numbers, are the same for all bands inside each of the four sequences.

2.3 The one-loop quantum graph↩︎

The spectrum of the graphs in figure 1 can be evaluated in yet another way by absorbing the loop pierced by the flux \(\phi_1\), together with the vertex, into a single scatterer. The reduced quantum graph consists of a single loop threaded by a flux of \(\phi_2\), connecting the two ports of the scatterer, see figure 2, with a reduced 2-by-2 scattering matrix \(\hat{S}_2\) that depends on the flux \(\phi_1\) and \(k\).

To derive \(\hat{S}_2\), we write the original \(4\)-by-\(4\) scattering matrix and the amplitude vector in block form \[\begin{align} \label{Smatrix} S=\begin{pmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{pmatrix}\;,\qquad\Psi=\begin{pmatrix}\Psi_1\\\Psi_2\end{pmatrix}\;, \end{align}\tag{8}\] with 2-by-2 matrix blocks and 2-vector column blocks. The spectral equation 4 then implies that \[\Psi_1=\frac{1}{e^{-ik}\Lambda_{1} -S_{11}}S_{12}\Psi_2\;,\] and therefore \[\label{eq:2spectral2} \Lambda_2\hat{S}_2(k)\Psi_2=e^{-ik}\Psi_2\;, \qquad\hat{S}_2(k)= S_{21} \frac{1}{e^{-ik}\Lambda_{1} -S_{11}}S_{12}+S_{22}\;.\tag{9}\] The same reduction applied to the second loop yields a scatterer-loop graph with a flux of \(\phi_1\) and a reduced scattering matrix \(\hat{S}_1\), whose spectral equation, related to 9 by symmetry, is \[\label{eq:2spectral1} \Lambda_1\hat{S}_1(k)\Psi_1=e^{-ik}\Psi_1\;, \qquad\hat{S}_1(k)= S_{11}+S_{12} \frac{1}{e^{-ik}\Lambda_{2}-S_{22}}S_{21}\;.\tag{10}\]

It follows from the arguments leading to the reduced spectral problems (9 –10 ) that they are equivalent to the original one 4 unless \(e^{-ik}\Lambda_1-S_{11}\) or \(e^{-ik}\Lambda_2-S_{22}\) is singular. As a result, the energy bands of the one-loop graphs are obtained from the secular equations \[\label{graphSpec} \det( \Lambda_{1}\hat{S}_1 (k) -e^{-ik} I)=\det( \Lambda_{2}\hat{S}_2 (k) -e^{-ik} I)=0\;,\tag{11}\] which are equivalent to equation 6 , although in this case the \(k\)-dependent scattering matrices \(\hat{S}_1\) and \(\hat{S}_2\) are of reduced \(2\times 2\) size. As we show below, this dimensional reduction simplifies the analysis of the topology of the energy bands.

3 Topological phases of the quantum graph↩︎

In this section, we demonstrate that the quantum graph models exhibit two distinct topological phases: a trivial phase, where the Chern numbers of all energy bands are 0, and a nontrivial phase, where the Chern numbers alternate between \(\pm1\) across the bands. This result is established in two steps: first, we show that the sum of the absolute values of the Chern numbers is four or less, and then that topological transitions occur as simultaneous intersections of two pairs of bands.

3.1 The Chern numbers of energy bands↩︎

3.1.1 Chern numbers as degrees of maps↩︎

Sticlet et al.[42] showed that the Chern numbers of the bands of a 2-by-2 fiber Hamiltonian are equal to the Brouwer degree of a map from the base space to the two-sphere defined by the linear expansion of the Hamiltonian in terms of Pauli matrices. This result extends in a straightforward manner to our case, by writing \[\label{Hrepr} e^{ik}\Lambda_2\hat{S}_2=e^{i(I h_0+\Vec{\sigma}\cdot\Vec{h})}\;,\tag{12}\] where \(\Vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)\) stands for the vector of Pauli matrices and \(\Vec{h}=(h_1,h_2,h_3)\) is a vector whose components are real-valued functions of the band phases \(\phi_1,\phi_2\) and the spectral parameter \(k\).

In the remainder of this subsection, we briefly review the spectral and topological properties of bands defined implicitly by imposing 9 , or equivalently by requirinf that \(I h_0+\Vec{\sigma}\cdot\Vec{h}\) has an eigenvalue equal to an integer multiple of \(2\pi\), that is, \[\label{eq:1loopsproj} h_0(k)+s\|\Vec{h}(k)\|\equiv 0 \pmod{2\pi}\;, \quad (s=\pm1)\;.\tag{13}\] For given \(\phi_1,\phi_2\), four values of \(e^{-ik}\) can be attained by solutions of 13 , namely the four eigenvalues of \(\Lambda S\); see 4 . Labeling these four values \(k_j\), \(j=1,\ldots,4\), for example in increasing order for \(-\pi<k_j\le\pi\) at \(\phi_1=\phi_2=0\), and by continuity elsewhere, we obtain four energy bands \(k_j(\phi_1,\phi_2)\), each associated with a sign choice \(s_j\) in 13 .

Since \(\|\Vec{h}\|>0\) away from points of degeneracy, we can define there the spectral projections \[\label{eq:hproj} \mathcal{P}_j(\phi_1,\phi_2)=\frac{1}{2}(I+s_j\hat{h}(\phi_1,\phi_2,k_j)\cdot\Vec{\sigma})\;,\qquad \hat{h}=\frac{\Vec{h}}{\|\Vec{h}\|}\tag{14}\] onto band \(j\).

When the bands do not intersect, \(\|\Vec{h}\|>0\) everywhere on the torus \(0\le\phi_1,\phi_2\le2\pi\), so that the four bands define maps \(f_j=s_j \hat{h}(\phi_1,\phi_2,k_j)\) from the torus into the unit sphere, and by the arguments of [42], the Chern number of band \(j\) is equal to the degree of \(f_j\). Recall [43] that a point in the range of a map is regular if the Jacobian determinant of the map does not vanish at any of its preimages, and that the degree of a map is equal to the sum of the signs of its Jacobian determinant at all preimages of a regular point. Thus, for any pair \(\hat{h}\) and \(-\hat{h}\) of antipodal regular points \[\label{ChernInt3} C_j=\frac{s_j}{2}\sum\mathop{\mathrm{sign}}\bigl((\partial_{\phi_1}\Vec{h}_j\times\partial_{\phi_2}\Vec{h}_j)\cdot \vec{h}_j\bigr) ,\tag{15}\] where the sum is over all preimages \((\phi_1,\phi_2)\) of \(\hat{h}_j\equiv\hat{h}(k_j)=\pm\hat{h}\).

3.1.2 Calculation of the preimages↩︎

We next calculate the preimages of the north and south poles \(\hat{h}_1=\hat{h}_2=0,\,\hat{h}_3=\pm1\), of the unit sphere. The condition \(h_1=h_2=0\) is equivalent to the statement that the matrix \(\hat{S}_2\) is purely off-diagonal. Since \(\hat{S}_2\) is unitary, it is sufficient that the matrix element \(\hat{S}_{2,11}=0\). Using the explicit representation 9 , the latter condition can be expressed as \[\label{abcd} a e^{ik}+b e^{-ik}=c e^{i\phi_1}+d e^{-i\phi_1}\;,\tag{16}\] where the coefficients \(a,\,b,\,c,\,d\) depend only on the parameters of the \(S\) matrix (explicit expressions are provided in section 4.1.1 below).

As \(k\) is swept from \(0\) to \(2\pi\), the left-hand side of 16 describes an ellipse in the complex plane, centered at the origin with semi-major(minor) axis of length \(||a|\pm|b||\) (respectively), and a similar result holds for the right-hand side as \(\phi_1\) is swept over the same range. If the poles are regular points of the map \(\hat{h}\), the two ellipses either do not intersect at all, or intersect at four distinct pairs \((k,\phi_{1})\). If the poles are singular points of the map, the ellipses are tangent at two pairs \((k,\phi_{1})\). In section 4 below we derive the necessary and sufficient conditions on the coefficients of 16 for the ellipses to intersect, and derive expressions for the values of \(k\) and \(\phi_1\) at these intersections.

When the ellipses defined by the left- and right-hand sides of equation 16 do not intersect, there is no choice of fluxes for which the spectral projections are diagonal, so all the bands are trivial. When the ellipses do intersect, the off-diagonal elements of the matrix \(\hat{S}_2\) evaluated at the intersection points are pure phases, so that the eigenvectors \(\Psi_2\) in 9 are \((1,0)\) and \((0,1)\) for \(e^{i\phi_2}=e^{-ik}\hat{S}_{2,21}^*\) and \(e^{i\phi_2}=e^{ik}\hat{S}_{2,12}\), respectively. It follows that, in total, the sums in 15 are over eight torus points. Since each of these preimages makes a \(\pm1/2\) step in the Chern number of one of the four bands, we can bound the sum of the absolute values of the Chern numbers \[\label{eq:sumc} \sum_{j=1}^4|C_j|\le4\;.\tag{17}\] In the next section we show that whenever there are topologically nontrivial bands, two of the bands have \(C=1\) and the other two have \(C=-1\).

3.2 Band intersections and topological phases↩︎

The graph spectral problem has a built-in symmetry that is most evident in the two-loop formulation 4 : the matrix \(\Lambda\) of 5 obeys \(\Lambda(\phi_1+\pi,\phi_2+\pi)=-\Lambda(\phi_1,\phi_2)\), implying that if \(k_1,\ldots,\,k_4\) are the band wave numbers of the graph for flux values \((\phi_1,\phi_2)\), the the band wave numbers at \((\phi_1+\pi,\phi_2+\pi)\) are \(k_1+\pi,\ldots,\,k_4+\pi\) (not necessarily in the same order, and modulo \(2\pi\)). In particular, if two bands intersect at \((\phi^*_1,\phi^*_2)\), it follows that \((\phi^*_1+\pi,\phi^*_2+\pi)\) is another intersection point. However, as we now show, the bands that intersect at the first point are distinct from those that intersect at the second point.

To this end, we order the bands \(k_1,k_2,k_3,k_4\) in cyclically increasing order as above. Assume for concreteness that \(k_1(\phi^*_1,\phi^*_2)=k_4(\phi^*_1,\phi^*_2)\), i.e., the intersection is between the first and fourth bands. Consider a path \(\mathcal{C}\) in the \((\phi_1,\phi_2)\) plane connecting the two degeneracy points \((\phi^*_1,\phi^*_2)\) and \((\phi^*_1+\pi,\phi^*_2+\pi)\), such that there are no other degeneracy points along \(\mathcal{C}\), see figure 3, left panel. Note that since \(\det\Lambda=1\) independently of \(\phi_1,\phi_2\), the sum \(\sum k_j\) is constant. The spectral symmetry of the graph bands dictates that the set of band wave numbers should reach values shifted by \(\pi\) when the fluxes are varied along \(\mathcal{C}\). The only way to achieve this continuously while maintaining \(\sum k_j\) is by mapping \[k_1\to k_3-\pi\;,\qquad k_2\to k_4-\pi\;,\qquad k_3\to k_1+\pi\;,\qquad k_4\to k_2+\pi\;,\] see figure 3, right panel. It follows that the intersection at \((\phi^*_1+\pi,\phi^*_2+\pi)\) is between the second and third bands.

If the parameters of the scattering matrix \(S\) are varied from the trivial to the nontrivial phase so that \(C_4\) increases by one at the band-intersection point, then the Chern numbers in the nontrivial phase become \(-C_1 = C_2 = -C_3 = C_4 = 1\). Conversely, if the parameters are varied so that \(C_4\) decreases by one at the band-intersection point, Chern numbers with opposite signs are obtained for each band. In both cases, the nontrivial phase is characterized by alternating \(\pm 1\) Chern numbers. If the parameters of \(S\) are varied further from such a phase through another band-intersection point, the resulting Chern numbers could, in principle, either all become zero or take values \(\pm 2\). The latter possibility, however, is excluded, as it would violate the bound 17 on the sum of absolute Chern numbers.

In conclusion, the quantum graph model has two possible topological phases: a trivial phase with zero Chern numbers for all bands and a nontrivial phase characterized by an alternating \(\pm 1\) pattern of Chern numbers across the four bands. If a path in the parameter space of the matrix \(S\) passes through a point of spectral degeneracy, the gaps between adjacent bands close simultaneously, and the graph undergoes a transition from the trivial to nontrivial phase, or vice versa.

4 Topological transitions of the quantum graph↩︎

4.1 Band intersections↩︎

In this section, we establish a necessary and sufficient condition on the scattering matrix \(S\) for a transition to occur between the trivial and nontrivial phases of the quantum graph. Topological transitions occur when the gaps between the bands close and the spectrum of the matrix \(\Lambda S\) has a degeneracy for some values \(\phi_{1}\), \(\phi_{2}\) of magnetic flux, and therefore also for \(\phi_1+\pi,\phi_2+\pi\) as explained above in section 3.2.

According to the Wigner–von Neumann theorem [44], the manifold of degenerate unitary matrices has codimension-three in the manifold of all unitaries of a given size. Since the energy bands depend on two flux parameters, the manifold of scattering matrices for which a topological transition occurs has codimension-one. It is therefore convenient to choose one scattering parameter and express the band-intersection condition in terms of it. We choose for this purpose the scattering phase \[\gamma=\tfrac{1}{4}(\arg\det S_{11}-\arg\det S_{22})\;,\] where \(S_{11}\), \(S_{22}\) are the two-by-two blocks defined in 8 above, and consider the one-parameter family of scattering matrices with two-by-two block representation \[\label{eq:sgamma} S=\begin{pmatrix} e^{i\gamma}I&0\\0&e^{-i\gamma}I \end{pmatrix} S_0\tag{18}\] for some fixed \(S_0\) for which \(\gamma=0\).

We tackle the gap-closing problem in terms of the equivalent one-loop scattering matrix \(\hat{S}_2\). At a generic intersection, the eigenspaces of the three spectral equivalent problems (4 ), (9 ), and (10 ) are two-dimensional, so the matrices \(\Lambda_{1}\hat{S}_1, \Lambda_{2}\hat{S}_2\) are scalar multiples of the identity matrix, \[\begin{align} \tag{19} &\Lambda_{1}\hat{S}_1=e^{-ik}I\;,\\& \tag{20}\Lambda_{2}\hat{S}_2=e^{-ik}I\;. \end{align}\] We derive the equation for \(\gamma\) from 20 in three steps: First, we express \(\phi_1\) at the band intersection points in terms of \(\gamma\) and the spectral parameter \(k\) (section 4.1.1). We then eliminate \(\phi_1\) to obtain a relation between \(\gamma\) and \(k\) (section 4.1.2). Finally, we eliminate \(\phi_2\) between two further expressions to obtain a second relation between \(\gamma\) and \(k\), from which the equation for \(\gamma\) is reached by eliminating \(k\) (section 4.1.3). In section 4.1.4 we obtain an independent necessary condition for spectral degeneracy by partially imposing both 19 and 20 .

4.1.1 The flux values at points of degeneracy↩︎

Each of equations 19 , 20 is a necessary and sufficient condition for the degeneracy of the spectrum of the graph matrix \(\Lambda S\). Since the matrices \(\Lambda_j\hat{S}_j\) are unitary, these matrix equations impose four real conditions, which we solve for the real variables \(\gamma,k,\phi_1,\phi_2\). In this subsection and the following two, we solve equation 20 , and then reconsider 19 in subsection 4.1.4.

It is convenient to start with the diagonal part of 20 , which is satisfied if and only if \(\hat{S}_2\) is purely off-diagonal, since this condition does not involve \(\phi_2\). A straightforward but lengthy calculation implies \(\hat{S}_{2,11}=\hat{S}_{2,22}=0\) if and only if \[\label{eq:abcd1} a e^{i(k+\gamma)}+b e^{-i(k+\gamma)}=c e^{i\phi_1}+d e^{-i\phi_1}\;,\tag{21}\] where the coefficients depend polynomially on the elements of \(S\): \[\begin{align} a&=S_{31}(S_{12}S_{23}-S_{22}S_{13})+S_{32}(S_{21}S_{13}-S_{11}S_{23})+S_{33}(S_{11}S_{22}-S_{12}S_{21})\;,\\ b&=-S_{33}\;,\\ c&=S_{31}S_{23}-S_{33}S_{21}\;,\\ d&=S_{32}S_{13}-S_{33}S_{12}\;. \end{align}\] Next we note that equation 21 and its complex conjugates are independent over the reals, allowing us to express \(e^{i\phi_1}\) as \[\label{eq:ABC1} Ae^{i(k+\gamma)}+B e^{-i(k+\gamma)}=Ce^{i\phi_1}\;,\tag{22}\] with \[\begin{align} A&=ac^*-b^*d\tag{23}\;,\\ B&=bc^*-a^*d\;,\\ C&=|c|^2-|d|^2\;.\tag{24} \end{align}\]

4.1.2 The wave number at points of degeneracy↩︎

It follows from equation 22 that \[\label{eq:ABCk} AB^*e^{2i(k+\gamma)}+A^*Be^{-2i(k+\gamma)}={C}^2-|A|^2-|B|^2\;;\tag{25}\] this equation implies the solvability condition \[\label{eq:solv} 2|A||B|\ge |C^2-|A|^2-|B|^2|\;,\tag{26}\] which is necessary and sufficient for the ellipses described by the left- and right-hand sides of 21 to intersect (as explained in section 3.1.2 above).

The strict inequality form of 26 is equivalent to the three inequalities \[\label{eq:triabc} |C|<|A|+|B|\;,\qquad |B|<|C|+|A|\;,\qquad |A|<|B|+|C|\;,\tag{27}\] which are necessary and sufficient conditions for the existence of a triangle with side lengths \(|A|,|B|,|C|\).

If we write 25 as \[\label{eq:k43gamma} e^{2i(k+\gamma)-i\delta}+e^{i\delta-2i(k+\gamma)}=g\;,\tag{28}\] with \[\label{eq:gd1} ge^{i\delta}= \frac{C^2-|A|^2-|B|^2}{AB^*}\;,\qquad g>0\;,\tag{29}\] then 26 is equivalent to \(g\le2\), in which case, the possible wave numbers at the points of degeneracy are \[\label{eq:kpsi} k=-\gamma+\frac{\delta\pm\psi}{2}+n\pi\tag{30}\] with an integer \(n\), where \[\label{eq:psi1} \psi=\arccos\Bigl(\frac{g}{2}\Bigr)\;.\tag{31}\]

Geometrically, \(\pi-\psi\) is the angle opposite the side of length \(|C|\) in the triangle with side lengths \(|A|^2+|B|^2>C^2\), see figure 4. The analogous argument that starts by eliminating \(k\) between 21 and its conjugate leads to an equivalent inequality for a triangle with sides \(|A|\), \(|B|\), and \(|D|\), where \(D=|a|^2-|b|^2\), and an expression for \(\phi_1\) analogous to 30 , see appendix 8.

4.1.3 Scattering phase for band intersection↩︎

The arguments of the previous subsections followed from the off-diagonal elements of the matrix equation 20 (for \(\Lambda_2\hat{S}_2\)), and allowed us to express \(k\) and \(\phi_1\) at degeneracy points in terms of the elements of the matrix \(S\) and the scattering phase \(\gamma\). We proceed to impose the diagonal part of 20 that determines \(\phi_2\) at the point of intersection, and the \(\gamma\) values for which the bands intersect.

Additional lengthy but straightforward algebra implies that the first and second diagonal elements of 20 are equivalent, respectively, to \[\begin{align} &q_1e^{i(k+\gamma)}+p_1e^{-i(k+\gamma)}-r_1e^{i\phi_1}-s_1e^{-i\phi_1}\nonumber\\&\qquad \qquad \qquad =(u e^{2i\gamma}+ve^{i(-k+\gamma+\phi_1)}+we^{i(-k+\gamma-\phi_1)}-e^{-2ik})e^{-i\phi_2}\;,\tag{32}\\ &q_2e^{i(k+\gamma)}+p_2e^{-i(k+\gamma)}-r_2e^{i\phi_1}-s_2e^{-i\phi_1} \nonumber\\&\qquad \qquad \qquad=(u e^{2i\gamma}+ve^{i(-k+\gamma+\phi_1)}+we^{i(-k+\gamma-\phi_1)}-e^{-2ik})e^{i\phi_2}\;,\tag{33} \end{align}\] with \[\begin{align} q_1&=S_{41}(S_{12}S_{23}-S_{22}S_{13})+S_{42}(S_{21}S_{13}-S_{11}S_{23})+S_{43}(S_{11}S_{22}-S_{12}S_{21})\\ p_1&=-S_{43}\\ r_1&=S_{41}S_{23}-S_{43}S_{21}\\ s_1&=S_{42}S_{13}-S_{43}S_{12}\\ q_2&=S_{31}(S_{12}S_{24}-S_{22}S_{14})+S_{32}(S_{21}S_{14}-S_{11}S_{24})+S_{34}(S_{11}S_{22}-S_{12}S_{21})\\ p_2&=-S_{34}\\ r_2&=S_{31}S_{24}-S_{34}S_{21}\\ s_2&=S_{32}S_{14}-S_{34}S_{12}\\ u&=S_{11}S_{22}-S_{12}S_{21}\\ v&=S_{21}\\ w&=S_{12}\;. \end{align}\]

Substituting \(e^{\pm i\phi_1}\) from 22 and its complex conjugate in 32 –33 gives \[\begin{align} &Q_1e^{i(k+\gamma)}+P_1e^{-i(k+\gamma)}=(U e^{2i\gamma}+Ye^{-2ik})e^{-i\phi_2}\;,\tag{34}\\ &Q_2e^{i(k+\gamma)}+P_2e^{-i(k+\gamma)}=(U e^{2i\gamma}+Ye^{-2ik})e^{i\phi_2}\;,\tag{35} \end{align}\] with \[\begin{align} Q_n&=q_nC-r_nA-s_nB^*\;,\\ P_n&=p_nC-r_nB-s_nA^*\;,\\ U&=uC+vA+wB^*\;,\\ Y&=-C+vB+wA^*\;; \end{align}\] eliminating \(e^{i\phi_2}\) between 34 and 35 and using 30 we obtain \[\label{eq:QPUV} e^{i(4\gamma-(\delta\pm\psi))}=\frac{Q_1Q_2e^{i(\delta\pm\psi)}+Q_1P_2+P_1Q_2+P_1P_2e^{-i(\delta\pm\psi)}}{U^2e^{i(\delta\pm\psi)}+2UY +Y^2e^{-i(\delta\pm\psi)}}\;.\tag{36}\] Since \(\gamma\) is defined modulo \(\pi\), equation 36 has four inequivalent solutions \[\label{eq:gamma4} \gamma=\frac{\chi_\pm+\delta\pm\psi}{4}\,,\quad \frac{\chi_\pm+ \delta\pm\psi}{4}+\frac{\pi}{2}\;,\tag{37}\] with \[\begin{align} &\chi_\pm=\arg\bigl(Q_1Q_2e^{i(\delta\pm\psi)}+Q_1P_2+P_1Q_2+P_1P_2e^{-i(\delta\pm\psi)}\bigr)\\&-\arg\bigl(U^2e^{i(\delta\pm\psi)}+2UY +Y^2e^{-i(\delta\pm\psi)}\bigr)\end{align}\]

Note that the ambiguous signs in 37 and 30 must be chosen upper or lower consistently. Thus, as discussed in section 3.2, for each \(\gamma\) for which the bands cross, there are two points of intersection with both flux values shifted by \(\pi\), obtained by choosing \(n=0\) and \(n=1\) (or any other combination of even and odd values of \(n\)) in 30 . The fluxes at the intersections points are then given by 22 and, for example, 32 . There is also a simple relation between the intersection points for \(\gamma\) values that differ by \(\pi/2\): they occur at points where exactly one of the fluxes is shifted by \(\pi\) for \(k\) values that are shifted by \(\pi/2\).

4.1.4 A necessary condition for band intersection↩︎

Recall that equation 21 follows from the diagonal part of the two-by-two matrix equation 20 and yields the relation 30 between \(k\) and \(\gamma\). Analogously, we can derive a second relation between these variables from the diagonal part of the dual matrix equation 19 , \[\label{eq:abcd2} \bar a e^{i(k-\gamma)}+ \bar b e^{-i(k-\gamma)}= \bar c e^{i\phi_2}+ \bar d e^{-i\phi_2}\;,\tag{38}\] with \[\begin{align} \bar a&=S_{13}(S_{34}S_{41}-S_{44}S_{31})+S_{14}(S_{43}S_{31}-S_{33}S_{41})+S_{11}(S_{33}S_{44}-S_{34}S_{43})\\ \bar b&=S_{11}\\ \bar c&=S_{13}S_{41}-S_{11}S_{43}\\ \bar d&=S_{14}S_{31}-S_{11}S_{34}\;. \end{align}\]

Following the steps that lead to 30 we define \(\bar A,\,\bar B,\,\bar C,\,\bar g,\,\bar\delta,\) and \(\bar\psi\) by the analogs of 23 –24 , 29 , and 31 with each variable replaced by its barred counterpart, to obtain \[\label{eq:kpsi2} k=\gamma+\frac{\bar\delta\pm \bar\psi}{2}+n\pi \;.\tag{39}\] Eliminating \(k\) between 30 and 39 gives \[\label{eq:gamma8} \gamma=\frac{\delta-\bar\delta\pm\psi\mp\bar\psi}{4}+\frac{n\pi}{2}\;.\tag{40}\] Since the two sign choices in this expression are independent, it yields eight inequivalent values modulo \(\pi\), compared with four choices in 37 . Thus, although the diagonal elements of 19 and 20 form a system of four real scalar equations for four real variables, some of the discrete solutions are not actual crossings. This system is therefore a necessary but not sufficient condition for band intersections.

The actual crossing values of \(\gamma\) obtained from 40 can be identified by noting (see equations 29 and 31 ) that \(\psi\) and \(\bar\psi\) change sign when \(C\) and \(\bar C\), respectively, change sign. Using this observation, we find the four inequivalent values of \(\gamma\): \[\label{gammaSol} \left\{\frac{\delta-\bar\delta\pm(\psi\mathop{\mathrm{sign}}C-\bar\psi\mathop{\mathrm{sign}}\bar C)}{4}\,,\quad \frac{\delta-\bar\delta\pm(\psi\mathop{\mathrm{sign}}C-\bar\psi\mathop{\mathrm{sign}}\bar C)}{4}+\frac{\pi}{2} \right\}\;,\tag{41}\] which are equal to those of 37 .

4.2 The orientation of the topological transitions↩︎

In the previous subsection we derived a necessary and sufficient condition for the crossing of the energy bands of the quantum graph, that is satisfied for a codimension-one family of scattering matrices \(S\). When \(S\) is varied continuously across the intersection manifold, the Chern number of the crossing bands jumps, so that the band topologies transition from trivial to nontrivial or vice versa. We set the sign of the transition according to the jump in the Chern number of, say, the upper of the two intersecting bands. We next derive an expression for the sign of the transition that allows us to draw the topological phase diagram of the model.

For this purpose, assume that \((\phi_{1,\times},\phi_{2,\times})\) is a degeneracy point of \(\Lambda_\times S_{\times}\) with a double eigenvalue \(e^{-ik_{\times}}\) so that \(\Lambda_{2,\times}\hat{S}_{2,\times}=e^{-ik_{\times}}I\), and write \(e^{ik_{\times}}\Lambda_{2}\hat{S}_{2}=\hat{S}_{2,\times}e^{i(h_0+\vec{\sigma}\cdot\vec{h})}\) in terms of Pauli matrices as in 12 . Since the band-crossing manifold has codimension one, we can simplify the description by explicitly displaying the dependence of \(S\) on a single parameter, denoted here as \(\phi_3\) (corresponding to the phase \(\gamma\) in Section 4). Viewing \(h_0\) and \(\vec{h}\) as functions of \(\phi = (\phi_1, \phi_2, \phi_3)\) and \(k\), the crossing conditions become \(S(\phi_{3,\times}) = S_\times\), or equivalently \(h_0(\phi_\times; k_\times)=\vec{h}(\phi_\times; k_\times)= 0\).

For \(\phi_j\) close to \(\phi_{j,\times}\), two of the solutions of 13 , \(k_\pm\), with opposite choices of \(s\), are close to \(k_\times\), and we can approximate \[\label{eq:k43-} \mathring{k}_\pm\frac{\partial h_0}{\partial k}+\mathring{\phi}\cdot\frac{\partial h_0}{\partial\phi}=\pm{\Bigl|\mathring{k}_\pm\frac{\partial\vec{h}}{\partial k}+\mathring{\phi}\cdot\frac{\partial\vec{h}}{\partial\phi}\Bigr|}\;,\tag{42}\] where \(\mathring{k}_\pm=k_\pm-k_\times\), \(\mathring{\phi}=\phi-\phi_\times\), and the derivatives are evaluated at the crossing point. Letting \(h_{j,\pm}(\phi)=h_j(\phi;k_\pm)\), we now use 15 to calculate the contributions \(C_\text{loc}\) to the Chern numbers from the neighborhood of the crossing point, \[\label{eq:clocpm} C_\text{loc}=\pm\frac{1}{2}\mathop{\mathrm{sign}}\Bigl(\frac{\partial\Vec{h}_\pm}{\partial{\phi_1}}\times\frac{\partial\Vec{h}_\pm}{\partial{\phi_2}}\cdot \vec{h}_\pm\Bigr)\;,\tag{43}\] where the expression is evaluated for any \(\phi\) close to \(\phi_\times\).

To proceed, we calculate \(\vec{h}_\pm\) for \(\mathring{\phi}=\eta\phi_k\), \(\eta\ll1\), where \(\phi_k=(\partial\vec{h}/\partial\phi)^{-1}(\partial\vec{h}/\partial k)\), obtaining \[\label{eq:k43-eta} \mathring{k}_\pm\frac{\partial h_0}{\partial k}+\eta\phi_k\cdot\frac{\partial h_0}{\partial\phi}=\pm|\mathring{k}_\pm+\eta|\Bigl|\frac{\partial\vec{h}}{\partial k}\Bigr|\;.\tag{44}\] Assuming \(|{\partial h_0}/{\partial k}|>|{\partial\vec{h}}/{\partial k}|\), \[\label{eq:h43-eta} \vec{h}_\pm=(\mathring{k}_\pm+\eta)\frac{\partial\vec{h}}{\partial k}\;,\tag{45}\] where, by 44 , \[\mathring{k}_\pm+\eta=\frac{\displaystyle \Bigl(\frac{\partial h_0}{\partial k}-\phi_k\cdot\frac{\partial h_0}{\partial\phi}\Bigr)\eta}{\displaystyle \frac{\partial h_0}{\partial k}\mp \Bigl|\frac{\partial\vec{h}}{\partial k}\Bigr|}\;.\]

Since \[\frac{\partial\vec{h}_\pm}{\partial \phi}=\frac{\partial\vec{h}}{\partial \phi}+\frac{\partial k}{\partial \phi}\frac{\partial\vec{h}}{\partial k}\;,\] 43 , 44 , and 45 give \[C_\text{loc}=\pm\mathop{\mathrm{sign}}\det\Bigl(\frac{\partial\vec{h}}{\partial{(\phi_1,\phi_2,k)}}\Bigr)\mathop{\mathrm{sign}}\Bigl(\frac{\partial h_0}{\partial k}-\phi_k\cdot\frac{\partial h_0}{\partial\phi}\Bigr)\mathop{\mathrm{sign}}(\eta)\;.\] The result is that, when \(\eta\) is swept upwards through the crossing point, the Chern number of the upper (higher \(k\)) band jumps by \[\label{eq:crossign} \mathop{\mathrm{sign}}\det\Bigl(\frac{\partial\vec{h}}{\partial{(\phi_1,\phi_2,k)}}\Bigr)\mathop{\mathrm{sign}}\Bigl(\frac{\partial h_0}{\partial k}-\phi_k\cdot\frac{\partial h_0}{\partial\phi}\Bigr)\;.\tag{46}\]

5 Real scattering matrix↩︎

So far we have studied the topological phases of a quantum graph with a general scattering matrix. The results above can be simplified if the matrix \(S\) in 18 is real. In this case, the phases \(\delta=\bar\delta=0\) and equation (41 ) reduces to \[\label{gammaEQ} \gamma=\pm\frac{1}{2}(\psi+\Bar{\psi}).\tag{47}\] The resulting phase diagram is symmetric under the reflection \(\gamma \longleftrightarrow -\gamma\), see figures 6 and 7. The intersection between branches occurs when \(\psi=-\bar{\psi}\pm\pi/2\) which gives \(\gamma=\pm{\pi}/{4}\). For these values of \(\gamma\) the Chern number is zero independently of the choice of other parameters.

Furthermore, by taking the real and imaginary parts of (16 ) we get \[(a+b)\cos(k+\gamma)=(c+d)\cos\phi_1 \, ,\, (a-b)\sin(k+\gamma)=(c-d)\sin\phi_1.\] From these equations, we have \[\bigg(\frac{a+b}{c+d}\bigg)^2\cos^{2}(k+\gamma)+\bigg(\frac{a-b}{c-d}\bigg)^2\sin^{2}(k+\gamma)=1\] and \[\label{EqEPSI} \tan(k+\gamma)=\mathcal{E}, \qquad \mathcal{E}=\pm\left( { -}\,\frac{ \left(\frac{a+b}{c+d}\right)^2-1}{ \left( \frac{a-b}{c-d}\right)^2-1 }\right)^{1/2}\;;\tag{48}\] a similar set of equations is obtained from (38 ), \[\label{EqEPSI2} \tan (k-\gamma)=\mathcal{\bar E}, \qquad \mathcal{\bar E}=\pm\left( {-}\, \frac{ \left(\frac{\bar a+\bar b}{\bar c+\bar d}\right)^2-1}{ \left( \frac{\bar a-\bar b}{\bar c-\bar d}\right)^2-1 }\right)^{1/2}.\tag{49}\] Eliminating \(k\) between (48 , 49 ), the condition for spectral degeneracies can be cast in the form \[\begin{align} \label{EqGamma} \tan(2\gamma)=\frac{\bar{\mathcal{E}}-\mathcal{E}}{1-\bar{\mathcal{E}}\mathcal{E}}. \end{align}\tag{50}\]

The above result can be simplified further in the cases where either \(\bar{\mathcal{E}}=\mathcal{E}=0\), or \(\bar{\mathcal{E}}=\mathcal{E}=\infty\), so that \(k=0,\pi\) and \(k=\pi/2,3\pi/2\), respectively, and \(\gamma=0\), so that \(S\) is entirely real. In these cases the band degeneracy conditions reduce to \[a+b=\pm(c+d)\,\, ,\,\, a-b=\pm(c-d)\;,\] which leads to \[\begin{align} \label{pd32eq} \frac{\cos(\frac{\alpha-\beta}{2})}{\cos(\frac{\alpha+\beta}{2})}=\pm\frac{\cos{\frac{\epsilon_+}{2}}}{\cos{\frac{\theta_+}{2}}} &, \qquad \frac{\cos(\frac{\alpha-\beta}{2})}{\cos(\frac{\alpha+\beta}{2})}=\pm\frac{\sin{\frac{\epsilon_+}{2}}}{\sin{\frac{\theta_+}{2}}}\\ \frac{\sin(\frac{\alpha-\beta}{2})}{\sin(\frac{\alpha+\beta}{2})}=\pm\frac{\cos{\frac{\epsilon_-}{2}}}{\cos{\frac{\theta_-}{2}}} &, \qquad \frac{\sin(\frac{\alpha-\beta}{2})}{\sin(\frac{\alpha+\beta}{2})}=\pm\frac{\sin{\frac{\epsilon_-}{2}}}{\sin{\frac{\theta_-}{2}}}, \end{align}\tag{51}\] where \(\theta_+=\theta_1+\eta_1+\theta_2+\eta_2\), \(\epsilon_+=-\theta_1-\eta_1+\theta_2+\eta_2\), \(\theta_-=\theta_1-\eta_1+\theta_2-\eta_2\) and \(\epsilon_-=-\theta_1+\eta_1+\theta_2-\eta_2\), and where \(\theta_{1,2},\eta_{1,2}, \alpha,\beta\) are scattering matrix parameters defined in appendix 7. The set of equations above explicitly defines the boudnary lines between different topological phases for a real \(S\). Equations (51 ) and (50 , 47 ) were verified by numerical Chern-number calculations shown in figures 7 (a) and 7 (b).

6 Conclusions↩︎

In this study, we investigated the Chern number of the energy bands of a two-dimensional periodic square lattice quantum graph with a general four-port scatterer at each nodes. This graph is equivalent by a Bloch transformation to a figure-eight graph with two loops pierced by Aharonov-Bohm magnetic fluxes. A direct calculation of Chern numbers is challenging due to the need to find roots of quartic polynomials. To circumvent this problem, we introduced two open quantum graphs with reduced 2-by-2 scattering matrices \(\hat{S}_1\), \(\hat{S}_2\) depending on the magnetic fluxes \(\phi_1\) and \(\phi_2\), respectively.

We showed that there are two possible topological phases: a trivial one with vanishing Chern numbers for all bands, and a nontrivial phase, in which the Chern numbers of the bands alternate between \(\pm1\). Transitions between the two topological phases occur precisely when either \(\hat{S}_{1}\) or \(\hat{S}_{2}\) is proportional to the identity matrix. At the transition point, the Chern number of each band changes by \(\pm 1\), where the sign can be deduced from the derivatives of the matrices \(\hat{S}_{1,2}\) with respect to the parameters \(\phi_1,\phi_2\). This condition enabled us to delineate the boundary in the topological phase diagram between trivial and nontrivial phases, which is a codimension-one submanifold in the space of unitary scattering matrices.

Our analytical results were compared with numerical calculations of the Chern numbers of the bands, which, unlike the naturally ordered real-valued bands of Hamiltonians, are valued in the unit circle in the complex plane and lack an inherent order. We addressed this issue by developing a method to establish and maintain a consistent ordering of the eigenvalues of the scattering matrix throughout our calculations.

It is worth emphasizing that in our model the Chern numbers are restricted to the three values \(\pm 1,0\). This is in sharp contrast with the quantum-graph lattices in the presence of a magnetic field investigated in [37], where the range of possible Chern numbers depends on the applied magnetic field. This example suggests that, in general, increasing the complexity of the unit cell of the quantum graph lattice expands the range of possible Chern numbers.

Finally, although our methodology provides a robust framework for understanding topological phases in general lattice quantum graphs, the analysis in this paper was restricted to the case of equal edge lengths. This assumption generally implies a periodic Chern number structure in the energy bands. Such periodicity is broken in rectangular quantum graph lattices with incommensurate edge lengths in the two directions. The investigation of the resulting Chern number statistics for an infinite number of bands is left for future research.

Acknowledgments↩︎

This work was funded by the German–Israeli Foundation within the project Statistical Topology of Complex Quantum Systems, grant number GIF I-1499-303.7/2019

References↩︎

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1. The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel 919040↩︎

2. School of Mathematical Sciences, Holon Institute of Technology, Holon 5810201, Israel↩︎