Quantum anholonomy with topology change

We study a family of closed quantum graphs described by one singular vertex of order n=4. By suitable choice of the parameters specifying the singular vertex, we can construct a closed sequence of paths in the parameter space that physically corresponds to the smooth interpolation of different topologies - a ring, separate two lines, separate two rings, two rings with a contact point. We find that the spectrum of a quantum particle on this family of graphs shows the quantum anholonomy.


INTRODUCTION
The idea of quantum mechanics on graphs has been introduced in the last century. During the last 30 years, it has been developed into a powerful tool for the study of particles on tiny graph-like structures [5]. Since a graph is a one-dimensional variety, the analysis of a particle on a graph consists in solving a system of ordinary differential equations, which makes the graph models simple from the mathematical point of view. Therefore, quantum mechanics on graphs can serve also as a useful laboratory for the study of various complex quantum phenomena in a simple setting.
Here we examine the influence of the topology of quantum graphs on spectral properties, taking inspiration from the works [1] and [7]. We may regard that the topology change mimics a macroscopic or violent variation of systems, e.g., the rupture and connection of quantum wires. We will show that a parametric evolution of an eigenenergy may form an open trajectory along a cycle involving the topology changes of graphs. If this is the case, the trajectory of the corresponding eigenspace also forms an open cycle. Such a discrepancy of eigenobjects is called quantum anholonomy [2,4,9]. Let us consider a singular node with four outgoing onedimensional lines of finite lengths. Let the first two lines be sticked together at their outer endpoints to form a ring of length L 1 , and likewise, the second two lines be sticked together at their outer endpoints to form a ring of length L 2 , giving the whole object the shape of the character "∞" (Fig. 1). We assume there is no potential on the lines outside the central vertex. Therefore, a quantum particle confined to this object moves freely on the lines, and the only non-trivial part of the system is the connection condition of the wave function at the node.
We assign the coordinates x 1 ∈ [0, L 1 ] and x 2 ∈ [0, L 2 ] to the left and the right ring. The values x 1 = 0 and x 2 = 0 correspond to the inner endpoints 1 and 3, respectively, whereas the values L 1 and L 2 correspond to the inner endpoints 2 and 4, see Fig. 1. The behavior of a particle in the central vertex of degree 4 is determined by a boundary condition in the vertex. The general form of a boundary condition in a vertex of degree n follows from the study of self-adjoint extensions of the Laplacian operator on a graph. The boundary condition consists of n equations connecting the boundary values and derivatives that can be written in the form [6] AΨ + BΨ = 0, where Ψ, Ψ are vectors formed from the values of the wave function in the vertex and from its derivatives (taken in the outgoing sense), respectively, and A and B are n × n complex matrices with the property with (A|B) denoting the n × 2n matrix composed from the columns of the matrices A and B. For the graph depicted in Figure 1, we have In our model, we focus on a specific subset of all possible where t and s are positive constants, and a(θ), b(θ), b (θ), c(θ) and d(θ) are real periodic functions of θ with period 2π that vary within the range between 0 and 1. In addition, it can be easily shown that the functions must satisfy for all θ ∈ [0, 2π] in order that the matrices A, B involved in the connection condition (4) obey (2).

Long Cycle
At first we consider the boundary conditions (4) with functions a(θ), b(θ), b (θ), c(θ) and d(θ) chosen as The graphs of these functions for θ ∈ [0, 2π] are plotted in Fig. 2. The interval [0, 2π] can be divided into six regions, in each of them the vector (a(θ), b(θ), b (θ), c(θ), d(θ)) has a different structure of zero and nonzero entries. As we show below, these regions correspond to mutually different topologies of the system. It extends the basic idea of parametric systems with a varying topology introduced in works of Balachandran et. al. [1] and also in [7]. Our use of a single n = 4 singular vertex makes the treatment more systematic and unified.  (7) for θ ∈ [0, 2π].

FIG. 2. Graphs of functions defined by
where ta(θ) takes the value t at θ = 0 and it continuously decreases down to 0 at θ = π 3 . With regard to the structure of (11), the endpoint 1 is joined to the endpoint 2 and the endpoint 3 is joined to the edpoint 4. On the other hand, the endpoints 1 and 2 are disjoint, so are 3 and 4. Therefore, the system is topologically equivalent to a single ring, see Fig. 3.
where we start with the value c( π 3 ) = 0 and continuously increase it up to c( 2π 3 ) = 1. Since all four inner endpoints are disjoint, the system is topologically equivalent to two separate lines.
where tb(θ) continuously increases from tb( 2π 3 ) = 0 up to tb(π) = t, while tb (θ) takes the value tb (θ) = 0 at θ = 2π 3 , continuously increases up to t at θ = 5π 6 and comes down back to 0 at θ = π. The connection condition (13) implies disjoints endpoints 1 and 3, as well as 2 and 4, making the system topologically equivalent to two separate rings. Region where tb(θ) continuously decreases from tb( π 2 ) = t to tb( 4π 3 ) = 0, while sd(θ) continuously increases from sd( π 2 ) = 0 up to sd( 4π 3 ) = s. The system is topologically equivalent to a ring and a line connected at a single point since three of four endpoints, namely 1, 2 and 4, are connected in the node. Note that at the juncture of regions III and IV , the system is topologically equivalent to a separated line and a ring.
where c continuously decreases from c( 4π 3 ) = 1 down to c( 5π 3 ) = 0. The system is topologically equivalent to a single line, since only endpoints 1 and 4 are connected, whereas 2 and 3 are disjoint.
where ta(θ) continuously increases from ta( 5π 3 ) up to ta(2π) = t, while sd(θ) continuously decreases from sd( 5π 3 ) = s down to sd(2π) = 0. In this region, the system is topologically equivalent to two rings connected at a single point since all four endpoints are connected together at the central node. Therefore, with the parameter change θ = 0 → 2π, seven distinct topologies are traversed in a continuous manner. The situation is illustrated in Fig. 3. The system is never identical for any two different values of θ in [0, 2π), because the the functions a(θ) to d(θ) form district combinations of patterns in all six regions.

Short Cycle
As an additional model, we consider a simplified parametric cycle represented by the boundary condition (4) with The graphs of these functions are plotted in Fig. 4. This system traverses four distinct segments in the parameter space of θ, as θ runs from 0 to 2π. The segments correspond to four topologies, cf. Fig. 5. Namely, a ring, two disjoint lines, a line attached to a ring, and two rings connected at a point.

QUANTUM ANHOLONOMY
The system wave functions can be written in the form ψ 1 (x 1 ) = α 1 sin kx 1 + β 1 cos kx 1 , ψ 2 (x 2 ) = α 2 sin kx 2 + β 2 cos kx 2 . (18) In this notation, we have The spectra of the system can be calculated from the secular equation It turns out that certain choices of s, t and ratio L 1 /L 2 result in a quantum anholonomy. Below we present the results, separately for the long and the short cycle.
We explain the numerical result shown in Figure 6. The spectra have period 2π as a function of θ. This is because all the Hamiltonians are 2π periodic by construction. On the other hand, however, some wavenumber do not have the 2π periodicity. For example, let us keep track of the wavenumber (say k 0 ) of the ground state at θ = 0 in the uppermost figure. Around π/3 < θ < 2π/3, k 0 increase suddenly and form three crossings with another eigenstates. Because the graphs consist of two disjoint parts, these crossings are not avoided. The eigenfunction corresponding to k 0 is localized within a disjoint fragment, and the other three eigenfunctions involving the crossings are localized in the other part. Hence, as a result of a 2π-cycle of θ, k 0 becomes the wavenumber of the third excited state. This is an example of anholonomy in eigenvalue and eigenspace. Some other wavenumbers, e.g., 4-th excited state at θ = 0 in the leftmost figure, are not involved by the anholonomy. By changing the ratio L 1 /L 2 , we see various patterns of the quantum anholonomy. Also, the anholonomies appear in various choice of the parameters, cf. Figure 7.

Short Cycle
Let us proceed to the simplified cycle represented by (17), in which four distinct topologies are connected in a single sequence. The spectra as functions of the angle parameter θ ∈ [0, 2π) have been calculated for the choice of parameters s = 0.5, t = 0.5 and L 1 /L 2 attaining the value of the golden mean, the silver mean and the bronze mean. The results are depicted in Fig. 8. 9. Finite approximation of the physical meaning of the boundary condition (4) for θ belonging to the sectors I and II. The strengths of the δ potentials in the vertices 1,2,3,4 are given by the following formulas. Left: Finite approximation of the physical meaning of the boundary condition (4) for θ belonging to the sectors III and IV. The strengths of the δ potentials in the vertices 1,2,3,4 are given by the following formulas. Left: v1 = (tb) 2 −tb , v2 = 1−tb , v3 = (tb ) 2 −tb , v4 = 1−tb . Right: 11. Finite approximation of the physical meaning of the boundary condition (4) for θ belonging to the sectors V and VI. The strengths of the δ potentials in the vertices 1,2,3,4 are given by the following formulas.
The boundary condition (1) represents a certain singular potential in the vertex that has in general a highly nontrivial nature. As it has been shown in [3,8], the physical content of a generic boundary condition corresponds to a set of infinitely strong δ potentials sited in points in an infinitely small web located in the vertex. We sketch how the webs look like for the boundary conditions of our model in Figures 9, 10 and 11. In all of the finite approximations, the parameter has to be chosen small with respect to the wavelength of the particle. For → 0, the approximation effectively produces the boundary conditions (4). The result has been obtained using the formulas from [3,8]. It illustrates in a simple way the relation between the global topology of the system and the topology of the internal infinitely small web.

DISCUSSION
The creation of the anholonomy in our model can be explained in the following way. In regions I, IV, V and VI, when the graph consists of a single component, the eigenvalues as functions of the system parameter θ can change their values, but they never cross each other, because generically there will be avoided crossings. However, in regions II and III, the graph is separated into two components (two lines or two rings), i.e., the system is in fact made up of two independent entities, both having its own energy levels that can independently vary with change of the system parameter θ. Therefore, if the parametrization of boundary conditions is conveniently chosen, the eigenvalues as functions of θ can cross each other, which leads to the anholonomy effect.
We thank Professor P. Exner and Professor I. Tsutsui for stimulating discussions. This research was supported by the Japan Ministry of Education, Culture, Sports, Science and Technology under the Grant numbers 24540412 and 22540396.