Quantum wire networks with local ${\mathbb{Z}}_2$ symmetry

For a large class of networks made of connected loops, in the presence of an external magnetic field of half flux quantum per loop, we show the existence of a large local symmetry group, generated by simultaneous flips of the electronic current in all the loops adjacent to a given node. Using an ultralocalized single particle basis adapted to this local ${\mathbb{Z}}_2$ symmetry, we show that it is preserved by a large class of interaction potentials. As a main physical consequence, the only allowed tunneling processes in such networks are induced by electron-electron interactions and involve a simultaneous hop of two electrons. Using a mean-field picture and then a more systematic renormalization-group treatment, we show that these pair-hopping processes do not generate a superconducting instability, but they destroy the Luttinger liquid behavior in the links, giving rise at low energy to a strongly correlated spin-density-wave state.

Figure 1

This picture presents two examples of ${\mathbb{Z}}_2$ networks: a chain of loops (upper part), and a square lattice of loops (lower part). All these networks are obtained from any planar lattice after replacing each link by a loop. To get a local ${\mathbb{Z}}_2$ symmetry, each loop is required to be perfectly invariant under the reflection through the long axis defined by its two adjacent nodes.

Figure 2

Different realizations of a local ${\mathbb{Z}}_2$ symmetry shown successively for a single loop (upper part), and then around sites present in a 1D geometry (chain of loops, lower left) or a two-dimensional (2D) geometry (square lattice of nodes connected by loops, lower right). Plus or minus signs keep track of the relative sign between initial and transformed wave functions $\psi$ and $T\psi \equiv {\psi }^{\prime }$.

Figure 3

Illustration of the requirements on the scattering matrix at a node present in a chain of loops (upper part), in order to impose a local ${\mathbb{Z}}_2$ symmetry. The most direct design (a) is likely to be incompatible with these constraints, since intuitively, the amplitude for an electron to go from A to B is different from the amplitude to go from A to C. In a slightly modified design (b), these amplitudes are more likely to be equal, and this is more appropriate for an experimental implementation of the local ${\mathbb{Z}}_2$ symmetry.

Figure 4

In the case of Hubbard-like interacting electrons, the only possible cage transitions preserving local ${\mathbb{Z}}_2$ symmetry can be plotted on the graph (thin lines), where $(n,m)$ and $(n^{\prime },m^{\prime })$ indicate initial and final cage positions, corresponding to a term $c_{n^{\prime }}^{†}{c}_{m^{\prime }}^{†}{c}_m{c}_n$. Dashed lines refer to possible nonzero amplitude transitions, compatible with the local nature of a two-electron interaction, but that are forbidden since they violate electron number-parity conservation resulting from the local ${\mathbb{Z}}_2$ symmetry.

Figure 5

(Color online) Schematic representation of the four interactions compatible with the local ${\mathbb{Z}}_2$ symmetry, given in Eq. (23). Each dashed ellipse represents a cage. An empty (full) circle represents a destruction (creation) operator. Coupling $A$ is an intercage interaction, coupling $B$ a cage exchange, coupling $C$ a pair-hopping, and coupling $D$ an intracage interaction.

Figure 6

Graphs renormalizing the pair-hopping term $C_{\gamma }$. The left and middle graphs are particle-hole bubbles, and the right one is a particle-particle bubble. For the latter, $\beta$ is summed over, and when $\beta =0$ or $\gamma$, the quantity $C_0$ that appears should be understood as $D$, so as to yield Eq. (29).

Figure 7

(Color online) The numerical results of $\ln \mid C_{\gamma }^{\mathrm{c}}(l=1)\mid$ as a function of $\gamma$, for $N=128$ cages and initial interaction $U=0.1$, are perfectly fitted by a straight line.

Figure 8

(Color online) Evolution of the strength of the interaction $C_{\gamma }^{\mathrm{c}}\left(l\right)$ as a function of the RG scale $l$, for $N=128$ cages and initial interaction $U=0.1$. The flow stops at the critical scale $l_{\mathrm{c}}$ denoted by a dashed line.

Figure 9

(Color online) Evolution of the range of the interaction $C_{\gamma }^{\mathrm{c}}\left(l\right)$ as a function of the RG scale $l$, for $N=128$ cages and interaction strength $U=0.1$. The flow stops at the critical scale $l_{\mathrm{c}}$ denoted by a dashed line.