Topological interfaces of Luttinger liquids

Topological interfaces of two-dimensional conformal field theories contain information about symmetries of the theory and exhibit striking spectral and entanglement characteristics. While lattice realizations of these interfaces have been proposed for unitary minimal models, the same has remained elusive for the paradigmatic Luttinger liquid, i.e., the free, compact boson model. Here, we show that a topological interface of two Luttinger liquids can be realized by coupling special one-dimensional superconductors. The gapless excitations in the latter carry charges that are specific integer multiples of the charge of Cooper pairs. The aforementioned integers are determined by the windings in the target space of the bosonic fields — a crucial element required to give rise to nontrivial topological interfaces. The latter occur due to the perfect transmission of certain number of Cooper pairs across the interface. The topological interfaces arise naturally in Josephson junction arrays with the simplest case being realized by an array of experimentally demonstrated $0-\pi$ qubits, capacitors and ordinary Josephson junctions. Signatures of the topological interface are obtained through entanglement entropy computations. In particular, the subleading contribution to the so-called interface entropy is shown to differ from existing field theory predictions. The proposed lattice model provides an experimentally realizable alternative to spin and anyon chains for the analysis of several conjectured conformal fixed points which have so far eluded ab initio investigation.

Figure 1

(a) The bosonic fields ${\phi }_l,{\phi }_r$ are glued at the interface located at $x=0$. The Luttinger parameters (winding numbers in target space) on either side of the interface are denoted by $K_l,K_r$ ($m_l,m_r$). The topological interface occurs for $m_l^2{K}_r=m_r^2{K}_l$. (b) One-dimensional Josephson-junction array that realizes the interface of (a) in the scaling limit. The superconducting granules (gray circles), with charging energy to ground plane $E_c$, are separated by generalized Josephson junctions. The latter allow tunneling of only $m_{l\left(r\right)}$ Cooper pairs with charge $2m_{l\left(r\right)}e$ on the left (right) of the interface with rate $E_{J_{l\left(r\right)}}$. (c) Schematics of an ordinary Josephson junction and a $0-\pi$ qubit [23, 24, 25, 26] required to realize the simplest nontrivial topological interface with $m_l=1,m_r=2$.

Figure 2

DMRG results in the folded configuration with identical boundary condition on both ends. The length of the folded array, $L=40$ and the winding numbers are chosen as: $m_l=1,m_r=2$. (a) EE $\left(S_{\mathrm{s}}\right)$ as a function of subsystem size for varying boundary conditions. Here, $N_l{N}_r$ and $D_l{D}_r$ and T correspond to Neumann (free), Dirichlet (fixed), and topological boundary conditions on both ends of the folded array. In particular, $N_l{N}_r$ ($D_l{D}_r$) correspond to free (fixed) boundary condition on both ${\phi }_l$ and ${\phi }_r$ with boundary g-function $g_b=g_{N_l}{g}_{N_r} \left(g_{D_l}{g}_{D_r}\right)$. The parameters $E_{J_l},E_{J_r}$ are chosen such that the corresponding Luttinger parameters on the two halves of the array are $K_r\approx 4K_l$. To emphasize the role of winding numbers, $S_{\mathrm{s}}$ is also computed for another case: $m_l=m_r^{\prime }=1$ with $K_r^{\prime }\approx 4K_l$. The corresponding free $\left(N_l{N}_r^{\prime }\right)$, fixed $\left(D_l{D}_r^{\prime }\right)$ and non-T (nontopological) boundary conditions are shown. The EE for the $N_l{N}_r^{\prime }\left(D_l{D}_r^{\prime }\right)$ case coincides with that for $N_l{N}_r\left(D_l{D}_r\right)$ [Eq. (4)]. However, the EE is not the same for the non-T and T boundary conditions, reflected in the difference between the orange and violet curves. This is due to a difference in the corresponding g functions. (b) Schematic of the configuration analyzed. (c) The Luttinger parameters $K_l,K_r$ and $K_r^{\prime }$ are obtained using infinite DMRG computations of correlation functions: $C_{\alpha }\left(d\right)=\langle {\mathrm{e}}^{\mathrm{i}\alpha {\varphi }_j}{\mathrm{e}}^{-\mathrm{i}\alpha {\varphi }_{j+d}}\rangle , \alpha =1,2$.

Figure 3

DMRG results for the ground-state EE for different bipartitionings of the array in Fig. 1 with open boundary conditions at the ends. The parameters $K_l,K_r$ and $K_r^{\prime }$ are chosen as in Fig. 2. (a) The blue (cyan) circles (squares) correspond to the EE for the ground state if only the left (right) free boson theory occupied the entire array. The violet (orange) curve shows the case of the topological (non-topological) interface, denoted by T (non-T). For the topological (nontopological) interface, the Luttinger parameters and winding numbers for the bosonic field on the right are chosen to be $K_r\approx 4K_l$($K_r^{\prime }\approx 4K_l$) and $m_r=2(m_r^{\prime }=1)$, respectively. Only in the case of $m_r=2$ is the topological interface realized. This is manifest in the EE seamlessly turning in to that for the bosonic field on the right as well as in the correlation functions $C_2\left(d\right)=\langle {\mathrm{e}}^{2\mathrm{i}{\varphi }_j}{\mathrm{e}}^{-2\mathrm{i}{\varphi }_{j+d}}\rangle$ (inset). The latter exhibit a continuous (abruptly changing) behavior across the interface for the topological (non-topological) case. (b) The topological (nontopological) nature of the interface for the violet (orange) curves of (a) is verified by the variation of the interface EE with the logarithm of the subsystem-size (expected values in the parenthesis). (c) EE as a function of subsystem size for the interface between bosonic fields with winding numbers $m_l=2,m_r=3$. The parameters $E_{J_l},E_{J_r}$ are chosen such that $K_l\approx 0.086,K_r\approx 0.196\approx 9K_l/4$. In both (a) and (c), the change in the EE at the interface $\approx ln(m_r/m_l)$, which is different from earlier predictions [45].