We construct a generalized transfer matrix corresponding to noninteracting tight-binding lattice models, which can subsequently be used to compute the bulk bands as well as the edge states. Crucially, our formalism works even in cases where the hopping matrix is non-invertible. Following Hatsugai [PRL 71, 3697 (1993)], we explicitly construct the energy Riemann surfaces associated with the band structure for a specific class of systems which includes systems like Chern insulator, Dirac semimetal and graphene. The edge states can then be interpreted as non-contractible loops, with the winding number equal to the bulk Chern number. For these systems, the transfer matrix is symplectic, and hence we also describe the windings associated with the edge states on Sp(2, R) and interpret the corresponding winding number as a Maslov index.

We theoretically study topological phase transitions in four generalized versions of the Kane-MeleHubbard model with up to 2 × 18 2 sites. All models are free of the fermion-sign problem allowing numerically exact quantum Monte Carlo (QMC) calculations to be performed to extremely low temperatures. We numerically compute the Z2 invariant and spin Chern number Cσ directly from the zero-frequency single-particle Green's functions, and study the topological phase transitions driven by the tight-binding parameters at different on-site interaction strengths. The Z2 invariant and spin Chern number, which are complementary to each another, characterize the topological phases and identify the critical points of topological phase transitions. Although the numerically determined phase boundaries are nearly identical for different system sizes, we find strong system-size dependence of the spin Chern number, where quantized values are only expected upon approaching the thermodynamic limit. For the Hubbard models we considered, the QMC results show that correlation effects lead to shifts in the phase boundaries relative to those in the non-interacting limit, without any spontaneously symmetry breaking. The interaction-induced shift is non-perturbative in the interactions and cannot be captured within a "simple" self-consistent calculation either, such as Hartree-Fock. Furthermore, our QMC calculations suggest that quantum fluctuations from interactions stabilize topological phases in systems where the one-body terms preserve the D3 symmetry of the lattice, and destabilize topological phases when the one-body terms break the D3 symmetry.

The proximity effect is a central feature of superconducting junctions that plays a key role in many devices and can be exploited in the design of new systems with quantum functionality [1][2][3][4][5][6][7][8][9][10][11][12] . Recently, exotic proximity effects have been observed in various systems, including superconductor-metallic nanowires 5-7 and graphenesuperconductor structures 4 . However, it is still not clear how superconducting order propagates spatially in a heterogeneous superconductor system. Here we report on intriguing junction geometry effects in a heterogeneous system consisting of electronically two-dimensional superconducting islands on a metallic substrate. Depending on the local geometry, the superconducting gap induced at the metallic surface sometimes decays within ∼20 nm of the superconductor, and sometimes survives at distances that are several coherence lengths from a superconductor. We show in particular that the curvature of the junction plays an essential role in the proximity effect.The sample system comprises superconducting two-dimensional (2D) Pb islands on top of a single-atomic-layer surface metal, the striped incommensurate (SIC) phase of the Pb overlayer on Si(111) (refs 13-16). The scanning tunnelling microscopy (STM) image shown in Fig. 1 captures a variety of junction configurations. Figure 1a shows an interesting 'π'-shaped Pb island five monolayers (ML) thick on top of the SIC surface. Previous scanning tunnelling spectroscopy (STS) studies have shown that the SIC phase is superconducting with T C_SIC = ∼1.8 K (ref. 17) and the 2D Pb islands have a T C around 6 K (ref. 18), although the actual T C value also depends on the lateral size as well as its thickness 19 . At 4.3 K, the SIC template is in the normal state. At locations far from the Pb islands, the tunnelling spectrum exhibits no gap (spectrum no. 2 in Fig. 1b), whereas the spectrum acquired at the 2D Pb island shows a clear superconducting gap (spectrum no. 3). In the SIC region near the 2D Pb island, a superconducting gap can also be observed (spectrum no. 1), indicative of a proximity effect. To address the spatial dependence, we performed spectroscopic mapping over the same area, whose differential conductance at zero bias (zero-bias conductance (ZBC)) is shown in Fig. 1c. As the ZBC directly correlates with the size of the tunnelling gap (the smaller the value of ZBC, the larger the tunnelling gap), the landscape of ZBC is a good representation of the landscape of the superconducting gap.The ZBC image reveals a rich landscape. In some regions (for example, region α), the induced superconducting gap decays very quickly within a very short distance from the SIC/superconductor (S) interface, whereas in region β where the SIC wetting layer is surrounded by Pb islands from both sides, the induced Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA. *e-mail: shih@physics.utexas.edu.superconducting gap is quite uniform. Similarly, in region γ where the SIC region is near the 'recess'...

We study the quantum entanglement of the spin and orbital degrees of freedom in the onedimensional Kugel-Khomskii model, which includes both gapless and gapped phases, using analytical techniques and exact diagonalization with up to 16 sites. We compute the entanglement entropy, and the entanglement spectra using a variety of partitions or "cuts" of the Hilbert space, including two distinct real-space cuts and a momentum-space cut. Our results show the Kugel-Khomski model possesses a number of new features not previously encountered in studies of the entanglement spectra. Notably, we find robust gaps in the entanglement spectra for both gapped and gapless phases with the orbital partition, and show these are not connected to each other. We observe the counting of the low-lying entanglement eigenvalues shows that the "virtual edge" picture which equates the low-energy Hamiltonian of a virtual edge, here one gapless leg of a two-leg ladder, to the "low-energy" entanglement Hamiltonian breaks down for this model, even though the equivalence has been shown to hold for similar cut in a large class of closely related models. In addition, we show that a momentum space cut in the gapless phase leads to qualitative differences in the entanglement spectrum when compared with the same cut in the gapless spin-1/2 Heisenberg spin chain. We emphasize the new information content in the entanglement spectra compared to the entanglement entropy, and using quantum entanglement present a refined phase diagram of the model. Using analytical arguments, exploiting various symmetries of the model, and applying arguments of adiabatic continuity from two exactly solvable points of the model, we are also able to prove several results regarding the structure of the low-lying entanglement eigenvalues.

We study an exactly solvable quantum spin model of Kitaev type on the kagome lattice. We find a rich phase diagram which includes a topological (gapped) chiral spin liquid with gapless chiral edge states, and a gapless chiral spin liquid phase with a spin Fermi surface. The ground state of the current model contains an odd number of electrons per unit cell which qualitatively distinguishes it from previously studied exactly solvable models with a spin Fermi surface. Moreover, we show that the spin Fermi surface is stable against weak perturbations.