C. L. Kane’s research while affiliated with University of Pennsylvania and other places

A Fermi gas of noninteracting electrons, or ultracold fermionic atoms, has a quantum ground state defined by a region of occupancy in momentum space known as the Fermi sea. The Euler characteristic χF of the Fermi sea serves to topologically classify these gapless fermionic states. The topology of a D-dimensional Fermi sea is physically encoded in the D+1 point equal time density correlation function. In this work, we first present a simple proof of this fact by showing that the evaluation of the correlation function can be formulated in terms of a triangulation of the Fermi sea with a collection of points, links, and triangles and their higher dimensional analogs. We then make use of the topological D+1 point density correlation to reveal universal structures of the more general M point density correlation functions in a D-dimensional Fermi gas. Two experimental methods are proposed for observing these correlations in D=2. In cold atomic gases imaged by quantum gas microscopy, our analysis supports the feasibility of measuring the third-order density correlation, from which χF can be reliably extracted in systems with as few as around 100 atoms. For solid-state electron gases, we propose measuring correlations in the speckle pattern of intensity fluctuations in nonlinear x-ray scattering experiments.

We develop the theory of an Andreev junction, which provides a method to probe the intrinsic topology of the Fermi sea of a two-dimensional electron gas (2DEG). An Andreev junction is a Josephson π junction proximitizing a ballistic 2DEG, and exhibits low-energy Andreev bound states that propagate along the junction. It has been shown that measuring the nonlocal Landauer conductance due to these Andreev modes in a narrow linear junction leads to a topological Andreev rectification (TAR) effect characterized by a quantized conductance that is sensitive to the Euler characteristic χF of the 2DEG Fermi sea. Here we expand on that analysis and consider more realistic device geometries that go beyond the narrow linear junction and fully adiabatic limits considered earlier. Wider junctions exhibit additional Andreev modes that contribute to the transport and degrade the quantization of the conductance. Nonetheless, we show that an appropriately defined rectified conductance remains robustly quantized provided large momentum scattering is suppressed. We verify and demonstrate these predictions by performing extensive numerical simulations of realistic device geometries. We introduce a simple model system that demonstrates the robustness of the rectified conductance for wide linear junctions as well as point contacts, even when the nonlocal conductance is not quantized. Motivated by recent experimental advances, we model devices in specific materials, including InAs quantum wells, as well as monolayer and bilayer graphene. These studies indicate that for sufficiently ballistic samples observation of the TAR effect should be within experimental reach.

We show that the topology of the Fermi sea of a two-dimensional electron gas (2DEG) is reflected in the ballistic Landauer transport along a long and narrow Josephson π junction that proximitizes the 2DEG. The low-energy Andreev states bound to the junction are shown to exhibit a dispersion that is sensitive to the Euler characteristic of the Fermi sea (χF). We highlight two important relations: one connects the electron or hole nature of Andreev states to the convex or concave nature of Fermi surface critical points, and one relates these critical points to χF. We then argue that the transport of Andreev states leads to a quantized conductance that probes χF. An experiment is proposed to measure this effect, from which we predict an I−V characteristic that not only captures the topology of the Fermi sea in metals, but also resembles the rectification effect in diodes. Finally, we evaluate the feasibility of measuring this quantized response in graphene, InAs and HgTe 2DEGs.

The ac conduction of epitaxially grown SmB6 thin films and superconducting heterostructures of SmB6/YB6 are investigated via time-domain terahertz spectroscopy. A two-channel model of thickness-dependent bulk states and thickness-independent surface states accurately describes the measured conductance of bare SmB6 thin films, demonstrating the presence of surface states in SmB6. While the observed reductions in the simultaneously measured superconducting gap, transition temperature, and superfluid density of SmB6/YB6 heterostructures relative to bare YB6 indicate the penetration of proximity-induced superconductivity into the SmB6 overlayer; the corresponding SmB6-thickness independence between different heterostructures indicates that the induced superconductivity is predominantly confined to the interface surface state of the SmB6. This study demonstrates the ability of terahertz spectroscopy to probe proximity-induced superconductivity at an interface buried within a heterostructure, and our results show that SmB6 behaves as a predominantly insulating bulk surrounded by conducting surface states in both the normal and induced-superconducting states in both terahertz and dc responses, which is consistent with the topological Kondo insulator picture.

We develop the theory of an Andreev junction, which provides a method to probe the intrinsic topology of the Fermi sea of a two-dimensional electron gas (2DEG). An Andreev junction is a Josephson $\pi$ junction proximitizing a ballistic 2DEG, and exhibits low-energy Andreev bound states that propagate $\textit{along}$ the junction. It has been shown that measuring the nonlocal Landauer conductance due to these Andreev modes in a narrow linear junction leads to a topological Andreev rectification (TAR) effect characterized by a quantized conductance that is sensitive to the Euler characteristic $\chi_F$ of the 2DEG Fermi sea. Here we expand on that analysis and consider more realistic device geometries that go beyond the narrow linear junction and fully adiabatic limits considered earlier. Wider junctions exhibit additional Andreev modes that contribute to the transport and degrade the quantization of the conductance. Nonetheless, we show that an appropriately defined $\textit{rectified conductance}$ remains robustly quantized provided large momentum scattering is suppressed. We verify and demonstrate these predictions by performing extensive numerical simulations of realistic device geometries. We introduce a simple model system that demonstrates the robustness of the rectified conductance for wide linear junctions as well as point contacts, even when the nonlocal conductance is not quantized. Motivated by recent experimental advances, we model devices in specific materials, including InAs quantum wells, as well as monolayer and bilayer graphene. These studies indicate that for sufficiently ballistic samples observation of the TAR effect should be within experimental reach.

We show that the topology of the Fermi sea of a two-dimensional electron gas (2DEG) is reflected in the ballistic Landauer transport along a long and narrow Josephson $\pi$-junction that proximitizes the 2DEG. The low-energy Andreev states bound to the junction are shown to exhibit a dispersion that is sensitive to the Euler characteristic of the Fermi sea ($\chi_F$). We highlight two important relations: one connects the electron/hole nature of Andreev states to the convex/concave nature of Fermi surface critical points, and one relates these critical points to $\chi_F$. We then argue that the transport of Andreev states leads to a quantized conductance that probes $\chi_F$. An experiment is proposed to measure this effect, from which we predict an I-V characteristic that not only captures the topology of Fermi sea in metals, but also resembles the rectification effect in diodes. Finally, we evaluate the feasibility of measuring this quantized response in graphene, InAs and HgTe 2DEGs.

We show that the topology of the Fermi sea of a D-dimensional Fermi gas is reflected in the multipartite entanglement characterizing D+1 regions that meet at a point. For odd D we introduce the multipartite mutual information and show that it exhibits a logDL divergence as a function of system size L with a universal coefficient that is proportional to the Euler characteristic χF of the Fermi sea. This provides a generalization, for a Fermi gas, of the well-known result for D=1 that expresses the logL divergence of the bipartite entanglement entropy in terms of the central charge c characterizing a conformal field theory. For even D we introduce a charge-weighted entanglement entropy that is manifestly odd under a particle-hole transformation. We show that the corresponding charge-weighted mutual information exhibits a similar logDL divergence proportional to χF. Our analysis relates the universal behavior of the multipartite mutual information in the absence of interactions to the D+1 order equal-time density correlation function, which we show exhibits a universal behavior in the long wavelength limit proportional to χF. Our analytic results are based on the replica method. In addition, we perform a numerical study of the charge-weighted mutual information for D=2 that confirms several aspects of the analytic theory. Finally, we consider the effect of interactions perturbatively within the replica theory. We show that for D=3 the log3L divergence of the topological mutual information is not perturbed by weak short-ranged interactions, though for D=2 the charge-weighted mutual information is perturbed. Thus, for D=3 the multipartite mutual information provides a robust classification that distinguishes distinct topological Fermi liquid phases.

We show that the topology of the Fermi sea of a D-dimensional Fermi gas is reflected in the multipartite entanglement characterizing D+1 regions that meet at a point. For odd D we introduce the multipartite mutual information, and show that it exhibits a $\log^D L$ divergence as a function of system size L with a universal coefficient that is proportional to the Euler characteristic $\chi_F$ of the Fermi sea. This generalizes the well known result for D=1 that expresses the $\log L$ divergence of the bipartite entanglement entropy in terms of the central charge c characterizing a conformal field theory. For even D we introduce a charge-weighted entanglement entropy that is manifestly odd under a particle-hole transformation. We show that the corresponding charge-weighted mutual information exhibits a similar $\log^D L$ divergence proportional to $\chi_F$. Our analysis relates the universal behavior of the multipartite mutual information in the absence of interactions to the D+1'th order equal-time density correlation function, which we show exhibits a universal behavior in the long wavelength limit proportional to $\chi_F$. Our analytic results are based on the replica method. In addition we perform a numerical study of the charge-weighted mutual information for D=2 that confirms several aspects of the analytic theory. Finally, we consider the effect of interactions perturbatively within the replica theory. We show that for D=3 the $\log^3 L$ divergence of the topological mutual information is not perturbed by weak short-ranged interactions, though for D=2 the charge-weighted mutual information is perturbed. Thus, for D=3 the multipartite mutual information provides a robust classification that distinguishes distinct topological Fermi liquid phases.

We introduce a two-dimensional electronic insulator that possesses a toric-code topological order enriched by translation symmetry. This state can be realized from disordering a weak topological superconductor by double-vortex condensation. It is termed the toric-code insulator, whose anyonic excitations consist of a charge-e chargon, a neutral fermion, and two types of visons. There are two types of visons because they have constrained motion as a consequence of the fractional Josephson effect of one-dimensional topological superconductor. Importantly, these two types of visons are related by a discrete translation symmetry and have a mutual semionic braiding statistics, leading to a symmetry enrichment akin to the type in Wen's plaquette model and Kitaev's honeycomb model. We construct this state using a three-fluid coupled-wire model, and analyze the anyon spectrum and braiding statistics in detail to unveil the nature of symmetry enrichment due to translation. We also discuss potential material realizations and present a band-theoretic understanding of the state, fitting it into a general framework for studying fractionalizaton in strongly interacting weak topological phases.

The AC conduction of epitaxially-grown SmB$_6$ thin films and superconducting heterostructures of SmB$_6$/YB$_6$ are investigated via time domain terahertz spectroscopy. A two-channel model of thickness-dependent bulk states and thickness-independent surface states accurately describes the measured conductance of bare SmB$_6$ thin films, demonstrating the presence of surface states in SmB$_6$ via an an optical probe for the first time. While the observed reductions in the simultaneously-measured superconducting gap, transition temperature, and superfluid density of SmB$_6$/YB$_6$ heterostructures relative to bare YB$_6$ indicate the penetration of proximity-induced superconductivity into the SmB$_6$ overlayer; the corresponding SmB$_6$-thickness independence between different heterostructures indicates that the induced superconductivity is predominantly confined to the interface surface state of the SmB$_6$. These results represent the first study of a purported superconductor-topological insulator heterostructure by a terahertz/optical probe, support the existence of surface conducting states in SmB$_6$ susceptible to the superconducting proximity effect, and demonstrate the ability of terahertz spectroscopy to probe proximity-induced superconductivity at the interface buried within a heterostructure via an optical probe.