We present a quantum algorithm to compute the entanglement spectrum of arbitrary quantum states. The interesting universal part of the entanglement spectrum is typically contained in the largest eigenvalues of the density matrix which can be obtained from the lower Renyi entropies through the Newton-Girard method. Obtaining the $p$ largest eigenvalues (${\lambda }_1>{\lambda }_2\cdots >{\lambda }_p$) requires a parallel circuit depth of $O\left[p{({\lambda }_1/{\lambda }_p)}^p\right]$ and $O[plog(N\left)\right]$ qubits where up to $p$ copies of the quantum state defined on a Hilbert space of size $N$ are needed as the input. We validate this procedure for the entanglement spectrum of the topologically ordered Laughlin wave function corresponding to the quantum Hall state at filling factor $\nu =1/3$. Our scaling analysis exposes the tradeoffs between time and number of qubits for obtaining the entanglement spectrum in the thermodynamic limit using finite-size digital quantum computers. We also illustrate the utility of the second Renyi entropy in predicting a topological phase transition and in extracting the localization length in a many-body localized system.

The optical conductivity of a metal near a quantum critical point (QCP) is expected to depend on frequency not only via the scattering time but also via the effective mass, which acquires a singular frequency dependence near a QCP. On the other hand, the quasiparticle residue $Z$, no matter how singular, does not appear in the conductivity as the latter probes quasiparticles rather than bare electrons. In local theories of QCPs, however, the ratio of band and renormalized masses, $m^{*}/m_b$, coincides with $1/Z$, and it is not straightforward to separate the two quantities. In this work, we use a direct diagrammatic approach and compute the optical conductivity, ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$, near two-dimensional (2D) nematic and spin-density wave (SDW) QCPs, using the local approximation in which $Z=m_b/m^{*}$. If renormalization of current vertices is not taken into account, ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$ is expressed via $Z=m_b/m^{*}$ and the transport scattering rate ${\gamma }_{\text{tr}}$ as ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)\propto Z^2{\gamma }_{\text{tr}}/{\mathrm{\Omega }}^2$. For a nematic QCP (${\gamma }_{\text{tr}}\propto {\mathrm{\Omega }}^{4/3}$ and $Z\propto {\mathrm{\Omega }}^{1/3}$), this formula suggests that ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$ would tend to a constant at $\mathrm{\Omega }\to 0$. We explicitly demonstrate that the actual behavior of ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$ is different due to strong renormalization of the current vertices, which cancels out a factor of $Z^2$. As a result, ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$ diverges as $1/{\mathrm{\Omega }}^{2/3}$, as earlier works conjectured. In the SDW case, we consider two contributions to the conductivity: from hot spots and from “lukewarm” regions of the Fermi surface. The hot-spot contribution is not affected by vertex renormalization, but it is subleading to the lukewarm one. For the latter, we argue that a factor of $Z^2$ is again canceled by vertex corrections. As a result, ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$ at a SDW QCP scales as $1/\mathrm{\Omega }$ down to the lowest frequencies, up to possible multiplicative logarithmic factors.

We identify an intrinsic mechanism of the anomalous Hall effect for nonsymmorphic chiral superconductors. This mechanism relies on both a nontrivial multiband chiral superconducting order parameter, which is a mixture of pairings of even and odd angular momentum channels, and a complex normal-state intersublattice hopping, both of which are consequences of the nonsymmorphic group symmetry of the underlying lattice. We apply this mechanism to the putative chiral superconducting phase of the heavy-fermion superconductor ${\mathrm{UPt}}_3$ and calculate the anomalous ac Hall conductivity in a simplified two-band model. From the ac Hall conductivity and optical data we estimate the polar Kerr rotation angle and compare it to the measured results for ${\mathrm{UPt}}_3$ [Schemm et al., Science 345, 190 (2014)].

The topological insulator ${\mathrm{BiSbTeSe}}_2$ corresponds to a compensated semiconductor in which strong Coulomb disorder gives rise to the formation of charge puddles, i.e., local accumulations of charge carriers, both in the bulk and on the surface. Bulk puddles are formed if the fluctuations of the Coulomb potential are as large as half of the band gap. The gapless surface, in contrast, is sensitive to small fluctuations but the potential is strongly suppressed due to the additional screening channel provided by metallic surface carriers. To study the quantitative relationship between the properties of bulk puddles and surface puddles, we performed infrared transmittance measurements as well as scanning tunneling microscopy measurements on the same sample of ${\mathrm{BiSbTeSe}}_2$, which is close to perfect compensation. At 5.5 K, we find surface potential fluctuations occurring on a length scale $r_s=40–50$ nm with amplitude $\mathrm{\Gamma }=8–14$ meV, which is much smaller than in the bulk, where optical measurements detect the formation of bulk puddles. In this nominally undoped compound, the value of $\mathrm{\Gamma }$ is smaller than expected for pure screening by surface carriers, and we argue that this arises most likely from a cooperative effect of bulk screening and surface screening.

Majorana fermions can be realized as quasiparticles in topological superconductors, with potential applications in topological quantum computing. Recently, lattices of magnetic adatoms deposited on the surface of $s$-wave superconductors—Shiba lattices—have been proposed as a new platform for topological superconductivity. These systems possess the great advantage that they are accessible via scanning-probe techniques and thus enable the local manipulation and detection of Majorana modes. Using a nonequilibrium Green's function technique we demonstrate that the topological Majorana edge modes of nanoscopic Shiba islands display universal electronic and transport properties. Most remarkably, these Majorana modes possess a quantized charge conductance that is proportional to the topological Chern number, $\mathcal{C}$, and carry a supercurrent whose chirality reflects the sign of $\mathcal{C}$. These results establish nanoscopic Shiba islands as promising components in future topology-based devices.

We present a generalization of Bloch's theorem to finite-range lattice systems of independent fermions, in which translation symmetry is broken solely due to arbitrary boundary conditions, by providing exact, analytic expressions for all energy eigenvalues and eigenstates. Starting with a reordering of the fermionic basis that transforms the single-particle Hamiltonian into a corner-modified banded block-Toeplitz matrix, a key step is a Hamiltonian-dependent bipartition of the lattice, which splits the eigenvalue problem into a system of bulk and boundary equations. The eigensystem inherits most of its solutions from an auxiliary, infinite translation-invariant Hamiltonian that allows for nonunitary representations of translation—hence complex values of crystal momenta with specific localization properties. A reformulation of the boundary equation in terms of a boundary matrix ensures compatibility with the boundary conditions, and determines the allowed energy eigenstates in the form of generalized Bloch states. We show how the boundary matrix quantitatively captures the interplay between bulk and boundary properties, leading to the construction of efficient indicators of bulk-boundary correspondence. Remarkable consequences of our generalized Bloch theorem are the engineering of Hamiltonians that host perfectly localized, robust zero-energy edge modes, and the predicted emergence, for instance, in Kitaev's Majorana chain, of localized excitations whose amplitudes decay in space exponentially with a power-law prefactor. We further show how the theorem may be used to construct numerical and algebraic diagonalization algorithms for the class of Hamiltonians under consideration, and use the proposed bulk-boundary indicator to characterize the topological response of a multiband time-reversal invariant $s$-wave topological superconductor under twisted boundary conditions, showing how a fractional Josephson effect can occur without entailing a fermionic parity switch. Finally, we establish connections to the transfer matrix method and demonstrate, using the paradigmatic Kitaev's chain example, that a defective (nondiagonalizable) transfer matrix signals the presence of solutions with a power-law prefactor.

We outline here how strong light-matter interaction can be used to induce quantum phase transition between normal and topological phases in two-dimensional topological insulators. We consider the case of a HgTe quantum well, in which band inversion occurs above a critical value of the well thickness, and demonstrate that coupling between electron states and the $E$ field from an off-resonant linearly polarized laser provides a powerful tool to control topological transitions, even for a thickness of the quantum well that is below the critical value. We also show that topological phase properties of the edge states, including their group velocity, can be tuned in a controllable way by changing the intensity of the laser field. These findings open up the possibility for new experimental means with which to investigate topological insulators and shed new light on topological-insulator-based technologies that are under active discussion.

Weyl semimetals have been intensely studied as a three-dimensional realization of a Dirac-like excitation spectrum where the conduction bands and valence bands touch at isolated Weyl points in momentum space. Like in graphene, this property entails various peculiar electronic properties. However, recent theoretical studies have suggested that resonant scattering from rare regions can give rise to a nonzero density of states even at charge neutrality. Here, we give a detailed account of this effect and demonstrate how the semimetallic nature is suppressed at the lowest scales. To this end, we develop a self-consistent $T$-matrix approach to investigate the density of states beyond the limit of weak disorder. Our results show a nonvanishing density of states at the Weyl point, which exhibits a nonanalytic dependence on the impurity density. This unusually strong effect of rare regions leads to a revised estimate for the conductivity close to the Weyl point and emphasizes possible deviations from semimetallic behavior in dirty Weyl semimetals at charge neutrality even with very low impurity concentration.

We report the direct spectroscopic observation of a metal to correlated-insulator transition in the family of iron-based superconducting materials. By means of optical spectroscopy we demonstrate that the excitation spectrum of ${\mathrm{NaFe}}_{1-x}{\mathrm{Cu}}_x\mathrm{As}$ develops a large gap with increasing copper substitution. Dynamical mean-field theory calculations show a good agreement with the experimental data and suggest that the formation of the charge gap requires an intimate interplay of strong on-site electronic correlations and spin-exchange coupling, revealing the correlated Slater-insulator nature of the antiferromagnetic ground state. Our calculations further predict the high-temperature paramagnetic state of the same compound to be a highly incoherent correlated metal. We verify this prediction experimentally by showing that the doping-induced weakening of antiferromagnetic correlations enables a thermal crossover from an insulating to an incoherent metallic state. Redistribution of the optical spectral weight in this crossover uncovers the characteristic energy of Hund's-coupling and Mott-Hubbard electronic correlations essential for the electronic localization. Our results demonstrate that ${\mathrm{NaFe}}_{1-x}{\mathrm{Cu}}_x\mathrm{As}$ continuously transitions from the typical itinerant phases of iron pnictides to a highly incoherent metal and ultimately a correlated insulator. Such an electronic state is expected to favor high-temperature superconductivity.

We investigate the out-of-equilibrium properties of a simple quantum impurity model, the interacting resonant level model. We focus on the scaling regime, where the bandwidth of the fermions in the leads is larger than all the other energies, so that the lattice and the continuum versions of the model become equivalent. Using time-dependent density matrix renormalization group simulations initialized with states having different densities in the two leads, we extend the results of Boulat, Saleur, and Schmitteckert [Phys. Rev. Lett. 101, 140601 (2008)] concerning the current-voltage $\left(I\text{−}V\right)$ curves, for several values of the interaction strength $U$. We estimate numerically the Kondo scale $T_B$ and the exponent $b\left(U\right)$ associated to the tunneling of the fermions from the leads to the dot. Next, we analyze the quantum entanglement properties of the steady states. We focus in particular on the entropy rate $\alpha$, describing the linear growth with time of the bipartite entanglement in the system. We show that, as for the current, $\alpha /T_B$ is described by some function of $U$ and of the rescaled bias $V/T_B$. Finally, the spatial structure of the entropy profiles is discussed.

We study a $T=0$ quantum phase transition between a quantum paramagnetic state and a magnetically ordered state for a spin $S=1$ XXZ Heisenberg antiferromagnet on a two-dimensional triangular lattice. The transition is induced by an easy-plane single-ion anisotropy $D$. At the mean-field level, the system undergoes a direct transition at a critical $D=D_c$ between a paramagnetic state at $D>D_c$ and an ordered state with broken $\text{U}\left(1\right)$ symmetry at $D<D_c$. We show that beyond mean field the phase diagram is very different and includes an intermediate, partially ordered chiral liquid phase. Specifically, we find that inside the paramagnetic phase the Ising $\left(J_z\right)$ component of the Heisenberg exchange binds magnons into a two-particle bound state with zero total momentum and spin. This bound state condenses at $D>D_c$, before single-particle excitations become unstable, and gives rise to a chiral liquid phase, which spontaneously breaks spatial inversion symmetry, but leaves the spin-rotational $\text{U}\left(1\right)$ and time-reversal symmetries intact. This chiral liquid phase is characterized by a finite vector chirality without long-range dipolar magnetic order. In our analytical treatment, the chiral phase appears for arbitrarily small $J_z$ because the magnon-magnon attraction becomes singular near the single-magnon condensation transition. This phase exists in a finite range of $D$ and transforms into the magnetically ordered state at some $D<D_c$. We corroborate our analytic treatment with numerical density matrix renormalization group calculations.

We have investigated the damping in the ferromagnetic resonance (FMR) of yttrium iron garnet (YIG) caused by spin pumping into adjacent conducting materials, namely, Pt and the conducting polymer poly(3,4-ethylenedioxythiophene):poly(4-styrenesulfonate) (PEDOT:PSS). By a systematic study which also includes multilayers in which the conducting layer is separated from YIG by an insulator, we can show that a considerable part of the damping can be attributed to the so-called radiation damping which originates from the interaction of the magnetic fields caused by the precessing magnetization with the conducting layer. Especially, when PEDOT:PSS is used as a spin sink, the observed damping must be attributed completely to radiation damping, and no contribution from spin pumping can be identified. These results demonstrate that the Gilbert damping as a measure of spin pumping can only be used when careful control experiments accompany the investigation.

Recent theories have proposed a concept of valley photonic crystals as an analog of gapped valleytronic materials such as $\mathrm{Mo}{\mathrm{S}}_2$ and bilayer graphene. Here, we further extend the applicability of valley photonic crystals to surface electromagnetic waves and experimentally demonstrate a valley surface-wave photonic crystal on a single metal surface as a photonic duplicate of $\mathrm{Mo}{\mathrm{S}}_2$. Both bulk transport and edge transport are directly mapped with a near-field microwave imaging system. The photonic valley pseudospins are demonstrated, together with the photonic valley Hall effect that splits the opposite photonic valley pseudospins into two opposite directions. The valley edge transport in $\mathrm{Mo}{\mathrm{S}}_2$ or other transition-metal dichalcogenide monolayers, which is different from bilayer graphene but still stays unrealized in condensed-matter systems, is demonstrated on this $\mathrm{Mo}{\mathrm{S}}_2$-like photonic platform. Our study not only offers a tabletop platform to study the valleytronic physics, but also opens a venue for on-chip integrated photonic device applications using valley-polarized information.

We study the classification of symmetry-protected topological (SPT) phases with crystalline symmetry (cSPT phases). Focusing on bosonic cSPT phases in two and three dimensions, we introduce a simple family of cSPT states, where the system is comprised of decoupled lower-dimensional building blocks that are themselves SPT states. We introduce a procedure to classify these block states, which surprisingly reproduces a classification of cSPT phases recently obtained by Thorngren and Else (arXiv:1612.00846) using very different methods, for all wallpaper and space groups. The explicit constructions underlying our results clarify the physical properties of the phases classified by Thorngren and Else, and expose additional structure in the classification. Moreover, the states we classify can be completely characterized by point-group SPT (pgSPT) invariants and related weak pgSPT invariants that we introduce. In many cases, the weak invariants can be visualized in terms of translation-symmetric stacking of lower-dimensional pgSPT states. We apply our classification to propose a Lieb-Shultz-Mattis–type constraint for two-dimensional spin systems with only crystalline symmetry, and establish this constraint by a dimensional reduction argument. Finally, the surprising matching with the Thorngren-Else classification leads us to conjecture that all SPT phases protected only by crystalline symmetry can be built from lower-dimensional blocks of invertible topological states. We argue that this conjecture holds if we make a certain physically reasonable but unproven assumption.

The modification of a circularly polarized vacuum field in three-dimensional chiral photonic crystals was measured by spontaneous emission from quantum dots in the structures. Due to the circularly polarized eigenmodes along the helical axis in the GaAs-based mirror-asymmetric structures we studied, we observed highly circularly polarized emission from the quantum dots. Both spectroscopic and time-resolved measurements confirmed that the obtained circularly polarized light was influenced by a large difference in the photonic density of states between the orthogonal components of the circular polarization in the vacuum field.

We combine low energy muon spin rotation ($\mathrm{LE}\text{−}\mu \mathrm{SR}$) and soft-x-ray angle-resolved photoemission spectroscopy (SX-ARPES) to study the magnetic and electronic properties of magnetically doped topological insulators, (${\mathrm{Bi},\mathrm{Sb})}_2{\mathrm{Te}}_3$. We find that one achieves a full magnetic volume fraction in samples of ${(\mathrm{V}/\mathrm{Cr})}_x{(\mathrm{Bi},\mathrm{Sb})}_{2-x}{\mathrm{Te}}_3$ at doping levels $x\gtrsim 0.16$. The observed magnetic transition is not sharp in temperature indicating a gradual magnetic ordering. We find that the evolution of magnetic ordering is consistent with formation of ferromagnetic islands which increase in number and/or volume with decreasing temperature. Resonant ARPES at the V $L_3$ edge reveals a nondispersing impurity band close to the Fermi level as well as V weight integrated into the host band structure. Calculations within the coherent potential approximation of the V contribution to the spectral function confirm that this impurity band is caused by V in substitutional sites. The implications of our results on the observation of the quantum anomalous Hall effect at mK temperatures are discussed.

It is well established that topological insulators sustain Dirac fermion surface states as a consequence of band inversion in the bulk. These states have a helical spin polarization and a linear dispersion with large Fermi velocity. We report on a set of experimental observations supporting the existence of additional massive surface states. These states are also confined by the band inversion at a topological-trivial semiconductor heterojunction. While first introduced by Volkov and Pankratov (VP) before the understanding of the topological nature of such a junction, they were not experimentally identified. Here we identify their signatures on transport properties at high electric field. By monitoring the ac admittance of HgTe topological-insulator field-effect capacitors, we access the compressibility and conductivity of surface states in a broad range of energies and electric fields. The Dirac states are characterized by a compressibility minimum, a linear energy dependence, and a high mobility persisting up to energies much larger than the transport band gap of the bulk. At higher energies, we observe multiple anomalous behaviors in conductance, charge metastability, and Hall resistance that point towards the contribution of massive surface states in transport scattering and charge transfer to the bulk. The spectrum of these anomalies agrees with predictions of a phenomenological model of VP states in a smooth topological heterojunction. The model accounts for the finite interface depth, the effect of electric fields including Dirac screening, and predicts the energy of the first VP state. The massive surface states are a hallmark of topological heterojunctions, whose understanding is crucial for transport studies and applications.

In a fluid subject to a magnetic field, the viscous stress tensor has a dissipationless antisymmetric component controlled by the so-called Hall viscosity. We here propose an all-electrical scheme that allows a determination of the Hall viscosity of a two-dimensional electron liquid in a solid-state device.

We study a topological superconductor island with spatially separated Majorana modes coupled to multiple normal-metal leads by single-electron tunneling in the Coulomb blockade regime. We show that low-temperature transport in such a Majorana island is carried by an emergent charge-$e$ boson composed of a Majorana mode and an electronic excitation in leads. This transmutation from Fermi to Bose statistics has remarkable consequences. For noninteracting leads, the system flows to a non-Fermi-liquid fixed point, which is stable against tunnel couplings anisotropy or detuning away from the charge-degeneracy point. As a result, the system exhibits a universal conductance at zero temperature, which is a fraction of the conductance quantum, and low-temperature corrections with a universal power-law exponent. In addition, we consider Majorana islands connected to interacting one-dimensional leads, and find different stable fixed points near and far from the charge-degeneracy point.

We demonstrate full charge control, narrow optical linewidths, and optical spin pumping on single self-assembled InGaAs quantum dots embedded in a $162.5-\text{nm}$-thin diode structure. The quantum dots are just $88\text{nm}$ from the top GaAs surface. We design and realize a $p-i-n-i-n$ diode that allows single-electron charging of the quantum dots at close-to-zero applied bias. In operation, the current flow through the device is extremely small resulting in low noise. In resonance fluorescence, we measure optical linewidths below $2\mu \text{eV}$, just a factor of 2 above the transform limit. Clear optical spin pumping is observed in a magnetic field of $0.5\text{T}$ in the Faraday geometry. We present this design as ideal for securing the advantages of self-assembled quantum dots—highly coherent single-photon generation, ultrafast optical spin manipulation—in the thin diodes required in quantum nanophotonics and nanophononics applications.