Here we present an efficient and numerically stable procedure for compressing a correlation matrix into a set of local unitary single-particle gates, which leads to a very efficient way of forming the matrix product state (MPS) approximation of a pure fermionic Gaussian state, such as the ground state of a quadratic Hamiltonian. The procedure involves successively diagonalizing subblocks of the correlation matrix to isolate local states which are purely occupied or unoccupied. A small number of nearest-neighbor unitary gates isolate each local state. The MPS of this state is formed by applying the many-body version of these gates to a product state. We treat the simple case of compressing the correlation matrix of spinless free fermions with definite particle number in detail, although the procedure is easily extended to fermions with spin and more general BCS states (utilizing the formalism of Majorana modes). We also present a density matrix renormalization group–like algorithm to obtain the compressed correlation matrix directly from a hopping Hamiltonian. In addition, we discuss a slight variation of the procedure which leads to a simple construction of the multiscale entanglement renormalization ansatz of a fermionic Gaussian state, and present a simple picture of orthogonal wavelet transforms in terms of the gate structure we present in this paper. As a simple demonstration, we analyze the Su-Schrieffer-Heeger model (free fermions on a one-dimensional lattice with staggered hopping amplitudes).

The momentum-resolved Eliashberg function (ELF) ${\alpha }^{2}F\left(\omega ,\stackrel{\rightharpoonup }{k}\right)$ for the $\mathrm{Be}\left(0001\right)$ zone-center surface state was extracted from the high-quality angle-resolved photoemission spectroscopy (ARPES) data at the Fermi energy in the $\overline{\mathrm{\Gamma }}\to \overline{M}$ direction, displaying ten peaks. A comparison of the peaks in the ELF to the bulk phonon density of states (DOS) and the bulk and surface phonon dispersion allows for an identification of the origin of all but two of the peaks. The five high energy peaks $(>52\mathrm{meV})$ are associated with the coupling of the surface state to bulk phonon modes. The peaks at 44.5 and at 49.0 meV have contributions from both the bulk and surface phonons. The most intense peak at 37.5 meV is evidently having a contribution from electron-phonon coupling (EPC) of the surface state to the surface Rayleigh phonon mode. Surprisingly, the two lowest energy modes, which must be associated with surface Rayleigh phonon, cannot be attributed to a high phonon DOS at the surface nor to any Fermi surface nesting. After detail analysis, the three lowest energy peaks are associated with momentum dependence in the EPC matrix, reflected in the phonon linewidth changes. As a result of the broken symmetry at the surface, coupling of the initial surface state due to the presence of the surface phonons contributes $\sim 48.5\pm 12.5\%$ of the spectral weight in the ELF and $\sim 66.5\pm 10.5\%$ to the mass enhancement $\left(\lambda \right)$.

We present a Raman-scattering study of optical phonons in zinc-blende (cubic) GaN for temperatures ranging from 80 to 750 K. The experiments were performed on high-quality, cubic GaN films grown by molecular-beam epitaxy on GaAs (001) substrates. The observed temperature dependence of the optical phonon frequencies and linewidths is analyzed in the framework of anharmonic decay theory, and possible decay channels are discussed in the light of density-functional-theory calculations. The longitudinal-optical (LO) mode relaxation is found to occur via asymmetric decay into acoustic phonons, with an appreciable contribution of higher-order processes. The transverse-optical mode linewidth shows a weak temperature dependence and its frequency downshift is primarily determined by the lattice thermal expansion. The LO phonon lifetime is derived from the observed Raman linewidth and an excellent agreement with previous theoretical predictions is found.

We investigate the properties of electron states localized at the edge of a superconducting island placed on the surface of a topological insulator in a magnetic field. In such systems, Majorana fermions emerge if an odd number of vortices (or odd multivortex vorticity) is hosted by the island; otherwise, no Majorana states exist. Majorana states emerge in pairs: one state is localized near the vortex core, and another at the island edge. We analyze in detail the robustness of Majorana fermions at the edge of the island threaded by a single vortex. If the system parameters are optimized, the energy gap between the Majorana fermion and the first excited state at the edge is of the order of the superconducting gap induced on the surface of the topological insulator. The stability of the Majorana fermion state against a variation of the gate voltage and its sensitivity to the magnetic field allows one to experimentally distinguish the edge Majorana fermion from conventional Dirac fermions.

We study the spectra of finite-length carbon nanotubes (CNTs) of arbitrary chirality. They divide into two classes, which arise because of different rotational symmetries of the low-energy eigenstates. In one of them (the zigzag class), the spinless spectrum is doubly degenerate and the two states can be assigned to different values of the valley degree of freedom. In the other (armchair class), the valley degeneracy is removed and the eigenstates are combinations of both valley states. Recent experimental observation of the valley mixing in ultraclean CNT quantum dots is consistent with our theory for armchair-class CNTs.

Applications of the negatively charged nitrogen-vacancy center in diamond exploit the center's unique optical and spin properties, which at ambient temperature are predominately governed by electron-phonon interactions. Here, we investigate these interactions at ambient and elevated temperatures by observing the motional narrowing of the center's excited-state spin resonances. We determine that the center's Jahn-Teller dynamics are much slower than currently believed and that they do not solely account for the broadening of the center's optical resonance above cryogenic temperatures. We show that the inclusion of interactions with symmetric phonon modes can explain the observed broadening and resolve the current inconsistencies in the literature. However, our model also reveals unexpected features of the electron-phonon processes that coincide with other poorly understood vibronic features of the center and require further investigation.

The electronic properties of epitaxial graphene grown on SiC(0001) are known to be impaired relative to those of freestanding graphene. This is due to the formation of a carbon buffer layer between the graphene layers and the substrate, which causes the graphene layers to become strongly $n$-doped. Charge neutrality can be achieved by completely passivating the dangling bonds of the clean SiC surface using atomic intercalation. So far, only one element, hydrogen, has been identified as a promising candidate. We show, using first-principles density functional calculations, how it can also be accomplished via the growth of a thin layer of silicon nitride on the SiC surface. The subsequently grown graphene layers display the electronic properties associated with charge neutral graphene. We show that the surface energy of this structure is considerably lower than that of others with intercalated atomic nitrogen and determine how its stability depends on the ${\mathrm{N}}_2$ chemical potential.

The renormalization of the band structure at zero temperature due to electron-phonon coupling is explored in diamond, BN, LiF, and MgO crystals. We implement a dynamical scheme to compute the frequency-dependent self-energy and the resulting quasiparticle electronic structure. Our calculations reveal the presence of a satellite band below the Fermi level of LiF and MgO. We show that the renormalization factor $\left(Z\right)$, which is neglected in the adiabatic approximation, can reduce the zero-point renormalization (ZPR) by as much as $40\%$. Anharmonic effects in the renormalized eigenvalues at finite atomic displacements are explored with the frozen-phonon method. We use a nonperturbative expression for the ZPR, going beyond the Allen-Heine-Cardona theory. Our results indicate that high-order electron-phonon coupling terms contribute significantly to the zero-point renormalization for certain materials.

Using Wannier-function-based interpolation techniques, we present compelling numerical evidence for the presence of a saddle-point Van Hove singularity at the $\mathrm{\Gamma }$ point near the phosphorene Fermi energy. We show that in proximity of the Van Hove singularity the spin susceptibility presents the logarithmic temperature dependence typical of Liftshitz phase transitions. Furthermore, we demonstrate that the critical temperature for the ferromagnetic transition can be greatly increased (up to 0.05 K) if strain along the zigzag ridges is applied. Although the ferromagnetic state would be very difficult to experimentally reach, the logarithmic temperature behavior of the spin susceptibility due to the Van Hove singularity is found to persist at much higher temperatures (up to $\sim 97$ K).

In order to clarify the physics of the gating of solids by ionic liquids (ILs) we have gated lightly doped $p$-Si, which is so well studied that it can be called the “hydrogen atom of solid state physics” and can be used as a test bed for ionic liquids. We explore the case where the concentration of induced holes at the Si surface is below ${10}^{12}{\text{cm}}^{-2}$, hundreds of times smaller than record values. We find that in this case an excess negative ion binds a hole on the interface between the IL and Si becoming a surface acceptor. We study the surface conductance of holes hopping between such nearest neighbor acceptors. Analyzing the acceptor concentration dependence of this conductivity, we find that the localization length of a hole is in reasonable agreement with our direct variational calculation of its binding energy. The observed hopping conductivity resembles that of well studied ${\text{Na}}^{+}$ implanted Si MOSFETs.

Ab initio pseudopotential calculations have been made to calculate the structural, electronic, vibrational, and superconducting properties of the borocarbide superconductors ${RM}_2{\mathrm{B}}_2\mathrm{C}(R=\text{Lu},\mathrm{La},\mathrm{Y};M=\text{Ni},\mathrm{Pd},\mathrm{Pt})$. Using the electronic and vibrational spectra, the average electron-phonon coupling parameter is calculated to be 0.78, 0.83, 0.96, and 1.48 for ${\mathrm{LaPt}}_2{\mathrm{B}}_2\mathrm{C}, {\mathrm{LuNi}}_2{\mathrm{B}}_2\mathrm{C}, {\mathrm{YPt}}_2{\mathrm{B}}_2\mathrm{C}$, and ${\mathrm{YPd}}_2{\mathrm{B}}_2\mathrm{C}$, respectively. From a detailed spectral analysis throughout the irreducible Brillouin zone (IBZ), we found that the highest contribution to the electron-phonon coupling parameter for these materials comes from the acoustic and low-frequency optical modes characterized with anomalous dispersion, instead of the $A_{1g}$ optical-phonon mode claimed in previous studies. Using the calculated electron-phonon coupling parameter values, the superconducting critical temperatures for ${\mathrm{LaPt}}_2{\mathrm{B}}_2\mathrm{C}, {\mathrm{LuNi}}_2{\mathrm{B}}_2\mathrm{C}, {\mathrm{YPt}}_2{\mathrm{B}}_2\mathrm{C}$, and ${\mathrm{YPd}}_2{\mathrm{B}}_2\mathrm{C}$ are found to be 10.40, 15.94, 11.30, and 20.60 K, respectively, which are in good accordance with their experimental values of 10.50, 16.10, 10.00, and 21.00 K.

The structurally simplest Fe-based superconductor FeSe with a critical temperature $T_c\approx 8.5\mathrm{K}$ displays a breaking of the fourfold rotational symmetry at a temperature $T_s\approx 87\mathrm{K}$. We investigate the electronic properties of FeSe using scanning tunneling microscopy/spectroscopy (STM/STS), magnetization, and electrical transport measurements. The results indicate two energy scales: (i) $T^{*}\approx 75\mathrm{K}$, denoted by an onset of electron-hole asymmetry in STS, enhanced spin fluctuations, and increased positive magnetoresistance; and (ii) $T^{**}\approx 22–30 \mathrm{K}$, marked by the opening up of a partial gap of about 8 meV in STS and a recovery of Kohler's rule. Our results reveal the onset of an incipient ordering mode at $T^{*}$ and its nucleation below $T^{**}$. The ordering mode observed here, both in spin as well as charge channels, suggests a coupling between the spins with charge, orbital, or pocket degrees of freedom.

A combined experimental (superconductor-insulator-superconductor tunneling spectra) and theoretical (density functional theory) study of the two-gap superconductor ${\mathrm{MgB}}_2$ is reported. The calculations confirm that the small gap is associated with a $\pi$ band mostly based on the boron $p_z$ orbitals leading to the three-dimensional band component of the Fermi surface. This channel almost completely dominates the tunneling images and spectra for $c$-axis-oriented samples and not the two-dimensional $\sigma$ band. The origin of this effect is due to the faster decay of the electronic states associated with the boron $p_x$ and $p_y$ orbitals compared to those associated with the boron $p_z$ orbitals, together with the symmetry properties of the wave functions. The calculated tunneling channels and partial density of states for each band agree with the values deduced from precise fits of experimental tunneling spectra. The present approach provides a framework for the understanding of tunneling spectra and the nature of superconducting gaps of other multigap superconductors.

In a recent paper [B. P. van Zyl et al., Phys. Rev. A 89, 022503 (2014)], the average density approximation (ADA) was implemented to develop a parameter-free, nonlocal kinetic energy functional to be used in the orbital-free density functional theory of an inhomogeneous, two-dimensional (2D) Fermi gas. In this work, we provide a detailed comparison of self-consistent calculations within the ADA with the exact results of the Kohn-Sham density functional theory and the elementary Thomas-Fermi (TF) approximation. We demonstrate that the ADA for the 2D kinetic energy functional works very well under a wide variety of confinement potentials, even for relatively small particle numbers. Remarkably, the TF approximation for the kinetic energy functional, without any gradient corrections, also yields good agreement with the exact kinetic energy for all confining potentials considered, although at the expense of the spatial and kinetic energy densities exhibiting poor pointwise agreement, particularly near the TF radius. Our findings illustrate that the ADA kinetic energy functional yields accurate results for both the local and global equilibrium properties of an inhomogeneous 2D Fermi gas, without the need for any fitting parameters.

We consider integer quantum Hall states and calculate the bulk entanglement spectrum by formulating the correlation matrix in the guiding center representation. Our analytical approach is based on the strategy of redefining the inner product of states in the Hilbert space, via a projection operator, to take care of the restriction imposed by the (rectangle-pixeled) checkerboard partition. The resultant correlation matrix contains various couplings between states of different guiding centers parameterized by the magnetic length and the pixel size. Given a fixed magnetic field, we find various patterns of band crossings by tuning the pixel size (quantified by the flux $\mathrm{\Phi }$ threading each pixel) and by changing the filling factor $\nu \in \mathbb{N}$ (determined by the Fermi level). When $\nu =1$ and $\mathrm{\Phi }=2\pi$, or $\nu =2$ and $\mathrm{\Phi }=\pi$, one Dirac band crossing is found. For $\nu =1$ and $\mathrm{\Phi }=\pi$, the band crossings are in the form of a nodal line, enclosing the Brillouin zone. As for $\nu =2$ and $\mathrm{\Phi }=2\pi$, the doubled Dirac point, or the quadratic point, is seen. Additionally, we infer that the quadratic point is protected by the ${\mathrm{C}}_4$ symmetry of the pixel since it evolves into two separate Dirac points when the symmetry is lowered to ${\mathrm{C}}_2$. We also identify the emerging symmetries responsible for the symmetric bulk entanglement spectra, which are absent in the underlying quantum Hall states.

We report the evolution of the surface electronic structure and surface material properties of a topological crystalline insulator (TCI), ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$, as a function of various material parameters including composition $x$, temperature $T$, and crystal structure. Our spectroscopic data demonstrate the electronic ground-state condition for the saddle point singularity, the tunability of surface chemical potential, and the surface states' response to circularly polarized light. Our results show that each material parameter can tune the system between the trivial and topological phase in a distinct way, unlike that seen in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and related compounds, leading to a rich topological phase diagram. Our systematic studies of the TCI ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$ are a valuable materials guide to realize new topological phenomena.

We study the effect of electron-electron interaction in graphene quantum dots defined by an external electrostatic potential and a high magnetic field. To account for the electron-electron interaction, we use the Thomas-Fermi approximation and find that electron screening causes the formation of compressible strips in the potential profile and the electron density. We numerically solve the Dirac equations describing the electron dynamics in quantum dots, and we demonstrate that compressible strips lead to the appearance of plateaus in the electron energies as a function of the magnetic field. Finally, we discuss how our predictions can be observed using the Kelvin probe force microscope measurements.

The compound ${\mathrm{SmB}}_6$ is a Kondo insulator, where the lowest-energy bulk electronic excitations are spin-excitons. It also has surface states that are subjected to strong spin-orbit coupling. It has been suggested that ${\mathrm{SmB}}_6$ is also a topological insulator. Here we show that, despite the absence of time-reversal symmetry breaking and the presence of strong spin-orbit coupling, the chiral spin texture of the Weyl cone is not completely protected. In particular, we show that the spin-exciton-mediated scattering produces features in the surface electronic spectrum at energies separated from the surface Fermi energy by the spin-exciton energy. Despite the features being far removed from the surface Fermi energy, they are extremely temperature dependent. The temperature variation occurs over a characteristic scale determined by the dispersion of the spin-exciton. The structures may be observed by electron spectroscopy at low temperatures.

We study charge-ordered solutions for fermions on a square lattice interacting with dynamic antiferromagnetic fluctuations. Our approach is based on real-space Eliashberg equations, which are solved self-consistently. We first show that the antiferromagnetic fluctuations can induce arc features in the spectral functions, as spectral weight is suppressed at the hot spots; however, no real pseudogap is generated. At low temperature, spontaneous charge order with a $d$-form factor can be stabilized for certain parameters. As long as the interacting Fermi surfaces possess hot spots, the ordering wave vector corresponds to the diagonal connection of the hot spots, similar to the non-self-consistent case. Tendencies towards observed axial order only appear in situations without hot spots.