Fermi Points Research Articles

Hyperdeterminantal point processes

As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to simulate and provide a quite broad class of models that exhibit repulsion between points. The fundamental ingredient used to construct a determinantal point process is a kernel giving the pairwise interactions between points: the joint distribution of any number of points then has a simple expression in terms of determinants of certain matrices defined from this kernel. In this paper we initiate the study of an analogous class of point processes that are defined in terms of a kernel giving the interaction between 2M points for some integer M. The role of matrices is now played by 2M-dimensional “hypercubic” arrays, and the determinant is replaced by a suitable generalization of it to such arrays—Cayley’s first hyperdeterminant. We show that some of the desirable features of determinantal point processes continue to be exhibited by this generalization.

Bound states in gapped graphene with impurities: Effective low-energy description of short-range interactions

We obtain a bound-state spectrum of low-energy excitations near the Fermi points of gapped graphene in the presence of a charge impurity. The effects of possible short-range interactions induced by the impurity are modeled by suitable boundary conditions. The spectrum in the subcritical region of the effective Coulomb coupling is labeled by a parameter which characterizes the boundary conditions and determines the inequivalent quantizations of the system. In the supercritical region we obtain a renormalization-group flow for the effective Coulomb coupling.

Alternative description of the 2D Blume–Capel model using Grassmann algebra

We use Grassmann algebra to study the phase transition in the two-dimensional ferromagnetic Blume–Capel model from a fermionic point of view. This model presents a phase diagram with a second-order critical line which becomes first order through a tricritical point. In particular, we are able to map the spin-1 system of the BC model onto an effective fermionic action from which we obtain the exact mass of the theory. The condition of vanishing mass defines the critical line. This effective action is actually an extension of the free fermion Ising action with an additional quartic interaction term. The effect of this term is merely to render the excitation spectrum of the fermions unstable at the tricritical point. The results are compared with recent numerical Monte Carlo simulations.

Edge states of zigzag bilayer graphite nanoribbons

The electronic structures of zigzag bilayer graphite nanoribbons (Z-BGNRs) with various ribbon widthsN are studied within the tight binding approximation. Neglecting the inter-layer hopping amplitudeγ4, whichis an order of magnitude smaller than the other inter-layer hopping parameters, there exist two fixed Fermipoints ± k* independent of the ribbon width with a peculiar energy dispersion neark* asε(k)∼ ± (k−k*)N. By investigating the edge states of Z-BGNRs, we notice that the trigonal warping of thebilayer graphene sheets is reflected in the edge state structure. With the inclusion ofγ4, theabove two Fermi points are not fixed but drift toward the vicinity of the Dirac point with increasingwidth N, as shown by the finite scaling method, and the peculiar dispersions change to parabolicones. The edge magnetism of Z-BGNRs is also examined by solving the half-filled HubbardHamiltonian for the ribbon using the Hartree–Fock approximation. We have shown thatwithin the same side of the edges, the edge spins are aligned ferromagnetically for theexperimentally relevant set of parameters.

Correlation functions of the openXXZ chain: II

We derive compact multiple integral formulae for several physical spin correlation functions in thesemi-infinite XXZ chain with a longitudinal boundary magnetic field. Our formulae follow from severaleffective resummations of the multiple integral representation for the elementary blocksobtained in our previous paper (I). In the free fermion point we compute the localmagnetization as well as the density of energy profiles. These quantities, in addition totheir bulk behavior, exhibit Friedel-type oscillations induced by the boundary; theiramplitudes depend on the boundary magnetic field and decay algebraically in terms of thedistance to the boundary.

PT symmetry of the non-Hermitian XX spin-chain: non-local bulk interaction from complex boundary fields

The XX spin-chain with non-Hermitian diagonal boundary conditions is shown to be quasi-Hermitian for special values of the boundary parameters. This is proved by an explicit construction of a new inner product employing a ‘quasi-fermion’ algebra in momentum space, where creation and annihilation operators are not related via Hermitian conjugation. For a special example, when the boundary fields lie on the imaginary axis, we show the spectral equivalence of the quasi-Hermitian XX spin-chain with a non-local fermion model, where long-range hopping of the particles occurs as the non-Hermitian boundary fields increase in strength. The corresponding Hamiltonian interpolates between the open XX and the quantum group invariant XXZ model at the free fermion point. For an even number of sites the former is known to be related to a CFT with central charge c = 1, while the latter has been connected to a logarithmic CFT with central charge c = −2. We discuss the underlying algebraic structures and show that for an odd number of sites the superalgebra symmetry can be extended from the unit circle along the imaginary axis. We relate the vanishing of one of its central elements to the appearance of Jordan blocks in the Hamiltonian.

Magnetism as a Mass Term of the Edge States in Graphene

The magnetism by the edge states in graphene is investigated theoretically. An instability of the pseudo-spin order of the edge states induces ferrimagnetic order in the presence of the Coulomb interaction. Although the next nearest-neighbor hopping can stabilize the pseudo-spin order, a strong Coulomb interaction makes the pseudo-spin unpolarized and real spin polarized. The magnetism of the edge states makes two peaks of the density of states in the conduction and valence energy bands near the Fermi point. Using a continuous model of the Weyl equation, we show that the edge-induced gauge field and the spin dependent mass terms are keys to make the magnetism of the edge states. A relationship between the magnetism of the edge states and the parity anomaly is discussed.

Superconducting proximity effect through graphene and graphite films

We discuss the superconducting proximity effect in graphite films (or graphene) by studying the critical current through superconductor-graphite film (or graphene)-superconductor Josephson junction. The two leads are assumed to be conventional s-wave superconductors. We especially pay attention to the dispersions of electrons in graphite films: because of the delicate band structure of graphite, the electron dispersion in the film, which undergoes the effects of various external factors such as leads, gate electrodes and sample inhomogeneities, can show a wide variety. We introduce several models for electron dispersion near Fermi energy: 1) graphene-like Fermi points, 2) semi-metal, and 3) semiconducting gap. The superconducting critical current Ic through the junction is usually expressed as Ic α exp(-d/ξ(T)}, where d is the distance between two leads and ξ(T) is the proximity length. We show that the temperature dependence of ξ(T) is largely affected by the band structure of the graphite film and by examining this dependence some insight into the electronic properties of the graphite film can be obtained.

Phase diagram of the one-dimensional Hubbard model with next-nearest-neighbor hopping

We study the one-dimensional Hubbard model with nearest-neighbor and next-nearest-neighbor hopping integrals by using the density-matrix renormalization group method and Hartree-Fock approximation. Based on the calculated results for the spin gap, total-spin quantum number, and Tomonaga-Luttinger-liquid parameter, we determine the ground-state phase diagram of the model in the entire filling and wide parameter region. We show that, in contrast to the weak-coupling regime where a spin-gapped liquid phase is predicted in the region with four Fermi points, the spin gap vanishes in a substantial region in the strong-coupling regime. It is remarkable that a large variety of phases, such as the paramagnetic metallic phase, spin-gapped liquid phase, singlet and triplet superconducting phases, and fully polarized ferromagnetic phase, appear in such a simple model in the strong-coupling regime.

Electronic instabilities of a Hubbard model approached as a large array of coupled chains: Competition betweend-wave superconductivity and pseudogap phase

We study the electronic instabilities in a two-dimensional (2D) Hubbard model where one of the dimensions has a finite width, so that it can be considered as a large array of coupled chains. The finite transverse size of the system gives rise to a discrete string of Fermi points, with respective electron fields that, due to their mutual interaction, acquire anomalous scaling dimensions depending on the point of the string. Using bosonization methods, we show that the anomalous scaling dimensions vanish when the number of coupled chains goes to infinity, implying the Fermi liquid behavior of a 2D system in that limit. However, when the Fermi level is at the Van Hove singularity arising from the saddle points of the 2D dispersion, backscattering and Cooper-pair scattering lead to the breakdown of the metallic behavior at low energies. These interactions are taken into account through their renormalization group scaling, studying in turn their influence on the nonperturbative bosonization of the model. We show that, at a certain low-energy scale, the anomalous electron dimension diverges at the Fermi points closer to the saddle points of the 2D dispersion. The $d$-wave superconducting correlations also become large at low energies, but their growth is cut off as the suppression of fermion excitations takes place first, extending progressively along the Fermi points toward the diagonals of the 2D Brillouin zone. We stress that this effect arises from the vanishing of the charge stiffness at the Fermi points, characterizing a critical behavior that is well captured within our nonperturbative approach.

On the emptiness formation probability of the openXXZ spin-1/2 chain

This paper is devoted to the study of the emptiness formation probabilityτ(m) of the openXXZ chain. We derivea closed form for τ(m) at Δ = 1/2 when the boundary field vanishes. Moreover we obtain its leading asymptotics for an arbitraryboundary field at the free fermion point. Finally, we compute the first term of the asymptotics ofln(τ(m)) for the wholemassless regime −1<Δ<1.

Bosonization of strongly interacting one-dimensional electrons

Strong repulsive interactions in a one-dimensional electron system suppress the exchange coupling $J$ of electron spins to a value much smaller than the Fermi energy ${E}_{F}$. The conventional theoretical description of such systems based on the bosonization approach and the concept of Tomonaga-Luttinger liquid is applicable only at energies below $J$. In this paper, we develop a theoretical approach valid at all energies below the Fermi energy, including a broad range of energies between $J$ and ${E}_{F}$. The method involves bosonization of the charge degrees of freedom, while the spin excitations are treated exactly. We use this technique to calculate the spectral functions of strongly interacting electron systems at energies in the range $J⪡\ensuremath{\epsilon}⪡{E}_{F}$. We show that in addition to the expected features at the wave vector $k$ near the Fermi point ${k}_{F}$, the spectral function has a strong peak centered at $k=0$. Our theory also provides analytical description of the spectral function singularities near $3{k}_{F}$ (the ``shadow band'' features).

Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films

Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films

On the correlation structure of some random point processes on the line

The correlation structure of some remarkable point processes on the one-dimensional real line is investigated. More specifically, focus is on translation invariant determinantal, permanental and/or renewal point processes. In some cases, anomalous (non-Poissonian) fluctuations for the number of points in a large window can be observed. This may be read from the total correlation function of the point process. We try to understand when and why this occurs and what are the anomalous behaviors to be expected. From examples, it is shown that determinantal (fermion) point processes can be super-homogeneous (the number variance grows slower than the number mean) and even hyper-uniform (when variance growth saturates). Renewal point processes with bounded spacings variance are essentially Poissonian (the number variance grows like the number mean as in Poisson models). Under certain conditions, permanental (boson) point processes can be sub-homogeneous or critical (in the sense that the number variance grows faster than the number mean). We give several detailed examples illustrating these properties of interest together with unexpected behaviors.

RPAE versus RPA for the Tomonaga model with quadratic energy dispersion

Recently, the damping of the collective charge (and spin) modes of interacting fermions in one spatial dimension was studied. It results from the nonlinear correction to the energy dispersion in the vicinity of the Fermi points. To investigate the damping, one has to replace the random-phase approximation (RPA) bare bubble by a sum of more complicated diagrams. It is shown here that a better starting point than the bare RPA is to use the (conserving) linearized time-dependent Hartree-Fock equations, i.e., to perform a random-phase approximation (with) exchange (RPAE) calculation. It is shown that the RPAE equation can be solved analytically for the special form of the two-body interaction often used in the Luttinger liquid framework. While (bare) RPA and RPAE agree for the case of a strictly linear dispersion, there are qualitative differences for the case of the usual nonrelativistic quadratic dispersion.

Topological Transition in the Fermi Surface of Cuprate Superconductors in the Pseudogap Regime

The angle-resolved photoemission spectroscopy studies on cuprates in the pseudogap region reveal an extraordinary topological transition in which the ground state changes from one with a normal Fermi surface to one with four Fermi points. Such a state is not possible without some symmetry breaking, which allows interference between one-particle basis states that is normally forbidden. We also show that the experimental results are quantitatively given without any free parameters by a theory and discuss the implications of the results.

Existence and topological stability of Fermi points in multilayered graphene

We study the existence and topological stability of Fermi points in a graphene layer and stacks with many layers. We show that the discrete symmetries (spacetime inversion) stabilize the Fermi points in monolayer, bilayer and multilayer graphene with orthorhombic stacking. The bands near $k=0$ and $\epsilon=0$ in multilayers with the Bernal stacking depend on the parity of the number of layers, and Fermi points are unstable when the number of layers is odd. The low energy changes in the electronic structure induced by commensurate perturbations which mix the two Dirac points are also investigated.

Diamagnetism of nodal fermions

Free nodal fermionic excitations are simple but interesting examples of fermionic quantum criticality in which the dynamic critical exponent $z=1$, and the quasiparticles are well defined. They arise in a number of physical contexts. We derive the scaling form of the diamagnetic susceptibility, $\chi$, at finite temperatures and for finite chemical potential. From measurements in graphene, or in $\mathrm{Bi_{1-x}Sb_{x}}$ ($x=0.04$), one may be able to infer the striking Landau diamagnetic susceptibility of the system at the quantum critical point. Although the quasiparticles in the mean field description of the proposed $d$-density wave (DDW) condensate in high temperature superconductors is another example of nodal quasiparticles, the crossover from the high temperature behavior to the quantum critical behavior takes place at a far lower temperature due to the reduction of the velocity scale from the fermi velocity $v_{F}$ in graphene to $\sqrt{v_{F}v_{\mathrm{DDW}}}$, where $v_{\mathrm{DDW}}$ is the velocity in the direction orthogonal to the nodal direction at the Fermi point of the spectra of the DDW condensate.

On the correlation measure of a family of commuting Hermitian operators with applications to particle densities of the quasi-free representations of the CAR and CCR

Let X be a locally compact, second countable Hausdorff topological space. We consider a family of commuting Hermitian operators a ( Δ ) indexed by all measurable, relatively compact sets Δ in X (a quantum stochastic process over X). For such a family, we introduce the notion of a correlation measure. We prove that, if the family of operators possesses a correlation measure which satisfies some condition of growth, then there exists a point process over X having the same correlation measure. Furthermore, the operators a ( Δ ) can be realized as multiplication operators in the L 2 -space with respect to this point process. In the proof, we utilize the notion of ⋆-positive definiteness, proposed in [Y.G. Kondratiev, T. Kuna, Harmonic analysis on the configuration space I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002) 201–233]. In particular, our result extends the criterion of existence of a point process from that paper to the case of the topological space X, which is a standard underlying space in the theory of point processes. As applications, we discuss particle densities of the quasi-free representations of the CAR and CCR, which lead to fermion, boson, fermion-like, and boson-like (e.g. para-fermions and para-bosons of order 2) point processes. In particular, we prove that any fermion point process corresponding to a Hermitian kernel may be derived in this way.

A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an Application

Given a positive definite, bounded linear operator A on the Hilbert space $$\mathcal{H}$$ 0≔l 2(E), we consider a reproducing kernel Hilbert space $$\mathcal{H}$$ + with a reproducing kernel A(x,y). Here E is any countable set and A(x,y), x,y∊ E, is the representation of A w.r.t. the usual basis of $$\mathcal{H}$$ 0. Imposing further conditions on the operator A, we also consider another reproducing kernel Hilbert space $$\mathcal{H}$$ − with a kernel function B(x,y), which is the representation of the inverse of A in a sense, so that $$\mathcal{H}$$ −⊃ $$\mathcal{H}$$ 0⊃ $$\mathcal{H}$$ + becomes a rigged Hilbert space. We investigate the ratios of determinants of some partial matrices of A and B. We also get a variational principle on the limit ratios of these values. We apply this relation to show the Gibbsianness of the determinantal point process (or fermion point process) defined by the operator A(I+A)−1 on the set E.