Quasihole States Research Articles

Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect

We analyze pairing of fermions in two dimensions for fully gapped cases with broken parity (P) and time reversal (T), especially cases in which the gap function is an orbital angular momentum (l) eigenstate, in particular $l=\ensuremath{-}1$ (p wave, spinless, or spin triplet) and $l=\ensuremath{-}2$ (d wave, spin singlet). For $l\ensuremath{\ne}0,$ these fall into two phases, weak and strong pairing, which may be distinguished topologically. In the cases with conserved spin, we derive explicitly the Hall conductivity for spin as the corresponding topological invariant. For the spinless p-wave case, the weak-pairing phase has a pair wave function that is asympototically the same as that in the Moore-Read (Pfaffian) quantum Hall state, and we argue that its other properties (edge states, quasihole, and toroidal ground states) are also the same, indicating that nonabelian statistics is a generic property of such a paired phase. The strong-pairing phase is an abelian state, and the transition between the two phases involves a bulk Majorana fermion, the mass of which changes sign at the transition. For the d-wave case, we argue that the Haldane-Rezayi state is not the generic behavior of a phase but describes the asymptotics at the critical point between weak and strong pairing, and has gapless fermion excitations in the bulk. In this case the weak-pairing phase is an abelian phase, which has been considered previously. In the p-wave case with an unbroken $U(1)$ symmetry, which can be applied to the double layer quantum Hall problem, the weak-pairing phase has the properties of the 331 state, and with nonzero tunneling there is a transition to the Moore-Read phase. The effects of disorder on noninteracting quasiparticles are considered. The gapped phases survive, but there is an intermediate thermally conducting phase in the spinless p-wave case, in which the quasiparticles are extended.

Quantum Monte Carlo calculations of the one-body density matrix and excitation energies of silicon

Quantum Monte Carlo (QMC) techniques are used to calculate the one-body density matrix and excitation energies for the valence electrons of bulk silicon. The one-body density matrix and energies are obtained from a Slater-Jastrow wave function with a determinant of local density approximation (LDA) orbitals. The QMC density matrix evaluated in a basis of LDA orbitals is strongly diagonally dominant. The natural orbitals obtained by diagonalizing the QMC density matrix resemble the LDA orbitals very closely. Replacing the determinant of LDA orbitals in the wave function by a determinant of natural orbitals makes no significant difference to the quality of the wave function's nodal surface, leaving the diffusion Monte Carlo energy unchanged. The Extended Koopmans' Theorem for correlated wave functions is used to calculate excitation energies for silicon, which are in reasonable agreement with the available experimental data. A diagonal approximation to the theorem, evaluated in the basis of LDA orbitals, works quite well for both the quasihole and quasielectron states. We have found that this approximation has an advantageous scaling with system size, allowing more efficient studies of larger systems.

Center-of-mass effects on the quasihole spectroscopic factors in the16O(e,e′p)reaction

The spectroscopic factors for the low-lying quasihole states observed in the O-16(e,e'p)N-15 reaction are reinvestigated with a variational Monte Carlo calculation for the structure of the initial and final nucleus. A computational error in a previous report is rectified. It is shown that a proper treatment of center-of-mass motion does not lead to a reduction of the spectroscopic factor for p-shell quasihole states, but rather to a 7% enhancement. This is in agreement with analytical results obtained in the harmonic oscillator model. The center-of-mass effect worsens the discrepancy between present theoretical models and the experimentally observed single-particle strength. We discuss the present status of this problem, including some other mechanisms that may be relevant in this respect. [S0556-2813(98)03005-2].

Quasiholes and fermionic zero modes of paired fractional quantum Hall states: The mechanism for non-Abelian statistics.

The quasihole states of several paired states, the Pfaffian, Haldane-Rezayi, and 331 states, which under certain conditions may describe electrons at filling factor \ensuremath{\nu}=1/2 or 5/2, are studied analytically and numerically in the spherical geometry, for the Hamiltonians for which the ground state are known exactly. We also find all the ground states (without quasiparticles) for these systems in the toroidal geometry. In each case, a complete set of linearly independent functions that are energy eigenstates of zero energy is found explicitly. For fixed positions of the quasiholes, the number of linearly independent states is ${2}^{\mathit{n}\mathrm{\ensuremath{-}}1}$ for the Pfaffian, and ${2}^{2\mathit{n}\mathrm{\ensuremath{-}}3}$ for the Haldane-Rezayi state; these degeneracies are needed if these systems are to possess non-Abelian statistics, and they agree with predictions based on conformal field theory. The dimensions of the spaces of states for each number of quasiholes agree with numerical results for moderate system sizes. The effects of tunneling and of the Zeeman term are discussed for the 331 and Haldane-Rezayi states, as well as the relation to Laughlin states of electron pairs. A model introduced by Ho, which was supposed to connect the 331 and Pfaffian states, is found to have the same degeneracies of zero-energy states as the 331 state, except at its Pfaffian point where it is much more highly degenerate than either the 331 or the Pfaffian. We introduce a modification of the model which has the degeneracies of the 331 state everywhere including the Pfaffian point; at the latter point, tunneling reduces the degeneracies to those of the Pfaffian state. An experimental difference is pointed out between the Laughlin states of electron pairs and the other paired states, in the current-voltage response when electrons tunnel into the edge. An appendix contains results for the permanent state, in which the zero modes can be occupied by composite bosons, rather than by composite fermions as in the other cases; the system is found to have an incipient instability toward a spin-polarized state. \textcopyright{} 1996 The American Physical Society.

Variational description of a quasihole excitation in a quantum antiferromagnet.

An accurate variational ansatz is proposed for a single quasihole excitation in a two-dimensional quantum antiferromagnet. The trial state includes spin-hole and spin-hole-spin correlations, in addition to background antiferromagnetic spin-spin correlations. Background spin-spin correlations are induced by the Heisenberg antiferromagnetic exchange interaction and describe the zero-point motion of spin waves around the Ne\ifmmode\acute\else\textasciiacute\fi{}el state. Spin-hole and spin-hole correlations describe ``strings'' of spins displaced by one lattice site along the hole path. This is a generalization of the Brinkman-Rice approach to include quantum spin fluctuations. We have used the Monte Carlo method to compute the variational energy and the spectral weight of the quasihole state in the t-J model on the square lattice, with the proposed ansatz. Comparison with exact results for the 4\ifmmode\times\else\texttimes\fi{}4 lattice shows very good agreement for both energy and spectral weight, at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) in the range 0\ensuremath{\le}t/J\ensuremath{\le}5. We report results for lattices of several sizes up to 16\ifmmode\times\else\texttimes\fi{}16 in the above range of t/J and for the two values of k where the hole energy band attains its minimum [k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2)] and maximum [k=(0,0)].

Spectral function at finite doping in the t-J model.

We investigate the effect of doping in the hole spectral function by exact diagonalization of a 16-site cluster with two holes. It is shown that for more than one hole the spectral function differs qualitatively from the one in the single-hole limit. In particular, we find no evidence for the formation of a Fermi surface having hole ``pockets'' localized around (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) in momentum space. On the contrary, a large low-energy peak appears in the spectral function of the hole propagator at k=(\ensuremath{\pi},0) or (0,\ensuremath{\pi}) which is then the best candidate for the existence of those pockets. A (\ensuremath{\pi}/2,\ensuremath{\pi}/2) quasihole state is obtained as a small energy excitation of the two-hole system. We discuss some consequences for photoemission and optical-conductivity experiments in high-${\mathit{T}}_{\mathit{c}}$ materials at intermediate doping.

Single-particle orbitals in liquid-helium drops.

Drops containing 20 to 240 atoms of Bose liquid /sup 4/He and Fermi liquid /sup 3/He are studied by variational Monte Carlo methods as simple examples of correlated inhomogeneous quantum systems. In the present work we report wave functions of natural, quasiparticle, and mean-field orbitals in these drops. The wave functions of natural orbitals are determined by diagonalizing the one-particle density matrix. The condensate fraction and wave function in the Bose-liquid drops are calculated. The quasihole states are defined in the spirit of Landau's theory, and their wave functions are calculated from the overlaps between states containing N and N-1 atoms. In Bose-liquid drops the wave function of the quasihole orbital is similar to that of the condensate; however, in Fermi-liquid drops the quasihole wave functions are different from those of natural or mean-field orbitals. We find that a simple local-density approximation provides an accurate relation between the mean-field and quasihole wave functions for both Bose- and Fermi-liquid drops. The wave functions of natural orbitals are very localized and simple methods to construct them from mean-field wave functions are also discussed. The momentum distributions of atoms in the drops are calculated and compared with those of extended liquids. The deviationsmore » of the momentum distributions from mean-field distributions are also discussed.« less

Monte Carlo evaluation of trial wave functions for the fractional quantized Hall effect: Disk geometry.

Monte Carlo methods have been employed to evaluate the energy of two previously proposed trial wave functions for the quasiparticle at the $\ensuremath{\nu}=\frac{1}{3}$ quantized Hall state of the two-dimensional electron system. The two wave functions have the same energy within our statistical accuracy, and are consistent with a value ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\varepsilon}}}_{+}(\frac{1}{3})\ensuremath{\approx}\frac{(0.073\ifmmode\pm\else\textpm\fi{}0.008){e}^{2}}{\ensuremath{\epsilon}{l}_{o}}$, where ${l}_{0}$ is the magnetic length, and $\ensuremath{\epsilon}$ the background dielectric constant. Simulations of the quasihole state confirm previous estimates of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\varepsilon}}}_{\ensuremath{-}}(\frac{1}{3})\ensuremath{\approx}\frac{0.026{e}^{2}}{\ensuremath{\epsilon}{l}_{0}}$. We have also studied the charge distributions of the quasiparticle and quasihole states, and we have evaluated the energies of a previously proposed microscopic trial wave function for the ground state at $\ensuremath{\nu}=\frac{2}{5}, \frac{2}{3}, \mathrm{and} \frac{2}{7}$.

Exact results for the fractional quantum Hall effect with general interactions.

Laughlin's state ${\ensuremath{\psi}}_{m}$ is shown to be the exact nondegenerate ground state for repulsive interactions of vanishing range. In this limit, his quasihole states are not exact, and some previously proposed states are ruled out. An experimental prediction is made concerning the competition of ${\ensuremath{\psi}}_{m}$ with charge-density waves. Several exact properties are shared by systems with interactions of shorter range than ${r}^{\mathrm{\ensuremath{-}}2}$, for which the center-of-mass motion separates from the other degrees of freedom. These include new invariant subspaces, an operator that creates exact eigenstates, and a subset property of the energy eigenvalues.

Particle-hole symmetry in the anomalous quantum Hall effect

This paper explores the uses of particle-hole symmetry in the study of the anomalous quantum Hall effect. A rigorous algorithm is presented for generating the particle-hole dual of any state. This is used to derive Laughlin's quasihole state from first principles and to show that this state is exact in the limit $\ensuremath{\nu}\ensuremath{\rightarrow}1$, where $\ensuremath{\nu}$ is the Landau-level filling factor. It is also rigorously demonstrated that the creation of $m$ quasiholes in Laughlin's state with $\ensuremath{\nu}=\frac{1}{m}$ is precisely equivalent to creation of one true hole. The charge-conjugation procedure is also generalized to obtain an algorithm for the generation of a hierarchy of states of arbitrary rational filling factors.

Deep-lying hole states in nuclei: Microscopic approach

The strength function for deep-lying hole states in a nucleus is examined from a many-body point of view. Due to their interaction with the compound state background, such single hole excitations are interpreted as quasihole states that are not eigenstates of the nuclear Hamiltonian. These states show up as giant resonances in the strength function, with position and width determined by the real and imaginary parts of the quasihole energy. A formal theory of the strength and fragmentation of such states is developed by splitting the self-energy into background and doorway state contributions. The theory is applied to the calculation of the strength function for the isotopes of Sn using doorway states of a collective nature that consist of a hole plus collective vibrations of the target nucleus. A microscopic description of both the collective excitations and the hole state that it dresses is given in terms of a modified random phase approximation procedure that uses Green's functions for the individual single particle and single hole states that have been dressed by their interaction with the nuclear background. Specific calculations for the isotopes of Sn show good agreement with experiment.

Single-particle states in nuclear matter and in finite nuclei

The complex single-particle field in nuclear matter is computed in the framework of Brueckner's theory, in the case of the realistic nucleon-nucleon interaction of Hammann and Ho-Kim. The Hartree-Fock, the core polarization, and the correlation contributions are calculated; renormalization corrections are also evaluated. The results are compared with those derived from a dispersion relation approach. Particular attention is paid to the behavior of the effective mass near the Fermi surface, where it presents a narrow enhancement. Following a recent suggestion by Orland and Schaeffer, the nuclear matter results are adapted to finite nuclei by assuming that the relevant variable is the difference between the single-particle energy and the Fermi energy. This yields fair agreement between the strength functions for quasihole states calculated here and those measured from (e,e'p) knockout processes.

Analysis of theEr168(d,t)Er167reaction at 17 MeV

Angular distributions have been measured for transitions populating residual states of $^{167}\mathrm{Er}$ through the $^{168}\mathrm{Er}(d,t)$ reaction at ${E}_{d}=17$ MeV. Many of these angular distributions have shapes which are well reproduced by distorted-wave Born-approximation calculations. A small but significant number of angular distributions are anomalous; i.e., either they do not resemble any reasonable distorted-wave Born-approximation calculation or they can only be fitted with a calculation which assumes an $l$ value incompatible with the previous Nilsson model assignment for the residual state. While the summed spectroscopic factors are in good agreement with Nilsson model expectations, there is considerable evidence of fragmentation of the single-quasihole strength and the spectroscopic factors for several levels deviate significantly from Nilsson model predictions, even though Coriolis coupling has been included in the model calculation. The observation of several strongly anomalous angular distributions almost certainly indicates that the assumption of a simple one-step direct reaction mechanism breaks down for some of these transitions. The discrepancies between the model prediction and the experimental spectroscopic factors could arise either from a multistep mechanism or from mixing of the simple quasihole states with more complicated states of the same spin-parity. The observed fragmentation of strength is strong evidence for such mixing.NUCLEAR REACTIONS $^{168}\mathrm{Er}(d,t)$, ${E}_{d}=17$ MeV measured $\ensuremath{\sigma}(\ensuremath{\theta})$; DWBA analysis, deduced levels, $l$ values, spectroscopic factors. Enriched targets.

Deep Hole States and (e, e'N) Reactions

Quasielastic electron scattering by nuclei is considered on the basis of the many-bucly theory. Properties of the hole states are described. Cross sections for the (e, e'p) and (e, e 1n) reactions are obtained. The role of nucleon correlations in these reactions is also discussed. § l. Introduction The hole state in the many-body system is a basic type of excitation. Analysis of the hole states in an atomic nucleus yields information on the single-particle properties of nuclei and on how these are affected by the correlation interaction betvveen nucleons. Deep-lying hole states are created by reactions of the types (jJ, 2p), (e, e'jJ), (P, d) and (d', He),n~sJ etc. In the analysis one assumes that such reactions, initiated with bombarding energies of several hundred MeV, can be described as direct or knock-out process involving only a single collision between an incident projectile and one of the target nucleous. Usually multiple scattering and compound nucleus formation are only taken into account through a complex optical model potential describing the motion of the projectiles before and after the collision. Deep-lying quasi-hole states (dressed holes) are there­ fore expected to lead to peaks in the cross section as a function of transferred energy. vVith quasi-free secattering one can measure the energ1es, lifetimes and momentum distributions of the dressed hole states. Experiments ha\·e pointed to a complicated nature for the deep hole states of the nucleus. The peaks observed in the summed energy spectra have large widths and the centre of these peaks lie much lower than those predicted by the independent-particle model. In this paper, we investigate the hole states and (e, e' N) reactions within the framework of the many-body theory! 1 ~ 61 Our approach has an advantage that it does not depend on any particular model from the begin­ ning.71~111 In § 2 the properties of the mass operator for the hole states are con­ sidered. In this way the mass operator, which corresponds to the nuclear potential for quasihole states, is nonlocal, complex and energy-dependent due to correlation interactions. This leads to a qualitative explanatio11- for the singularities in the hole-excitation spectrum. In § 3 the (e, e'lJ) reactions are investigated by using Green's function method. The cross~section obtained has a resonance form. In

Lifetime of nuclear hole states caused by phonon-hole coupling

Quasihole states in the 1d shells of 36,38,40Ar and 40,42,44,48Ca have been excited with the (d, 3He) reaction at 52 MeV. A phonon-hole coupling model reproduces the measured spreading widths within typically 10 percent.

The quasihole states in the model for singly filled orbitals

The quasihole states in the model for singly filled orbitals

Cumulant Expansion of Localized-Electron Model for Antiferromagnetic Insulators

A many-body perturbation theory is developed for the calculation of the antiferromagnetic ground state and lower excited states of one-electron atoms in the insulating state. It is assumed that the interatomic distances are moderately small and that the interactions are well described by exchange coupling, but that distortion of atoms in the lattice is still negligible. The method is essentially the Heitler-London approach: A set of Slater determinants is constructed from nonorthogonal atomic orbitals with all possible spin arrangements. The unperturbed ground state $|0〉$ has an alternating spin configuration, while excited states are generated one by one from $|0〉$ by interchanges of spins and by projecting out the lower excited states. The resulting orthonormal states introduce the concept of quasielectron (or quasihole) states and define creation and destruction operators of those quasiparticles. Although the quasiparticles are correlated with each other through the overlap integrals in the projected wave functions, the energy matrix elements can be decomposed into linked clusters of localized electrons and holes. It is found that the clusters thus generated are cumulants. The expression for the energy matrix is then transformed into the form of an effective Hamiltonian consisting of cumulants and of creation and destruction operators of quasifermions, and the concept of the exchange interaction is defined in terms of those operators. The new formulation not only facilitates the use of the familiar many-body perturbation theory, but also eliminates difficulties in handling spin variables in the Heisenberg theory of magnetism. If only the two-body Coulomb and exchange diagrams are retained and the higher-order diagrams are neglected, however, the effective Hamiltonian takes a form reminiscent of the anisotropic exchange Hamiltonian, ${H}_{\mathrm{spin}}=\ensuremath{-}{\ensuremath{\Sigma}}_{k,h}{J}_{\mathrm{kh}}[{{S}_{k}}^{z}{{S}_{h}}^{z}+\ensuremath{\lambda}({{S}_{k}}^{x}{{S}_{h}}^{x}+{{S}_{k}}^{y}{{S}_{h}}^{y})]$, which is used in the generalized Heisenberg theory and which includes the Ising model in its limit $\ensuremath{\lambda}\ensuremath{\rightarrow}0$. Finally, it is shown that excitations accompanying electron transfers are readily included in the present method as additional diagrams by an extension of the prescriptions for the generation of excited spin states. This yields, in its first approximation, the intra-atomic interaction and the electron-transfer interaction, leading to the Hamiltonian assumed by Hubbard and Anderson.