## Explorations in Dirac Fermions and Spin Liquids A dissertation presented by (2011)

### BibTeX

@MISC{Biswas11explorationsin,

author = {Rudro Rana Biswas and Subir Sachdev and Rudro Rana Biswas},

title = {Explorations in Dirac Fermions and Spin Liquids A dissertation presented by},

year = {2011}

}

### OpenURL

### Abstract

A significant portion of this dissertation is devoted to the study of the effects of impurities in substances whose low energy modes can be described by fermions obeying the gapless Dirac equation in 2+1 dimensions. First, we examine the case of a spin vacancy in the staggered flux spin liquid whose excitations are Dirac fermions coupled to a U(1) gauge field. This vacancy leads to an anomalous Curie susceptibility and does not induce any local orders. Next, a Coulomb charge impurity placed on clean graphene is considered. We find that the Dirac quasiparticles in graphene do not screen the impurity charge, to all orders in perturbation theory. However, electronic correlations are found to induce a cloud of charge having the same sign as the impurity charge. We also analyze the case of a local impurity in graphene in the presence of a magnetic field and derive the spatial fourier transform of tunneling spectroscopy data obtained on an almost-clean

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Citation Context ...ion of scattering of Dirac quasiparticles in graphene. One of the striking features discussed in the past is the so called Klein paradox where particles can tunnel without backscattering under barrier=-=[95, 65, 10]-=-. The situation we address here is similar yet different: the additional symmetry which is present in our analysis but absent in graphene is the exact suppression of backscattering for any time revers... |

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Citation Context ...sChapter 1: Introduction 22 proposed in Chapter 7 a spin liquid theory composed of deconfined spin 1 Majorana fermion spinons. The analysis is based on the following parton representation of spin half=-=[85, 146, 24, 133, 126]-=- S µ = i 4 ɛµαβ γ α γ β , (1.21) where the various indices run over x, y, z and the γ α is a Majorana fermion (γ α ) † = γ α , { γ α , γ β } = 2δαβ (1.22) There is a Z2 gauge redundancy in this formul... |

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Citation Context ...nd I cannot wait to travel down this path with her.To my lovely wife, Srividya. xiiiChapter 1 Introduction 1.1 Dirac Fermions The Dirac Equation In 1926, Erwin Schrödinger wrote down a wave equation=-=[125]-=- which described the time evolution of the quantum mechanical wave function of a non-relativistic particle: i� ∂ ( ψ(x, t) = − ∂t �2∇2 ) + V (x) ψ(x, t) (1.1) 2m Using the correspondence E → i�∂/∂t an... |

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Citation Context ...ontrivial phase with extended (spin-polarized) edge modes if the well depth was tuned above a critical value. This prediction was realized just a year later by Konig and others [70]. In a later paper =-=[64]-=-, Kane and Mele gave the first argument that the special phase of graphene discussed above is one of two ‘topologically-classified’ states that are possible for an electronic band-structure – the ‘nor... |

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Citation Context ...the Fermi surface. The second effect arises due to the wave nature of the electrons and leads to an oscillatory component of the response. This phenomenon was discovered by and is named after Friedel =-=[37]-=-. These oscillations occur at the Fermi wave vector kF . In graphene, at the neutrality point, both kF and the density of states at the Fermi level ∝ λ −1 D are zero. As a result, there is no conventi... |

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Citation Context ... exchangeChapter 7: SU(2)-invariant spin liquids on the triangular lattice with spinful Majorana excitations 106 interactions in equation (7.5) constitute the gauge fluctuations of a Z2 gauge theory =-=[113, 129]-=-. These have a finite range of stability without a transition to confinement, when our theory as described above is valid. Next, we consider the influence of terms quartic in the Majorana fermions. Th... |

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Citation Context ...is reason, materials were sought which contained extended electronic states that are forced to be spin-polarized because of spin-orbit coupling. One such possibility was shown in 2005 by Kane and Mele=-=[63]-=- – they argued that there existed a nontrivial phase in graphene with strong enough spin-orbit coupling where extended edge modes had to exist that were spin-polarized. For the edge states to exist, g... |

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Citation Context ...ts appears thrice on the boundary, contributing one-third of the Dirac quasiparticles in each case. The ARPES data on the right showing the measured Dirac dispersion at the K point is reproduced from =-=[102]-=-, while the three-dimensional band structure is reproduced from the review [163]. Coulomb Impurity To understand the response to charged impurities in metals, one takes into account two main aspects o... |

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Citation Context ...ith either a finite [51, 100, 60] or vanishing [38, 149] density of states at the Fermi energy. More recently, the impurity responses of a variety of ‘non-Fermi-liquid’ bulk states have been studied. =-=[62, 32, 33, 120, 127, 148, 121, 118, 182, 177, 138, 145, 122, 52, 35]-=- The S = 1/2 antiferromagnetic spin chain generically has a critical ground state, and displays interesting universal characteristics in its response to impurities or boundaries [32, 33]. Universality... |

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Citation Context ...tudy in some detail the physics associated with a single impurity carrying electrical charge Z. Nanoscale studies of the electronic properties of a single graphene sheet have recently become possible =-=[84, 141]-=-, and so it should eventually be possible to observe the variation in the charge density and the local density of states as a function of distance from the impurity. We shall show here that this spati... |

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Citation Context ... the staggered configuration of the winding fluxes given by the hopping ansatz. The spinon dispersion in this phase has gapless Dirac nodes at the points (±π/2, ±π/2) in the Brillouin zone. Wen showed=-=[156]-=- that this state partly breaks the SU(2) symmetry down to a U(1) symmetry group. Rantner and Wen[111] showed that these U(1) gauge fluctuations strongly modify the properties of the Dirac quasiparticl... |

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Citation Context ... kx = 0, π because reversing them brings one back to the same point in the (periodic) Brillouin zone.Chapter 1: Introduction 12 In three dimensional band insulators an analogous classification exists=-=[42, 40, 89, 116]-=- which is similar in spirit in that it relies upon the concepts of Kramers degeneracy and the switching/retaining of Kramers pairs as one moves between the four TRIMd of the surface Brillouin zone. We... |

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Citation Context ...ith either a finite [51, 100, 60] or vanishing [38, 149] density of states at the Fermi energy. More recently, the impurity responses of a variety of ‘non-Fermi-liquid’ bulk states have been studied. =-=[62, 32, 33, 120, 127, 148, 121, 118, 182, 177, 138, 145, 122, 52, 35]-=- The S = 1/2 antiferromagnetic spin chain generically has a critical ground state, and displays interesting universal characteristics in its response to impurities or boundaries [32, 33]. Universality... |

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Citation Context ...tudy in some detail the physics associated with a single impurity carrying electrical charge Z. Nanoscale studies of the electronic properties of a single graphene sheet have recently become possible =-=[84, 141]-=-, and so it should eventually be possible to observe the variation in the charge density and the local density of states as a function of distance from the impurity. We shall show here that this spati... |

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Citation Context ... kx = 0, π because reversing them brings one back to the same point in the (periodic) Brillouin zone.Chapter 1: Introduction 12 In three dimensional band insulators an analogous classification exists=-=[42, 40, 89, 116]-=- which is similar in spirit in that it relies upon the concepts of Kramers degeneracy and the switching/retaining of Kramers pairs as one moves between the four TRIMd of the surface Brillouin zone. We... |

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Citation Context ...edom. This can be very useful as we shall see below. Recently, a couple of organic charge-transfer salts have been shown to possess interesting spin liquid-like ground states – [κ-(BEDT-TTF)2Cu2(CN)3]=-=[134]-=- and EtMe3Sb[Pd(dmit)2]2[58]. Both of these feature triangular lattices with spin 1/2s with strong antiferromagnetic Heisenberg interactions. In the latter case, which we shall henceforth refer to as ... |

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Citation Context ...avior have been written down and there is a consensus that this phenomenon is controlled by scattering from charged impurities in the vicinity of the graphene sheet, possibly embedded in the substrate=-=[3]-=-. Understanding the effects of charged impurities in graphene near the neutrality point is thus a very important step towards understanding its transport properties in the lightly-doped regime.scale ... |

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Citation Context ...y at leastChapter 6: Scattering from Surface Step Edges in Strong Topological Insulators 90 as fast as (Figure 6.1) δρ(ω, x) ρ (0) (ω) ∼ 1 x (6.1) In contrast, in 2DEGs this decay is slower ∼ 1/ √ x =-=[25]-=-. (iii) We predict the existence of the ‘perfectly reflecting’ step edge (6.15) by using the scattering matrix restrictions. This perfectly reflecting wall produces LDOS modulations of the kind mentio... |

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Citation Context ...tion enjoyed by this kind of surface band crossing and makes these materials an attractive candidate for spintronics applications[91] as well as a possible platform for topological quantum computation=-=[41]-=-. In this context an important issue is the stability of the STI surface nodes to the presence of impurities[124, 94, 46]. We contribute to this discussion by looking at the modification of surface st... |

6 |
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Citation Context ...pological insulators in materials with inversion symmetry[40]. Following that, numerous materials have been experimentally discovered both preceding (like Bi2Se3[166]) and following (like Bi-Sb alloys=-=[53]-=- and Bi2Te3[21]) their theoretical prediction. The surfaces of Bi2Se3 and Bi2Te3 host single Dirac cones at the Brillouin zone center (k = 0, the Γ point). The equation for the effective Hamiltonian n... |

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Citation Context ...Scienta R4000 electron energy analyzer. The graphene’s valence bands, which exhibit a linear dispersion were cooled to 30 K by liquid He. The photo degenerate near the so-called Dirac point energy ED =-=[10]-=-. was 94 eV with the overall energy resolution of These unconventional Chapter 1: properties Introduction of graphene offer a new for Figs. 1 and 2(a)–2(d). 5 route to room temperature, molecular-scal... |

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Citation Context ...of the band structure to the charge distribution brought about by HDirac = the interplay between strong interlayer hopping and weak interlayer screening, neither of which is currently well understood =-=[13,14]-=-. In order to evaluate the interlayer screening, stacking order,andinterlayer coupling, Graphene we band have structure systematically studied cσ · k The band structures of a single [Fig. 1(a)] and [1... |

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Citation Context ... — i.e, V(r) is a general hermitian 2×2 matrix function of r (V(r) = V †(r)). 4.4 Results Scanning Tunneling Microscopes (STMs) give a signal corresponding to the local value of the spectral function =-=[19]-=-. We can use the Green’s function derived above to obtain the spatial FT of the change in the LDOS (given by the change in the spectralChapter 4: Quasiparticle Interference and Landau Level Spectrosc... |

5 |
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Citation Context ... kx = 0, π because reversing them brings one back to the same point in the (periodic) Brillouin zone.Chapter 1: Introduction 12 In three dimensional band insulators an analogous classification exists=-=[42, 40, 89, 116]-=- which is similar in spirit in that it relies upon the concepts of Kramers degeneracy and the switching/retaining of Kramers pairs as one moves between the four TRIMd of the surface Brillouin zone. We... |

5 |
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Citation Context ...t prominent example is the Kondo effect, which describes the interplay between a variety of impurities with a spin and/or ‘flavor’ degree of freedom and a system of free fermions with either a finite =-=[51, 100, 60]-=- or vanishing [38, 149] density of states at the Fermi energy. More recently, the impurity responses of a variety of ‘non-Fermi-liquid’ bulk states have been studied. [62, 32, 33, 120, 127, 148, 121, ... |

5 |
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Citation Context ...lationship of our results to these works in Section 2.3.2. There are a number of experimental motivations for our work. A large number of experiments have studied Zn and Ni impurities in the cuprates,=-=[18, 104, 173, 105, 55]-=- and much useful information has been obtained on the spatial and temperature dependence of the induced moments around the impurity. It would clearly be useful to compare these results with the corres... |

5 |
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Citation Context ...per site, and J = 4t 2 /U. Here S areChapter 1: Introduction 18 the spin operators. One method to try and solve this model is to use the following representation of the electron annihilation operator=-=[4, 159]-=-. We begin by introducing auxiliary fermionic fields fs(x) and bosonic fields bs(x), with s = 1, 2. These can be combined as follows Ψ1 = ⎛ ⎜ ⎝ f1 f † 2 ⎞ ⎟ ⎠ , Ψ2 = ⎛ ⎜ ⎝ f2 −f † 1 ⎞ ⎟ ⎠ , b = ⎛ ⎜ ⎝ ... |

4 |
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Citation Context ...used above, �, vF and W are unity. 5.2 Introduction The Dirac spectrum of chiral excitations are realized in a wide range of materials including d-wave superconductors[7], graphene[95], semiconductors=-=[36]-=- and superfluid 3He-A[150]. The Dirac spectrum brings in substantial similarities in electronic properties – like response to defects as well as low energy and low temperature properties. It is thus n... |

4 |
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Citation Context .... The screening charge is then n(q) = −Z λ(q) 1 + λ(q) , λ(q) = g2N . (3.21) 32v(q) The problem is now reduced to the problem of finding v(q) [or, equivalently, λ(q)]. This problem has a long history =-=[44, 45]-=-; most recently it has been revisited in Ref. [139] (see also below). To find the spatial charge distribution n(r) one needs to take Fourier transform of Eq. (3.21). First one notice that if the veloc... |

4 |
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Citation Context ...rong topological Insulators (STI) are three-dimensional band insulators that have an odd number of gapless chiral modes on their surfaces[42, 89, 178]. These have recently been realized experimentally=-=[166, 115, 54, 43]-=- and are an active area of current research. The chiral states on the STI surfaces consist of time-reversed pairs of states propagating in opposite directions, between which backscattering by time-rev... |

4 | Classification of Topological Insulators and Superconductors, American institute of physics conference series
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Citation Context ...the surface while being fully gapped in the bulk. In addition, STIs are unique because the topology of their bulk band structure constrains their surface states to possess an odd number of Dirac nodes=-=[42, 124]-=-. Suppressed backscattering inside the odd Dirac cone guarantees that the Dirac dispersion remain essentially unperturbed for any perturbation to the Hamiltonian that preserves time reversal symmetry.... |

4 |
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Citation Context ...number of S = 1/2 spins per unit cell, and consequently their bulk quantum criticality is described within the conventional Landau-GinzburgWilson (LGW) framework of a fluctuating Néel order parameter.=-=[130, 128, 131]-=- Away from the impurity, such systems only have excitations which carry integer spin. It is the purpose of this chapter to extend the above theory [120, 148, 122] to fractionalized ‘spin liquid’ state... |

4 | Deconfined quantum critical points
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Citation Context ...number of S = 1/2 spins per unit cell, and consequently their bulk quantum criticality is described within the conventional Landau-GinzburgWilson (LGW) framework of a fluctuating Néel order parameter.=-=[130, 128, 131]-=- Away from the impurity, such systems only have excitations which carry integer spin. It is the purpose of this chapter to extend the above theory [120, 148, 122] to fractionalized ‘spin liquid’ state... |

4 | Quantum orders and symmetric spin liquids
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Citation Context ...ears – requiring spin rotation invariance and adherence to all lattice symmetries, modulo a gauge transformation. This procedure is the same as that of using the projective symmetric group (PSG – see =-=[157]-=-). It turns out that any quadratic Majorana Hamiltonian breaks time reversal and inversion symmetries, and our theory is restricted to have the symmetry of combined time reversal and an elementary lat... |

4 |
Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat
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Citation Context ...ion that they discovered for finding topological insulators in materials with inversion symmetry[40]. Following that, numerous materials have been experimentally discovered both preceding (like Bi2Se3=-=[166]-=-) and following (like Bi-Sb alloys[53] and Bi2Te3[21]) their theoretical prediction. The surfaces of Bi2Se3 and Bi2Te3 host single Dirac cones at the Brillouin zone center (k = 0, the Γ point). The eq... |

3 |
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Citation Context ...he effects of charged impurities in graphene near the neutrality point is thus a very important step towards understanding its transport properties in the lightly-doped regime.scale coherence length =-=[7,8]-=-, and novel many-body cou- Electronic Structure Factory end station at beam plings [9].Theseeffectsoriginatefrom the effectively mass- of the Advanced Light Source (ALS), equippe less Dirac fermion ch... |

3 |
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(Show Context)
Citation Context ... state energy 9N J 4 g ( g − 2 π ) 1 e is minimized when g = : π E0 = − 9 JN = −0.22JN (7.11) 4π2 As expected, E0 is higher than the numerically calculated ground state energy of about −0.54J per site=-=[15]-=- for the best candidate spin-ordered ground state[56]. Additional interactions should be added to the Heisenberg Hamiltonian in equation (7.5) to stabilize the spin liquid state described by the non-i... |

3 |
Experimental realization of a three-dimensional topological insulator
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(Show Context)
Citation Context ...ators in materials with inversion symmetry[40]. Following that, numerous materials have been experimentally discovered both preceding (like Bi2Se3[166]) and following (like Bi-Sb alloys[53] and Bi2Te3=-=[21]-=-) their theoretical prediction. The surfaces of Bi2Se3 and Bi2Te3 host single Dirac cones at the Brillouin zone center (k = 0, the Γ point). The equation for the effective Hamiltonian near those Dirac... |