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Dirac Fermions on a 2D lattice in the presence of defects : with possible applications to quantum computation Weeks, William Conan 2007

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Dirac Fermions on a 2D lattice in the presence of defects With possible applications to quantum computation by William Conan Weeks B . S c , Carleton University, 2004 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FORTHE DEGREE OF Master of Science  in The Faculty of Graduate Studies (Physics)  The University Of British Columbia January 2007 © William Conan Weeks 2007  A b s t r a c t Dirac Fermions on 2 D lattice are studied in the presence of a single flux tube, a pair of flux tubes and an edge dislocation/anti-dislocation pair. N u merical studies have been carried out using exact diagonalization for each case, and the results indicate that the model supports fractionally charged particles with anyonic statistics. A n analytical calculation is also provided, predicting a zero energy bound state in the continuum for Dirac Fermions in the presence of a single flux tube carrying one-half flux quantum, as well as a brief discussion of anyonic quasiparticles and their possible application to quantum computation.  ii  Table of Contents Abstract T a b l e of C o n t e n t s L i s t of F i g u r e s Acknowledgements 1  ii iii v vi  Introduction  1  1.1 1.2 1.3 1.4 1.5  1 1 2 5 6  Context of Current Project Topological Quantum Computation Anyons and Braiding The Fractional Quantum Hall Effect P-wave superconductivity  2  Dirac Fermions on the Lattice 2.1 Motivation 2.2 Real Space 2.3 Momentum space  3  Continuum Limit  14  4  Numerical Results 4.1 Method 4.2 Clean Lattice 4.3 Flux Tube 4.4 Flux Pair 4.5 Dislocations  18 18 18 19 20 23  5  Discussion and Outlook  28  Bibliography  7 7 7 8  29  iii  Table of Contents  Appendices A Useful relations  31  A.l  B l o c h relations  31  A.2  T r i g Identities  31  B  Supercell Method  32  C  Gauge Invariance  33  D Code  35  List of Figures 1.1  Example of braids for two particle exchange  4  2.1 2.2  2D lattice in Landau Gauge with N N N hopping Bloch electron dispersion over first Brillouin zone for N N & N N N hopping DOS for k-space dispersion Conical dispersion around Dirac nodes in Brillouin zone . . .  9  2.3 2.4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15  D O S : N N Hopping D O S : N N N Hopping, t' = 0.05 l ^ l of the Edge state: Clean 40 x 40 lattice with open boundary conditions Flux tube threaded through 40 x 40 lattice with open boundary conditions Fractional Charge for bound state on 30 x 30 lattice, with a single flux tube Energy levels and density plots for flux pair, 20 x 20 lattice with d=10 Asymptotic trend towards zero energy for half-quantum flux pair Various Defects on a 2D lattice Edge Dislocation Movement of an edge dislocation Density plots for 20 x 20 lattice, d = W,<p = { and t' = 0.05 Asymptotic trend towards zero energy for dislocation pair . . Periodicity of Dislocation Bound state in d Density plots for 80 x 40 lattice, with d = 38 Density plots for 40 x 20 lattice, with d = 20  11 11 12 18 18  2  c  19 19 20 21 22 23 24 24 25 26 26 27 27  v  Acknowledgements Primarily, I would like to acknowldege the enormous input from my supervisor, Dr. Marcel Franz, who generated all of the ideas for the research found within, while also providing knowledge and assistance to your humble narrator here throughout the duration of this project. I am grateful to him for giving me the opportunity to work in his group. I would also like to acknowledge our previous group members, Tom Davis and Tami Pereg-Barnea, for helping me get the numerical code off the ground, and more recently, Babak Saradjeh, for helpful discussions and technical expertise. Special thanks to Garth Kroecker for meeting with me weekly and making sure that sanity wasn't accidentally left behind, and to my dear old dad for helping me along the way here to U B C .  vi  C h a p t e r  1  I n t r o d u c t i o n  1.1  Context of Current Project  The ability to create topologically protected non-abelian anyonic quasiparticles in 2D would be a significant step towards realizing a quantum computer. A number of condensed phases of matter in 2D have been studied, which can produce a topological quantum ground state and are topologically invariant at low temperatures and energies. The two most promising avenues here are the fractional quantum hall effect, and p-wave superconductivity. In this project, the effect of introducing defects into a system of Dirac Fermions on a lattice is explored in hopes that a further possibility will emerge that could be of some general use in the field.  1.2  Topological Quantum Computation  A Quantum computer would, in principle, be able to solve certain problems exponentially faster than a classical computer as well as providing the means for much more effective simulations of quantum systems, and as such would be of great value to the scientific community in general. Assuming one could build physical qubits (quantum two level systems with states |0) and |1)), quantum computation would then require the initialization, evolution and measurement of these qubits, which in turn would need to be manipulated, coupled and entangled with other qubits by external means [12]. Now, the proliferation of errors and quantum decoherence i n such a system would require some method of quantum error correction, which at first sight seems a ghastly impossibility. In classical computers, errors can be effectively dealt with by keeping multiple copies of the information and then checking against those copies. In a quantum system, if the state is measured during a calculation to see whether any errors have occured, a quantum superposition may be destroyed and the calculation would be ruined. O n top of this, errors need not only be a discrete bit flip, but can instead be a continous phase error, that is to say: a |0) + b |1) —» a |0) + be |1) for some arbitrary 10  1  Chapter 1.  Introduction  angle 6. For these reasons, and the fact that error correction itself can be somewhat noisy, estimates of the threshold error rate above which error correction is impossible typically fall in the range 1 0 to 10~ , meaning that a quantum computer would be required to flawlessly execute 10 to 10 operations before producing an error[15]. In essence then, one would require fault tolerant quantum gates, whose action on protected information would remain invariant when the implementation of the gate is deformed by noise. Topology, which concerns geometric configurations that are left unaltered by elastic deformations, or global properties of an object that remain unchanged when we deform an object locally, comes to the rescue here in spectacular fashion. Alexei Kitaev, a pioneer in the field, showed that if a physical system has topological degrees of freedom, which are inherently robust against small local perturbations, then information contained in those degrees of freedom would be automatically protected against errors caused by local interactions with the environment [7]. Kitaev's vision of a topological quantum computer is a device in which quantum information is encoded in the quantum numbers carried by anyonic quasiparticles (low level excitations with particle like properties such as spatial localization, well defined momentum, charge, etc.) residing on a 2D surface that have long-range Aharonov-Bohm interactions with one another [8]. The Aharonov-Bohm effect itself concerns the topological property of the path traversed by a charged particle around a perfectly shielded magnetic solenoid, whereby the wavefunction of the particle acquires a phase e ^ (q being the amount of charge and 4> being the magnetic flux enclosed by the solenoid) irrespective of the actual path itself. However, in order that non-trivial computations be allowed, the Aharonov-Bohm phenomena in the system must be non-abelian in nature so that one can build up complex unitary transformations by performing many particle exchanges in succession[12]. Nature does supply some fundamental non-abelian gauge fields, but none of these appear to be suitable for quantum computation, and so the hope is that these non-Abelian effects under interchange and braiding can arise as collective phenomena in 2D systems. - 4  6  4  6  n  1.3  Anyons and Braiding  fn quantum theory, indistinguishable particles in three-dimensional space behave in only two distinct ways upon interchange, that is to say, they can be either fermions or bosons. Anyons, however, are particles in 2D having exchange statistics that can be described by a continuous variable cu, such 2  Chapter 1.  Introduction  that upon interchange the wavefunction of the two particles picks up the phase e [12]. To establish a more physical picture of an anyon, we introduce the concrete example of a flux-charge composite. In this situation, we imagine a spinless particle of charge q orbiting around a thin solenoid along the z-axis enclosing flux 0[13]. A more formal way to achieve this end is via ChernSimons field theory, where magnetic flux is attached to matter charge density in such a way that it follows the matter charge density wherever it goes [3], but we shall make do by studying the following Hamiltonian involving two simple charge-flux composites: ia  H  where  {pi-qaif 2m  =  -. a i  2  +  <t> Z .  2  o  =  , (p -ga ) 2m  T  X  (fi,  2  .  2  2  (  - f ,i)  1  )  .  2  — ia 1^1,2-^2,11  .  (  —  Z l x  . L  '  are the vector potentials at the positions of the composites 1 and 2, due to the fluxes in the composites 2 and 1 respectively. Now, going over to centre of mass (CM) and relative co-ordinates (rel), we define R=—-—,  P = pi+P2  r = n - r , p =  and  2  —  (1.3)  which then allows to recast the Hamiltonian as  *=f  M  +  ( 1  Am  .4)  m  with <t> zxr a  =  2 ^ ~ W  (  L  5  )  The C M motion, translating both of the particles rigidly, is independent of statistics and no longer relies on the gauge field, and the relative motion has reduced to the system of a single particle of mass m/2 orbiting around a flux tube of magnitude 0 at a distance r. Now, since our system is bosonic, the wavefunction of the two composite systems will be symmetric under exchange, which is equivalent to the boundary condition VJrel  0  ,  0 +  TT) =  VVei  0  ,  0)  (1.6)  where we have moved to cylindrical co-ordinates, f = (r,9). Following our notes on gauge transformations in appendix C . l , we set a^a'  = a-  V A (r, 0)  (1.7) 3  Chapter 1. Introduction where A (r, 9) = ^9. In the primed gauge, we then see that our wavefunction is now given by rp'  rd  (r, 9) = e - ^ r e l (r, 69) = e'^Urel  fa  9)  (1.8)  This is clearly no longer symmetric under interchange, since yj'  rel  (r, 9 + Tr) = e ^ ^ e i  (r, 9) =  fa  *)  (1-9)  Thus, we have shown that two interacting bosonic charge-flux composites are equivalent to two free particles whose wavefunctions develop a phase -iq4>/2 j exchange, that is to say, they obey fractional statistics. So, armed now with a concrete model of anyons, a computation might look as follows: First, pairs of anyons, representing qubits, would be created and then moved around one another in a carefully predetermined sequence. The worldlines of these particles would then create an effective braid in spacetime, which would encapsulate the desired quantum computation. The output, depending solely on the topology of the particular braiding, would be minimally affected by stray electric or magnetic fields, and thus inherently protected from outside disturbances. For a simple picture of braiding see fig. 1.1[2]. e  u n c  e r  CLOCKWISESWAP  RESULTING BRAID  COUNTERCLOCKWISE SWAP  RESULTING BRAID  Figure 1.1: Example of braids for two particle exchange Now, as stated above, an additional complication is that these anyons must be non-abelian, that is, the order in which the particles are swapped is important. If, however, one was capable of creating these non-abelian anyons, a physical realization of the braid group could then allow for any desired non-trivial unitary transformation up to some desired accuracy. The braid group itself is a mathematical structure describing all of the possible ways of braiding a specific number of strings together. A n y braid can be created out of elementary operations whereby two strings are moved in either a clockwise or counterclockwise direction, and thus every possible sequence of manipulations on a set of anyons has a corresponding braid [2]. A t the  4  Chapter 1.  Introduction  end of a calculation, or braiding if you will, the state of the system could be read out using a non-abelian generalization of the Aharonov-Bohm effect, whereby a non-abelian anyonic test quasiparticle could be sent into the system, encircling the quasiparticles containing the desired quantum information. The result would then be manifested in the interference effects from the different trajectories of the test particle.[12] Of course, these are all beautiful ideas that hold much promise, but the reality is that there are a bewildering number of technical hurdles that must be met before a physically realizable quantum computer can be created. Moving on to what can actually be achieved, we briefly introduce the two avenues of research alluded to in section 1.1.  1.4  The Fractional Quantum Hall Effect  Experimentally, electrons can be confined to move in a two-dimensional plane between gallium arsenide heterostructure layers, where movement in the third direction is essentially frozen out at low enough temperatures because of the discrete nature of the lowest energy modes[4]. When placed in a perpendicular magnetic field, the 2D electron gas will then arrange itself into a topologically ordered state, a manifestation of which is the quantization of the Hall resistance, R = along with the simultaneous vanishing of the longitudinal resistance R , where h is Planck's constant, the filling factor, v is a rational number, and e is the electronic charge. The original experiments by von Klitzing in 1980 found that the resistance increased in plateaus characterized by an integer filling fraction u — n, the effect which is now known as the Integer Quantum Hall Effect (IQHE). Further experiments carried out in f982 by Tsui, Stormer and Gossard, with cleaner samples, lower temperatures and stronger magnetic fields, found plateaus at filling fractions v that were not integers but instead simple fractions, giving us the Fractional Quantum Hall Effect ( F Q H E ) . Theoretically, it was found by Robert Laughlin that for a fractional quantum Hall state at v = ^ with m odd, the strongly correlated, collective behaviour of the electrons resulted in a gapped ground state supporting quasiparticles with fractional charge [9]. For instance, the Laughlin wave function for v = \ predicts that an electron added to the system would break up into three quasiparticles, each with charge e/3. Furthermore the state is found to have Abelian anyonic quasiparticle exchange statistics. Now, for the filling fraction v = 5/2, theoretically explored by Moore and Read using conformal field theory techniques [11], the quasiparticles are found to have xy  xx  5  Chapter 1.  Introduction  non-abelian exchange statistics, and thus there is the possibility that one could exploit this behaviour for quantum compuational purposes.  1.5  P-wave superconductivity  It has also been suggested that the quantum vortex matter in spin-triplet superconductors, the textbook example being Strontium ruthenate (Sr2RuOi), could supply the non-abelian braiding statistics required for topological quantum computing. The physical basis underlying the proposal is the Vx + ipy order parameter, which admits half-quantum vortices having zeroenergy Majorana fermions in their cores that endow them with the requisite non-abelian statistics[6].  6  C h a p t e r  2  Dirac Fermions on the Lattice 2.1  Motivation  The motivation for exploring the following theoretical model was based on a desire to exploit the properties of a regular two dimensional electron gas (2DEG) confined inside a diluted magnetic semiconductor (DMS) with a superconducting film deposited on top, under the influence of an external magnetic field. This experimentally realizable system uses superconducting vortices to generate confining potentials in the DMSs, which can induce the localization of charge carriers, where the repulsion between vortices can lead to regular vortex lattices and the distance between them can be varied by adjusting the external magnetic field[1]. In the limit where the vortices are well separated, the bound states under each vortex broaden into energy bands, typical of ordered crystals, and the charge carriers become delocalized. The width of the bands can be defined by the hopping amplitude t between neighbouring bound states, and for certain regions where t is small compared to the energy spacing between consecutive bound states, one can model the problem with a simple tight-binding model[14]. The aim then, is to study a system of electrons on a lattice in a perpendicular magnetic field, which is a good approximation to the system described above in certain limits, and add a number of defects into the model in hopes of discovering some interesting new physics.  2.2  Real Space  So, we shall now focus on the problem of tight-binding electrons on a square 2D lattice in a uniform magnetic field producing one-half flux quantum per plaquette, with hopping between N N (nearest neighbour) and N N N (next-  7  Chapter 2. Dirac Fermions on the  Lattice  nearest neighbour) sites. The Hamiltonian in this case is written [5] = - E  + H.c,  tijC^He '^ 1  (2.1)  (ij) where cj- the creation operator for a fermion on the lattice, the phase factor Oij is defined on a link (ij), and Uj is the hopping integral, ff we identify the phase with the line integral of the vector potential (taken counter-clockwise):  then the quantity  $ = —  V  0~;  - J -  =  • dl -  IA  / B • dS  (2.3)  is the magnetic flux through the area S in units of the magnetic flux quantum $o = i which in our case is simply \. Choosing the Landau gauge A = (0, Bx, 0) and using the integers n and m to represent the x and y coordinates of the square lattice, respectively, the various Peierls phase factors, Oij, and hopping elements for an arbitrary plaquette can be seen in figure 2.1 Substituting in the above relations then yields H — —t^ c|j n,m ~ * E i n,m C  +  l ,  +1]rn  m  +  c  — t ^c\ n,m  n)7n  l ^ «  m  (  n  +  l  )  c ^ e ' l  i m + l  n  K n  m  - f E clm lCn l,me < ^ n,m l  +  +  n  + H.C  (2.4)  where the lattice spacing is taken to be unity. To proceed from here we can either solve the Hamiltonian using exact diagonalization or, since the system is translationally invariant with respect to the lattice basis, we could also Fourier transform everything and then solve the system in momentum space. As we are interested in creating defects in our lattice, we shall lose this translational invariance and thus will have to proceed numerically. However, we shall solve the system analytically in order to verify the preliminary numerical results for a clean lattice.  2.3  Momentum space  We shall start by mapping our original Hamiltonian onto one with two inequivalent lattice sites c and d , which allows us to ignore the spatial nm  nm  8  Chapter 2. Dirac Fermions on the Lattice  Figure 2.1: 2D lattice in Landau Gauge with N N N hopping  dependence of the phase, since for one-half flux quantum the phases are the same on each respective sublattice. In this case we have  H—  t ~y  ^  ( n,rn,dn,m + n,m n,m+l c  c  c  +  d\i,m n+l,m c  <4i,m^n,7n+l)  (2.5)  Substituting in the general relation above and the lattice vector for our new unit cell R = (2ax, ay) yields  n,m  u £/ i f ( c ^ - , + e ^ c ^ , + e ^ d t c j , - e^dUj;,)]  + H.c  (2.6)  where we have taken the lattice spacing, a, to be unity. Utilizing the delta function and incorporating the Hermitian conjugate then gives  9  Chapter 2. Dirac Fermions on the Lattice  c  = Y,[- {{ + ~ ) k — it' {^e  H  t  1  e  i2kx  h+  — e~  iky  2 c o sk  +e' (  iky  i  _ ^'(V^ - e  -  v h + (i+* c  v ^  k  +2k  _ *( »-  ^ - ^y k  f c  e  l 2 k x  2 f e  )  i)^  h -  d  c  2 c  o  s  v l k)  k  d  t_„  + e-^*"" *-)) dtc ] (2.7)  +2kx)  2  e  £  Now, if we let %t = (ct, d t ) , then we can re-arrange the above so that '-  k k  v  where  1  (2.8)  „ M  .a b t = W -<  (2.9)  with  (2.10)  a = —2£cos fc,. and b=  -t(l + e~ ^ - it (2ismk i2k  + e-^ y ^  1  k  y  +2k  - e^ y- ^ k  2k  (2.11)  The eigenvalues of H can now be found by evaluating the determinant of Mjg. So, collecting the non-zero terms with similar coefficients and dealing with each one individually, we have for t: At cos ky + t (l + e~ 2  2  2  i2kl  ) (l+e  i2fe  *) (2.12)  = At cos k + 2 i (1 + cos (2k )) 2  2  2  y  x  — At (cos k + cos fcz) 2  2  2  y  and V: -1'  2  [-8 sin k + A sin k (sin (fc + 2k ) + sin (/c + 2/^))] 2  y  y  y  y  x  = -t' [-8 sin k + 8 sin fc cos 2A; ] 2  2  2  y  y  (2.13)  X  = 16i' sin k sin fc 2  2  y  2  x  where we have made use of the identities A . 4 and A . 5 . Combining the two terms then leads to the desired dispersion, namely Ek = ±2i-v/cos k + cos k + 4£sin k sin k 2  x  2  y  x  y  (2.14) 10  d  Chapter 2. Dirac Fermions on the Lattice where £ = j and the momenta are taken over the first magnetic Brillouin zone, namely, ^ < k < 5, —7r < k < n. Plotting equation 2.14 over the first magnetic brillouin zone and also using it to calculate the DOS (density of states), seen in figures 2.2 and 2.3 respectively, it is clear that the effect of the N N N terms is to open up a gap between the two bands, which was the initial motivation for including these terms. A n y zero energy state will be effectively isolated from the rest of the band. x  y  (b) £ = 0.05  (a) £ = 0  Figure 2.2: Bloch electron dispersion over first Brillouin zone for N N Si N N N hopping  Q  o  Energy (E/t)  Figure 2.3: D O S for k-space dispersion In order to qualify the use of the term 'Dirac Fermions', we would now like to show that a low energy expansion around the nodes K± = ± (5, f ) results in a Hamiltonian analogous to that of a relativistic Dirac Hamiltonian. We also note that the conical dispersion around the nodes, enlarged in 11  Chapter 2. Dirac Fermions on the Lattice fig. 2.4, is normally referred to as a 'Dirac Cone' because of the similarity to a normal relativistic dispersion, the difference being that the Fermi velocity of electrons or holes replaces the speed of light.  Figure 2.4: Conical dispersion around Dirac nodes in Brillouin zone So, making the replacement fe = K± + p in equations 2.10 and 2.11, for small p, and using the relations A.7 and A.8 we have a± (p) = ±2tp  (2.15)  + O (p ) 3  y  b± (p) = -2t[i ±2Z}  +  Px  (2.16)  0{tip,p ) 2  Now, in order to transform our matrix M  (fe) = a (fe) o~  z  + Re (b) o — Im (b) o~ x  y  (2.17)  into the regular form for the Dirac equation, that is 2  H =}  o- i + tr m iP  3  (2.18)  i=l  we need to use the following unitary transformation G =  e-^e-^*' 1 — io„ 1 — id. y/2  (2.19)  y/2 e  4  e 4  %/2 12  Chapter 2. Dirac Fermions on the Lattice to permute (o- ,cr ,a ) x  y  z  — i > (o ,a ,a ). z  x  y  T h e resulting m a t r i x  G M {k} G = -Im (b) a + a (jfe) a + Re ]  x  y  « 2t (p^o-x T ^ ( 7 ^ ± 2£a ) z  (6)  a  z  (2.20)  clearly gives us a H a m i l t o n i a n of D i r a c form.  13  C h a p t e r  3  Continuum Limit Going over to the continuum limit, we shall now solve the Dirac equation in the presence of a single flux tube in two spatial dimensions analytically, in order to have some idea of what to expect from our numerical simulations. The Hamiltonian in eqn. 2.18 above is modified as follows (charge -e) [16] 2 H = c ]P a* (pi + -A^j i=i  + cT mc 3  (3.1)  2  C  where we have used the minimal coupling prescription to include the effects of the electromagnetic gauge field. Substituting in the Pauli matrices  <*=(!  o ) - - = ( °  "o ) • " - ( «  - i )  and re-arranging then yields the following \ J (3.3) Now, the vector potential for a single line of flux passing through the origin in cylindrical co-ordinates is H  =  (  rnc V c [(p + i ) + \ (A 2  x  Py  ^  c [(p - ip ) + \(A -mc x  + iA ))  x  y  =  ^  y  iA )]  X  y  2  = 2  ^  ( 3  '  4 )  ff we make the substitutions p —> - i W , (p = (— sin <p, cos (p) and use the relations: (d ± idy) = x  (id ± r  -d^j  l  (3.5)  so that ex a \ ( — sin <p-—$o ± i cos y?-—$o )  zirr  27rr  27rr  /  14  Chapter  3.  Continuum  Limit  then we have /  oa  mc  1  r  r  H = -ikce *  o + -0<p r  1  V  r  ao  v  H,  a  r  (3.6)  J  -mc  r  Since the system is r o t a t i o n a l l y symmetric, angular m o m e n t u m is a good q u a n t u m number, a n d we can factor the wavefunction so t h a t y  a  =  ^ - ^  e  u  ( r ) ^  =  b  (3.7)  e *v{r) il  or, i n a more convenient form, o  i(l-l)H>  e  0  e^ 1  \  fu(r)  J  \v(r)  (3.8)  M u l t i p l y i n g the equation  (3.9) from the left by U  1  , where the wavefunction is given i n the sublattice basis,  then gives us the modified form U-'HUx  = H  E x p a n d i n g to find H, using h = he (—d mc  e  2  e^ih*  -mc m  =  IJ-  itp i(l-l)tp  e  -i{l-l)y  ijf  -i(;-i)<p^* i(/-i)<p  ihe  or  i{l-l)ip  e  —mc  e  mc  e  e  e- fihe f il  il  mc e v 2  il  e-^ihe^  mc e  e (1)<P -iip il<p  V  e  e  we find;  0  0  2 i ( J - l ) e  + ^j,  v  e  J V  c  X  i(l-l)v  ih  lip  2  (3.10)  E  + i^d  r  H = U~  =  X  .2  ihe r  dr  ,2  r 9  -mc  (3.11) 15  Chapter 3. Continuum Limit Our original motivation was to find a zero-energy state, and so we would now like to find if there is a non-trivial solution to the equation #("")=<>  (3.12)  v(r)  K  Making a further factorization Mr)\ (u(ry =ra  v (r) J  (  3  1  3  )  \v (r)  and using the identity (3.14)  £ r ° / = r« (? + !-) / dr \ r d r we arrive at the form mc  2  Hr«X = r«\  i  _  (  = 1 ± s = z  ,  n (B_t+^a\ ^ c  r  2  )  , \X  (3-15)  To eliminate the terms in 1/r we require both £+£, + a = 0 and I—l+£—a = 0, which are equivalent to the relations £ = \ — £ and a — — (£ + 0 ~TThus we shall have a tractable equation when £ = 3, where =  and Xo= ( . ) e 1  m r  (3.17)  Check:  =  e  mc ihcm 2  -mr  iftcm \ / 1 -mc J \-iJ 2  (3.18)  '0 ,0 = 0 16  Chapter 3. Continuum Limit T h e conclusion t h e n is that, for £ = | , the original H a m i l t o n i a n has a zero-energy b o u n d state; itp\ -%  )  p—mr y/f  (3.19)  17  C h a p t e r  4  Numerical Results 4.1  Method  The following simulations were all carried out using the exact diagonalization of the matrix derived from the tight-binding Hamiltonian above, eqn. 2.1, where the various defects have been introduced accordingly. The numerical code, seen in Appendix D , uses the L A P A C K routine zheev, which is a generic program for extracting the eigenvalues and eigenvectors of an m x n hermitian matrix.  4.2  Clean Lattice  Figures 4.1 and 4.2 show the DOS for a clean 2D lattice i n a uniform magnetic field for N N and N N N hopping respectively, matching the results shown in figure 2.3. The figures were created using a finite 10 x 10 lattice with periodic boundary conditions, and the necessary resolution for the DOS was achieved by using a supercell method as described in Appendix B . The dimensions of the lattice were chosen to ensure an even number of plaquettes and thus an integer number of flux threading the lattice as a whole, so that the wavefunctions would be single valued across the boundary.  Energy (E/t)  *  Figure 4.1: D O S : N N Hopping  Energy (E/t)  Figure 4.2: D O S : N N N Hopping, t' = 0.05  18  Chapter 4. Numerical Results  4.3  Flux Tube  To simulate a single flux tube threading the 2D lattice, the only modification required to the clean system was to send some extra flux through an individual plaquette, which was achieved using the string gauge, as described in Appendix C . l . The Density Plots for the lowest energy edge-state in a clean 2D system are given in Fig.4.3 , and the results for the flux tube, resulting in a near zero energy bound state, are given in F i g . 4.4. The use of open boundary conditions was necessary to enforce a single tube of flux. We note that the finite size of the lattice, causing some overlap between the edge and core states, as well as the necessity of having particle hole symmtery, has given us two states close to zero energy here.  (a) 2D  (b) 3D  Figure 4.3: \ip\ of the Edge state: Clean 40 x 40 lattice with open boundary conditions 2  (a) Energy levels  (b) 2D  (c) 3D  Figure 4.4: F l u x tube threaded through 40 x 40 lattice with open boundary conditions  19  Chapter 4. Numerical Results  (a) Chemical Potential  (b) Charge Density  Figure 4.5: Fractional Charge for bound state on 30 x 30 lattice, with a single flux tube. The charge density, relative to the density at half filling, around the bound state core was also plotted as a function of the number of shells from the core, and it was found that it converged to a fractional charge of ± | , depending on where the chemical potential was set. Figure 4.5 shows the results for three values of the chemical potential: red, green and blue representing values below, at and above half filling respectively. To cut down on computational time a slightly smaller lattice, of dimension 30 x 30, was used in this simulation.  4.4  Flux Pair  Threading two flux tubes through the background of dirac fermions, achieved on the lattice using periodic boundary conditions and a single string, resulted, as hoped, in a pair of near zero-energy bound states. The density plot for the positive eigenvalue of the initial test can be seen in figure 4.6 and in figure 4.7, we can see the result of varying the length between the flux pairs along the y-direction while keeping the x-direction fixed to make it computationally feasable, namely, an asymptotic trend towards zero energy. This was expected since the finite lattice size necessarily causes some overlap between the two degenerate zero-energy states, which drops off as they are separted. We are currently using the proposal outlined by Levin and Wen [10] to calculate, numerically, the exchange phase of these fractionally charged particles. For a 40 x 40 lattice, the preliminary results give an exchange phase of agreeing with the model in section 1.3, when using particles of charge | having fermionic exchange statistics coupled to one-half flux quantum. However, an analytic calculation is also under way using Chern20  Chapter 4. Numerical Results Simons theory, in order to verify these findings in a more rigorous fashion.  (a) Energy levels  (b) 2D  (c) 3D  Figure 4.6: Energy levels and density plots for flux pair, 20 x 20 lattice with d = 10  21  Chapter 4. Numerical Results  Figure 4.7: Asymptotic trend towards zero energy for half-quantum flux pair  Chapter 4. Numerical Results  4.5  Dislocations  Aside from the introduction of extra localized flux, there are a variety of possible defects that can arise in a 2D system, the majority of which are exemplified in fig. 4.8. Our main concern is with the type b) style defects, namely, edge dislocations. These are characterized by their Burgers vector, which is the fundamental quantity defining an arbitrary dislocation. If a closed circuit is made around a dislocation from lattice point to lattice point, as in fig. 4.9(a), and the exact same path is taken around a perfect reference lattice, as in fig. 4.9(b), then the vector needed to close the circuit in the reference crystal is by definition the Burgers vector b. A dislocation can actually propagate through the lattice due to the application of shear stress, as seen in Fig. 4.10, but we shall focus here solely on the static case of a single dislocation/anti-dislocation pair on a finite 2D lattice. To simplify things, the lattice distortions created by the dislocation were ignored in the numerical routine, as we hope the end result should be topological in nature, that is to say, subtle modifications of the hopping integrals in the vicinity of the dislocation core should not affect the topological stability of the expected anyonic quasiparticles.  •  t  , t!  t!  tl  a  b  cd  e f  gh  Figure 4.8: Various Defects on a 2D lattice  23  Chapter 4. Numerical Results  1 i i  \ t u  i  i i .  n  i  (a) Dislocation  (b) Reference lattice with Burgers vector  F i g u r e 4.9: Edge D i s l o c a t i o n  24  Chapter 4. Numerical Results The initial simulation was run on a single 20 x 20 lattice with periodic boundary conditions, where a dislocation pair having a distance d = 10 between the cores was inserted along the y-direction, resulting in an effective flux of <f> = \ threading each dislocation core. The result, similar to the flux pair, showed two eigenvalues of the same magnitude close to zero energy, one positive and one negative, which, from figure 4.f f (a), are seen to be well inside the gap. The density plots for the positive eigenvalue are shown in figures 4.11(b) and 4.11(c), and, as above, simulations were carried out on dislocation cores separated by increasing distance, all threaded by 4> = \, yielding figure 4.f 2, again showing an asymptotic trend towards zero energy. c  c  (a) Energy Levels  (b) 2D  (c) 3D  Figure 4.11: Density plots for 20 x 20 lattice, d = 10, 4> = \ and t' = 0.05 c  Further simulations also found that the resulting density plots for the 20 x 20 lattice were periodic in the distance between the dislocation cores, due to the effective flux threading the cores in each case. Figure 4.13 shows a series of plots from d = 10 to d = 14, covering each possible outcome. The cases where an odd number of atoms remain in the lattice, as in 4.f 3(b) and 4.13(d), produce only one state at exactly zero energy, which is localized over the dislocation core being threaded by <f> = 4. Figure 4.13(c) where d is again even, but with 0 = | over each core, produces two energy states either side of zero near the edge of the gap, the result being a more uniform distribution with very weak localization over the cores. A number of other simulations were carried out to ensure the results outlined above carried over to larger lattice sizes, and as expected they appear to be general. Figure 4.f4 shows the density plots for increased lattice size where the dislocation cores are threaded by <p = \, and 4.15 shows again the weak localization for a 32 x 32 lattice. c  C  c  25  Chapter 4. Numerical Results  40 Distance (a)  60  Figure 4.12: Asymptotic trend towards zero energy for dislocation pair  (d) d = 13  (e) d= 14  Figure 4.13: Periodicity of Dislocation Bound state in d  26  Chapter 4. Numerical Results  27  C h a p t e r  5  Discussion and Outlook A l t h o u g h not complete, the research o u t l i n e d i n this project shows some p r o m i s i n g results t h a t indicate a fractionally charged zero-energy quasiparticle w i t h abelian anyonic exchange statistics arises from i n s e r t i n g a half-flux q u a n t u m t h r o u g h a b a c k g r o u n d lattice of D i r a c fermions. T h e i n i t i a l m o t i vation, w h i c h was to find a non-abelian quasiparticle, has not been realized, but perhaps future research might be able to b u i l d on this simple m o d e l . F u r t h e r studies could include the s i m u l a t i o n of other lattice geometries, w h i c h might arise i n various experiments, as well as a more t h o r o u g h j o b of calculating the exchange statistics numerically. It w o u l d also be useful to have an a n a l y t i c a l m o d e l for the dislocation, so t h a t we c o u l d understand more clearly w h y the b o u n d state appears for a value of <f>Q = \.  28  B i b l i o g r a p h y [1] Mona Berciu, Tatiana G . Rappoport, and Boldizsar Janko. Manipulating spin and charge in magnetic semiconductors using superconducting vortices. Nature, 435:71, 2005. [2] Graham P. Collins. Computing with quantum knots. Scientific American, April:57-63, 2006. [3] Gerald V . Dunne. Aspects of chern-simons theory. Springer Berlin Les Houches, 69:177, 1999, hep-th/9902115. [4] Steven M . Girvin. The quantum hall effect: Novel excitations and broken symmetries. Springer Berlin - Les Houches, 69:53, 1999, condmat/9907002. [5] Y . Hatsugai and M . Kohmoto. Energy spectrum and the quantum hall effect on the square lattice with next-nearest-neighbor hopping. Phys. Rev. B, 42:8282, 1990. [6] D . A . Ivanov. Non-abelian statistics of half-quantum vortices in p-wave superconductors. Phys. Rev. Lett., 86:268, 2001. [7] A . Kitaev. Fault-tolerant quantum computation by anyons. Ann. 303:2, 2003.  Phys.,  [8] A . Y u Kitaev. Fault tolerant quantum computation by anyons. cond-mat/9707021.  1997,  [9] R. B . Laughlin. Anomalous quantum hall effect: A n incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett., 50:1395, 1983. [10] Michael Levin and Xiao-Gang Wen. Fermions, strings and gauge fields in lattice spin models. 2003, cond-mat/0302460. [11] G . Moore and N . Read. Anomalous quantum hall effect: A n incompressible quantum fluid with fractionally charged excitations. Nucl. Phys. B, 360:362, 1991. 29  Bibliography [12] R. Walter Ogburn and John Preskill. Topological quantum computation. Springer Verlag - LNCS, 1509:341-356, 1999. [13] Sumathi Rao. A n anyon primer. 1992, cond-mat/9209066. [14] Tatiana G . Rappoport, Mona Berciu, and Boldizsar Janko. The effect of the abrikosov vortex phase on spin and charge states in magnetic semiconductor-superconductor hybrids. 2006, cond-mat/0605469. [15] S. Das Sarma, M . Freedman, and C. Nayak. compuation. Physics Today, July: 32-38, 2006.  Topological quantum  [16] B . Thaller. Dirac Particles in Magnetic Fields. Kluwer Academic Publishers, Amsterdam, 1991.  30  Appendix A  Useful relations A.l  Bloch relations  Fourier T r a n s f o r m  D e l t a F u n c t i o n - Discrete  J2e« - >  = N8 ,  k k  kk  Delta Function - Continuous  f  f  L  -^-k')x'  dxe  =  5 k k /  LJo  A.2  Trig Identities r,  s i nA +s i nB  A n  = 2s  +  B  i n —-— c o s  s i n A - ^ (1 - cos 2A) 2  cos A = ^ (f + cos 2A) 2  cos  A  + pj = T s i n p  sin ( ± 7 ^ + p ) = i c o s p  Appendix B  Supercell Method As the computing time for exact diagonalization goes as L where L is the size of the lattice, it is not practical to use an infinitely large lattice to calculate the D O S . In order to achieve the desired resolution then, one must turn to other methods. Applying Bloch's theorem to a supercell with periodic boundary conditions is one such method, the one used to arrive at Figures 4.1 & 4.2. W i t h this method, the computing time scales linearly with each new value of the effective Brillouin zone that is sampled. So, for our system, the Schrodinger equation is 3  Y tij^i  = ^i  J  (B-l)  3  Inserting an TV x N unit matrix in between the hopping parameter and the wave function so that Y . l - i j e ^ e ^ ^ j  (B.2)  = e^i  3  multiplying from the left by e~ ' , lk  J2  (e  _ i f c  - %e r i  r(k)  i f c  Ti  and then using Bloch's theorem, we find  - >) (e r  - i f c  - 'tfj) = e ( e " ^ ^ ) r  (k)  (B.3)  (k)  The effect of the supercell can then be thought of as adding a phase factor to the hopping parameter tij such that  32  Appendix  C  Gauge Invariance We first note that the vector potential given in equation 2.2 cannot be fixed completely, as it can be transformed by an arbitrary scalar function x ( r ) , such that A ^ A - A '  (C.l)  + Vx  while leaving Maxwell's equation B = V x A  (C.2)  unchanged. We can see this as follows: Substituting eqn. C . l into eqn. C.2, B = V x A = V X ( A ' + VY)  x A' +  = V  V  x (Vx)  '  = B'  since V x (Vx) = 0, and thus A and A ' describe the same magnetic field. Such a transformation is called a 'gauge transformation', fn order for the Schrodinger equation  m  { r )  =  (  ~  V  +  ( r )  )  =  ^  W  ( C  '  4 )  to remain invariant under the transformation, the wavefunction of the electron must also transform in the following manner rj)' (r) = vb (r) e ' f  (C.5)  x ( r )  As the wavefunction only picks up a phase, the probability distribution and thus all physical predictions will remain identical, so we can safely say that the theory is gauge invariant. As an example, suppose we had picked the symmetric gauge for our system  ' \  X =  Xf  )  =  \  (_jBy  ' > BX  0)  (C  -  6)  33  Appendix C. Gauge Invariance To transform back to the Landau gauge, we would choose x = \Bxy so that V% = \B(y, i , 0 ) , yielding A = (0, Bx, 0) as desired. The wavefunction in this case would then transform as ip' (r) = ip (r) e ~n ^. Let's now look at the string gauge. If we attempt to gauge away the gauge field given above in cylindrical co-ordinates, we discover that we can do so everywhere except on a cut. This new gauge is called the string gauge. The gauge transformation is given as follows: x  Bxy  Ac -> A - Vx = A,  (C.7)  X = * — (P-Vo)  (C8)  c  where Z7T  Thus Vx = ^ — <p + $6(tp-<p )<p >  0  0  (C.9)  and the cut, or string is then found for <p = <^o-  34  Appendix D Code #include " f 2 c . h " ^include < c o m p l e x . h > #include < s t d i o . h > #include " c l a p a c k . h " ^include <math . h> #define P I 3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 4 /*Program to calculate the Eigenvalues for Field with 1/2 flux quantum per unit cell of a single dislo cation/anti — disl o cation  a 2D lattice in a Uniform Magnetic in the Landau gauge, in the presence pair.*/  /* Functions used inside main program*/ double c o m p l e x ** d c m a t r i x (int , i n t ) ; d o u b l e c o m p l e x * d c v e c t o r (int ) ; doublereal *drvector(int); void f r e e _ d c m a t r i x (double c o m p l e x ** , i n t ) ; void f r e e . d c v e c t o r ( d o u b l e c o m p l e x * ) ; void f r e e _ d r v e c t o r ( d o u b l e r e a l * ) ;  const const const const const const const const const const const  int X = 1 0 ; int Y = 1 0 ; double t = 1 . 0 ; double t 2 = 0 . 0 5 ; int B = 2 0 0 ; int K =150 ; double a l p h a = 0 . 5 ; int D = 4 ; int Z = 5; int s = 4 ; int f l u x o n = 9 9 ;  /* Width of lattice*/ /* Height of lattice*/ /* Hopping parameter for NN*/ /* Hopping parameter for NNN*/ /*Number of bins for DOS*/ /*Number of Points in k—space*/ /* Flux—Quantum per unit cell*/ /*Length of Dislocation Pair*/ /* Position of Dislocation Pair along /*symmetry factor*/ /* Postion of Eigenvalue for DOS*/  /* Create necessary matrices/indices int n [ X ] [ Y ] ; int i , j , k x , k y , i n d e x ; double s i g m a , P s i , e i g ; doublecomplex p l , p 2 , p 3 ;  ,  etc.*/  35  x—axis*/  Appendix D. Code i n t e g e r L D A , LWORK,INFO, N = ( X * Y - D ) ; double V [ N ] , d o s [ N ] , d p [ X * Y ] ; double c o m p l e x * * H ; d o u b l e c o m p l e x * M , *WCRK; d o u b l e r e a l *RWORK, *W; /* Allocate Memory*/ H = dcmatrix (N,N) ; M = d c v e c t o r (N*N) ; W = drvector(N); WCRK = d c v e c t o r ( 2 * N ) ; RWORK = d r v e c t o r ( 3 * N - 2 ) ; /*Zero H* / for ( i = 0 ; i < N ; i + + )  { for(j=0;j<N; j++) { H[ i ] [ j ] = 0 . 0 +  0.0*I;  } } /*Zero Vectors*/ for ( i = 0 ; i < N ; i + + ) d o s [ i ] = 0; f o r ( i = 0 ; i < ( X * Y ) ; i + + ) d p [ i ] = 0; /* Create an array for ( i = 0 ; i < X ; i + + )  to index  the  atoms*/  { for(j=0;j<Y; j++) { if ( i < ( Z - l ) ) {  }  n[i][j]  else  = i*Y+  (j+1);  i f ( i = = ( Z —1))  { if(j<(Y-D)/2) {  }  " n[i][j]  else  = i*Y+  (j+1);  if (j > ( ( Y + D ) / 2 - l ) )  {  }  n[i][j]  = U Y+ (j+1) - D ;  else { n[i][j] }  =  10000;  36  Appendix D. Code } else  { n[i][j]  = i * Y + (j+1) - D ;  }  } /*Set up the  Hamiltonian*/  f o r ( k x =0;kx<K; kx++)  { f o r ( k y = 0 ; k y < K ; ky++) {  /*Common Phase  Factors*/  pl=I *2*PI*alpha ; p2=I*(2*PI/s)*kx/K; p3=I*(2*PI/s)*ky/K;  /*fill  the links  in y—dir e ction*/  f o r ( i = 0 ; i < X ; i++)  { if (i <(Z-1)  ||  i>(Z-l))  { for(j=0;j<(Y-l);j++) { H[n[i][j]-l][n[i][j+l]-l] H[n[i][j+l]-l][n[i][j]-l]  }  = =  t*cexp(pl*(i+l))*cexp(p3); t*cexp(-pl*(i+l))*cexp(-p3);  }  else  i f ( i=Z—1)  { for(j=0;j<(Y-l);j++) { if(j<((Y-D)/2-l)  ||  j>((Y+D)/2-l))  { H[n[i][j]-l][n[i][j+l]-l] H[n[i][j+l]-l][n[i][j]-l]  }  = =  t*cexp(pl*(i+l))*cexp(p3); t*cexp(-pl*(i+l))*cexp(- 3); P  }  }  } /*fill  the links  in the  x—direction*/  for(i=0;i<(X-l);i++) { f o r ( j = 0 ; j < Y ; j++) { if(j<((Y_D)/2) {  ||  j>((Y+D)/2-l))  37  Appendix D. Code H[n[i][j]-l][n[i+l][j]-l] H[n[i+l][j]-l][n[i][j]-l]  = =  t*cexp(p2); t*cexp(- 2);  = =  t*cexp(p2); t*cexp(-p2);  P  } else  i f ( i <(Z—2)  ||  i >(Z —1))  { H[n[i][j]-l][n[i+l][j]-l] H[n[i+l][j]-l][n[i][j]-l]  } else  if  (i=Z-2)  { if(j>(Y-D)/2) { H[n[ i ] [j ] - l][n[ i+2][j] -1] H [ n [ i +2][j ] - l ] [ n [ i ] [ j] - 1 ]  }  }  }  }  = t *cexp (I * ( P I / 2 ) * (2* j +1))* cexp (2*p2 ) ; = t * c e x p ( - 1 * ( P I / 2) * ( 2 * j + 1 ) ) * c e x p ( - 2 * p 2 ) ;  }  /*fill the links for for(i=0;i<(X-l);i++)  next—nearest  neighbours*/  { for(j=0;j<(Y-l);j++) { i f (i <(Z-2) || i > ( Z - l ) ) H [ n [ i ] [ j ] - l ] [ n [ i +1] [ j + 1 ] - 1 ] H[n[i+l][j+l]-l][n[i ] [j ] - l ] H [ n [ i + l ] [ j ] - 1 ] [n [ i ] [ j + 1 ] - 1 ] H[n[i ] [j + l ] - l ] [ n [ i + l ] [ j ] - l ] else  if(j<((Y-D)/2-l)  H[n[i H[n[ i H[n[ i H[n [ i else  if  H[n[ i H[n[ i H[n[ i H[n[i  ||  t 2 * c e x p ( p i *( i + 1 + 0 . 5 ) ) * c e x p ( 3 + p 2 ) ; 1 2 * c e x p ( - p l * ( i +1 + 0 . 5 ) ) * c e x p ( - p 3 - p 2 ) ; 12 * c e x p ( p i * ( i +1 + 0.5)) * c e x p ( p 3 - p 2 ) ; t 2 * c e x p ( - p l *( i + 1 + 0.5)) * c e x p ( - p 3 + p 2 ) ; P  j >((Y+D)/2-l))  ] [j] - l ] [ n [ i + l ] [ j + l ] - l ] + l ] [ j +1] - l ] [ n [ i ] [ j ] - 1 ] +l][j ] - l ] [ n [ i ] [j + l ] - l ] ] [ j +1] - l ] [ n [ i + l ] [ j ] - 1 ] (i=  = = = =  = = = =  t 2 * c e x p ( p i *( i + 1 + 0 . 5 ) ) * c e x p ( 3 + p 2 ) ; t 2 * c e x p ( - p l *( i + l + 0 . 5 ) ) * c e x p ( - p 3 - p 2 ) ; t 2 * c e x p ( p i *( i + 1 + 0 . 5 ) ) * c e x p ( p 3 - p 2 ) ; 1 2 * c e x p ( - p l *( i + 1 + 0 . 5 ) ) * c e x p ( - p 3 + p 2 ) ;  = = = =  t2 t2 t2 t2  P  (Z-2))  ] [ j ] - l][n[ i + 2 ] [ j + l ] - l ] + 2 ] [ j +1] - l ] [ n [ i ] [ j ] - 1 ] + 2][j ] - l ] [ n [ i ] [ j + l ] - l ] ][j+l]-l][n[i+2][j]-l]  * c e x p ( p i *( i + 1 + 0 . 5 ) ) * c e x p ( 3 + p 2 ) ; *cexp(-pl *(i + l + 0.5))*cexp(-p3-p2); * c e x p ( p i *( i + 1 + 0 . 5 ) )* c e x p ( p 3 - p 2 ) ; * c e x p ( - p i *( i + 1 + 0 . 5 ) ) * c e x p ( - p 3 = p 2 ) ; P  } } }  38  Appendix D. Code /* Apply Periodic Boundary f o r ( i = 0 ; i < X ; i++)  Conditions*/  { H[n[ i ] [ Y - l ] - l ] [ n [ i ] [ 0 ] - 1 ] H[n[ i][0] - l ] [ n [ i ] [ Y - l ] - l ]  = U c e x p ( p i *( i + l ) ) * c e x p ( p 3 ) ; = t * c e x p ( - p l * ( i + 1 ) ) * c e x p ( - p 3 );  } for( i=0;i<X-l;i++) { H [ n [ i ] [ Y - l ] - l ] [ n [ i +1][0] - 1 ] H [ n [ i + 1 ] [ 0 ] - 1 ] [n[ i ] [ Y - l ] - l ] H [ n [ i + 1 ] [ Y - 1 ] - l ] [ n [ i ] [0] - 1 ] H[n[i][0] - l ] [ n [ i + l ] [ Y - l ] - l ]  = = = =  t2 t2 t2 t2  * c e x p ( p i * ( i + 1 + 0.5)) * c e x p ( p 3 ) * c e x p (p2 ) ; * c e x p ( - p l *( i +1 + 0 . 5 ) ) * c e x p ( - p 3 )* c e x p ( - p 2 ) ; * c e x p ( p i *( i + 1 + 0 . 5 ) ) * c e x p (p3 ) * c e x p ( - p 2 ) ; * c e x p ( - p l *( i + 1 + 0 . 5 ) ) * c e x p ( - p 3 )* c e x p (p2 ) ;  = = = =  t 2 * c e x p ( p i * (0 + 1 + 0.5) )* c e x p ( p 3 ) * c e x p (p2 ) ; t 2 * c e x p (—pi * ((Y— l ) + l + 0 . 5 ) ) * cexp(—p3) * cexp(—p2 ) ; t2 * cexp ( p i * ( ( Y - l ) + l + 0.5))* cexp ( p 3 ) * c e x p ( - p 2 ) ; t 2 * c e x p ( - p l * ( 0 + 1 + 0 . 5 ) ) * c e x p ( - p 3 ) * c e x p (p2 ) ;  } for(j=0;j<Y-l;j++) { H[n[X-l][j] -l][n[0] [j+l]-l] H [ n [0] [ j +1] - l ] [ n [ X - l ] [ j ] - 1 ] H [ n [0] [ j ] - l ] [ n [ X - l ] [ j + l ] - l ] H [ n [ X - l ] [ j +1] - l ] [ n [0] [ j ] - 1 ]  } for(j=0;j<Y;  j++)  { H[n[0][j]-l][n[X-l][j]-l] H[n[X-l][j]-l][n[0][j]-l]  = =  t*cexp(-p2); t*cexp(p2);  /*Diagonal Contribution*/ H[n[X-l][Y-l]-l][n[0][0] -1] H[n[0][0] - l ] [ n [ X - l ] [ Y - l ] - l ] H [ n [0] [ Y - l ] - l ] [ n [ X - l ] [ 0 ] - 1 ] H[n[X-l][0] -l][n[0] [Y-l]-l]  = = = =  12 * c e x p ( p i * ( 0 + 1 + 0 . 5 ) ) * c e x p ( p 3 ) * c e x p (p2 ) ; t2 * c e x p ( - p l * ( Y - l + l + 0 . 5 ) ) * c e x p ( - p 3 ) * c e x p ( - p 2 ); t 2 * c e x p ( p i * ( Y - 1 + 1+0.5))*cexp (p3)* c e x p ( - p 2 ); t 2 * c e x p ( - p l *(0 + l + 0 . 5 ) ) * c e x p ( - p 3 ) * c e x p (p2 ) ;  }  /* Transform Hamiltonian f o r ( i = 0 ; i < N ; i++)  to  Fortran  Mode  (i . e column—major  mode)*/  { for(j=0;  j<N;  j++)  { M[i+N*j].r = creal(H[i][j]); M[ i+N*j ] . i = c i m a g ( H [ i ] [ j j) ;  }  }  /*Send Hamiltonian through CLAPACK*/ LDA=N; LWORK=2*N; z h e e v . ( " V " , " U " , & N , M , & L D A , W, WORK, ^ W O R K , RWORK, /*parameters in the no left eigenvectors  order as they appear , no right eigenvectors  in ,  MNFO);  the function call: order of input matrix  39  A,  Appendix D. Code input matrix A, array for left array for right workspace array workspace array  leading dimension of A, array for eigenvalues , eigenvalue , leading dimension of DUMMB, eigenvalues , leading dimension of DUMMB, dim>=2*order of A, dimension of WORK dim=2*order of A, return value */  }  }  /* Calculate  the Density  of States for SD density  plot*/  f o r ( j = 0 ; j < N ; j++)  { P s i = (M[ j + N * f l u x o n ] . r ) * ( M [ j+N* f l u x o n ] . r) + (M[ j+N* f l u x o n ] . i ) * (M[ j+N* f l u x o n ] . i ) ; dos [ j]+=Psi ;  } /* Fill  in missing  sites from  dislocation*/  f o r ( i = 0 ; i < ( X * Y ) ; i++)  { if (i < ( ( Z - 1 ) * Y + dp[i] else  =  (Y-D)/2))  dos[i];  i f (i > ( ( Z - 1 ) * Y + (Y+D)/2 - 1 ) )  dp[i]  = dos[i-D];  else dp[i]  = 0.0;  } /*  Create Output Array for Grace*/ f o r ( i = 0 ; i <(X*Y) ; i++) p r i n t f ("%f\n" ,  dp [ i ] ) ;  /* Free Memory*/ free.dcmatrix (H,N); free_dcvector(M) ; f r , e e _ d r v e c t o r (W) ; f r e e . d c v e c t o r (WORK); f r e e - d r v e c t o r (RWORK) ; }  d o u b l e complex ** d c m a t r i x ( i n t  nrows , i n t  ncols)  { double complex * * a ; int i ;  40  Appendix D.  Code  a = ( d o u b l e c o m p l e x **) m a l l o c ( n r o w s * s i z e o f ( d o u b l e c o m p l e x * ) ) ; f o r ( i = 0 ; i < n r o w s ; i + + ) a [ i ] = ( d o u b l e c o m p l e x *) m a l l o c ( n c o l s * s i z e o f ( d o u b l e complex ) ) ; r e t u r n a;  } doublecomplex  * dcvector (int  n)  { doublecomplex *a; a=(doublecomplex *) m a l l o c ( n * s i z e o f ( d o u b l e c o m p l e x ) ) ; i f (!a) p r i n t f ( " a l l o c a t i o n „ f a i 1 u r e - i n - d v e c t o r ( ) " ) ; r e t u r n a; } doublereal  *drvector(int  n)  { doublereal *a; a = ( d o u b l e r e a l *) m a l l o c ( n * s i z e o f ( d o u b l e r e a l ) ) ; i f (!a) printf("allocat ion-fai 1u re-in-drvector"); r e t u r n a;  } void  free.dcmatrix (double  complex  **a,int  nrows)  { int i ; f o r ( i = 0 ; i <nrows ; i++) free(a[i]); free(a) ;  } void  free_dcvector (doublecomplex  *a)  { free(a); } void  free_drvector ( doublereal  *a)  { free ( a ) ; }  41  

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