Field-driven Instabilities of the Non-Abelian Topological Phase in the Kitaev Model

Transcription

1 Field-driven Instabilities of the Non-Abelian Topological Phase in the Kitaev Model Masterarbeit zur Erlangung des akademischen Grades Master of Science vorgelegt von Sebastian Fey geboren in Herborn Lehrstuhl für Theoretische Physik I Fakultät Physik Technische Universität Dortmund 2013

2 1. Gutachter : Dr. Kai P. Schmidt 2. Gutachter : Dr. Julien Vidal Datum des Einreichens der Arbeit: 30. September 2013

3 Abstract In this thesis the phase transition between the non-abelian topological phase of Kitaev s honeycomb model and the polarized phase present at large fields is studied. To this end, perturbative Continuous Unitary Transformations (pcut) about the high-field limit and exact diagonalization (ED) for several finite systems with both, open and periodic boundary conditions, are performed. Afterwards pcut results for the thermodynamical limit are presented. For a positive interaction parameter the validity range of the series expansion cannot capture the phase transition. This is different for ED. A comparison to DMRG for systems up to 64 spins by Jiang et al. shows a deviation that might be due to finite-size effects. For a negative interaction parameter the ED spectra display very complex behavior. Series expansion of the one-particle gap and the second-derivative of the ground-state energy obtained in ED indicate a second-order phase transition out of the polarized phase. Nevertheless, a first-order phase transition and the presence of more than two phases in the phase diagram cannot be excluded.

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5 i Contents Contents Contents List of Figures List of Tables i iii vii 1 Introduction 1 2 Methods Fourier transformation Exact Diagonalization (ED & Lanczos algorithm) Programs Lanczos algorithm Finite-size effects in the Kitaev model for h = Perturbative Continuous Unitary Transformations (pcut) Continuous Unitary Transformations (CUT) Perturbative CUT Application to the model Extrapolations Padé extrapolation dlog Padé extrapolation & critical parameters Model Systems Ground-state energy per site of small systems Isolated dimer in a uniform field Three spins with x- and y-coupling Four spins with x-, y-, and z-coupling Gap for positive interaction parameter J Isolated dimer in a uniform field Three spins with x- and y-coupling Four spins with x-, y-, and z-coupling Gap for negative interaction parameter J Isolated dimer in a uniform field Three spins with x- and y-coupling Four spins with x-, y-, and z-coupling

6 Contents ii 3.4 Summary of the results for small open systems Large Periodic Systems & Thermodynamic Limit Phase transition for positive J Periodic 8-site system Periodic 18-site system Periodic 24-site system Thermodynamic limit Phase transition for negative J Periodic 8-site system Periodic 18-site system Periodic 24-site system Thermodynamic limit Conclusion and Outlook 71 A Transformation from Antiferromagnetic to Ferromagnetic Interaction. 75 B Hamiltonian in Field Basis 77 C Eigenvalues of Plaquette Operators W p 79 D pcut Results 81 D.1 Overview of pcut series D.1.1 Ground-state energy per site D.1.2 Gap for J D.1.3 Gap for J D.2 One-qp hopping amplitudes in the thermodynamic limit E ED Results 89 E.1 Eigenvalues of an isolated z-dimer in the (1,1,1)-magnetic field F Periodic Lattices 91 G Kitaev s Three-Spin Interaction 93 H Phase Transition for Various J x = J y with Fixed J z 95 Bibliography 97

7 iii List of Figures List of Figures 1.1 Honeycomb and brickwall lattice for the Kitaev model (a) Honeycomb lattice (b) Brickwall lattice Phase diagram of the original Kitaev model Field after transformation to opposite interaction sign Lattice vectors and unit cell of the honeycomb lattice Difference between the energy spectrum of a finite system and the thermodynamical limit at the Kitaev point (a) Thermodynamic limit (b) Finite system The flow of the pcut method Dimer: Full ED spectrum Dimer: Convergence of the series coefficients of the ground state Dimer: Different orders of the series of the ground-state energy site system site system: Full ED spectrum site system: Different orders of the series of the ground-state energy site system site system: Different orders of the series of the ground-state energy Dimer: Different orders of the series of the gap for J > site system: Different orders of the series of the gap for J > site system: Different orders of the series of the gap for J > Dimer: Different orders of the series of the gap for J < site system: Different orders of the series of the gap for J < site system: Different orders of the series of the gap for J < Two periodic lattices (a) 8 spins (b) 18 spins site system: Series and extrapolations for J > (a) pcut series (b) Extrapolations site system: Low part of the spectrum and ground-state energy derivatives. 50

8 List of Figures iv 4.3.(a) Spectrum, J = d 4.3.(b) dh E 0 and d2 E dh 2 0, J = site system: Number of vortices for J > (a) J = (b) h = site system: Series and extrapolations for J > (a) pcut series (b) Extrapolations site system: 2nd derivative of ground-state energy for J > site system: Number of vortices for J > (a) J = (b) h = Periodic 24-site system site system: Series and extrapolations for J > (a) pcut series (b) Extrapolations site system: 2nd derivative of ground-state energy for J > site system: Number of vortices for J > (a) J = (b) h = Thermodynamic limit: Series and extrapolations for J > (a) pcut series (b) Extrapolations and various 24-site systems: 2nd derivative of ground-state energy for J > Curves of 24, 32, 36, and 64 spin systems from [JGQT11] site system: Series and extrapolations for J < (a) pcut series (b) Extrapolations site system: Low part of the spectrum with pcut curves site system: 2nd derivative of ground-state energy for J < site system: Number of vortices for J < (a) J = (b) h = site system: Series and extrapolations for J < (a) pcut series (b) Extrapolations site system: 2nd derivative of ground-state energy and vortex number for J < (a) d 2 E/dh 2, J = (b) d 2 E/dJ 2, h = (c) Vortex number, J = (d) Vortex number, h =

9 v List of Figures site system: Series and extrapolations for J < (a) pcut series (b) Extrapolations site system: 2nd derivative of ground-state energy and vortex number for J < (a) d 2 E/dh 2, J = (b) d 2 E/dJ 2, h = (c) Vortex number, J = (d) Vortex number, h = Thermodynamic limit: Series and extrapolations for J < (a) pcut series (b) Extrapolations Comparison of the gap calculated with ED and pcut and various 24-site systems: 2nd derivative of ground-state energy for J < (a) J = (b) h = A.1 Transformation of the interaction sign of the Kitaev model A.2 Application of the transformation that changes the interaction sign on the Kitaev model in a uniform field A.2.(a) Transformation pattern A.2.(b) Transformed field C.1 Plaquette of the honeycomb lattice for definition of W p D.1 9-site system: Series for J > D.2 9-site system D.3 9-site system: Series for J < D.4 Illustration of the site labeling F.1 Overview of periodic lattices I F.1.(a) Periodic 8-site system F.1.(b) Periodic 18-site system F.1.(c) Periodic 24-site system F.2 Overview of periodic lattices II F.2.(a) Periodic 24-site system (diamond) F.2.(b) Periodic 24-site system (4x3 plaquettes) H.1 Illustration of the path in the phase diagram H.2 18-site system: 2nd derivative of ground-state energy for various J x = J y. 96 H.3 24-site system: 2nd derivative of ground-state energy for various J x = J y. 96

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11 vii List of Tables List of Tables 4.1 Tables of the critical parameter value J c and the critical exponent zν for various (N/M) extrapolants. Defective approximants are marked with an asterisk (*) (a) Critical parameters of the families (M/M 2), (M/M 1), and (M/M) (b) Critical parameters of the families (M/M + 1) and (M/M + 2). 68

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13 1 Introduction In the last tens of years there has been a lot of interest in the area of quantum computing in both, the scientific [DiV95, LD98, MLL03, SFN06, Sta13] and the public world [spe09, Aar08]. In 1982 Richard Feynman showed that classical computers would never be able to simulate a quantum mechanical systems consisting of some hundred particles exactly on the full Hilbert space. [Fey82] Instead a solution would be to use quantum computers - universal quantum simulators. Besides the simulation of quantum systems, quantum algorithms have been developed that could perform specific calculations much faster than their classical counterparts on non-quantum computers. Examples would be the Deutsch-Josza algorithm [DJ92] and an algorithm for the fast Fourier transform by Coppersmith [Cop94] which proved very useful for a prime factoring algorithm for large numbers introduced by Peter Shor [Sho95]. Systems proposed to provide properties necessary for quantum computation include e.g. superconductors [MSS01], liquid-state NMR (Nuclear Magnetic Resonance) [Ger97, CFH97], and atomic ions in electromagnetic traps [SKHR + 03, LDM + 03]. Another possible realization has been seen in solid-state systems that exhibit exotic particles due to their topological structure [Kit03, FKLW02]. Topological memories rely on the fact that encoded quantum states are protected against errors. These systems are stable in regard to local perturbations, such as e.g. small magnetic fields. As stated in [BP08] quantum computation with anyons are intrinsically error-free. Studying these systems has led to the discovery of an algorithm that provides a solution to the Jones polynomials only a few years ago [AJL06]. They have several applications in biology (DNA reconstruction) and statistical physics [BP08].

14 2 A nice overview over the history of quantum computation, algorithms, and implementations can be found e.g. in [SS08]. A quantum spin model displaying topological order, introduced by Kitaev in 2006, allows the existence of non-abelian anyons [Kit06]. Freedman et al. showed that universal quantum computation is possible using particular sorts of these particles [FLW02]. The main focus of the present thesis is on this model, widely known as Kitaev s honeycomb model. It consists of spin-1/2 degrees of freedom located on the vertices of a honeycomb lattice. The original Hamiltonian proposed by Kitaev reads H = J x σi x σj x J y σ y i σy j J z σi z σj z. (1.1) x links y links z links Spins located at opposing sites (i and j) of a honeycomb s edge are interchanging via the Pauli matrices σi x, σy i, or σz i on site i, respectively. This model falls into the category of compass models (for an overview see [NvdB13]) since the type of the link depends on its orientation. Each honeycomb consists of two links of each type. An illustration of the lattice structure can be found in Fig. 1.1.(a). x y z y z x 1.1.(a) Honeycomb lattice. 1.1.(b) Brickwall lattice. Figure 1.1. The lattice of the Kitaev model consists of honeycombs (a) with three different types of links between two neighboring sites. The letter α {x, y, z} represents a J α σ α i σα j coupling. Because of the bipartite nature of the lattice, sites belonging to the two sublattices are shown as full and empty circles, respectively. The honeycomb lattice can also be represented by a brickwall lattice (b) for an easier perception. The Kitaev model draws a lot attraction from its interesting properties while being exactly solvable [Kit06]. To this end, Majorana operators c j can be be introduced that fulfill c 2 j = 1, c j c l = c l c j if j l. (1.2)

15 3 From these Majorana operators, new Pauli-like operators in an extended space can be constructed that obey the same algebraic relations. Applying this to the Hamiltonian (1.1) leads to a model of free Majorana fermions in a static Z 2 gauge potential. For finding the ground state it proves helpful to introduce the plaquette operators W p that commute with the Hamiltonian: W p = 5 i=0 σ α(i) p,i,. (1.3) They are acting on plaquette p with α(i) being the type of the outgoing link and have eigenvalues w p = ±1. The equation is discussed in more detail in appendix C. The ground state is determined by the condition that w p = 1 for all plaquettes p, which is called vortex-free [Kit06, LL93]. States containing a vortex (an eigenvalue of w p = 1 is identified with a vortex) have a higher energy than the ground state. The Kitaev model has two phases called A and B as shown in the phase diagram in figure 1.2. The system realizes phase B if the inequalities J x J y + J z, J y J x + J z, J z J x + J y, (1.4) are fulfilled by the interaction parameters J α, α {x, y, z}. If one of the inequalities is violated, the system is in the gapped phase A. For a strong anisotropy J z J x,y J z A A B A J x J y Figure 1.2. The phase diagram for the Kitaev model (without a field). In the corners of the triangle two couplings vanish and only the third one exists. On the edges one coupling is zero. Phase A is gapped while phase B is gapless and acquires a gap in presence of an infinitesimal magnetic field [Kit06]. the two spins on each z-bond are strongly correlated and therefore, the system in figure 1.1.(a) may be replaced by dimers on a square lattice. The model is then adiabatically connected to the toric code model [Kit03]. Order four of degenerate perturbation theory yields the toric code. Higher orders have been calculated in [SDV08] and describe an

16 4 interaction of vortices while particles are static. The ground-state degeneracy only depends on the topology of the system. For example, on a torus it has four ground states that can be represented by a superposition of loops. For topologies with genus g the number increases to 4 g [Kit03]. The dependency only on the topology reflects the topological order of the system. There are two types of elementary excitations (electric and magnetic charges) with anyonic statistics. Moving two particles around each other (which is called braiding) leads to a non-trivial phase factor. Since the limit J z J x,y is adiabatically connected to all points in the phase A z, these particles also exist for all other points of phase A. The use of such topologically ordered states as a quantum memory has been suggested by Kitaev. As long as all anyonic excitations are localized, the states are topologically protected against decoherence [Kit03]. While in phase B the fermionic excitations are gapless, it has been shown [Kit06] that the addition of an infinitesimal field pointing in (1,1,1) direction to the Hamiltonian in Eq. (1.1) breaks the time reversal symmetry in order three and opens a gap with the leading term h xh y h z J 2, (1.5) with the magnetic field (h x, h y, h z ). The system is still exactly solvable and Kitaev has shown that the excitations are non-abelian Ising anyons. For the calculations in this thesis, the Hamiltonian H h = i (h x σ x i + h y σ y i + h zσ z i ) α-links α=x,y,z <i,j> J α σ α i σ α j (1.6) is introduced. Since the main focus here lies on phase B, the interchange values are set to J : = J x = J y = J z (thus the center of the middle triangle in Fig. 1.2). Additionally, the magnetic field will be set to point in (1,1,1)-direction, so the Hamitonian finally reads H h = h i (σ x i + σ y i + σz i ) J α-links α=x,y,z <i,j> σ α i σ α j. (1.7)

17 5 For a strong field all spins point in the direction of the magnetic field. The aim of this thesis is to analyze the phase transition between the polarized and the topological phase and to examine how well this topological phase is protected against external perturbations. For positive exchange parameters, this has already been studied by Jiang et al. [JGQT11] using density-matrix renormalization group (DMRG) calculations on finite systems. The critical point shows a good convergence for different system sizes up to 64 spins at h Jiang c for J = 2 while the nature of the phase transition remains unclear. It is argued that, while in the presence of a vortex gap, the system is vortex free, the number of vortices quickly increases at the phase transition. The deconfined vortices might transition to a confined phase. field Figure 1.3. The Kitaev model in the transformed field which is depicted here, is identical to the Kitaev model with a flipped interaction sign in a (1, 1, 1)-field. The Hamiltonian (1.1) has been proposed by Kitaev. In the present thesis the model is analyzed for both ferromagnetic and antiferromagnetic spin-spin interaction (i.e. J R) in contrast to the DMRG calculations by Jiang et al. who only studied the case of J > 0. While Kitaev argues that the global ground state energy does not depend on the signs of the exchange constants J x, J y, J z [Kit06, p. 18], things may change when applying an additional magnetic field. In appendix A it is shown how the Hamiltonian (1.7) can be mapped to a Hamiltonian with opposite interaction sign via a local unitary transformation. The result is a staggered twisted magnetic field (c.f. Fig. 1.3), which would have to be applied to the Kitaev model with ferromagnetic coupling to get the same behavior as for the Kitaev model with antiferromagnetic coupling in a field pointing in (1, 1, 1)-direction. The different signs of J are therefore not identical and treated

18 6 separately in this thesis, but still the order-three term of the magnetic field opens a gap due to equation (1.5). In the next chapter an overview of the various methods used in the present thesis is given. I explain how the methods used for calculations are working and how they are applied to the model. These include series expansion (perturbative Continuous Unitary Transformation (pcut)) and exact diagonalization (in particular the Lanczos algorithm)). Using these methods in the third chapter, I calculate quantities for the ground-state energy and the energy of the gap between the ground state and first excited state for small systems with open boundary conditions. In chapter 4 larger periodic systems are treated and pcut results for the thermodynamic limit are presented. The last chapter contains a summary of the results I obtained and an outlook of what can be done further.

19 2 Methods In this chapter I give an overview over the various methods I used to obtain and analyze results for the model introduced in the previous chapter. In section 2.1 the Fourier transformation, commonly used when treating translational invariant systems, is described. In the following two sections 2.2 and 2.3 methods for calculating the energy eigenvalues of Hamiltonians on finite clusters (ED/Lanczos & pcut) and in the thermodynamic limit (pcut) up to high orders of an expansion parameter are presented. One might ask the question why it is desirable to achieve a high order series for the quantities of interest. The Rayleigh-Schrödinger perturbation theory, taught to every undergraduate student of quantum mechanics, could easily give second order approximations that might lead to quite accurate results for some problems. But imagine problems where the perturbation parameter is not small because an easy description around the point of interest is not available. Higher orders will then contribute a lot and the necessity for higher order terms arises. Another problem might be that the series terms do not converge fast enough to ensure that higher orders can be neglected. Sometimes there are also problems where e.g. the first order term even vanishes (as will be the case for the Kitaev model studied in later chapters) due to symmetry reasons. To further stretch the range where the series expansion leads to meaningful results, I deal with extrapolation techniques (Padé & dlog Padé) in the last part of this chapter. With extrapolation of the series the possibility to extend the validity area for the pcut results and to overcome the restriction to small perturbations exists to some extent.

20 2.1. Fourier transformation Fourier transformation The discrete translational symmetry of an arbitrary lattice can be used to simplify the Hamiltonian. Let us consider a two-dimensional lattice consisting of lattice vectors a 1 and a 2 and unit cells containing n sites. The sites inside of a unit cell are counted via m i {1, 2,..., n}. A vector pointing to an arbitrary unit cell can be constructed by multiples of the lattice vectors as r = l a 1 + m a 2, with l, m Z. The distance between two unit cells is denoted by δ = r i r j. Since I will look at a quasi particle conserving Hamiltonian, no processes where a particle decays into two (or more) new particles with a different momentum or where the particle is destroyed are part of such a Hamiltonian. The derivation of this Hamiltonian can be reached via perturbative Continuous Unitary Transformations (pcuts) [KU00] that will be introduced later in this chapter. The one-particle sector of the effective particle conserving Hamiltonian can be written as H eff = δ ˆt m 1m 2 δ (2.1) m 1,m 2 where the action of ˆt m 1m 2 δ is determined by ˆt m 1m 2 δ r, m = δ m1,mt m 1m 2 δ r + δ, m 2. (2.2) This can be seen as the hopping of a particle from site ( r, m) to site r + δ, m 2 with the v 1 v 2 Figure 2.1. A unit cell (dashed) consisting of two sites and lattice vectors v 1 and v 2 of the honeycomb lattice. amplitude t m 1m 2 δ. The Fourier transform from the spatial basis states r to momentum basis states k, that are a superposition of the spatial states weighed with an exponential factor, is

21 Exact Diagonalization (ED & Lanczos algorithm) given by r, m k, m : = 1 e i k r r, m (2.3) N with the number of lattice sites N. For transformation of the Hamiltonian into momentum space one would calculate the matrix elements in the new basis k, m i H eff k, m j. k, m i H eff k, m j = k, m i ˆt m 1m 2 1 e i k r r, m j δ N δ m 1,m 2 = k 1, m i N δ m 1,m 2 = k 1, m i N = k, m i δ,m2 = δ k, k m i δ,m2 = δ k, k δ δ,m2 r r r r δ mj,m 1 t m 1m 2 δ e i k r r + δ, m 2 t m jm 2 δ e i k( r δ) r, m 2 t m jm 2 e i k δ 1 e i k r r, m 2 δ N t m jm 2 δ e i k δ m 2 r }{{} k,m 2 t mjmi δ e i k δ. (2.4) The effective Hamiltonian becomes a block diagonal matrix with blocks of size n n the matrix elements of which are given by Ω ij = δ t ji δ e i k δ. In case of the honeycomb lattice, that is the main subject of the present thesis, the lattice vectors and the unit cell are depicted in Fig Applying the Fourier transform to the lattice will lead to a 2 2 matrix the eigenvalues of which can be calculated analytically. 2.2 Exact Diagonalization (ED & Lanczos algorithm) For the purpose of a comparison of the results obtained by the pcut calculations I want to introduce the method of Exact Diagonalization (ED & Lanczos algorithm).

22 2.2. Exact Diagonalization (ED & Lanczos algorithm) 10 The idea is to write the Hamiltonian of a finite system as a Hamilton matrix with numerical values for the parameters. For example one would have to fix the magnetic field value, the interaction strength between the particle(s), etc. depending on the problem that should be addressed. For the Kitaev model in an isotropic field (h : = h x = h y = h z ) with symmetric interactions (J : = J x = J y = J z ) there are two parameters that have to be set, namely the two spin interaction strength and the value of the magnetic field. This would lead to a, in general, complex 2 N 2 N Hermitian matrix the eigenvalues of which then can be computed numerically if there is enough RAM available for storing the matrix and doing the calculations. In the present work for the ED calculations a maximum of nine spins is considered which translates to a dimension of 2 9 = 512 of the Hilbert space. While these eigenvalues can still easily be computed, it becomes clear that the computational limits are reached quite fast hence the dimension of the Hilbert matrix grows exponentially with the system size. It is therefore not possible to calculate e.g. critical values of phase transitions for very large systems that give an insight into what might be correct for the thermodynamic limit. What can be achieved using this method is the possibility to compare the pcut results for small systems to numerically exact known results. To push this limitation a bit, the Lanczos algorithm is implemented for calculating the eigenvalues of interest. The algorithm introduced in Sec is one that (hopefully) quickly converges towards the extreme eigenvalues of a matrix without having the same restrictions regarding memory as the full ED. Given that these eigenvalues are required, larger system sizes can be reached Programs For the smallest systems (2 spins, 3 spins) I used Wolfram Mathematica [Wol03] to perform the ED. The matrices were calculated by hand and inserted into a Mathematica notebook. With a call to the function Eigenvalues[x] the Mathematica algorithm chooses an appropriate method to compute the eigenvalues by itself. For the larger systems a C++ program incorporating Intel R s Math Kernel Library (MKL) (see [Int]) is used to generate the Hamilton matrix. The Hilbert space is generated using the local spin basis on which the action of the Hamiltonian is known. Matrix elements that connect two different states are spin flips and refer to the σ x and σ y Pauli matrices while diagonal elements stem from the σ z matrices in H. After generating the

23 Exact Diagonalization (ED & Lanczos algorithm) Hamilton matrix the MKL/Lapack [ABB + 99] routine zheevd is called. Depending on the options given to that routine either only the eigenvalues or both eigenvalues and eigenvectors are calculated Lanczos algorithm The Lanczos algorithm [Lan50] gives a possibility to perform exact diagonalization for larger systems than those reachable with the brute force method of directly diagonalizing the Hamilton matrix. The limiting effect for this method would be the memory necessary to store the large matrix which has the dimension 2 N for an N spin system without using any symmetries or other simplifications. Within the Lanczos algorithm the Hamilton matrix does not need to be explicitly known, but it is sufficient to know the action of the Hamiltonian on a Hilbert space vector in a chosen basis. What takes up the most memory are the Hilbert space vectors that have to be stored but in theory they can be limited to a number of three. For some information and a short analysis of the Lanczos algorithm one could read e.g. chapter Numerical Simulations of Frustrated Systems by A. Läuchli of [L 11]. The idea of the Lanczos algorithm is to change the basis of the Hamilton matrix to one where the matrix is reduced to a tridiagonal form. The basis chosen for the Lanczos algorithm lies in the Krylov space K = { φ, H φ, H 2 φ,... } and starting with a random, normalized vector φ 1 the other normalized basis vectors φ n can be computed as β 1 φ 2 = H φ 1 α 1 φ 1 (2.5) β n φ n+1 = H φ n α n φ n β n 1 φ n 1 (2.6) where α n = φ n H φ n and (2.7) β n = φ n+1 H φ n (2.8) are the elements of the resulting tridiagonal matrix. This can easily be seen by

24 2.2. Exact Diagonalization (ED & Lanczos algorithm) 12 multiplying equation (2.6) by φ n+1 and φ n, respectively. The multiplication yields β n φ n+1 φ n+1 }{{} 1 β n φ n φ n+1 }{{} 0 = φ n+1 H φ n α n φ n+1 φ n }{{} 0 = φ n H φ n α n φ n φ n }{{} 1 β n 1 φ n+1 φ n 1 }{{} 0 β n 1 φ n φ n 1 }{{} 0 (2.9) (2.10) since three succeeding basis vectors are mutually orthonormal by construction if the β n are chosen so that the φ n become normalized vectors. For a proof of orthogonality of the basis vectors see e.g. Stoer/Bulirsch [SB05, p. 38f.]. The resulting matrix, called T-matrix, then reads α 1 β 1 β 1 α 2 β 2 T n = β 2 α 3 β 3. (2.11)... βn 1 β n 1 α n The strength of the algorithm lies in the fact that one does not need to calculate the whole Hamiltonian in the new basis to get the extreme eigenvalues. The eigenvalues of the intermediate matrix T n are quickly converging towards the extreme eigenvalues of H. The Lanczos algorithm for different n can be performed repeatedly. During the process the convergence of the eigenvalues should be observed to make sure that the energy can be computed with the desired precision. While analytically all computed vectors are orthogonal to each other, numerical rounding errors can occur that lead to a loss of orthogonality. A study of the error behavior has e.g. been conducted by C.C. Paige [Pai70]. To avoid this errors re-orthogonalization of the vectors should be performed during the calculations. A deviation of the calculated vectors can be detected within the algorithm [Pai70, p.1], namely by calculating the scalar product of the currently calculated vector with the previously calculated ones. The Lanczos process (2.6) would then be extended to γ n 1 ψ n = β n 1 φ n n α in ψ i (2.12) i

25 Exact Diagonalization (ED & Lanczos algorithm) with α in = ψ n β n 1 φ n. (2.13) That means that the non-orthogonal part α in of the last calculated vector in direction ψ i are subtracted from the vector and the orthogonality is preserved. The sum runs over all previously calculated vectors. The ψ n are now spanning the Krylov subspace of H and the T n matrix has the normalization factors γ n on the secondary diagonal. Implementation of reorthogonalization As mentioned before, a problem that may arise while performing the Lanczos algorithm is that due to numerical errors the eigenvectors loose their orthogonality in the process of the computation. I addressed this problem by additionally orthogonalizing the vectors that have lost their orthogonality (c.f. the scalar product is larger than a given value) with respect to some of (but not all of!) the previously calculated vectors. As an example consider the newly calculated vector φ n+1 and the previously calculated vector φ j. If φ j φ n+1 > ɛ, where ɛ defines an accuracy for the calculations, I choose the vector φ n+1 as φ n+1 = φ n+1 φ j φ n+1 φ j. (2.14) The two vectors are now mutually orthogonal as they are supposed to be φ j φ n+1 = φ j φ n+1 φ j φ j φ n+1 φ j = 0. (2.15) An advantage of the Lanczos algorithm is that the knowledge of only two vectors is necessary to calculate the next one, hence only three vectors need to be stored. For orthogonalization with respect to all vectors calculated before, it would be necessary to store all of the vectors and the advantage of saving memory is somewhat lost. The number of vectors with respect to I perform the orthogonalization is therefore restricted to the first few. In most cases I chose about five to ten vectors depending on the memory necessary for the calculations (i.e. the system size).

26 2.2. Exact Diagonalization (ED & Lanczos algorithm) 14 Eigenvectors The algorithm described above can be used to obtain the extreme eigenvalues of a large Hamilton matrix H. But for analyzing the system s state belonging to the computed eigenvalues it is necessary to get the eigenvectors, too. Letting d be the dimension of the Hilbert space, the Lanczos algorithm transforms the (d d) Hamilton matrix to a k dimensional tridiagonal matrix T }{{} k k = }{{} V k d }{{} H d d }{{} V d k, (2.16) where k is the number of Lanczos steps performed. The transformation matrix V consists of the Lanczos vectors spanning the Krylov space. With the aim of calculating the eigenvalues, the tridiagonal matrix T is diagonalized numerically: D L }{{} k k = }{{} S k k }{{} T k k }{{} S k k. (2.17) S is the matrix transforming T into the diagonal matrix D L. It consists of the eigenvectors of T that are also computed numerically. If we knew all eigenvectors of the Hamiltonian it could also be diagonalized with unitary (d d) matrix U consisting of all eigenvectors. D H = U HU (2.18) Looking again at the diagonal matrix obtained by the Lanczos algorithm D L = S T S = S V HV S = U L H U }{{}}{{} L k d d k (2.19) it becomes obvious that the rows of U L = V S are the eigenvectors of H belonging to the eigenvalues on the diagonal of D L.

27 Exact Diagonalization (ED & Lanczos algorithm) Finite-size effects in the Kitaev model for h = 0 For the Kitaev model described by equation (1.1) there are two possible excitations in the system. On the one hand it is possible to have different vortex-free states with fermionic excitations. On the other hand there are different vortex sectors with varying energies. While in the thermodynamic limit the phase B is gapless for h = 0, the system gains a gap if finite systems are considered. Even on finite systems the ground state may be in the vortex-free sector of the Hamiltonian (c.f. chapter 4). But since there is a finite number of sites the fermionic excitations do not form a continuous energy band above the different N-vortex states (see Fig. 2.2 for illustration). h (a) Thermodynamic limit. h (b) Finite system. Figure 2.2. This is an exemplary illustration of the different spectra of the Kitaev model in phase B. For the thermodynamic limit (a) the system is gapless. While the ground state lies in the vortex-free sector, free fermions may be created with zero energy. The different vortex sectors have an overlap, so no gap appears in the spectrum. An arbitrary finite system (b) gains a gap due to the loss of continuity in the fermionic excitation energies. The vortex sectors consist of points rather than a band. For ED (Lanczos) calculations in chapter 4 it needs to be checked if the ground state lies in the vortex-free sector and is therefore comparable to the ground state in the thermodynamic limit. To this end, the plaquette operators W p introduced in chapter 1 are used to define the number of vortices for an arbitrary state ψ as the expectation

28 2.3. Perturbative Continuous Unitary Transformations (pcut) 16 value N vortex = ψ p 1 W p ψ. (2.20) 2 The relative number of vortices per plaquette N vortex/plaquette is referred to as the vortex filling. For h 0 the plaquette operators do not commute with the Hamiltonian anymore and therefore the Hamiltonian and the W p do not have a common eigenbasis. But as is shown in chapter 4 that this quantity seems correlated to a phase transition. Additionally, the quantity has also been calculated in [JGQT11] and they have observed that vortices appear to condense and their number quickly increases around the phase transition point. 2.3 Perturbative Continuous Unitary Transformations (pcut) The main focus of this thesis is to investigate the quantum phase transition between the B-phase of the Kitaev model where an external magnetic field gives rise to non-abelian anyonic excitations (vortices) [Kit06] and the conventional polarized phase at large field values. In this section I will shortly describe Continuous Unitary Transformations before focusing on the pcut method. This method has been introduced by Knetter and Uhrig in 2000 [KU00] and is based on the CUT method which uses flow equations for the Hamiltonian to derive an effective model and has been proposed by F. Wegner in 1994 [Weg94]. To perform the perturbative calculation with pcut it is necessary to have a nondegenerate ground-state with well defined excitations. Hence I will consider the system in the high-field limit where the ground-state is the one where all spins are polarized and excitations are local spin flips. If the number of these excitations are counted by the operator Q the original Hamiltonian H can be transformed into a block diagonal, particle number conserving Hamiltonian H eff that fulfills the relation [Q, H eff ] = 0. The effective Hamiltonian is in general a lot easier to treat than the original one which usually does not have the same block diagonal structure. In this work I intend to track the ground-state energy and the energy of a single excitation in the system for a parameter range as large as possible. With pcut it is possible to calculate high-order series for these energies in the thermodynamic limit.

29 Perturbative Continuous Unitary Transformations (pcut) Continuous Unitary Transformations (CUT) Unitary transformations can be seen as a rotation in the Hilbert space of a Hamiltonian. They are often performed to simplify a given Hamiltonian for a better accessibility of its eigenvalues. A general unitary transformation of a matrix A with the unitary matrix U would read A B : = U AU. (2.21) The spectrum of a matrix does neither change by a single unitary transformation nor during a series of unitary transformations. If the matrix U was built using the eigenvectors of A, B would even be diagonal with A s eigenvalues on its diagonal. For a given Hamiltonian the idea is now to find a suitable transformation that leads to a less complex effective Hamiltonian H eff with the same eigenvalues. Going to infinitesimal small transformations, that are successively performed on the operator H, one speaks of Continuous Unitary Transformations (CUT). The method has been introduced by Wegner and is detailed in [Weg94]. The Hamiltonian during the continuous sequence of transformations (or the flow) can be described by the continuous parameter l (called flow parameter), where H = H(l = 0) is the Hamiltonian at the beginning without a transformation applied and H eff = H(l = ) is the resulting effective Hamiltonian. The Hamiltonian during the flow can be written as H(l) = U(l)H(l = 0)U (l). (2.22) The derivative of 2.22 with respect to l gives the flow equation dh(l) dl = [η(l), H(l)] (2.23) where η(l) is the generator of the unitary transformation. η is an anti-hermitian operator that depends on the flow Hamiltonian H(l) and therefore implicitly on the parameter l η(l) = η (l). (2.24) For simplification of a difficult problem it is now necessary to find a suitable generator that makes the effective Hamiltonian H eff as easy as possible. The generator proposed by Wegner in [Weg94] is η = [H d, H] with H d being the diagonal

30 2.3. Perturbative Continuous Unitary Transformations (pcut) 18 part of the Hamiltonian. This choice of generator has some disadvantages since it does not retain a possible band diagonal structure of the original Hamiltonian but rather generates higher interactions. Furthermore there are problems with degeneracies where the Hamiltonian might not be diagonalized because off-diagonal matrix elements do not vanish. In [Mie98] Mielke pointed out these disadvantages and proposed a different choice of generator. If the generator is written down (for real, symmetric matrices H = (h nm )) as η nm = η mn = sgn(n m)h nm, η nn = 0 (2.25) the off-diagonal elements vanish and also if the Hamiltonian has initially a banded structure, this form is preserved. [Mie98, p. 610] Perturbative CUT In this section I present a summary of the perturbative Continuous Unitary Transformation (pcut) method. The method is based on the CUT described in the previous section and also uses a generator which allows to keep the band block diagonal structure of the original problem [KU00, p. 210] but takes up perturbative elements to allow computation of series expansions of the energies for problems of a special type. In the present thesis pcut is the main method to tackle the Kitaev model in a field with the aim of calculating a phase transition point between the polarized phase in the high-field limit and the vortex-free phase calculated by Kitaev [Kit06]. This method can be implemented directly on a computer and I used a C++ program that has kindly been made available by the group of Kai P. Schmidt. The idea is to map the Hamiltonian H to an effective Hamiltonian H eff that conserves the number of excitations in a system. The effective Hamiltonian in the Fock basis will then have a block diagonal structure with blocks corresponding to i particle subspaces U i. As a result one can get quantities like the dispersion ω( k) and the ground-state energy per site as a polynomial of a small perturbation parameter x. Several conditions regarding the Hamiltonian are necessary for the following perturbation scheme. First the considered Hamiltonian should be of the form H = H 0 + xv, (2.26)

31 Perturbative Continuous Unitary Transformations (pcut) meaning that it consists of an unperturbed part H 0 the eigenvalues of which are well known. As an addition to other methods of perturbation theory the spectrum of H 0 needs to be equidistant and bounded from below. Therefore it is possible to write the energies as E i with i {0, 1, 2,... }. They can be understood as the energies of i quanta (excitations above the ground-state E 0 ) in the system. Also the system is divided into subspaces U i that have no connecting matrix elements in the simple Hamiltonian H 0. Now a perturbation xv with a small parameter x, linking the subspaces U i, is added to H 0. One needs to be able to write the perturbation as N V = T n (2.27) n= N where the action of T n is to increase (n > 0) or decrease (n < 0) the number of excitation quanta in the system by n or keep the number of particles (n = 0). This can be written as [H 0, T n ] = nt n (2.28) since [H 0, T n ] i = (H 0 T n T n H 0 ) i = H 0 i + n T n i i = (i + n i) i + n = nt n i (2.29) with the i-particle state i. In the following these are referred to as (quasi) particles, while the word quasi might often be omitted. The full Hamiltonian written in terms of the T n thus reads N H = H 0 + x T n. (2.30) n= N Introducing a particle language to the Kitaev model in a field treated in this work (cf. section 2.3.3) leads to N = 2. These particles live on the vertices of the honeycomb lattice and the T n act on the edges between the sites. In [KU00] the use of a generator η(l), shown in Eq. (2.33), similar to the one introduced

32 2.3. Perturbative Continuous Unitary Transformations (pcut) 20 by Mielke [Mie98], is proposed. The evolving CUT Hamiltonian can then be written as H(l) = H 0 + xθ(l) (2.31) with the most general form for θ(l) θ(l) = The generator is given by η(l) = where k=1 x k k=1 x k 1 F (l; m)t ( m) (2.32) m =k sgn (M( m)) F (l; m)t ( m) (2.33) m =k m = (m 1, m 2,..., m k ), m i { N, N + 1,..., N 1, N} (2.34) m = k (2.35) T ( m) = T m1 T m2 T m3... T mk (2.36) M( m) = k m i. (2.37) i=1 In order k all possible sequences of k T n operators defined in (2.27) appear. The sgn function is defined so that sgn(0) = 0. The F (l; m) are real valued functions for which nonlinear, but recursive, differential equations [KU00, p. 211] are derived in [KU00] on page 211f. while a proof that this choice of generator leads to block diagonality of the effective Hamiltonian is given in appendix A of the cited paper. A graphical illustration of the method is given in figure 2.3 In the end the coefficients of H eff = H(l = ) are the relevant values for the computation. These coefficients C( m) = F (l = ; m) (2.38) are ratios that are model independent and can be computed by computer programs. Knetter and Uhrig show that only terms with M(m) = 0 will not vanish for l [KU00, p. 212], so H eff commutes with H 0 hence it does not change the number of

33 Perturbative Continuous Unitary Transformations (pcut) 1 qp 1 qp 2 qp 2 qp l 3 qp 3 qp H(l = 0) H(l = ) Figure 2.3. For the Kitaev model in a field a particle language is introduced in As shown on the left side there are only processes that create (annihilate) one or two particles (colored green and red) or processes that conserve the number of particles (gray). The flow of the pcut method leads to block diagonality of a Hamiltonian that can be described by energy quanta in the limit l (right side). The block diagonal elements and the excitations ((quasi) particles) denoted by 1qp, 2qp,... on both sides are different ones. quasi particles. The final effective Hamiltonian then reads H eff = H 0 + k=1 x k m =k M( m)=0 C( m)t ( m). (2.39) Cluster additivity As has been stated before, the pcut results are valid even for the thermodynamic limit where the number of lattice sites N approaches infinity. This would be clear if the calculations where done on an infinitely large lattice but one strength of the method lies in the fact that only clusters of a finite size are needed for calculations up to a given order. The reason can be found in the linked cluster theorem [DKS + 10] which states that only linked clusters contribute to the quantity of interest. This means only processes that are linked are contributing and terms that describe processes at far away points

34 2.3. Perturbative Continuous Unitary Transformations (pcut) 22 on an infinite lattice (c.f. are not linked) can be dismissed. The size of clusters needed for the calculation of a given order therefore only grows proportional to the order if the Hamiltonian is only acting locally. For further explanation and description of linked cluster expansions one might look e.g. in the book Series Expansion Methods for Strongly Interacting Lattice Models written by J. Oitmaa, C. Hamer and W. Zheng [OHZ10] Application to the model The original Kitaev model has three different spin-spin interactions on the trivalent honeycomb lattice. In this work a magnetic field is added so the Hamiltonian reads (c.f. (1.6)) H h = i (h x σ x i + h y σ y i + h zσ z i ) α-links α=x,y,z <i,j> J α σ α i σ α j. (2.40) with Pauli-matrices σi α acting on site i and the coupling strength J with a sum over next-neighbours where every link is touched once. Only focussing on the isotropic case (h : = h x = h y = h z ) the Hamiltonian simplifies to H h = h i (σ x i + σ y i + σz i ) α-links α=x,y,z <i,j> J α σ α i σ α j. (2.41) For the pcut method one needs to divide the Hamiltonian into an unperturbed part H 0 with an equidistant spectrum and a perturbation xv of some kind. Afterwards a picture is introduced where excitations above the ground-state of H 0 are described as (quasi) particles. To get an H 0 with a non-degenerate ground-state I start from the high-field limit h J where all spins are pointing in field direction. The field term therefore has an equidistant spectrum and the Hamiltonian can be transformed via a unitary transformation U HU into the basis where this term becomes diagonal. In appendix B further information on these calculations is given.