* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Kitaev Honeycomb Model [1]
Quantum machine learning wikipedia , lookup
Quantum entanglement wikipedia , lookup
Quantum dot cellular automaton wikipedia , lookup
Quantum group wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Path integral formulation wikipedia , lookup
Quantum field theory wikipedia , lookup
Wave function wikipedia , lookup
Second quantization wikipedia , lookup
Ferromagnetism wikipedia , lookup
History of quantum field theory wikipedia , lookup
Bra–ket notation wikipedia , lookup
Hidden variable theory wikipedia , lookup
EPR paradox wikipedia , lookup
Density matrix wikipedia , lookup
Coherent states wikipedia , lookup
Tight binding wikipedia , lookup
Scalar field theory wikipedia , lookup
Dirac bracket wikipedia , lookup
Quantum state wikipedia , lookup
Self-adjoint operator wikipedia , lookup
Spin (physics) wikipedia , lookup
Bell's theorem wikipedia , lookup
Compact operator on Hilbert space wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Ising model wikipedia , lookup
Oberseminar: Quantum Knots - Prof. Dr A. Rosch, Prof. Dr. S. Trebst - University of Cologne - May 6, 2014 Kitaev Honeycomb Model [1] Christopher Bartel - [email protected] Peter Gerliz - [email protected] In the following we will use the following notation: bx = c1 , by = c2 , bz = c3 , c = c4 . It turns out, that we can use The model consists of a honey- an operator D = bx by bz c to determine whether any state comb lattice with a spin sitting |ψi is physical or not. |ψi is physical ↔ D |ψi = |ψi. The on each of its vertices. The pauli operators σ x σ y σ z can be represented by σ̃ x = ibx c, y spins interact with their near- σ̃ y = iby c, σ̃ z = ibz c which act on the extended space. 3 4 2 est neighbors via three different z 1 5 types of links. The x-, y- and 6 Application To The Model x z-interaction. The interaction of the spins can be described us- If we replace the pauli operators with those that act on j ing the pauli spin operators σiα , the extended space we get σ˜ α σ˜ α = (ibα c )(ibα c ) = k j k j j k k where α = x, y, z and i indexing −iuˆ c c with uˆ = ibα bα which we associate with the jk j k jk j k the site. link (j,k). We now insert this into the Hamiltonian and get The Hamiltonian has the followFigure 1: The Lattice iX ing form: H= Âjk cj ck 4 j,k X X y y X H = −Jx σjx σkx − Jy σj σk − J z σjz σkz with Âjk = 2Jαjk ûjk if (j,k) are connected and Âjk = 0 else. x-links y-links z-links Remarkably, the operators Âjk commute with the HamilIn the lattice we can define a plaquette(hexagon) and the tonian and with each other and have the eigenvalues ±1. operator Wp = σ1x σ2y σ3z σ4x σ5y σ6z which commutes with the Remember the operators Wp did the same. Using a theorem Hamiltonian and itself. Thus, the Hamiltonian can be called ”Lieb’s Theorem”, we know that the groundstate solved individually for the eigenspaces of Wp . The original of the system lies in the subspace where all operators Wp Hilberspace is of the dimension 2n where n is the number of have the eigenvalue +1 (vortex free configuration). This lattice-points. The use of the eigenspaces only reduces our leads to the fact that we can replace the operators Âjk by problem to the dimension of 2n/2 . It turns out, that if we the eigenvalue +1 - a procedure Kitaev calls ”removing represent our spin operators with majorana operators we hats”. From this we finally obtain our Hamiltonian in the can simplify our model even more and get the Hamiltonian quadratic form H = i P Ajk cj ck where A is no more an 4 to a quadratic form. j,k operator but a number Ajk = 2Jαjk . It turns out, that the configuration uˆjk = +1 is translational invariant and we Representing Spins by Majorana can solve our problem using Fourier-Transformation. We Fermions will represent the site index j as (s, λ) where s refers to a unit cell and λ to a position inside the cell. The calculation A system with n fermionic modes is usually described by leads to the Fourier transformed Hamiltonian: the annihilation and creation operators ak and a†k . Instead, 1 X H= iÃλ,µ (~q)a−~q,λ aq~,µ one can use their linear combinations c2k−1 = ak + a†k and 2 ak −a†k q,λ,µ c2k = i which are called majorana operators. We now represent our spins by two fermionic modes, i.e. by four with the matrix majorana operators. 0 if (~q) iÃ(~q) = −if (~q)∗ 0 The Model Figure 2: Majorana representation of spin up and spin down and the function f (~q) = 2(Jx ei(~q,n~1 ) + Jy ei(~q,n~2 ) + Jz ). Out of this we get the energy dispersion (~q) = ±|f (~q)| We choose the state with no fermions to be spin up and the state with two fermions to be spin down. The states An important property of this function is, whether it has with one fermion dont have any physical relevance to us. zeros for some ~q or not. We say it is gapless if it has zeros Oberseminar: Quantum Knots - Prof. Dr A. Rosch, Prof. Dr. S. Trebst - University of Cologne - May 6, 2014 and say gapped if not. The equation f (~q) = 0 only has solutions if and only if |Jx |,|Jy |,|Jz | satisfy the triangle inequalities. That means if the parameters are in the range of phase B in figure 3. Phases In contrast to the described case there is also a highly frustrated case, where all the coupling parameters are roughly the same, which takes place in the gapless phase B. In this phase the system corresponds to a quantum spin liquid with Z2 topological order, which is disordered even at lowest temperature due to quantum fluctuation. J =(0,0,1) Az B Ax J =(1,0,0) Let us now consider the plane Jx + Jy + Jz = 1 in the parameter space. There are three equivalent gapped phases called phase A here and one gapless phase B. In phase A there is only one dominant interaction, while the other two types are weak. This leads to a simpler model, where pairs of spins live on a lattice of disjoint dimers. By adjusting the unit cell to a square lattice, this model can be reduced to the toric code [2]. References Ay [1] A. Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321(1):2 – 111, 2006. J =(0,1,0) Figure 3: Parameter space of the coupling strengths [2] A. Kitaev and A. Laumann. Topological phases and quantum computation.