Computational Physics

11.3 Two-dimensional spin models 343 11.3 Two-dimensional spin models We shall now describe the transfer matrix analysis for the ...

344 Transfer matrix and diagonalisation of spin chains s'i ssi s'i– 1 i+ 1 Figure 11.1. A step in the multiplication of a vector ...

11.3 Two-dimensional spin models 345 given in(11.25)is not correct fori=1ori=M, because the first new spin,s′ 1 , does not yet h ...

346 Transfer matrix and diagonalisation of spin chains programming exercise Write a program for calculating the largest eigenval ...

11.4 More complicated models 347 f 0.93 0.931 0.932 0.933 0.934 0.935 0.936 0 0.006 0.012 0.018 0.024 0.879 0.88 0.881 0.882 0.8 ...

348 Transfer matrix and diagonalisation of spin chains





12345 6 Figure 11.3. Vertex configurations of the six-vertex model ...

11.5 ‘Exact’ diagonalisation of quantum chains 349 11.5 ‘Exact’ diagonalisation of quantum chains So far in this chapter, we hav ...

350 Transfer matrix and diagonalisation of spin chains The real parametertrepresents the hopping rate, andU(which is also real) ...

11.5 ‘Exact’ diagonalisation of quantum chains 351 Before we proceed, we note that the Heisenberg Hamiltonian can be written in ...

352 Transfer matrix and diagonalisation of spin chains programming exercise Program a routine for multiplying the Hamiltonian by ...

11.5 ‘Exact’ diagonalisation of quantum chains 353 For the one-dimensional Heisenberg chain, the implementation of symmetry is v ...

354 Transfer matrix and diagonalisation of spin chains states is smaller than four, the correspondingk-values are spaced by more ...

11.6 Quantum renormalisation in real space 355 You have now seen group theory ‘at work’. The possible improvements in perform- a ...

356 Transfer matrix and diagonalisation of spin chains Figure 11.4. Two fixed-boundary solutions for a narrow box (solid triangl ...

11.6 Quantum renormalisation in real space 357 Calculate the eigenstates of these four matrices and keep the lowestM.Now the la ...

358 Transfer matrix and diagonalisation of spin chains Table 11.1. Energies for the particle in a box after N= 10 steps. Energie ...

11.7 The density matrix renormalisation group method 359 We have generated a new basis set of sizeM×MS(rememberMS= 2 S+1). The m ...

360 Transfer matrix and diagonalisation of spin chains Now we work outEq. (11.49)using the basis set|σ〉. We obtain 〈QS〉= ∑ σ,, ...

11.7 The density matrix renormalisation group method 361 S E lll+ 1 l+ 2 + 3 Figure 11.5. Schematic representation of the DMRG p ...

362 Transfer matrix and diagonalisation of spin chains We still have to address one more question: how can we calculate the Hami ...