Computational Physics

11.7 The density matrix renormalisation group method 363 total Hamiltonian is then HmU,sl+ 1 ,sl+ 2 ,n;m′,s′ l+ 1 ,s′l+ 2 ,n′ =H ...

364 Transfer matrix and diagonalisation of spin chains So far, we have considered the simplest DMRG method in fair detail. It is ...

Exercises 365 lll+ 1 l+ 2 + 3 S E S E Figure 11.6. The third step in the finite-size DMRG algorithm. Each time the sweep has rea ...

366 Transfer matrix and diagonalisation of spin chains
1
5
3
6  
n n
ν
ν Figure 11.7. Action of the transfer matrix ...

Exercises 367 This leads to the following code for the multiplication routine involving the first (leftmost) vertex: FORn=0TO2M− ...

368 Transfer matrix and diagonalisation of spin chains (b) Show thatSz= ∑L l= 1 Sz,lcommutes withH. Also show that S^2 = (L ∑ l= ...

Exercises 369 the last equation can be written in the more compact form: ρS|uα〉= ∑ α′ 〈uα|ρS|uα′〉|uα′〉. Show that the last equat ...

370 Transfer matrix and diagonalisation of spin chains References [1] H. W. J. Blöte and M. P. Nightingale, ‘Critical behaviour ...

References 371 [25] F. Aletet al., ‘The ALPS project: open source software for strongly correlated systems.’ ar Xiv: cond-mat/04 ...

12 Quantum Monte Carlo methods 12.1 Introduction In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for ...

12.2 The variational Monte Carlo method 373 on a strip, for cases where the matrix size renders even sparse matrix diagonalisati ...

374 Quantum Monte Carlo methods The loop stops when the minimum energy is reached according to some criterion. It is the second ...

12.2 The variational Monte Carlo method 375 The expectation value of the local energy is now calculated as an average over the s ...

Table 12.1. Variational Monte Carlo energies. Harmonic oscillator Hydrogen atom Helium atom α 〈E 〉 var (〈 E〉 ) Ev var (E )v α 〈E ...

12.2 The variational Monte Carlo method 377 The same can be done for the variance with the result var(E)v= ( 1 − 4 α^2 )^2 32 α^ ...

378 Quantum Monte Carlo methods of−2.83 a.u., and with the exact value of−2.903 7 a.u. The VMC value can obvi- ously be improved ...

12.2 The variational Monte Carlo method 379 ASis the Slater determinant (seeChapter 4) andφis a function which contains the two ...

380 Quantum Monte Carlo methods Forl>0, the radial wave function is written in the formrlρ(r)whereρdoes not vanish atr=0. Ana ...

12.2 The variational Monte Carlo method 381 probability 1− 2 α. This is clearly a Markov process as described inSection 10.3. We ...

382 Quantum Monte Carlo methods we move the walker to a new positionzat time 2 twith probability distribution G(y,z; t). We have ...