EVIDENCIAS Y DIRECCION DEL FLUJO DE GENES EN P. lunatus L

As pointed out in Chapter 1, the central topic of this thesis is the study of frustrated spin-1₂ models by a variational Monte Carlo approach based on a parton construction. We resort to the Abrikosov fermion representation of spins to introduce a trial wave function |Ψ0i of the form of Eq. (1.22), which is made of the product of a symmetric spin-spin

Jastrow factor1 and a Gutzwiller-projected fermionic state. Since the fermionic part of the trial state, |Φ0i, is a Slater determinant, the variational guess |Ψ0i is often referred to

as Jastrow-Slater wave function. Within the parton framework, the states |xi which are sampled by the Monte Carlo algorithm are fermionic configurations in real space (labelled by particle positions and spin components along z). Due to the presence of the Gutzwiller projector, only the configurations with no empty nor doubly occupied sites (i.e. the spin configurations) are considered. As emphasized in the previous section, the crucial quantity to compute in a variational Monte Carlo calculation is the ratio of the amplitudes of the wave function in two different configurations. We split the calculation of ratios in two parts, separately discussing the contributions coming from the Jastrow factor and the fermionic part of the variational state.

Jastrow factor

The Jastrow correlator included in the variational wave function is diagonal in the space of configurations, i.e. hx|Js|x′i = Js(x)δx,x′ A naive calculation of the ratio of two Jastrow factors, Js(x′)/Js(x), costs O(N2) operations. However, this computational burden can be

reduced to O(1) if the configurations |xi and |x′_{i differ only by few fermionic hoppings [86].}

This favourable condition is always verified in the numerical calculations considered in this thesis, because the Monte Carlo moves employed in the simulations consist of spin-flipping processes of the form

|x′i = Si+Sj−|xi = ci,↑† ci,↓c†j,↓cj,↑|xi. (3.16)

Therefore, the fast algorithm for the calculation of Jastrow ratios can be employed [86]. Without entering the details of this method, here we limit ourselves to observe that the implementation requires to store the following table at the beginning of the Monte Carlo run:

TJastrow(j) =

X

i

vi,jSiz, (3.17)

where vi,j is the pseudopotential entering Eq. (1.21). Then, any ratio of Jastrow factors of

the aforementioned form can be simply computed by reading out the appropriate element of TJastrow [86]. Since the above table needs to be updated every time a new configuration

is accepted, the resulting computational cost per Monte Carlo move scales as O(N ). The fermionic state

The fermionic part of the variational wave function is the ground state of a certain auxiliary Hamiltonian H0, defined in Eq. (1.15) and (1.20). If no pairing terms are included in

H0, the total number of fermions is conserved and the ground state of the system is a

Slater determinant. In order to construct this wave function, the auxiliary Hamiltonian is numerically diagonalized in real space to obtain a set of single-particle orbitals, whose creation operators are

φ†_I =X

i,σ

Ui,σ;Ic†i,σ. (3.18)

Here, capital letters are used to label the aforementioned orbitals (in ascending order according to the corresponding energies) and the coefficients Ui,σ;I are the entries of

the unitary matrix that diagonalizes H0. If the system contains N spins, the ground

state of H0 is built by filling the N single-particle orbitals with the lowest energies, i.e.

|Φ0i =QNI=1φ†I|0i [86]. The overlap of this state with a physical configuration |xi is the

determinant of a N ×N matrix. The numerical evaluation of such a Slater determinant has a computational cost that scales as O(N3). However, as already observed, the variational Monte Carlo technique involves the calculation of ratios of determinants, which can be performed with O(N ) operations by using the so-called fast update algorithm [86]. Since O(N ) ratios are required for the calculation of the local energy, the overall computational cost of the present variational Monte Carlo method scales as O(N2) [86].

We note that the numerical construction of |Φ0i requires the single-particle spectrum

of H0 to be a closed shell, in order to avoid ambiguities when filling up the lowest orbitals.

This condition, which ensures that the ground state of the system is uniquely defined, trivially occurs in the case of gapped auxiliary Hamiltonians. On the contrary, if H0

is gapless in the thermodynamic limit, its diagonalization on finite size clusters could return an open-shell spectrum. In this case, we can try to impose different (periodic or antiperiodic) boundary conditions to the fermionic degrees of freedom in order to open a finite-size gap. This is usually possible in the situations in which the fermionic spectrum displays gapless Dirac nodes: playing with the size and the shape of the finite cluster, and with the boundary conditions of the fermions, it is possible to avoid the k-points of the Brillouin in which the spectrum is gapless. On the other hand, if H0 is characterized

by an extended Fermi surface, the situation becomes more complicated and a closed-shell spectrum is not always achievable.

If the Hamiltonian H0 contains pairing terms, the number of particles of the system is

no more conserved and the fermionic ground state is generally represented by a Pfaffian wave function [86]. However, in the particular case in which only hopping and pairing terms are included (i.e., no magnetic field in the xy-plane), we can exploit a formal trick to reduce all the couplings of the Hamiltonian to hoppings, thus restoring the conservation of particles. This trick is based upon the definition of a different set of fermionic operators (labelled by d), which are related to the original Abrikosov fermions by a particle-hole transformation on down spins:

d_i,↑ = c_i,↑ d_i,↓ = c†_i,↓. (3.19)

We strongly emphasize that this particle-hole transformation is not a gauge transforma- tion, because it does not preserve the original form of the spin operators. The single-site

configurations of d-fermions are related to the ones of the c-fermions by the following equations:

|0ci = d†_↓|0di, c†_↑c†_↓|0ci = d†_↑|0di,

c†_↓|0ci = |0di, c†_↑|0ci = d†_↑d†_↓|0di.

(3.20)

Here |0ci and |0di label the vacuum states of the c- and d-fermions, respectively. The

Monte Carlo sampling of d-fermions is ruled by the transformed Gutzwiller projector, PGd =

Y

i



1 + d†_i,↑d_{i,↑− d}†_i,↓d_i,↓ _{1 − d}†_i,↑d_i,↑+ d†_i,↓d_i,↓, (3.21)

which kills all the configurations containing singly-occupied sites. As a consequence of the particle-hole transformation of Eq. (3.19), the pairing terms of H0 are transformed into

spin-flipping hoppings (i.e. c_i,↓c_{j,↑ 7→ d}†_i,↓d_j,↑), and the resulting Hamiltonian conserves the total number of d-fermions. Therefore, in the new fermionic language, the ground state of the system can be expressed as a Slater determinant, and the computational machinery previously described can be applied [86].

All the numerical results presented in this thesis are obtained by using a Slater wave function for the fermionic part of the variational state. In other words, no auxiliary Hamil- tonians involving both the in-plane magnetic field and singlet pairing terms are considered. This limitation does not affect significatively the accuracy of the variational calculations since, in general, the energy gain provided by pairing couplings is small within magnetically ordered phases.