Generalized diagonalization scheme for many-particle systems

Despite the advances in the development of numerical methods analytical approaches still play the key role on the way towards a deeper understanding of many-particle systems. In this regard, diagonalization schemes for Hamiltonians represent an important direction in the field. Among these techniques the method, presented here, might be that approach with the widest range of possible applications: We demonstrate that both stepwise and continuous unitary transformations to diagonalize the many-particle Hamiltonian as well as perturbation theory and also nonperturbative treatments can be understood within the same theoretical framework. The new method is based on the introduction of generalized projection operators and allows us to develop a renormalization scheme which is used to evaluate directly the physical quantities of a many-particle system. The applicability of this approach is shown for two important elementary many-particle problems.

Figure 1

Definition of the projectors ${\mathbf{P}}_{\lambda }$ and ${\mathbf{Q}}_{\lambda }$ for a simple toy model consisting of three different eigenstates $|n_{\lambda }\rangle$ of ${\mathcal{H}}_{0,\lambda }$. (a) Corresponding eigenvalues $E_n^{\lambda }$ and transition matrix elements of an observable $\mathcal{A}$. The energy range of the cutoff $\lambda$ is indicated by the blue shaded region. Zero-energy transitions of $\mathcal{A}$ between same eigenstates are not shown. (b) Arrangement of the matrix elements $A_{nm}=\langle n_{\lambda }\left|\mathcal{A}\right|m_{\lambda }\rangle$ shown in the respective colors distinguishing the different transition energies shown in panel (a). (c) Action of the projector ${\mathbf{P}}_{\lambda }$. It cuts out the matrix elements $A_{02}$ and $A_{20}$ with the energy difference larger than $\lambda$. All other matrix elements with energy differences smaller than $\lambda$ are kept unchanged. (d) The orthogonal projector ${\mathbf{Q}}_{\lambda }$ cuts out the low-energy transition matrix elements.

Figure 2

Illustration of the diagonalization scheme for the example system of Fig. 1. (a) Starting point is the original Hamiltonian $\mathcal{H}={\mathcal{H}}_0+{\mathcal{H}}_1$ with unperturbed eigenvalues $E_n$ and nonzero transitions between different eigenstates $|n\rangle$ caused by the interaction part ${\mathcal{H}}_1$ (indicated by blue arrows). The corresponding off-diagonal matrix elements are called $V_{nm}=\langle n|{\mathcal{H}}_1|m\rangle$. (b) Next, suppose the original Hamiltonian $\mathcal{H}$ can be mapped by a unitary transformation (6) to a renormalized Hamiltonian ${\mathcal{H}}_{\lambda }={\mathcal{H}}_{0,\lambda }+{\mathcal{H}}_{1,\lambda }$, which only contains transitions with energy differences smaller than $\lambda$ (blue shaded area). That is, in ${\mathcal{H}}_{\lambda }$ the particular transitions which combine the states with the highest and lowest energy (indicated by the red dotted arrows) have vanished ($V_{20}^{\lambda }=V_{02}^{\lambda }=0$). Here $V_{nm}^{\lambda }=\langle n_{\lambda }|{\mathcal{H}}_{1,\lambda }|m_{\lambda }\rangle$ denote the new renormalized off-diagonal matrix elements of ${\mathcal{H}}_{\lambda }$. Note that due to the unitary transformation all eigenvalues and matrix elements have become renormalized. (c) By further decreasing the cutoff $\lambda$ to a new value $\lambda -\mathrm{\Delta }\lambda$ all transitions between $\lambda$ and $\lambda -\mathrm{\Delta }\lambda$ will be eliminated. In the toy model of Fig. 1 the transitions with the next smaller energy difference are eliminated ($V_{10}^{\lambda -\mathrm{\Delta }\lambda }=V_{01}^{\lambda -\mathrm{\Delta }\lambda }=0$). (d) In the next steps a sequence of small transitions $\mathrm{\Delta }\lambda$ is performed by successively reducing the cutoff until finally $\lambda =0$ is reached. In our particular example the remaining transition with the lowest possible nonzero energy difference ($V_{12}$) is eliminated. The resulting Hamiltonian $\tilde{\mathcal{H}}$ is fully diagonal and its eigenvalues ${\tilde{E}}_n$ are the same as the ones of the original Hamiltonian.

Figure 3

Schematic behavior of some interaction matrix elements $V_{nm}^{\lambda }$ as a function of $\lambda$ for an infinitely large number of small renormalization steps ($\mathrm{\Delta }\lambda \to 0$) for the two choices of ${\mathbf{P}}_{\lambda -\mathrm{\Delta }\lambda }{X}_{\lambda ,\mathrm{\Delta }\lambda }$ discussed in the text. In Wegner's flow equation method (dashed lines) the part $\mathbf{P}X$ is chosen such that the matrix elements $V_{nm}^{\lambda }$ continuously flow to zero before $\lambda$ reaches their respective transition energies $|E_n^{\lambda }-E_m^{\lambda }|$. In this case the part $\mathbf{Q}X$ can be neglected. In the minimal transformation (solid lines) the reverse situation is found. Here all matrix elements $V_{nm}^{\lambda }$ remain constant for all steps down to their particular $\lambda$ values where they suddenly drop to zero. The $\lambda$ values where this happens are fixed by their respective transition energies (compare Figs. 1 and 2). That is, in the minimal transformation the ‘high-energy’ part $\mathbf{Q}X$ is solely responsible for the elimination of the matrix elements.

Figure 4

(a) Fully renormalized branches of the exactly solvable fermion model (20) both for the minimal transformation approach and the flow equation method. The solid lines in blue and red indicate the branches for the ‘$f$’ type and ‘$c$’ type electrons. The dispersions of the original model ${\mathcal{H}}_0$ are also shown (dotted lines). Note that within our method the branches are identical and keep their characters from the original dispersions (blue and red, respectively) accompanied by a discontinuity at the original crossing point. In contrast, the two branches (23) from the exact diagonalization change their character from ‘$f$’ to ‘$c$’ type (from blue to red) and vice versa at the crossing point. (b) Coupling parameter $V_{k,\lambda }$ as a function of $k$ for four different $\lambda$ values starting from the largest possible value $\mathrm{\Lambda }$ to a rather small value [compare corresponding transitions marked by green arrows in panel (a)]. The gray shaded areas indicate the minimal transformation (with $V_{k,\lambda }=V_k{\mathrm{\Theta }}_{k,\lambda }$) with a reduced extension for decreasing $\lambda$. Within the flow equation method a continuous shrinking is found (solid lines).

Figure 5

Results obtained from the numerical evaluation of the renormalization equations for the spinless Holstein model in the adiabatic (a)–(c), intermediate (d)–(f), and antiadiabatic case (g)–(i). Left: Bosonic quasiparticle energies ${\tilde{\omega }}_q/{\omega }_0$ at half filling as a function of $q$ for different values of the EP coupling $g$. Middle: Phonon distribution $n_q^{\mathrm{b}}=\langle b_q^{†}{b}_q\rangle$ as a function of $q$ for the same parameters as in the left panel. Right: Fermionic quasiparticle energies $({\tilde{\varepsilon }}_k-{\varepsilon }_k)/t$ as a function of $k$ for the same parameters as in the left and middle panel. Here ${\varepsilon }_k$ is the original electronic dispersion. For the phonon energy the following parameter values were used: ${\omega }_0/t=0.05$ in the adiabatic case, ${\omega }_0/t=2.8$ in the intermediate case, ${\omega }_0/t=6.0$ in the antiadiabatic case.

Figure 6

Schematic picture of the zero temperature results for the $1d$ spectral functions $A^c(k,\omega )$ (left panels) and $A^f(k,\omega )$ (right panels) at half filling for several characteristic values of $U$. Red and blue colors indicate whether the excitations are of ‘$c$’ or ‘$f$’ character. The actual numerical results can be found in Ref. [42]. For small $U\ll t_c$ the spectral functions are dominated by the ‘coherent’ contributions caused by the first term in Eq. (70) and in the equation for $A^f(k,\omega )$. ‘Incoherent’ contributions are negligible. For increasing values of $U\approx t_c$ a gap begins to open at the Fermi momentum caused by incoherent contributions. For even larger $U\gg t_c$ the gap broadens further. Of particular importance is a considerable admixture of $c$ electron contributions to the $f$ electron spectrum between $k=0$ and $k\approx k_F$. Note that the vertical dotted lines mark the chemical potential $\mu$. Further note that at medium values of $U\gtrsim t_c$ a gap opens at $\mu$.