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A renormalized excitonic method in terms of block excitations. Application to spin lattices

Mohamad Al Hajj, Jean‐Paul Malrieu, Nathalie Guihéry

Published Nov 1, 2004• 0 citations
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Abstract

Dividing the lattice into blocks with singlet ground state and knowing the exact low energy spectrum of the blocks and of dimers (or trimers) of blocks, it is possible to approach the lowest part of the lattice spectrum through an excitonic type effective model. The potentialities of the method are illustrated on the 1-D frustrated chain and the 1/5-depleted square and the plaquette 2-D lattices. The method correctly locates the phase transitions.

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Renormalized excitonic method in terms of block excitations: Application to spin lattices

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Dividing the lattice into blocks with a singlet ground state and knowing the exact low-energy spectrum of the blocks and of dimers (or trimers) of blocks, it is possible to approach the lowest part of the lattice spectrum through an excitonic type effective model. The potentialities of the method are illustrated on the one-dimensional (1D) frustrated chain and the $1∕5$-depleted square and the plaquette 2D lattices. The method correctly locates the phase transitions between gapped and nongapped phases.

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Dividing the lattice into blocks with singlet ground state and knowing the exact low energy spectrum of the blocks and of dimers (or trimers) of blocks, it is possible to approach the lowest part of the lattice spectrum through an excitonic type effective model. The potentialities of the method are illustrated on the 1-D frustrated chain and the 1/5-depleted square and the plaquette 2-D lattices. The method correctly locates the phase transitions.
Dividing the lattice into blocks with a singlet ground state and knowing the exact low-energy spectrum of the blocks and of dimers (or trimers) of blocks, it is possible to approach the lowest part of the lattice spectrum through an excitonic type effective model. The potentialities of the method are illustrated on the one-dimensional (1D) frustrated chain and the 1∕51∕51∕5-depleted square and the plaquette 2D lattices. The method correctly locates the phase transitions between gapped and nongapped phases.
The real-space renormalization group for the study of periodic systems goes through the reduction of the Hilbert space to products of lowest eigenstates of identical blocks. Instead of working with the Hamiltonian, it is possible to estimate effective interactions between blocks from the spectrum of dimers or trimers of blocks, according to the effective Hamiltonian theory of Bloch. Tests on a series of spin problems [cohesive energy of the one-dimensional (1D) chain and of the 2D square lattice, excitation energies of the dimerized 1D spin chain, and behavior of the frustrated chain in the low frustration regime] show the potentiality of the method.
The Hubbard Hamiltonian with infinite on-site repulsion and negative transfer is diagonalized numerically on 3×3 frustrated lattices, such as the square lattice with next-nearest-neighbor transfer and the triangular lattice. The number of electrons is chosen to be less by just one electron than that in the exactly half-filled band. When the strength of the frustration is sufficiently large, a singlet state appears as the ground state. Then, the total spin of the lowest excited state is unity. There is an excitation gap for the square lattice but not for the triangular lattice.
This work is devoted to the study, the development, and the application of a new systematic method yielding the dominant correlations that govern a quantum many-body state in an unbiased way. The dominant correlations between any two disjoint blocks of a system are extracted by performing a singular value decomposition of the correlation density matrix (CDM) between those blocks. We determine several mathematical properties and features of this method, in particular the consequences of the lattice symmetries or the symmetries intrinsic to the studied state on the singular values spectrum. We investigate the relation between the norm of the CDM – providing a natural measure of the total correlation between the two blocks – and the so-called mutual information, a quantity originally introduced in quantum information theory. This novel tool is utilized for sheding new light on the zero temperature physics of the spin-1/2 frustrated ferromagnetic J1–J2 Heisenberg chain in a magnetic field as well as on the low-energy physics of the spin-1/2 antiferromagnetic Heisenberg model on the two-dimensional kagomé lattice. The states are computed using exact diagonalization and the density matrix renormalization group procedure in the first case, and exact diagonalization only in the second case. This work is introduced in Chapter 1. The first model is then presented in Chapter 2. Chapter 3 introduces the CDM method, and Chapter 4 is devoted to the study of the kagomé antiferromagnet. In the J1–J2 chain, we reveal a vector chiral phase at low magnetic field and a sequence of multipolar Luttinger liquid phases at high field. We explicitly show that these multipolar phases result from the destabilization – driven by a locking mechanism – of the classical spiral ground state in the absence of magnetic field. This point of view is completely new: multipolar phases were known to be a possible destabilization of ferromagnetic phases, but they have never been reported as a destabilization of spiral states yet. Regarding the kagomé antiferromagnet, we address for the first time the question of the nature of the singlet states forming its quite dense singlet spectrum above the ground state. We show that some of these low-lying singlet states have large dimer correlations which do not seem to significantly decrease with the distance, moreover our CDM studies confirm that the dominant correlations in those singlet states are of the dimer-dimer type. Studies of Von Neumann block entropies reveal a very short correlation length on the one hand, and entropies that are roughly independent on the energy of the state under consideration on the other hand. The scenario of a valence bond crystal phase is investigated and the relevance of different kinds of crystals (from the literature or ad hoc) for reproducing the dimer correlations in the 36-site sample is probed.
We study a model of strongly correlated S=1/2S=1/2S=1/2 fermions on the planar pyrochlore, or checkerboard, lattice, at fractional (1/8) filling. Starting with the extended Hubbard model in the limit of strong Coulomb repulsion, low-energy configurations can be mapped onto hard-core dimer configurations whose dimers carry a spin degree of freedom. An effective Hamiltonian similar to the kinetic term of a quantum dimer model on the square lattice which rotates two parallel dimers (in a spin-singlet configuration) by $90\ifmmode^\circ\else\textdegree\fi{}$ naturally emerges. We also introduce an additional term in the Hamiltonian, a generalized dimer plaquette interaction, in order to realize a closer analogy to the latter model. For a strong dimer plaquette attraction stabilizing a columnar phase, a spontaneous dimerization takes place in the direction of the columns of (spin-carrying) dimers. Using exact diagonalizations of two-dimensional periodic clusters, the analysis of the low-energy spectrum and of several types of correlation functions gives indeed evidence for a new type of lattice symmetry breaking phase, the eightfold degenerate mixed columnar-plaquette crystal, and for a transition from this phase to a resonating singlet-pair crystal (found in previous studies) which restores the rotational symmetry of the lattice. Similar conclusions and phase diagram are also reached from a simple variational approach.