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As escadas de Heisenberg de N pernas

Flávia Regina Souza Ramos

Published Mar 4, 2013• 0 citations
Publisher

Abstract

Compounds such as vanadyl pirophosphate (VO2)P2O7 and some cuprate systems (example: SrCu2O3 and Sr2Cu3O5) are experimental realizations of the Heisenberg ladders. These compounds are well known examples of strongly correlated systems. The physics of these systems is described, correctly, only by non-perturbative methods. In this work, we investigate the spin-S N-leg Heisenberg ladders using the density matrix renormalization group (DMRG) technique. We obtain estimates of the ground state energy per site, in thermodynamic limit (e∞), as well as of the spin gap (ΔS) for the ladders with spin up to S = 5/2 . We found that for half-integer spin ladders the spin gap is zero (finite) for N odd (even); whereas for the integer spin ladders our results support a finite spin gap, independent of the number of legs. These results are in agreement with the Haldane-Sierra s conjecture [1 3]. We also estimate the value of e∞ for the two-dimensional spin-S Heisenberg model from the ground state energies, in the thermodynamic limit, of the Heisenberg ladders.

DOI
10.14393/ufu.di.2013.163

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Compounds such as vanadyl pirophosphate (VO2)P2O7 and some cuprate systems (example: SrCu2O3 and Sr2Cu3O5) are experimental realizations of the Heisenberg ladders. These compounds are well known examples of strongly correlated systems. The physics of these systems is described, correctly, only by non-perturbative methods. In this work, we investigate the spin-S N-leg Heisenberg ladders using the density matrix renormalization group (DMRG) technique. We obtain estimates of the ground state energy per site, in thermodynamic limit (e∞), as well as of the spin gap (ΔS) for the ladders with spin up to S = 5/2 . We found that for half-integer spin ladders the spin gap is zero (finite) for N odd (even); whereas for the integer spin ladders our results support a finite spin gap, independent of the number of legs. These results are in agreement with the Haldane-Sierra s conjecture [1 3]. We also estimate the value of e∞ for the two-dimensional spin-S Heisenberg model from the ground state energies, in the thermodynamic limit, of the Heisenberg ladders.
We investigate the NNN-leg spin-SSS Heisenberg ladders by using the density matrix renormalization group method. We present estimates of the spin gap ${\ensuremath{\Delta}}_{s}$ and of the ground-state energy per site ${e}_{\ensuremath{\infty}}^{N}$ in the thermodynamic limit for ladders with widths up to six legs and spin $S\ensuremath{\le}\frac{5}{2}$. We also estimate the ground-state energy per site ${e}_{\ensuremath{\infty}}^{2\mathrm{D}}$ for the infinite two-dimensional spin-SSS Heisenberg model. Our results support that for ladders with semi-integer spins the spin excitation is gapless for NNN odd and gapped for NNN even, whereas for integer spin ladders the spin gap is nonzero, independent of the number of legs. Those results agree with the well-known conjectures of Haldane and S\'en\'echal-Sierra for chains and ladders, respectively. We also observe edge states for ladders with NNN odd, similar to what happens in spin chains.
A comprehensive theoretical and experimental study is presented of the magnetic susceptibility versus temperature χ(T) of spin S = 1/2 two- and three-leg Heisenberg ladders and ladder oxide compounds. Extensive quantum Monte Carlo simulations of χ(T) were carried out for both isolated and coupled two-leg ladders with spatially anisotropic intraladder exchange. Accurate fits to these and related literature QMC data were obtained. We have also calculated the one- and two-magnon dispersion relations and the dynamical spin structure factor for anisotropic isolated 2 x 12 ladders. The exchange constants in the two-leg ladder compound SrCu2O3 are estimated from LDA+U calculations. We report the detailed crystal structure of SrCu2O3 and of the three-leg ladder compound Sr2Cu3O5. New experimental χ(T) data are reported for the two-leg ladder cuprates SrCu2O3 and LaCuO_{2.5}, and for the (nominally) two-leg ladder vanadates CaV2O5 and MgV2O5. The new and literature χ(T) data for these compounds and for Sr2Cu3O5 are modeled using our QMC χ(T) simulation fits, and the exchange coupling constants between the spins-1/2 are thereby estimated for each material. The surpisingly strong spatial anisotropy of the bilinear intraladder exchange constants in the cuprate compounds is discussed together with the results of other experiments sensitive to this anisotropy. Recent theoretical predictions are discussed including those which indicate that a four-spin cyclic exchange interaction within a Cu4 plaquette is important to determining the magnetic properties and which can significantly influence the exchange interactions estimated from χ(T) data assuming the presence of only bilinear exchange.
The goal of this project is to study the ground-state properties of frustrated spin-1/2 two- and three-leg Heisenberg ladders, and to investigate possible effects of diagonal couplings on spin gaps. The ground-state and the first-excited-state energies of these two systems are calculated systematically by using the density matrix renormalization group method. The ground state phase diagrams for these two systems are obtained. For the frustrated two-leg ladder, we found that the spin gap is insensitive to the change of the FM diagonal coupling constant. This is in agreement with experimental suggestions. For the frustrated three-leg ladder, we conclusively proved that introducing AF diagonal couplings can not change the gapless property of the ground state.
We present a density matrix renormalization group study of the ground-state properties of spin-1/2 frustrated J1-J3 Heisenberg n(l)-leg ladders (with n(l) up to 8). For strong frustration (J(3)/J(1) approximately 0.5), both even-leg and odd-leg ladders display a finite gap to spin excitations, which we argue remains finite in the two-dimensional limit. In this regime, on odd-leg ladders the ground state is spontaneously dimerized, in agreement with the Lieb-Schultz-Mattis prediction, while on even-leg ladders the dimer correlations decay exponentially. The magnitude of the dimer order parameter decreases as the number of legs increases, consistent with a two-dimensional spin-liquid ground state.
The antiferromagnetic Heisenberg spin systems on the three-leg ladder are investigated. Periodic boundary condition is imposed in the rung direction. The system has an excitation gap for all antiferromagnetic inter-chain coupling (J⊥>0J_{\perp}>0J⊥​>0). The estimated gap for the strong coupling limit (J⊥/J1→∞J_{\perp}/J_1 \to \inftyJ⊥​/J1​→∞) is 0.28J1J_1J1​. Although the interaction is homogeneous and only nearest-neighbor, the ground states of the system are dimerized and break the translational symmetry in the thermodynamic limit. Introducing the next-nearest neighbor coupling (J2J_2J2​), we can see that the system is solved exactly. The ground state wave function is completely dimer-ordered. Using density matrix renomalization group algorithm, we show numerically that the original model (J2=0J_2=0J2​=0) has the same nature with the exactly solvable model. The ground state properties of the ladder with a higher odd number of legs are also discussed.
Using the density matrix renormalization group method, we have obtained the ground-state phase diagram of a spin-1/2 two-leg antiferromagnetic Heisenberg ladder with a ferromagnetic diagonal interaction ${J}_{\ifmmode\times\else\texttimes\fi{}}.$ The boundary between the two phases found in the system, the Haldane phase and the singlet phase, lies in the frustrated area of the parameter space, i.e., both the rung coupling ${J}_{\ensuremath{\perp}}$ and the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ are ferromagnetic. The spin gap is found to be insensitive to the change of the ferromagnetic diagonal coupling strength when the ratio of the rung coupling constant ${J}_{\ensuremath{\perp}}$ to the leg coupling constant ${J}_{\ensuremath{\Vert}}$ is 0.5, in agreement with the analysis of recent experimental spin susceptibility data.