Magnetism of one-dimensional Wigner lattices and its impact on charge order

M. Daghofer, R. M. Noack, and P. Horsch

Phys. Rev. B 78, 205115 – Published 20 November 2008

Abstract

The magnetic phase diagram of the quarter-filled generalized Wigner lattice with nearest-neighbor and next-nearest-neighbor hoppings, $t_{1}$ and $t_{2}$ , is explored. We find a region at negative $t_{2}$ with fully saturated ferromagnetic ground states that we attribute to kinetic exchange. Such interaction disfavors antiferromagnetism at $t_{2} < 0$ and stems from virtual excitations across the charge gap of the Wigner lattice, which is much smaller than the Mott-Hubbard gap $\propto U$ . Remarkably, we find a strong dependence of the charge structure factor on magnetism even in the limit $U \to \infty$ , in contrast to the expectation that charge ordering in the Wigner lattice regime should be well described by spinless fermions. Our results, obtained using the density-matrix renormalization group and exact diagonalization, can be transparently explained by means of an effective low-energy Hamiltonian.

Received 23 October 2008

DOI:https://doi.org/10.1103/PhysRevB.78.205115

Authors & Affiliations

M. Daghofer^1,2,*, R. M. Noack³, and P. Horsch¹

¹Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
²Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA and Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
³Philipps Universität Marburg, D-35032 Marburg, Germany

^*m.daghofer@fkf.mpg.de

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Vol. 78, Iss. 20 — 15 November 2008

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Images

Figure 1
(Color online) Schematic depiction of relevant $\propto t_{1}^{2} t_{2} / ϵ_{0}^{2}$ kinetic exchange processes. We start from perfect charge order (a) and (e), where circles denote empty sites ( ${Cu}^{3 +}$ atoms in ${Na}_{1 + x} {CuO}_{2}$ ), and arrows denote occupied sites $({Cu}^{2 +})$ . NN hopping $t_{1}$ induces excitations ( $a \to b$ and $e \to f$ ) with two domain walls (dashed lines) and cost $ϵ_{0}$ . Two different $t_{2}$ processes, one with $(f \to g)$ and one without $(b \to c)$ electron exchange, become possible. For FM spins (triplet channel) or spinless fermions, however, these two processes $(a \to d)$ and $(e \to h)$ cancel exactly because of the relative Fermi sign in the next-nearest-neighbor hopping process.Reuse & Permissions
Figure 2
(Color online) Charge structure factor $N (q)$ for $q = π$ as a function of nearest-neighbor hopping $t_{1}$ . The dotted line for spinless fermions is from ED calculations with $L = 28$ ; symbols were calculated using the DMRG with $L = 24$ for electrons with spin. Analytic results are obtained from Eq. (4) with $Δ = ϵ_{0}$ (FM) and $Δ = ϵ_{0} - 2 t_{2}$ (AF), respectively.Reuse & Permissions
Figure 3
(Color online) The charge structure factor $N (π)$ versus $t_{2}$ . Note that it does depend on the sign of $t_{2}$ for fermions with spin and does not for spinless fermions. The results for spinless fermions were calculated using exact diagonalization $(L = 18)$ and the results for electrons with spin using the DMRG ( $t_{1} = 0.02 V$ , $U = 4 V$ , and $L = 24$ ; $t_{1} = 0.07 V$ , $U = 100 V$ , and $L = 32$ ).Reuse & Permissions
Figure 4
Phase diagram for spinless fermions determined from the charge structure factor $N (q)$ [see Eq. (2)]. WL (dark gray): strongly charge-ordered WL with $N (π) > 0.7$ . $π$ -CDW (light gray): CDW with periodicity $π$ but $N (π) < 0.7$ . (This choice corresponds approximately to the inflection point of $N (π)$ as a function of $t_{1}$ .) In the white area, $N (q)$ has its maximum at $π / 2 \leq q < π$ , at $π / 2$ for large $t_{2}$ . In the exact diagonalization, we take $N_{e} = 8$ fermions on $L = 16$ sites.Reuse & Permissions
Figure 5
(Color online) Ground-state energy $E_{0}$ for $L = 24$ versus $t_{2}$ for $U = 4 V$ with $t_{1} = 0.05 V$ (circles) and $t_{1} = 0.07 V$ (triangles). Filled symbols indicate a fully polarize ground state and horizontal lines the energy of the FM state at $t_{2} = 0$ . The dashed and dash-dotted lines are analytic results obtained from perturbation theory [Eq. (8)] see text.Reuse & Permissions
Figure 6
(Color online) Magnetic phases and charge structure in the WL regime. FM $(°)$ : $m > 0.99 M$ ; AF (black $\times$ ): $m < 0.01 M$ ; “almost AF” (gray $\times$ ): $0.01 M \leq m < 0.1 M$ , where $m = S_{tot} (S_{tot} + 1)$ was obtained by DMRG for $L = 24$ , $U = 4 V$ , and $M = S_{max} (S_{max} + 1) = 42$ . Gray lines give potential lines for $N (π)$ , i.e., the strength of the alternating charge order. The lines were interpolated from data points obtained at the same parameter values as the magnetic data.Reuse & Permissions

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