Synopsis - The effect of geometric frustration on some correlated electron systems

The effect of geometric frustration on some correlated electron systems

Synopsis¶

The notion of geometric frustration emerged originally in the context of Ising spins with antiferromagnetic nearest neighbour interaction on a triangular lattice. The classic solution by Wannier clarified how long range order is suppressed by frustration in this case. One can generalise the situation to frustrated structures in higher dimensions, e.g, the pyrochlore or the face centered cubic (FCC) lattices, where neighbouring spins live on a tetrahedral motif. Such models, and their classical and quantum Heisenberg versions, have been intensely studied.

Correlation physics, on the other hand, grew out of the continuing study of many body systems over the last several decades, with a fresh impetus given by the discovery of high $T_c$ superconductivity in the doped Mott insulator La $_{2-x}$ Sr $_x$ CuO $_4$ . This thrust the Mott transition and the doped Mott insulator centerstage. It quickly became apparent that a large family of oxides, including the magnetoresistive manganites, the high thermopower cobaltates, {\it etc}, owed their exotic properties to electron correlation. The development of powerful tools like dynamical mean field theory (DMFT) and its combination with {\it ab initio} methods has clarified many aspects of correlation physics over the last two decades.

Correlated systems involve metals systems with itinerant electrons, while traditional frustrated systems are insulating magnets with localised electrons. There are broadly two situations where they intersect:

One may have a `two species’ system, of electrons and local moments, where the local moments live on a frustrated structure and are Kondo (or Hund’s) coupled to itinerant electrons.
We could have a Mott insulator in a frustrated structure and the consider its metallisation, due to decreasing interaction. \end{itemize}

The first situation arises in Kondo lattice like, or ‘double exchange’, models, while the second is described by the Hubbard model. In both cases the ideal frustrated situation arises in the absence of itinerant electrons. The interest is in clarifying how the presence of electrons in the Kondo lattice, or the approach to the insulator-metal transition in Mott-Hubbard systems modifies the physics. The ‘two species’ description is appropriate for the pyrochlores (iridates, etc) and double perovskites, while the Hubbard model is relevant for materials like the cluster compound GaTa $_4$ Se $_8$ and A $_3$ C $_{60}$ .

The thesis addresses the interplay of correlation effects and geometric frustration in three cases: (i) the metallic double perovskites, (ii) the triangular lattice Hubbard model for the $\kappa$ -BEDT organics, and (iii) the Mott transition in the face centered cubic (FCC) lattice. It is organised as follows.

Chapter. 1 provides a quick review of the twin areas of (i) geometric frustration in magnets, and (ii) correlation driven phenomena in electron systems. We then review the intersection of these two in the context of materials like (a) the $\kappa$ -BEDT based triangular lattice organics, (b) the FCC based Mott materials like the Ga cluster compounds, GaTa $_4$ Se $_8$ , the fullerides A $_3$ C $_{60}$ , and the double perovskites, and (c) the pyrochlore iridates and molybdates. While each family has its peculiarity, we focus on the following generic features:

Magnetism: long range order, spin freezing, or a spin liquid state,
Resistivity and its temperature dependence near the IMT,
Charge dynamics and spectral weight transfer near the IMT,
Anomalous Hall response, where relevant.

The experimental summary is followed by a discussion of the minimal models that are used for the materials above. There are broadly two kinds of models:

\begin{align*} H_{KLM} &= H_0 + J \sum_{i} {\bf S}_i.{\vec \sigma}_i \cr \cr H_{Hubb} &= H_0 + U\sum_{i}n_{i\uparrow}n_{i\downarrow} \cr \nonumber \end{align*}

(1)

In both cases $H_0$ defines the non interacting (band) problem. ${\vec \sigma}$ is the electron spin operator. $H_{KLM}$ refers to a Kondo lattice model where the local moments ${\bf S}_i$ live on a frustrated lattice, while $H_{Hubb}$ refers to the Hubbard model with electron repulsion $U$ . The KLM is a `two species’ model, involving spins and fermions, while the Hubbard model just involves interacting electrons (at half filling in our case). For the KLM, the usual approach is to use Monte Carlo (assuming the spins to be classical) while the Hubbard model is solved via variational Monte Carlo, cellular DMFT, or some form of cluster perturbation theory. We review the major results, leaving the detailed discussion to later chapters.

Chapter. 2 describes the models that we use and the computational strategy. In the KLM case when the spins ${\bf S}_i$ are treated as classical, their correlations are controlled by the distribution

P\{{\bf S}_i \} \propto Tr_{c,c^{\dagger}} e^{-\beta H_{KLM}}

(2)

This trace is not analytically computable at strong coupling (large $J$ ) and we use a exact diagonalisation based Monte Carlo to sample $P\{{\bf S}_i \}$ . Electronic properties are computed by diagonalising $H_{KLM}$ in the equilibrium configurations of $P\{{\bf S}_i \}$ .

The Hubbard problem looks very different, and, beyond weak coupling, has traditionally been handled via quantum Monte Carlo and exact diagonalisation. Methods like DMFT also ultimately resort to these tools. These methods have a size limitation, despite the enormous increase in computing power over the decades, and are still not able to access complex magnetic states. We employed a Hubbard-Stratonovich (HS) transformation to decouple the Hubbard interaction in terms of auxiliary charge $(\phi_i)$ and spin $({\bf m}_i)$ fields, retaining full rotation invariance. We neglect the time dependence of these fields, keep the charge field $\phi_i$ at its saddle point value $U/2$ , but retain the full spatial fluctuations in the ${\bf m}_i$ . The Hubbard problem now looks like:

H_{Hubb}^{eff} = H_{0} - U \sum_{i} {\bf m}_i\cdot\vec{\sigma}_i + \sum_{i} {\bf m}_i^2

(3)

This can be solved by the same tools as for the KLM. Neglecting time dependence in the Hubbard case is an approximation and we check its validity against exact answers wherever available.

Chapter. 3 discusses our results on possible non ferromagnetic phases in the metallic double perovskites. These are materials of the form ABO $_3$ .AB’O $_3 \equiv$ A $_2$ BB’O $_6$ . They usually involve a transition metal ion, B, with a large magnetic moment, and a non magnetic ion B $'$ . While many double perovskites are ferromagnetic, studies on the underlying model reveal the possibility of antiferromagnetic phases as well driven by electron delocalisation. We present a comprehensive study of the magnetic ground state and $T_c$ scales of the minimal double perovskite model in three dimensions using a combination of spin-fermion Monte Carlo and variational calculations. In contrast to two dimensions, where the effective magnetic lattice is bipartite, three dimensions involves a geometrically frustrated face centered cubic (FCC) lattice. This promotes non-collinear spiral states and ‘flux’ like phases in addition to collinear anti-ferromagnetic order. We map out the possible magnetic phases for varying electron density, ‘level separation’ $\epsilon_B - \epsilon_{B'}$ , and the crucial B $^{\prime}$ B $^{\prime}$ (next neighbour) hopping $t'$ .

Chapter. 4 makes a transition to the half-filled Hubbard model in two dimensions, defined on an anisotropic triangular lattice. This structure is like a square with one diagonal. The hopping along the axes is $t$ , and along the diagonal it is $t'$ . While we have studied the entire phase diagram in terms of anisotropy, interaction strength, and temperature, we focus on the anisotropy regime appropriate to the $\kappa-$ BEDT based organics. We study the interaction driven crossover, which mimics the effect of pressure (or composition change) in the organics. We use bandstructure data and the measured transport gap to determine the electronic parameters and obtain a consistent description of magnetism and the resistivity. We uncover a pseudogap phase between the ‘ungapped’ metal and the ‘hard gap’ insulator, predict the momentum dependence of quasiparticle broadening and pseudogap formation, and clarify how these features arise from incommensurate magnetic fluctuations in this frustrated system. Beyond the organic Mott problem, we have mapped out the metal-insulator transition and pseudogap formation over the entire $U-t'-T$ parameter space.

Chapter. 5 discusses the Mott transition and magnetic properties of the Hubbard model on the FCC lattice. Like triangular motifs in two dimensions, the tetrahedral motifs on the FCC lattice frustrate Neel order in the Mott insulator. We discover that the low temperature state is a paramagnetic metal at weak interaction, an antiferromagnetic insulator (AFI) with flux like order at intermediate interaction, and an AFI with ‘C type’ order at very strong interaction. Remarkably, there is a narrow window between the paramagnetic metal and the AFI where the system exhibits spin glass behaviour arising from the presence of disordered but ‘frozen’ local moments. The spin glass state is metallic at weaker interaction but shows crossover to pseudogap behaviour and an insulating resistivity with growing interaction. We make a qualitative comparison of our results with trends observed in a broad class of FCC and pyrochlore based materials, and attempt a detailed quantitative match for the Mott transition in the FCC compound GaTa $_4$ Se $_8$ .

Chapter. 6 concludes the thesis. It collects together the general conclusions that one can draw about the interplay of correlation, itinerancy and geometric frustration from our results, and lists a few possibilities for further study in terms of models and materials.

The effect of geometric frustration on some correlated electron systems

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