We set out to study a model for a modulated magnetic state acting as an inter- mediate phase in a metamagnetic transition. We were inspired in this idea by experimental results on the material Sr3Ru2O7 and by a general principle of mod- ulated phases appearing as intermediate states in phase transitions. This is most clearly seen in the superconducting Fulde-Ferrell-Larkin-Ovchinnikov phase, which is analogous to the magnetic system we consider. Such phases are also found in cold-atomic gases [62], excitonic insulators [100], color superconductivity in quark matter [60] and neutron stars, where they have been invoked to explain glitches in pulsar rotation [60].
We showed that generic features in the band-structure could drive not only meta- magnetism, but magnetic spiral states. These states appear as an intermediate phase in the metamagnetic transition, causing the bifurcation of the parent transition to enclose the phase. We expect the same reasoning to carry across to more complex spin-textures made from superpositions of spirals. Having calculated the phase di- agram for such an intermediate phase we see that it matches closely the topology of the experimentally determined phase diagram of Sr3Ru2O7. In addition we have shown that the thermodynamic consequences of our proposal are in agreement with experiment. This suggests that the anomalous phase of Sr3Ru2O7 may be just such a modulated phase. The same effects may occur in other systems, such as NbFe2 and ZrZn2 which show metamagnetism associated with resistive anomalies [47, 48]. Our analysis is not specific to one material, being based on rather generic features of the bandstructure.
Our calculation of the phase diagram proceeds from a consideration of the over- all experimental phase diagram of Sr3Ru2O7. This is dominated by a metamagnetic transition. We therefore take the simplest theory which explains this transition - a Stoner model with a peak in the electronic density of states. We calculate the phase diagram via a Ginzburg-Landau expansion based both on phenomenological arguments and an expansion of the microscopic Hamiltonian. We then show that modulated states arise naturally in this model and calculate that modulated or-
9. Conclusion 131
der becomes more favourable as we tune along the metamagnetic transition away from the van Hove singularity. Using the Ginzburg-Landau expansion we calculate how the parent metamagnetic transition reconstructs to accomodate this modu- lated magnetisation. The transition bifurcates at a dislocated tricritical point, with the modulated phase lying between the two transitions. The phase is bounded from above by a sheet of continuous transitions that form a roof which encloses the phase. This phase diagram matches the topology of the experimentally determined phase diagram for Sr3Ru2O7 [52]. This material shows a metamagnetic transition, the critical endpoint of which can be tuned until the transition bifurcates. Between the wings is a phase with an anomalously high and anisotropic resistivity. We believe that this phase is the modulated magnetic order which we propose.
Following on from this we examined the thermodynamic properties of the par- ent metamagnetic transition, calculating how the specific heat and entropy evolve with applied field and temperature. We found subtlety even in this basic feature of the phase diagram, with the specific heat having a double-peak as a function of field and a non-trivial relationship of field and temperature scales. These results are a good match for the experimentally determined thermodynamic behaviour of Sr3Ru2O7 [74] and inform the interpretation of this data. The next step is a calcu- lation of the thermodynamic properties of the spiral phase. These calculations have not yet been completed but we note that our modulated phase provides possibilities for reconciling with experimental data on Sr3Ru2O7 which should be examined.
As well as thermodynamic data the phase is characterised by its transport prop- erties [26]. The formation of a modulated state would produce enhanced scattering of electrons and the wavevector dependence would result in anisotropy. We therefore believe that our proposed phase would produce the high and anisotropic resistivity of Sr3Ru2O7. This is another subject for future calculation.
There are several extensions of our model that will allow even closer connection to the experimental data on Sr3Ru2O7. These are the inclusion of the effect of the magnetic field angle and a more realistic picture of the electron dispersion. It seems likely that this angle dependence can be accounted for by introducing an orbital Zeeman coupling which modifies the band structure as the field angle is changed. Associated with this will be the choice of a minimal model of the band structure which will accurately reflect Sr3Ru2O7 while remaining amenable to calculation. These effects should be straightforward to introduce into our framework.
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criticality. Although we have not included the effect of quantum fluctuations in our model there are several possible areas of connection. The initial interest in Sr3Ru2O7 lay in the presence of a quantum critical endpoint. Although models of the anomalous phase do not include its effects quantum critical points are known to stabilize new phases and so may be important. Indeed recent work on non-analytic corrections to Hertz-Millis theory shows that itinerant quantum critical points are unstable to the formation of first-order transitions and spin-spiral states [32, 39, 40] - exactly the phenomena studied here in a lattice-driven context. The interplay of the two will be important to the behaviour of real systems.
As well as the modulated states which we consider there are intriguing proposals for nematic states which retain the orientational but not translational orientation of the modulated state [23]. These may be made by melting modulated states and have very similar energetics. The connection between these two types of order is another interesting extension of the ideas contained here.
It is clear that there is much subtlety in the phases of itinerant magnets. The crystal lattice can drive a number of ordering tendencies, like the ferromagnetism and the spiral magnetism considered here. We have shown how a complex magnetic phase diagram may come about by combining several simple ideas. These ideas present a compelling explanation for the anomalous phase of Sr3Ru2O7. The ideas may be extended further, continuing to explore the rich phenomenology of strongly correlated systems.