New approach to competing order
Competing order is a ubiquitous phenomenon in two-dimensional interacting electron systems. A most prominent example is the competition between antiferromagnetism and high temperature superconductivity in cuprate and iron pnictide compounds. Some of the ordering tendencies are fluctuation driven, and can therefore not be captured by mean-field (MF) theory. Numerical simulations of correlated electrons are still restricted to relatively small systems.
For weak and moderate interaction strengths, the functional renormalization group (fRG) has been developed as an unbiased and sensitive tool to detect instabilities toward any kind of order in interacting electron models . In that method, effective interactions, self-energies, and susceptibilities are computed from a differential flow equation, where the flow parameter Λ controls a scale-by-scale integration of fields in the underlying functional integral. Instabilities are signalled by divergences of effective interactions and susceptibilities at a critical energy scale Λc. To complete the calculation and compute, for example, the size of the order parameters, one has to continue the flow below the scale Λc, which requires the implementation of spontaneous symmetry breaking.
The flow in the symmetry-broken regime (Λ<Λc) is complicated considerably by the presence of anomalous interaction vertices. In complex problems, such as systems with several competing and possibly coexisting order parameters, or in multi-band systems, it can therefore be mandatory or at least desirable to simplify the integration of the scales below Λc. A natural possibility is to treat the low-energy degrees of freedom (below Λc) in mean-field theory. The generation of instabilities and also the possible reduction of the critical scale by fluctuations is not affected by such a simplification. In the ground state, fluctuations below the critical scale are expected to influence the size of order parameters only mildly. A combination of an fRG flow for Λ>Λc with a mean-field treatment of symmetry-breaking has been formulated and applied already for a particular fRG version based on Wick ordered generating functionals . However, for calculations beyond the lowest-order truncation, another fRG version, which is based on the effective action , turned out to be more efficient, as it avoids one-particle reducible contributions, and self-energy feedback can be implemented easily.
Recently we have derived a consistent combination of the one-particle irreducible fRG with MF theory for symmetry breaking. In that approach, the two-particle interaction vertex is computed from a renormalization group flow until a scale ΛMF slightly above Λc has been reached, that is, before the vertex diverges. Order parameters associated with spontaneous symmetry breaking are then obtained from mean-field theory with the irreducible two-particle vertex γΛMF as effective interaction. The latter is obtained from the full two-particle vertex ΓΛMF by inverting a Bethe-Salpeter equation.
Antiferromagnetism and superconductivity
To illustrate the performance of the fRG+MF approach in a situation of competing instabilities, we present an application to the two-dimensional Hubbard model, describing tight-binding electrons subject to a local repulsion U. The model is well-known for its intriguing competition between antiferromagnetism and superconductivity. Indeed the fRG flow of the vertex generically diverges either in the antiferromagnetic or in the d-wave pairing channel in that model . Hence we allow for antiferromagnetic and superconducting order, including the possibility of coexistence. Although the antiferromagnetic wave vector may deviate from (π, π), we consider only the case of conventional Neél order for simplicity.
We show and discuss results for the magnetic and superconducting order parameters in the ground state of the hole-doped Hubbard model with a small next-to-nearest neighbor hopping t'=-0.15t and a moderate Hubbard interaction U=3t, where t is the hopping amplitude between nearest neighbors on the square lattice. The fRG flow has been computed with a static vertex parametrized via a decomposition in charge, magnetic and pairing channels, with s-wave and d-wave form factors.
In Fig. 1 we show results for the amplitudes of the antiferromagnetic and superconducting gap functions, ΔAF=maxkΔkAF and ΔSC=maxk ΔkSC, as a function of the electron density. The coupled solution of both gap equations exhibits an extended region where magnetic and superconducting order coexist. In the major part of that region the pairing gap is smaller than the magnetic gap. Here superconductivity is a secondary instability within the antiferromagnetic phase, which naturally occurs as a Cooper instability of electrons near the reconstructed Fermi surface confining hole pockets in the antiferromagnetic state. The pairing gap decreases rapidly as the pockets shrink upon approaching half-filling. Magnetic order vanishes at a critical density ncAF situated slightly above Van Hove filling. Below that density the state is purely superconducting. The magnetic transition is continuous such that ncAF is a quantum critical point. Figure 1 also shows results for the gap amplitudes as obtained from solutions of the individual gap equations with either magnetic or superconducting order. A comparison with the coupled solution confirms that the two order parameters compete with each other. In particular, superconductivity is strongly suppressed by antiferromagnetism. In the absence of superconductivity, the antiferromagnetic regime extends to lower densities and terminates at a first order transition accompanied by a density jump, which opens a density window where no homogeneous solution exists.
For densities below n=0.95, the two-particle vertex diverges actually at incommensurate wave vectors, indicating a leading instability toward incommensurate antiferromagnetic order. The resulting ground state is probably an incommensurate magnetic state coexisting with superconductivity. Such states can also be treated by the fRG+MF theory. Since mean-field equations for incommensurate order are more involved, we leave this extension for future studies. For parameters where pairing is the leading instability, the results for ΔSC are very close to those from a full fRG calculation, which indicates that the fluctuations below the scale Λc have indeed limited impact on the size of the ground state order parameter.
The momentum dependence of the gap functions is shown in Fig. 2. The antiferromagnetic gap ΔkAF exhibits only a moderate modulation around a constant. The superconducting gap ΔkSC obeys the expected dx2-y2 symmetry, but with visible deviations from the simple cos kx−cos ky form. In the coexistence regime the (global) extrema of ΔkSC are shifted away from the axial directions, as a natural consequence of the Fermi surface truncation in the antinodal region.
The above results for the gap functions agree qualitatively with those obtained previously from the Wick ordered fRG+MF theory . However, the suppression of ΔkSC by antiferromagnetic order was stronger in that work. A relatively broad coexistence of antiferromagnetism and superconductivity as found here has also been obtained at stronger interactions by embedded quantum cluster methods. An interesting extension would be the computation of incommensurate magnetic order, in possible coexistence with superconductivity, which is very hard to study by other methods. More generally, competing instabilities in complex multi-band systems offer a wide field of fruitful applications for the fRG+MF theory.