We carefully study how the fermion-fermion interactions affect the low-energy states of a two-dimensional spin-1/2 fermionic system on the kagomé lattice with a quadratic band crossing point. With the help of the renormalization group approach, we can treat all kinds of fermionic interactions on the same footing and then establish the coupled energy-dependent flows of fermionic interaction parameters via collecting one-loop corrections, from which a number of interesting results are extracted in the low-energy regime. At first, various sorts of fermion-fermion interactions furiously compete with each other and are inevitably attracted by certain fixed point in the parameter space, which clusters into three qualitatively distinct regions relying heavily upon the structure parameters of materials. In addition, we notice that an instability accompanied by some symmetry breaking is triggered around different sorts of fixed points. Computing and comparing susceptibilities of twelve potential candidates indicates that charge density wave always dominates over all other instabilities. Incidently, there exist several subleading ones including the x-current, bond density, and chiral plus s-wave superconductors. Finally, we realize that strong fluctuations nearby the leading instability prefer to suppress density of states and specific heat as well compressibility of quasiparticles in the lowest-energy limit.
We carefully study how the fermion-fermion interactions affect the low-energy states of a two-dimensional spin-1/2 fermionic system on the kagomé lattice with a quadratic band crossing point. With the help of the renormalization group approach, we can treat all kinds of fermionic interactions on the same footing and then establish the coupled energy-dependent flows of fermionic interaction parameters via collecting one-loop corrections, from which a number of interesting results are extracted in the low-energy regime. At first, various sorts of fermion-fermion interactions furiously compete with each other and are inevitably attracted by certain fixed point in the parameter space, which clusters into three qualitatively distinct regions relying heavily upon the structure parameters of materials. In addition, we notice that an instability accompanied by some symmetry breaking is triggered around different sorts of fixed points. Computing and comparing susceptibilities of twelve potential candidates indicates that charge density wave always dominates over all other instabilities. Incidently, there exist several subleading ones including the x-current, bond density, and chiral plus s-wave superconductors. Finally, we realize that strong fluctuations nearby the leading instability prefer to suppress density of states and specific heat as well compressibility of quasiparticles in the lowest-energy limit.
Roles of fermion-fermion interactions are carefully studied in governing the low-energy fates of a two-dimensional spin-1/2 fermionic system on the kagome lattice, which features a quadratic band crossing point touched parabolically by up and down energy bands. In the framework of a renormalization group, we establish the coupled energy-dependent flows of fermionic interaction parameters by treating all kinds of fermionic interactions on the same footing and unbiasedly taking into account their one-loop corrections. Through a compressive analysis of these evolutions that carry the hierarchical information of physics, a number of outlandish behaviors are progressively decoded and addressed manifestly in the low-energy regime. At first, we find that various sorts of fermion-fermion interactions furiously compete and affect each other. With the decrease of energy scale, they are inevitably attracted by certain fixed point in the parameter space which clusters into three qualitatively distinct regions relying heavily upon structure parameters of materials. In addition, the leading instability accompanied by some symmetry breaking is attentively investigated in the proximity of different sorts of fixed points. Computing and comparing susceptibilities of twelve potential candidates indicates that charge density wave always dominates over others. Incidently, there exists several options for subleading ones including the x-current, bond density, and chiral plus s-wave superconductors. At last, we examine how physical observables behave nearby the dominant instability. Due to the strong fluctuations neighboring this instability linked to charge density wave, density of states and specific heat as well compressibility of quasiparticles are significantly suppressed, signalling the emergence of non-Fermi liquid behaviors.
We investigate the impacts of combination of fermion-fermion interactions and impurity scatterings on the low-energy stabilities of two-dimensional asymmetric materials with a quadratic band-crossing point by virtue of the renormalization group that allows us to treat distinct sorts of physical ingredients on the same footing. The coupled flow evolutions of all interaction parameters which carry the central physical information are derived by taking into account one-loop corrections. Several intriguing results are manifestly extracted from these entangled evolutions. At first, we realize that the quadratic band touching structure is particularly robust once the fermionic couplings flow toward the Gaussian fixed point. Otherwise, it can either be stable or broken down against the impurity scattering in the vicinity of nontrivial fixed points. In addition, we figure out two parameters $\ensuremath{\eta}$ and $\ensuremath{\lambda}$ that measure rotational and particle-hole asymmetries are closely energy dependent and exhibit considerably abundant behaviors depending upon the fates of fermion-fermion couplings and different types of impurities. Incidentally, as both $\ensuremath{\eta}$ and $\ensuremath{\lambda}$ can be remarkably increased or heavily reduced in the low-energy regime, an asymmetric system under certain restricted conditions exhibits an interesting phenomenon in which transitions either from rotational or particle-hole asymmetry to symmetric situation would be activated.
Spinless fermions on the honeycomb lattice with repulsive nearest-neighbor interactions are known to harbour a quantum critical point at half-filling, with critical behavior in the Gross-Neveu (chiral Ising) universality class. The critical interaction strength separates a weak-coupling semimetallic regime from a commensurate charge-density-wave phase. The phase diagram of this basic model of correlated fermions on the honeycomb lattice beyond half-filling is, however, less well established. Here, we perform an analysis of its many-body instabilities using the functional renormalization group method with a basic Fermi surface patching scheme, which allows us to treat instabilities in competing channels on equal footing also away from half-filling. Between half-filling and the Van Hove filling, the free Fermi surface is holelike and we again find a charge-density wave instability to be dominant at large interactions. Moreover, its characteristics are those of the half-filled case. Directly at the Van Hove filling, the nesting property of the free Fermi surface stabilizes a dimerized bond-order phase. At lower filling, the free Fermi surface becomes electronlike and a superconducting instability with f-wave symmetry is found to emerge from the interplay of intra-unit-cell repulsion and collective fluctuations in the proximity to the charge-density wave instability. We estimate the extent of the various phases and extract the corresponding order parameters from the effective low-energy Hamiltonians.
We analyze the effect of disorder on the weak-coupling instabilities of a quadratic band crossing point (QBCP) in two-dimensional Fermi systems, which, in the clean limit, display interaction-driven topological insulating phases. In the framework of a renormalization group procedure, which treats fermionic interactions and disorder on the same footing, we test all possible instabilities and identify the corresponding ordered phases in the presence of disorder for both single-valley and two-valley QBCP systems. We find that disorder generally suppresses the critical temperature at which the interaction-driven topologically nontrivial order sets in. Strong disorder can also drive a topological phase transition into a topologically trivial insulating state.
In the present thesis, we consider interaction induced instabilities in different electronic models consisting of multiple orbitals. We employ the functional renormalization group method, to tackle the many-particle problem. The main goal of this work is the identification of leading instabilities of different interacting electron systems and the relevant energy scales for the onset of correlations. First, we recapitulate the functional renormalization group formalism and discuss details of its application. We proceed with the investigation of two models, a continuum model and a model on a checkerboard lattice, which exhibit a single quadratic band crossing point at the Fermi level. Here, we find that electron-electron interactions can lead to ordering with non-trivial topological properties, i.e. the existence of topologically protected edge states. We further discuss the emergence of superconductivity, when the band crossing point is shifted away from the Fermi surface. Then, we turn to more realistic models featuring quadratic and cubic band crossing points, namely the honeycomb bi- and trilayer, which are used to describe the graphene layered systems. We obtain rich phase diagrams, with antiferromagnetic spin-density, two types of charge-density waves and a quantum spin Hall phase, which features in the case of the trilayer topologically protected edge states. With interaction parameters extracted from ab initio calculations, we observe a close competition between the antiferromagnetic and quantum spin Hall state. Here, the precise shape of the interaction decides which instability is the leading one. A comparison of the energy scales of the ordering phenomena in tri-, bi- and single layer reveals that when the interaction parameters from ab initio are appropriately rescaled, the functional renormalization group can qualitatively reproduce experimental gap sizes in the respective materials consistently. Afterwards, we study a multiband model of the copper-based unconventional superconductors. This is motivated by a puzzling material trend in copper-based high-Tc superconductors, pointed out by Pavarini et al. [PRL. 87, 047003 (2001)]: Compounds with larger next-nearest-neighbor hopping generally exhibit larger critical temperatures, which cannot be easily understood within spin-fluctuation theories of the one-band Hubbard model. Therefore, we consider the extension of the original model by including additional orbitals. We investigate a two-band model with an additional 4s orbital within a two-patch approximation and the N-patch-scheme. By this, a general understanding of the influence of the additional orbitals is established. Afterwards, we additionally include the 3dz2 orbital and show that the material trend, found in experiments, can, at least, qualitatively be reproduced. Lastly, we go back to the single band Hubbard model and examine the consequences of the widely used approximation of neglecting the self-energy and the frequency dependence of the interaction vertex. We establish that the phase diagram obtained from the functional renormalization group is only quantitatively affected. From this study, we conclude that neglecting self-energy and frequency dependence, is a sensible approach, in order to obtain a reliable qualitative picture of the low energy behavior of a weakly or moderately coupled system at reasonable expense.
Two-dimensional semi-Dirac fermions are quasiparticles that disperse linearly in one direction and quadratically in the other. We investigate instabilities of semi-Dirac fermions toward charge and spin density wave and superconducting orders, driven by short-range interactions. We analyze the critical behavior of the Yukawa theories for the different order parameters using Wilson momentum shell renormalization group. We generalize to a large number Nf of fermion flavors to achieve analytic control in 2+1 dimensions and calculate critical exponents at one-loop order, systematically including 1/Nf corrections. The latter depend on the specific form of the bosonic infrared propagator in 2+1 dimensions, which needs to be included to regularize divergencies. The 1/Nf corrections are surprisingly small, suggesting that the expansion is well controlled in the physical dimension. The order parameter correlations inherit the electronic anisotropy of the semi-Dirac fermions, leading to correlation lengths that diverge along the spatial directions with distinct exponents, even at the mean-field level. We conjecture that the proximity to the critical point may stabilize novel modulated order phases.