Antiferromagnetically driven electronic correlations in iron pnictides and cuprates

The iron pnictides and the cuprates represent two families of materials, where strong antiferromagnetic correlation drives three other distinct ordering tendencies: (1) superconducting pairing, (2) Fermi-surface distortion, and (3) orbital-current order. We propose that (1)–(3) and the antiferromagnetic correlation are the hallmarks of a class of strongly correlated materials to which the cuprates and pnictides belong. In this paper, we present the results of the functional renormalization-group studies to support the above claim. In addition, we show that as a function of the interlayer hopping parameter, the double-layer Hubbard model nicely interpolates between the cuprate and the iron pnictide physics. Finally, as a check, we will present the renormalization-group study of a ladder version of the iron pnictide and compare the results to those of the two-dimensional model.

Figure 1

(Color online) (a) The LDA FS of iron pnictides (Ref. 13). 1–3 are hole FSs and 4 and 5 label the electron FSs. (b) The LDA dispersion along the $\left(0,0\right)-\left(\pi ,0\right)$ line. Here the red, black, and blue curves represent the $\alpha$, $\beta$, and $\gamma$ bands, respectively. Dashed green line marks the Fermi level of undoped compound. The van Hove and the Dirac band singularities discussed in the text are indicated.

Figure 2

(Color online) A schematic real-space representation of the two types of SDW order.

Figure 3

(Color online) (a) The FRG form factors of the SDW order parameter. The red, black, and blue curves are the form factors associated with the (2,4), (1,4), and (3,5) Fermi pockets [i.e., when $\left(\mathbf{k},\mathbf{k}+\mathbf{Q}\right)∊\left(2,4\right),\left(1,4\right),\left(3,5\right)$ FS], respectively. (b) The FRG form factors of the SC order parameter. Here, different colored curves represent the order parameters associated with the five different FS [i.e., when $\left(\mathbf{k},-\mathbf{k}\right)∊1,\dots ,5$ FS]. For each pocket, $\theta$ is defined as the angle sustained by the vector connecting the center of the pocket to point $\left(k_x,k_y\right)$ on the FS and the vector $\left(\pi ,0\right)$. The $\mathbf{k}$ coordinates of the center of FS are (0,0) for pockets 1 and 2, $\left(\pi ,\pi \right)$ for pocket 3, $\left(0,-\pi \right)$ for pocket 4, and $\left(\pi ,0\right)$ for pocket 5. The parameters under which the above results are obtained are specified in the text.

Figure 4

(Color online) (a) A schematic representation of the Umklapp scattering processes which has the dual character of SDW and SC. (b) The amplitude of the scattering process shown in (a) as a function of the FRG energy cutoff $\Lambda$. (c) A schematic real-space representation of $\text{cos}{k}_x\text{cos}{k}_y$ singlet pairing.

Figure 5

(Color online) (a) A schematic representation of the scattering processes which have the dual character of SDW and PI. (b) The amplitude of the scattering process shown in (a) as a function of the FRG energy cutoff $\Lambda$. (c) The form factor, $f_{\mathbf{k}}$, in Eq. (25) associated with the most negative $w_i$. The angle $\theta$ is defined the same way as that in Fig. 3

Figure 6

(Color online) (a) The FS distortion triggered by the PI. After the distortion each electron pockets is split into two smaller electron pockets. (b) For very strong forward scattering, each electron pockets turns into two hole pockets after the FS distortion. In addition, the smaller hole pocket centered around $\left(\pi ,\pi \right)$ is gone.

Figure 7

(Color online) (a) The form factor, $f_{\mathbf{k}}$, in Eq. (25) associated with the subleading $w_i$ of Pomeranchuk instability. The angle $\theta$ is defined the same way as that in Fig. 3. (b) The FS distortion triggered by the $d_{x^2-y^2}$-type PI. After the distortion, one of the electron pockets is better nested with the hole pocket.

Figure 8

(Color online) (a) A schematic representation of the scattering processes which have the dual character of SDW and CDW. (b) The amplitude of the scattering process shown in (a) as a function of the FRG energy cutoff $\Lambda$. (c) The form factor $f_{\mathbf{k}}$ associated with the leading CDW instability. The angle $\theta$ is defined the same way as that in Fig. 3. (d) A schematic real-space representation of orbital current-density wave order.

Figure 9

(Color online) The grand FRG flow for the iron pnictides. Here the leading $w_i\left(\Lambda \right)$ for the SDW, SC, PI, and CDW channels are plotted as a function of $\Lambda$. (a) is constructed using the parameters describing a SC sample and (b) is the corresponding plot for a SDW sample. (a1) and (b1) cover the full energy scale and (a2) and (b2) zoom-in at the intermediate energy regime, with the competing order manifested.

Figure 10

(Color online) The overlap between the form factors predicted by the $J_1-J_2$ model and those obtained from the FRG. The black and blue curves are the overlaps associated with the SC SDW form factors, respectively. (a) and (b) corresponds to SC and SDW samples in Fig. 9. The overlap is set to zero when the symmetries of the form factors do not agree or when SDW and SC are not the leading instabilities of the $J_1-J_2$ model.

Figure 11

(Color online) The FRG results for the $t-t^{\prime }$ Hubbard model. (a) The SDW form factor and a schematic real-space representation of the $\left(\pi ,\pi \right)$ AFM order. (b) The $d$-wave SC pairing form factor and its schematic real-space representation. (c) The PI form factor and the distorted FS surface. The blue dashed line and the red solid line denote the original and the distorted FSs, respectively. (d) The imaginary CDW, or the orbital-current order, form factor and its schematic real-space representation.

Figure 12

(Color online) The dual scattering processes for the one-band $t-t^{\prime }$ Hubbard model. (a) The dual SDW-SC processes. (b) The dual SDW-PI processes. (c) The dual SDW-CDW processes. The left column is the schematic of the scattering processes and the right column shows the amplitude of the scattering process as a function of the FRG energy cutoff $\Lambda$.

Figure 13

(Color online) (a) The grand FRG flow for the $t-t^{\prime }$ Hubbard model. Here the $w_i\left(\Lambda \right)$ associated with the SDW, SC, PI, and CDW are plotted. Inset zooms in at the intermediate energy regime. (b) The overlap of the FRG form factors with those predicted by the $J_1-J_2$ model. The red and the blue curves are the overlaps for the SC and SDW form factors, respectively. The overlap is set to zero when the symmetries of form factors do not agree or when the $J_1-J_2$ model does not predict SC and SDW as the leading instabilities.

Figure 14

(Color online) (a) A schematic illustration of the distorted FS and the scattering processes that have the dual character of SC and incommensurate unidirection charge density wave or in short stripe order. The blue solid line is the original FS and the red dashed line is the distorted FS triggered by the PI. (b) The FRG flow of the scattering amplitude associated with (a).

Figure 15

(Color online) (a) A schematic illustration of hopping associated with the double-layer Hubbard model. (b) A schematic representation of the proposed phase diagram, where $\delta$ is doping.

Figure 16

(Color online) (a) The FS of the double-layer Hubbard model with $t_z=0.5$ at 4% hole doping. The two solid lines (blue and red) are the electron and hole FSs, respectively. The purple dotted line is the nodal line of $\text{cos}{k}_x-\text{cos}{k}_y$ and the black dashed line is the nodal line of $\text{cos}{k}_x+\text{cos}{k}_y$. (b) The $w_i\left(\Lambda \right)$ associated with SDW, extended $s$-wave SC, and $d$-wave SC. (c) The FRG-predicted $d$-wave pairing form factor. (d) The FRG-predicted form factor of the subleading extended $s$-wave pairing. Here for electron FS around (0,0), $\theta$ is the angle between $\left(k_x,k_y\right)$ and $\left(\pi ,0\right)$. For the hole FS around $\left(\pi ,\pi \right)$, $\theta$ is the angle between $\left(k_x-\pi ,k_y-\pi \right)$ and $\left(\pi ,0\right)$.

Figure 17

(Color online) (a) The FS of the double-layer Hubbard model with $t_z=1.0$ at 3% hole doped (b) The $w_i\left(\Lambda \right)$ associated with SDW, extended $s$-wave SC, and $d$-wave SC. (c) The FRG-predicted $d$-wave pairing form factor. (d) The FRG-predicted form factor of the nearly degenerate extended $s$-wave pairing. The angle is defined in the same way as Fig. 16.

Figure 18

(Color online) (a) The FS of the double-layer Hubbard model with $t_z=1.5$ at 8% hole doping. (b) The $w_i\left(\Lambda \right)$ associated with SDW, extended $s$-wave SC, and $d$-wave SC. (c) The FRG-predicted extended $s$-wave pairing form factor. (d) The FRG predicted form factor of the less-divergent $d$-wave pairing. The angle is defined in the same way as Fig. 16.

Figure 19

(Color online) (a) The FS of iron pnictide and the five pairs of Fermi points for the iron pnictide ladder. (b) The $w_i\left(\Lambda \right)$ associated with all singlet SC pairings. The red line is the $w_i$ associated with the leading SC instability.

Figure 20

(Color online) Left panels: the schematic representation of the strongest scattering processes (their scattering amplitude is at least 2 orders of magnitudes larger than all other scattering amplitudes) for the two-leg pnictide ladder at low energies: (a) $c_{24}^{\rho }$ (solid line) and $c_{22}^{\rho }$ (dashed line); (b) $c_{24}^{\sigma }$ and $c_{22}^{\sigma }$ (dashed line); (c) $f_{24}^{\rho }$ (solid line). Right panels: the RG flow associated with the processes associated with the right panels.