## Abstract

The nature of the nematicity in iron pnictides is studied with a proposed magnetic fluctuation. The spin-driven order in the iron-based superconductor has been realized in two categories: stripe SDW state and nematic state. The stripe SDW order opens a gap in the band structure and causes a deformed Fermi surface. The nematic order does not make any gap in the band structure and still deforms the Fermi surface. The electronic mechanism of nematicity is discussed in an effective model by solving the self-consistent Bogoliubov-de Gennes equations. The nematic order can be visualized as crisscross horizontal and vertical stripes. Both stripes have the same period with different magnitudes. The appearance of the orthorhombic magnetic fluctuations generates two uneven pairs of peaks at ± π 0 and 0 ± π in its Fourier transformation. In addition, the nematic order breaks the degeneracy of d xz and d yz orbitals and causes the elliptic Fermi surface near the Γ point. The spatial image of the local density of states reveals a d x2-y2-symmetry form factor density wave.

### Keywords

- magnetic fluctuation
- stripe SDW
- nematic order
- two-orbital
- elliptic Fermi surface
- LDOS maps

## 1. Introduction

The discovery of Fe-based superconductors with critical temperatures up to 55 K has begun a new era of investigations of the unconventional superconductivity. In common with copper-like superconductors (cuprate), the emergency of superconductivity in electron-doped Fe-pnictides such as _{4} (tetragonal) to C_{2} (orthorhombic).

At present, there are two scenarios for the development of nematic order through the electronic configurations [18]. One scenario is the orbital fluctuations [19, 20, 21, 22, 23]. The structural order is driven by orbital ordering. The orbital ordering induces magnetic anisotropy and triggers the magnetic transition at a lower temperature. The other scenario is the spin fluctuation [24, 25, 26, 27]. The magnetic mechanism for the structural order is associated with the onset of SDW.

Recently, Lu et al. [28] reported that the low-energy spin fluctuation excitations in underdoped sample _{4} symmetry to C_{2} symmetry in the nematic state. Zhang et al. [29] exhibited that the reduction of the spin-spin correlation length at

The partial melting of SDW has been proposed as the mechanism to explain the nematicity. The properties of the spin-driven nematic order have been studied in Landau-Ginzburg-Wilson’s theory [18, 24, 25, 26]. Meanwhile, the lack of the realistic microscopic model is responsible for the debates where the leading electronic instability, i.e., the onset of SDW, causes the nematic order. Recently, an extended random phase approximation (RPA) approach in a five-orbital Hubbard model including Hund’s rule interaction has shown that the leading instability is the SDW-driven nematic phase [30]. Although the establishment of the nematicity in the normal state has attracted a lot of attentions, the microscopic description of the nematic order and, particularly, the relation between SC and the nematic order are still missing.

The magnetic mechanism for the structural order is usually referred to the Ising-nematic phase where stripe SDW order can be along the x-axis or the y-axis. The nematic phase is characterized by an underlying electronic order that the Z_{2} symmetry between the *x*- and *y*- directions is broken above and the O(3) spin-rotational symmetry is preserved [25].

The magnetic configuration in FeSCs can be described in terms of two magnetic order parameters

where

In the stripe SDW state, the order parameters are set to _{2} symmetry indicating to the degenerate of spin stripes along the y-axis (corresponding to _{4} (tetragonal) to C_{2} (orthorhombic). In the nematic state, the order parameters are set to _{2} symmetry is broken, but the O(3) spin-rotational symmetry is not. In the real space configuration, the x- and y-directions of the magnetic fluctuations are inequivalent.

Recently, the reentrant C_{4} symmetry magnetic orders have been reported in hole-doped Fe-pnictide [27, 31, 32]. A double-Q order (choose both _{4} symmetry, it is not suitable to explain the nematicity.

The magnetic fluctuations trigger a transition from the tetragonal-to-orthorhombic phase. At very high temperature,

In this chapter, we will exploit a two-orbital model to study the interplay between SC and nematicity in a two-dimensional lattice. The two-orbital model has been successfully used in many studies such as quasiparticle excitation, the density of states near an impurity [35, 36] and the magnetic structure of a vortex core [37].

## 2. Model

Superconductivity in the iron-pnictide superconductors originates from the FeAs layer. The Fe atoms form a square lattice, and the As atoms are alternatively above and below the Fe-Fe plane. This leads to two sublattices of irons denoted by sublattices A and B. Many tight-binding Hamiltonians have been proposed to study the electronic band structure that includes five Fe 3d orbitals [38], three Fe orbitals [39, 40], and simply two Fe bands [41, 42, 43]. Each of these models has its own advantages and range of convenience for calculations. For example, the five-orbital tight-binding model can capture all details of the DFT bands across the Fermi energy in the first Brillouin zone. However, in practice, it becomes a formidable task to solve the Hamiltonian with a large size of lattice in real space even in the mean-field level. Several studies used five-orbital models in momentum space to investigate the single-impurity problem for different iron-based compounds such as

On the other hand, the two-orbital models apparently have a numerical advantage dealing with a large size of lattice while retaining some of the orbital characters of the low-energy bands. Among the two-orbital (

The multi-orbital Hamiltonian of the iron-pnictide superconductors in a two-dimensional square lattice is described as

where

with

Here, we adopt Tai’s phenomenological two-by-two-orbital model because it is able to deal with a large size of lattice in many aspects and the details of low-energy bands are similar to the results from DFT + LDA. In a two-orbital model, the hopping amplitudes are chosen as shown in Figure 1 [48] to fit the band structure from the first-principle calculations:

where

Figure 1 shows the hopping parameters between unit cells and orbitals. For the same orbital, the hopping parameters _{4} symmetry on the same orbital between different sublattices is broken. However, _{4} symmetry of the lattice structure.

In the mean-field level

the Hamiltonian is self-consistently solved accompanied with

The next nearest-neighbor intraorbital attractive interaction

In momentum space, the spin configuration is determined by the order parameters

where the wave vectors

In the case of the absence of both

As

In the nematic state, the presence of both

Figure 2 displays the Fermi surface and the band structure in the absence of SDW at the normal state, i.e., the superconductivity is set to zero. In the absence of SDW

where

The eigenvalues are

where

Figure 2(a) shows that two hole bands are around the _{4} symmetry of the lattice structure.

In the stripe SDW state, the spin configuration is shown as Figure 3 . The stripe SDW order enlarges the two-Fe unit cell to four-Fe unit cell as denoted by the blue dashed square in Figure 3 . The antiferromagnetic order is along the

where

According to the itinerant picture, the interactions between two sets of pockets give rise to a SDW order at the wave vector connecting them with *along the* _{2} symmetry of the Fermi surface resulting from the SDW gap is shown in Figure 4(b) .

In the nematic state, the antiparallel spins are along both the *and* *and* *directions, there is no gap in the band structure. In addition, as* *direction are lifted higher and* cause the bands to be asymmetric with respect to the _{2}-symmetry of the Fermi surface results from the broken degeneracy of two orbitals

Recently, Qureshi et al. [53], Wang et al. [54], Steffens et al. [55], and Luo et al. [56] pointed out that in-plane spin excitations exhibit a large gap and indicating that the spin anisotropy is caused by the contribution of itinerant electrons and the topology of Fermi surface. These experiments indicate that the elliptic spin fluctuations at low energy in iron pnictides are mostly caused by the anisotropic damping of spin waves within FeAs plane and the topology of Fermi surface. The degeneracy of orbitals will introduce the single-ion anisotropy in spin fluctuations.

## 3. Visualize nematicity in a lattice

To visualize the nematicity in a lattice, we self-consistently solve the Bogoliubov-de Gennes (BdG) equations for the nematic state in a two-dimensional square lattice:

where

and

Here,

In Figure 6 , we show the magnetic configuration in the coexisting state of the nematic order and SC. To view the detail of the structure, the slided profile along the peaks along the x- or y-direction is made (as shown on the sides of *a*. Since

Figure 7 shows the Fourier transformation of the spatial configuration of the nematic fluctuations. Two peaks appearing at

We further illustrate the electronic charge density *a* which is the half period of the magnetization.

Moreover, although the checked pattern of the CDW is twofold symmetry, the CDW exhibits a

## 4. The local density of states

The local density of states (LDOS) proportional to the differential tunneling conductance as measured by STM is expressed as

where

In the striped SDW state, spins are parallel in the y-direction and antiparallel in the x-direction and cause the gap and gapless features in the band structure, respectively. The SDW gap shifts toward negative energy, and the coherence peak at the negative energy is pushed outside the SDW gap and enhanced. The coherence peak at the positive energy is moved inside the SDW gap and suppressed. This is a prominent feature caused by the magnetic SDW order that the intensities of superconducting coherence peaks are obvious asymmetry (as shown in Figure 9(a) ) [57].

In the nematic state, spins are antiparallel in the x- and y-directions leading to a gapless feature in the band structure. The superconducting gap is the only gap that appears in the LDOS. Moreover, comparing to the state without SDW, the competition between the nematic order and the superconducting order causes the slightly suppression of the coherence peaks. The feature of the suppression results in a dip at the negative energy outside the coherence peaks (as shown in Figure 9(b) ).

Furthermore, Figure 10 displays a spatial distribution of LDOS, also known as LDOS map, at

It is worth to note that STM measurements by Chuang et al. [5] and Allan [58] reported that the dimension of the electronic nanostructure is around

## 5. Phase diagram

To further verify the spin configuration of the nematic order, a phase diagram is presented in Figure 11 . In the phase diagram, the stripe SDW order, nematic order, and

In the hole-doped region, the magnetization exhibits the stripe SDW order and drops dramatically around

In the electron-doped region, the stripe SDW order (green curve) has its maximal value at

There are two regions where the stripe SDW coexist with the SC and the nematic order coexists with the SC. In the region where the stripe SDW coexist with the SC, the magnetic structure is an orthorhombic uniaxial stripe state. The ordering vector is either

It is worth to note that the phase diagram of the electron-doped region is consistent with Figure 1.3 of Kuo’s thesis on

## 6. Conclusions

The two-orbital Hamiltonian used in the iron-based superconductors has always been questioned for its validity. Many studies have approved that a lot of phenomena are attributed to

The stripe SDW order opens a gap in the band structure and deforms the Fermi surface. However, the band structure of the nematic order is gapless, and the Fermi surface is deformed to an ellipse. The mechanism can be understood from the instability of SDW. The nematic order has visualized as a checked pattern formed by a crisscrossed modulated horizontal and vertical stripes. The inequivalent strengths of the horizontal and vertical stripes break the degeneracy of two orbitals *dxz *and *dyz *and cause an elliptic Fermi surface. The Fourier transformation of the orthorhombic structure of the magnetization shows two uneven pairs of peaks at (±*π*,0) and (0,±*π*). Moreover, the LDOS map shows a *d* _{x2-y2}-symmetry form factor density wave.

Finally, the nematic order is favored to exist in the electron-doped regime, but not the hole-doped regime.

## Acknowledgments

HYC was supported by MOST of Taiwan under Grant MOST 107-2112-M-003-002 and National Center for Theoretical Science of Taiwan.