A Short Review of the $S_4$ Symmetric Microscopic Model for Iron-Based High Temperature Superconductors

We briefly review the recently constructed two orbital microscopic model for iron-based superconductors based on $S_4$ symmetry (PRX 2 021009(2012)). With this faithful representation of the kinematics of the tri-layer FeAs or FeSe structure, the model provides answers and physical pictures to fundamental questions related to the robustness of superconductivity and pairing symmetry, unifies different families of iron-based superconductors, casts new insight into the connections to the other high $T_c$ superconductors, cuprates, and reveals why an s-wave pairing can be stabilized by repulsive interactions. Further progresses include that the model provides a clean understanding of band reconstruction observed in magnetically ordered states, which is a strong support to the kinematics of the $S_4$ model, and captures the essential low energy physics of iron-based superconductors based on numerical results from unbiased quantum Monte Carlo simulation.


Introduction
In the past four years, the main theoretical difficulty in understanding iron-based high temperature superconductors [2,3,4,5] stems from the fact that the materials are intrinsic multi-orbital systems due to the presence of all five 3d-orbitals of iron atoms. Several initial attempts to construct simplified models with fewer orbitals was made in an early stage of the research [6,7,8]. While these models could successfully explain some properties, they all had serious drawbacks, which resulted in a belief that any microscopic model composed of less than all five d-orbitals is insufficient [9,10].
Recently, very surprisingly, with a proper consideration of the S 4 symmetry, the symmetry of the building block-the FeAs or FeSe trilayer, we demonstrate that an effective two-orbital model is sufficient to capture the kinematics responsible for low energy physics [1]. The two orbital model includes two nearly degenerate and weakly coupled single-orbital parts that can be mapped to each other under the S 4 transformation. Not only has the model greatly simplified kinematics which essentially can be described by only three important tight-binding hopping parameters, but the model also has manifested a clear physical picture to explore many intriguing fundamental properties of the materials and has drawn a deep connection to the other high temperature superconductors, cuprates. In this paper, we give a quick review of the model by emphasizing its unique aspects and summarizing recent numerical and analytic results [11,12].

Motivation
Before proceeding to the S 4 model, we discuss several main initial motivations which lead to the construction of the model. The motivations stem from a few intriguing questions related to iron-based superconductors.
• What is the relation between iron-based superconductors and cuprates? This question has undoubtedly been asked by many researchers since iron-pnictides were discovered. In the past several years, it has been shown that the two classes of materials share many similar properties, including similar layered lattice structures, phase diagrams, strong antiferromagnetism in parental compounds, short superconducting coherent lengths and so on. However, it is still highly controversial that whether iron-based superconductors allow us to finally apply the induction method for high temperature superconductors. • How can the two different families of iron-based superconductors, iron-pnictides and iron chalcogenides, be unified? Iron-based superconductors include two families, ironpnictides [2,3,4] and iron-chalcogenides [5]. They share many intriguing common properties. They both have the highest T c s around 50K [13,3,14,15,16]. The superconducting gaps are close to isotropic around Fermi surfaces [17,18,19,20,21,22] and the ratio between the gap and T c , 2∆/T c , are much larger than the BCS ratio, 3.52, in both families. However, the electronic structures in the two families, in particular, the Fermi surface topologies, are quite different in the materials reaching the highest T c . The hole pockets are absent in iron-chalcogenides but present in iron-pnictides [17,20,21,22]. The presence of the hole pockets has been a necessity for superconductivity in the majority of studies and models which deeply depend on the properties of Fermi surfaces [9,13,23,24,25,26,27,28,29,30]. Therefore, the absence of the hole pockets in iron-chalcogenides causes a strong debate over whether both families belong to the same category that shares a common superconducting mechanism.
• Why and how can a s-wave pairing state be stabilized if the superconductivity is originated from a repulsive interaction? With a d-wave pairing symmetry, such as in cuprates, the positive and negative signs of the superconducting order parameter are equally distributed in both real and reciprocal space . Therefore, one can argue that the onsite repulsive interaction is naturally avoided in a d-wave pairing state. For a s-wave pairing symmetry, even in a sign changed s ± state [24], the argument can not be held in general. While many of us use repulsive interactions to obtain a s-wave pairing or s ± , in iron-pnictides, this fundamental problem is largely unaddressed theoretically. • Why is the superconductivity so robust and how can a single energy scale dominate in pairing gap functions in iron-based superconductors? Observed by angle-resolved photoemission microscopy (ARPES), a very intriguing property in the SC states of iron-pnictides is that the SC gaps around optimal doping on different Fermi surfaces are nearly proportional to a simple form factor cosk x cosk y in reciprocal space. This form factor has been observed in both 122 [17,31,18,32] and 111 [33,34] families of iron-pnictides. Just like the d-wave form factor cosk x − cosk y in cuprates, such a form factor indicates that the pairing between two next nearest neighbour (NNN) iron sites in real space dominates. In a multi-orbital model, many theoretical calculations based on weak coupling approaches have shown that the gap functions are very sensitive to detailed band structures and vary significantly when the doping changes [9,26,27,28,29,30]. The robustness of the form factor has been argued to favor strong coupling approaches which emphasize electron-electron correlation or the effective next nearest neighbour (NNN) antiferromagnetic (AF) exchange coupling J 2 [35,36,37,38,39,40,41,42] as a primary source of the pairing force. However, realistically, it is very difficult to imagine there is a single energy scale in local exchange interactions if a multi d-orbital model is considered.
In order to answer the above questions, it is clear that we have to identify microscopic parameters that control the essential physics of iron-based superconductors at low energy.
3. Gauge mapping between the symmetries of superconductivity pairing and kinematics The idea to address the robustness of possible s-wave pairing in iron-based superconductors stems from a paper by Berg, Scalapino and Kivelson [42], where they showed that the s-wave pairing in a two leg-ladder model for iron-pnictides is equivalent to a d-wave pairing in a similar model of cuprates. The s-wave and the d-wave can be mapped by a simple gauge transformation. Although their idea was carried out in an one-dimensional model, they raised an indispensable general point for any dimensional model, that is, the pairing symmetry and kinematics must be considered together. In particular, in a system where the short range pairing dominates, the phases or the signs of superconducting order parameters can be exchanged with those of the local hopping parameters.

Gauge Mapping
A gauge mapping can be generally described as follows. Given a general Hamiltonian including a kinetic energyĤ 0 and an interaction energyĤ I , we can perform a gauge mapping that changes the sign of local Fermionic operators, namely, where θ i = 0, or 1. It is clear that such a general gauge mapping can be described by an unitary operatorÛ . Under the mapping, the hopping terms inĤ 0 can change the sign so that the kinetic energy Hamiltonian is changed. However, the eigenvalues ofĤ 0 are not changed since it is an unitary transformation. It is also easy to see that the interaction HamiltonianĤ I , if it includes standard interaction terms, such as local density interaction, spin exchange interaction and so on, in general, is not changed under the gauge transformation. Therefore, under the mapping, If the original HamiltonianĤ has a superconducting ground state with an superconducting order parameter < c i c j >= ∆ ij , the ground state of the new HamiltonianĤ must have < c i c j >= (−1) θ i +θ j ∆ ij . Therefore, if the pairing is rather local short range, it is possible to change order parameter symmetry by choosing a proper gauge mapping.

Gauge mapping between d-wave and extended s-wave pairing symmetries in cuprates
Let's use cuprates as an example to illustrate the point. For cuprates, the kinetic energyĤ 0 is dominated by the nearest neighbour(NN) hopping in a square lattice, namely, where < ij > represents the links between two NN sites, and the order parameter of the d-wave pairing symmetry in real space is also dominated by coupling two NN sites which is corresponding to the d-wave gap form cosk x − cosk y in reciprocal space, namely, where x, y are lattice unit vectors along x and y directions respectively. As shown in fig.1, if we set a gauge mapping as  where i x labels the coordinate along the x direction of the ith site. After the mapping, we have and Therefore, the gauge mapping allows us to exchange the symmetries between the hopping parameters and the superconducting pairing order parameters. Before the gauge mapping, the hopping has a s-wave symmetry and the pairing has a d-wave symmetry. After the gauge mapping, the hopping has a d-wave symmetry and the pairing has a s-wave symmetry. It is also important to notice that the Fermi surface topology is changed after the gauge mapping as shown in fig.1. In a weak coupling approach, the Fermi surface topology has been the key ingredient to determine many properties, such as pairing symmetry. If the pairing is driven by repulsive interaction or antiferromagnetic fluctuations, the Fermi surface topology before the gauge mapping favors a d-wave symmetry while one can argue that the Fermi surface topology after the gauge mapping favors a s-wave symmetry. Therefore, the result obtained from a simple Fermi surface topology analysis is actually consistent with the gauge mapping result. We will revisit this point for iron-based superconductors in the following subsection.

Gauge mapping between extended s-wave and d-wave pairing symmetries in iron-based superconductors
The above analysis has been generalized to iron-based superconductors [1]. A similar gauge mapping which maps a s-wave to a d-wave pairing symmetry can be designed as shown in fig.2(1, 2, 1 , 2 ). We group four neighbour iron sites to form a super site and change the sign of Fermionic operators at the red iron sites in fig.2(1 ). This transformation maps the A 1g s-wave cos(k x )cos(k y ) pairing symmetry in the original Fe lattice to a familiar d-wave cosk x − cosk y pairing symmetry defined in the two sublattices of the orginal lattice, where (k x , k y ) and (k x , k y ) label momentum in Brillouin zones of the origin lattice and sublattice respectively. This is actually a straightforward generalization of the transformation used in the study of the one dimensional iron ladder model [42]. Following the same procedure, we can ask how the kinetic energy changes. In ref. [1], we have shown that all the different tight binding models constructed for iron-based superconductors, after the gauge mapping, display very striking simplicity: (1) all Fermi surfaces after the gauge mapping are relocated around X , the anti-nodal points in a standard d-wave superconducting state in the sublattice Brillouin zone; (2) the bands previously located at the different places are magically linked in the new gauge setting. In particular, the two bands that contribute to electron pockets are nearly degenerate and the bands that contribute to hole pockets are remarkably connected to the former ones.
The first feature seriously challenges the consistency of previous weak coupling analysis depending on the difference of Fermi surface topologies for iron-pnictides and iron-chalcogenides. After the gauge mapping, based on the new Fermi surface topology, a standard weak coupling analysis would naturally favor a robust d-wave superconducting state with respect to the sublattice in the presence of repulsive interaction or NN AF coupling [43,38]. Therefore, as shown in fig.3  The second feature is the key feature that leads us to believe that the low energy physics is controlled by a two orbital model which is characterized by two nearly degenerate bands. The emergence of the universal kinematics in different models suggest that the underlining electronic dynamics should be controlled by a much simplified set of parameters.
4. The construction of a two-orbital model with the S 4 symmetry With above observations and the following important facts in the tri-layer FeAs or FeSe structure, • The d-orbitals that form the bands near the Fermi surfaces are strongly hybridized with the p-orbitals of As(Se). Since the d x z and d y z have the largest overlap with the p x and p y orbitals, it is natural for us to use d x z and d y z to construct the model; • The local lattice symmetry at an iron site is D 2d in which S 4 symmetry is the key symmetry operation; • The two As(Se) planes are separated in space along c-axis in the tri-layer structure. Since there is little coupling between the p orbitals of the two planes and the hopping integrals through the p-orbitals are expected to dominate over the direct exchange couplings between the d-orbitals themselves, the two orbital model essentially could be decoupled into two nearly degenerate one orbital models; • The decoupling must be controlled by the S 4 symmetry since S 4 symmetry links two As(Se) planes; we draw a natural conclusion that the two-orbital model should be classified according to the S 4 symmetry and each of the two nearly degenerate one-orbital models should have a unit cell with two iron sites. A picture of the effective two-orbital model with the S 4 symmetry is illustrated in fig.4. The two orbital model is divided into two weakly and coupled one-orbital models. The orbital degree of freedom in the two one-orbital models can be called as two components of an S 4 iso-spin. The first one includes the d x z in the A sublattice and the d y z in the B sublattice if we use A and B to label two sublattices, and the second one includes the d x z in the B sublattice and the d y z in the A sublattice. The first strongly couples to the top As(Se) layer and the second couples to the bottom As(Se) layer. We denoteĉ iσ tod iσ to label Fermionic operators of the two models that are connected by the S 4 transformation. Then, a general Hamiltonian that describes the kinematics is given by, where H 0 includes the two decoupled one-orbital models and H 0.c which also obey the S 4 symmetry describes the weak coupling between the two models. As shown in [1], if we keep the leading hopping parameters, andĤ 0,c = k 2t c (cosk x + cosk y )(ĉ + kσd kσ + h.c.).
• Two weakly coupled electronic structures: Iso--spin degree of freedom linked by S 4 symmetry! • Intrinsic doubling of unit cell: An electron pocket and hole pocket are paired because of the doubling of unit cell. • Doping is near half -filling.
Ignoring t c , each S 4 iso-spin components provides the following two paired band energy dispersions due to unit cell doubling, where k = 4t 2s cosk x cosk y + 2t 3 (cos2k x + cos2k y ) − µ. E e can capture the electron pocket at M points and E h can capture the hole pocket at Γ point. t 1s , t 2s and t 2d are the key parameters. By just keeping these three parameters, the model is already good enough to capture the main characters of the bands contributing to the Fermi surface. Including interactions, the general Hamiltonian is given by where α = 1, 2 labels the S 4 iso-spin, U describes the effective Hubbard repulsion interaction within each iso-spin, U describes the one between them and J H describes the effective Hund s coupling. Since the two components couple weakly, we may expect U dominates over U and J H . Then, in the first order approximation, the model could become a single orbital-Hubbard model near half-filling.
In the original paper [1], we have suggested that t 1s breaks D 2d symmetry. This statement is not absolutely correct because t 1s and t 1d can be exchanged by choosing different orbital configurations. With the orbital configuration shown in fig.4, t 1s does not break D 2d symmetry. It is also important to note that the Γ point and the (π, π) point in Brillouin zone here are exchangeable because of the doubling of the unit cell.

Numerical results and supporting experimental evidence for the S 4 model
In fig.5, we summarize the important properties in the S 4 model. These properties include • For each S 4 iso-spin, there is a pair of bands generated by unit cell doubling: one contributes a hole pocket and the other contributes an electron pocket. • For each S 4 iso-spin, the filling factor is close to half-filling, an important property to allow strong spin fluctuation. • The unit cell is intrinsically doubled, which is a very different starting point of kinematics from previous models. • The hole pockets can be suppressed by adjusting t 1s or t 2s . Fixing t 2d and t 2s , reducing t 1s leads to flatter dispersion of the band responsible for the hole pockets. Thus, it suppresses the hole pockets. Therefore, the model unifies iron-pnictides and iron-chalcogenides. • From orbital configuration, we have t 2 > |t 1s | > |t 2 | and the signs of t 2 and t 2 are opposite.
These characters have important physical consequences. In the following, we discuss some recent progresses associated with them.

Quantum Monte Carlo (QMC) Simulation on the S 4 model
The microscopic understanding provides a new opportunity to make use of highly controllable and unbiased numerical methods to study iron-based superconductors. In a recent paper [11], we have demonstrated that the model in the presence of electron-electron interaction is characterized by strong (0, π) collinear antiferromagentic (C-AFM) fluctuations as observed in iron-based superconductors [44,45,46]. Moreover, the pairing with an extended s-wave symmetry robustly dominates over other pairings at low temperature in reasonable parameter region regardless of the change of Fermi surface topologies, namely, the presence or absence of hole pockets at Γ point. For both iron-pnictides and iron-chalcogenides, the pairing correlation, the effective pairing interaction and the antiferromagnetic correlation strongly increase as the on-site Coulomb interaction increases.
QMC results are summarized in fig.6. The competition between the s-wave and the d-wave in the S 4 model is manifested by t 1 and t 2 where t 1 favors the d-wave while t 2 favors the s-wave. A simple phase diagram from QMC simulation is given as the inset in fig. 6(b). For iron-based superconductors, t 1 should be roughly equal to t 2 /2 and |t 2 | is much smaller than t 1 and t 2 . Therefore, the s-wave should be robustly favored.
(a) Quantum Monte Carlo calculations about the static spin susceptibility for the S 4 model. The inset shows a significant large t 1 can change the peak from (0, π) to (π, π), which signals the competition between NN hopping and NNN hopping. (b) The effective pairing interaction as a function of temperature at < n >=1.1, U = 4.0, t 2 = 1.2, t 2 = −0.8 for different t 1 . Inset: the competition between s xy and d x 2 −y 2 depends on t 1 /t 2 and −t 2 /t 2 .

5.2.
Oriented gap opening in the C-AFM state: an impact of intrinsic unit cell doubling As we have discussed above, one unique feature in the S 4 symmetry construction is that the unit cell includes 2 irons. Most of previous studies are based on models with one iron per unit cell. In principle, one can argue that models based on both representations should be equivalent. However, since the search of a meanfield result depends on the starting point of kinematics, a proper starting point is crucial to obtaining a correct meanfield result.
In a recent paper, we have demonstrated an important impact of the intrinsic unit cell doubling [12]. Unlike in a conventional spin density wave (SDW) state where gaps at any two points on the Fermi surface that are connected by the ordered SDW wave vector should be opened, the band structure in the C-AFM state appears to be reconstructed by the development of the magnetic order. ARPES experiments [47,48,49,50,51,52] observe that small pockets or hot spots develop around Fermi surfaces. It is fair to say that the scenario of the gap opening due to a magnetic order in a multi-orbital system, such as iron-pnictides, is complex. For example, it has been argued that the entire Fermi surfaces can be gaped out [53]. Nevertheless, the surprising band reconstruction is never fully understood in previous models. As summarized in fig.7, the complicated band reconstruction near Fermi surfaces in the magnetically ordered state of iron-pnictides observed by ARPES can be understood in a meanfield level by the S 4 model. The (0,π) or (π,0) C-AFM order does not open gaps between two points on theFermi surfaces linked by the ordered wave vector but forces a band reconstruction involving four points in unfolded Brillouin zone and gives rise to small pockets or hot spots.
The doubling of the unit cell leads to an orientation-dependent gap opening in reciprocal space. In one Γ − M direction, the magnetic order splits two folded bands while in the other direction the gap can be opened between the pair of electron and hole pocket bands. Such an oriented gap opening generates the hot spots or small pockets as observed by ARPES as shown in fig.7(c)..

Sign change between the top and bottom As(Se) planes in the s-wave state
The NNN hopping parameters t 2 and t 2 have opposite signs in the S 4 model. This sign difference stems from the orbital symmetries. If we take the gauge mapping mentioned earlier, after the mapping, the two hopping parameters in the new model have the same signs. Therefore, the gauge mapping exactly switches the symmetries between the pairing and the underlining kinematics as shown in fig.2(1, 2, 1 , 2 ). In the view of the d-wave picture after the gauge mapping, the superconducting phase for each S 4 iso-spin alternates between neighboring squares, which is corresponding to the sign change between the top and bottom As/Se planes in the view of the S 4 symmetry. Such an intrinsic sign change between the top and bottom As planes along c-axis explains why a s-wave before gauge mapping can be stabilized even if the repulsive interaction is responsible for superconductivity.
The relative phase between the two S 4 iso-spins can take two different values, 0 or π, in principle. The difference can be classified as the irreducible representation of the S 4 symmetry. The S 4 point group has four one-dimensional representations, including A, B and 2E. In the A state, the S 4 symmetry is maintained. In the B state, the state changes sign under the S 4 transformation. Therefore, in the A phase, since the S 4 symmetry is not violated, the relative phase between the two iso-spin are equal to π in space, while in the B phase, the relative phase is zero. The A phase in the presence of the hole pockets is an sign-changed s-wave or s ± state. Therefore, the model predicts a new superconducting state, B phase state. However, it is still an open question whether iron-base superconductors can carry both states or not.
6. Discussion and perspectives 6.1. Comparison between cuprates and iron-based superconductors By revealing the underlining electronic structure, we can make a close comparison between cuprates and iron-based superconductors. In Table.1, we list the close relations between two high T c superconductors. From the table, it is clear that by determining these physical properties of iron-based superconductors listed in the table can help to determine the high T c superconducting mechanism.
6.2. More supporting evidence for the S 4 symmetry The S 4 model provides a microscopic understanding to two measured universal behaviors: the superconducting gap function that is dominated by the single cosk x cosk y form and the almost universal NNN antiferromagnetic exchange coupling strength J 2 measured by neutron scattering [45,54,55] . The existence of such two single energy scale quantities is also a strong support for the S 4 model. If we treat the S 4 model in the presence of strong electron-electron correlation, both J 2 and the gap function in the s-wave state is determined by the NNN hopping parameters.
The S 4 model also suggests that the properties of an iron-based superconductor can be understood by simply observing bands responsible for electron pockets. Since the physics is dominated by the pair of bands that are responsible for a pair of hole and electron pockets. The third hole pockets at the Γ point observed in iron-pnictides is not expected to play a significant role. However, the existence of the third hole pocket may change the doping concentration for the S 4 model in a stoichiometric material. Namely, in the view of the S 4 model, the material could be self electron doped. For example, if a large third hole pocket is present, we would expect that the S 4 model could be heavily electron-doped. This expectation naturally explains puzzled physical properties of LiF eAs where a large third hole pocket was observed [56]. No static magnetic order is observed in the stoichiometric LiF eAs with T c ∼ 18K [57,58]. In the view of the S 4 model, the LiF eAs is self electron doped so that many physical properties of the material should be understood from the heavy electron doped side of phase diagram [59].

Perspectives
The S 4 model provides a fresh starting point to explore novel physics in iron-based superconductors. Here, we discuss several perspectives for the future research.
First, since the S 4 symmetry stems from the fact that As stands out of Fe plane, obvious interesting physics may stem from c-axis coupling. Recently, ARPES has provided the existence of superconducting gap nodes along c-axis in BaF e 2 As 2−x P x [60,61]. It is very likely that the S 4 symmetry may be linked to such a development of nodes since the superconducting phase along c-axis in the S 4 model has very intriguing sign change as we discussed earlier.
Second, electron nematism observed in iron-pnictides [62] can be fully investigated in the S 4 model. Namely, the S 4 model provides a microscopic base to understand electron nematism. The S 4 symmetry breaking can lead to an electron nematic state. In the previous theoretical studies, electron nematism is obtained through phenomenological models [63,64,65,66].
Third, with only a set of three parameters describing kinematics, it becomes possible to obtain the dependence of these parameters on lattice parameters. Such an dependence may help us to explain empirical observations of the connection between the maximum T c s and the As-Fe-As angles [13].
Fourth, it is interesting to investigate physical properties on surfaces where the S 4 symmetry, in principle, is naturally broken. A natural consequence of such a symmetry breaking is that symmetry broken states may be more stable on surfaces than in bulk.
Finally, the doping phase diagram and its parameter dependence, as well as many other physical quantities need to be investigated in the S 4 model.

Summary
The S 4 model provides great simplicity to study and unify the physics of iron-based superconductors.
It reveals the deep connections between curpates and iron-based superconductors. Since the uniqueness of the model stems from the unique structure of FeAs or FeSe tri-layer, revealing such an underlining microscopic kinematics cast a new light on the search of new structures that can produce high superconducting transition temperatures.