Is there any hidden symmetry in the stripe structure of perovskite high temperature superconductors?

TL;DR

This paper explores the hidden symmetries in stripe structures of doped perovskite high-temperature superconductors, proposing a mathematical model based on superpositions of stripe pairs and four-color theorem mappings to understand their complex lattice modulations.

Contribution

It introduces a novel theoretical framework representing stripe modulations as superpositions of tilted stripe pairs and maps their arrangements using planar graphs and the four-color theorem.

Findings

Stripe structures can be modeled as superpositions of two tilted stripe types.

The spatial distribution of stripes relates to four-color theorem mappings.

Atomic sharpness of stripe boundaries is consistent with the model.

Abstract

Local and fast structural probes using synchrotron radiation have shown nanoscale striped puddles and nanoscale phase separation in doped perovskites.It is known that the striped phases in doped perovskites are due to competing interactions involving charge, spin and lattice degrees of freedom,but while many theoretical models for spin and charge stripes have been proposed we are missing theoretical models describing the complex lattice striped modulations in doped perovskites. In this work we show that two different stripes can be represented as a superposition of a pair of stripes, U(theta_n) and D(theta_n), characterized by perovskite tilts where one of the pair is rotated in relation to the other partner by an angle Delta(theta _n)=pi/2. The spatial distribution of the U and D stripes is reduced to all possible maps in the well-known mathematical four-color theorem. Both the periodic striped puddles and random structures can be represented by using $planar$ graphs with a chromatic number. To observe the colors in mapping experiments, it is necessary to recover variously oriented tilting effects from the replica. It is established that there is an interplay between the annihilation/creation of new stripes and ordering/disordering tilts in relation to the theta _n angle in the CuO2 plane, where the characteristic shape of the stripes coincides with the tilting-ordered regions. By their origin, the boundaries between the stripes should be atomically sharp.

Tables

Table 1. Table 1: The multiplication table for the ninth order abelian group G ​ ( φ ) 𝐺 𝜑 G\left(\varphi\right) , where the group unit is an untilted prototype O 𝑂 O

Group elements	$D\left(0\right)$	$D\left({90}^{\circ }\right)$	$D\left({180}^{\circ }\right)$	$D\left({270}^{\circ }\right)$	$U\left({45}^{\circ }\right)$	$U\left({135}^{\circ }\right)$	$U\left({225}^{\circ }\right)$	$U\left({315}^{\circ }\right)$	$O$
$D\left(0\right)$	$D\left({180}^{\circ }\right)$	$U\left({45}^{\circ }\right)$	$O$	$U\left({315}^{\circ }\right)$	$U\left({135}^{\circ }\right)$	$D\left({90}^{\circ }\right)$	$D\left({270}^{\circ }\right)$	$U\left({225}^{\circ }\right)$	$D\left(0\right)$
$D\left({90}^{\circ }\right)$	$U\left({45}^{\circ }\right)$	$D\left({270}^{\circ }\right)$	$U\left({135}^{\circ }\right)$	$O$	$U\left({315}^{\circ }\right)$	$U\left({225}^{\circ }\right)$	$D\left({180}^{\circ }\right)$	$D\left(0\right)$	$D\left({90}^{\circ }\right)$
$D\left({180}^{\circ }\right)$	$O$	$U\left({135}^{\circ }\right)$	$D\left(0\right)$	$U\left({225}^{\circ }\right)$	$D\left({90}^{\circ }\right)$	$U\left(0\right)$	$U\left({315}^{\circ }\right)$	$D\left({270}^{\circ }\right)$	$D\left({180}^{\circ }\right)$
$D\left({270}^{\circ }\right)$	$U\left({315}^{\circ }\right)$	$O$	$U\left({225}^{\circ }\right)$	$D\left({90}^{\circ }\right)$	$D\left(0\right)$	$D\left({180}^{\circ }\right)$	$U\left({135}^{\circ }\right)$	$U\left(0\right)$	$D\left({270}^{\circ }\right)$
$U\left({45}^{\circ }\right)$	$U\left({135}^{\circ }\right)$	$U\left({315}^{\circ }\right)$	$D\left({90}^{\circ }\right)$	$D\left(0\right)$	$U\left({225}^{\circ }\right)$	$D\left({270}^{\circ }\right)$	$O$	$D\left({180}^{\circ }\right)$	$U\left(0\right)$
$U\left({135}^{\circ }\right)$	$D\left({90}^{\circ }\right)$	$U\left({225}^{\circ }\right)$	$U\left(0\right)$	$D\left({180}^{\circ }\right)$	$D\left({270}^{\circ }\right)$	$U\left({315}^{\circ }\right)$	$D\left(0\right)$	$O$	$U\left({135}^{\circ }\right)$
$U\left({225}^{\circ }\right)$	$D\left({270}^{\circ }\right)$	$D\left({180}^{\circ }\right)$	$U\left({315}^{\circ }\right)$	$U\left({135}^{\circ }\right)$	$O$	$D\left(0\right)$	$U\left(0\right)$	$D\left({90}^{\circ }\right)$	$U\left({225}^{\circ }\right)$
$U\left({315}^{\circ }\right)$	$U\left({225}^{\circ }\right)$	$D\left(0\right)$	$D\left({270}^{\circ }\right)$	$U\left(0\right)$	$D\left({180}^{\circ }\right)$	$O$	$D\left({90}^{\circ }\right)$	$U\left({135}^{\circ }\right)$	$U\left({315}^{\circ }\right)$
$O$	$D\left(0\right)$	$D\left({90}^{\circ }\right)$	$D\left({180}^{\circ }\right)$	$D\left({270}^{\circ }\right)$	$U\left({45}^{\circ }\right)$	$U\left({135}^{\circ }\right)$	$U\left({225}^{\circ }\right)$	$U\left({315}^{\circ }\right)$	$O$

Equations

$$U\left(45^{\circ }\right)=D\left(0\right)\times D\left(90^{\circ }\right)=\left(0,\alpha ,0\right)+\left(\alpha ,0,0\right)$$

$$D\left(180^{\circ }\right)=D{\left(0\right)}^2=\left(0,\alpha ,0\right)+\left(0,\alpha ,0\right)$$

$$O=D\left(0\right)\times D\left(180^{\circ }\right)=\left(0,\alpha ,0\right)+\left(0,-\alpha ,0\right)$$

$$G$$

$$D_1=D\left(0\right),D\left(180^{\circ }\right),O;U_1=U\left(45^{\circ }\right),U\left(225^{\circ }\right),O;$$

$$D_2=D\left(90^{\circ }\right),D\left(270^{\circ }\right),O;U_2=U\left(135^{\circ }\right),U\left(315^{\circ }\right),O$$

Full text

superexchange, Dzyaloshinskii-Moriya interaction, optical excited states, Mott-Hubbard insulator

I Introduction

A characteristic feature of perovskite structures ABO3, known as Glazer’s systems, is a variety of tilting effects. Glazer (1972, 2011) The latter can be caused by external factors, for example, by pressure, Xiang et al. (2017) by temperature, Zhou and Kennedy (2004) by a discrepancy between the lattice parameters of a substrate and the material Vailionis (2015) or by the misfit strain between different layers in complex materials made of stacks of atomic layers. Bianconi et al. (2000a); Agrestini et al. (2003); Poccia et al. (2010); Harris et al. (2018)

The unique functionality of several quantum complex materials with perovskite structure, like cuprates, Bussmann-Holder et al. (2017); Egami (2017); Benedek and Muller (1993); Sigmund and Müller (2012); Keller and Bussmann-Holder (2010); Bianconi et al. (2001) manganites Dagotto et al. (2001); Burgy et al. (2004) and bismuthates Giraldo-Gallo et al. (2013) can be tuned by atomic substitutions, tolerance factor, misfit strain and pressure which control local structural tilts, bond disproportionation and nanoscale phase separation. A similar complex lattice fluctuations have been identified by EXAFS in BaPb1-xBixO3 perovskite superconductors showing bond disproportionation and correlated tilts. Menushenkov et al. (2013, 2016)

While the competition between disorder and superconductivity has been a topic of intense interest for decades there are several exemplary systems can be found in the literature that demonstrate how correlated disorder Imry and Ma (1975); Kresin et al. (2006); Zaanen (2010); Littlewood (2011); Carlson (2015) manifested through doping Fratini et al. (2010) vacancies and strain Ricci et al. (2013) in both two and three dimensions, Giraldo-Gallo et al. (2015); Duan et al. (2018) plays an important role in the pairing mechanism Bianconi et al. (2000b); Menushenkov et al. (2016); Sakakibara et al. (2012) and can favor the enhancement of critical temperature Bianconi (2013, 2012a, 2012b) toward room temperature. Bianconi and Jarlborg (2015a, b)

It is known that cuprates show a universal dome of Tc versus doping, little is known about the mechanism beyond the variation of the maximum critical temperature of the dome, Tc-max, which has been found to be controlled by pressure or strain. Using advanced scanning micro x-ray diffraction the space distribution of inhomogeneity in the nanoscale has been well understood in cuprates. It has been found that a correlated disorder induced by doping and strain leads to charge puddles creating an inhomogeneous lattice, and superconductivity is the result of percolation in a hyperbolic space through filamentary pathways defined by the puddles Campi et al. (2015); Ricci et al. (2015) which has been confirmed in iron based superconductors, Duan et al. (2018) in bismuth sulfide Athauda and Louca (2019) and it is expected also in oxyselenide Krzton-Maziopa et al. (2014) perovskite superconductors.

Since high-$T_{c}$ superconductors are superlattices of metallic CuO2 layers, an appropriate physical variable describing elastic effects is not a tolerance factor for three-dimensional ABO3 systems, but the superlattice misfit strain between different layers. Forgacs et al. (1991); Bak (1982) Using resonant Cu K-edge x-ray diffraction Bianconi et al. (1996a), Cu K-edge XANES and Cu K-edge EXAFS Bianconi (1994); Missori et al. (1994) it was shown that the local structure of the CuO2 plane of doped Bi2Sr2CaCu2O8+y (called BSCCO or Bi2212) Bianconi (1994); Missori et al. (1994); Bianconi et al. (1996a); Saini et al. (1998) and of doped La2-xSrxCuO4+y (called LSCO or La124)  Bianconi et al. (1996b); Lanzara et al. (1996); Saini et al. (2003a, b) perovskite families show a periodic anharmonic incommensurate lattice modulation described as a particular nanoscale phase separation due to the formation of alternate stripes of undistorted $U$ stripes and distorted $D$ stripes with different tilts (see Fig.1) and disproportionated Cu-O bonds in cuprates. Bianconi et al. (1996b) For the superlattice of CuO2 atomic layers separated by rocksalt LaO spacer layers in the hole-doped layered perovskite family La2CuO4 (LCO) the relevant physical quantity is the misfit strain between the active [CuO2] and the spacer rocksalt [La2O${}_{2}+y$] layers. For the BSCCO superlattice of perovskite bilayers CaCu2O4 separated by spacer layers [Sr2O2] and [Bi2O2] the relevant physical quantity is the misfit strain between the active CuO2 and the spacer [Bi2O2] layers.

Recent experimental studies using scanning tunneling conductivity (STM) over a wide range of temperatures and doping Zhao et al. (2019) evidence that there is a clear connection between the stripe structure and electronic structure of Cu oxides supported by the fact that the pseudogap state and the stripe structure detected by EXAFS Missori et al. (1994); Lanzara et al. (1996); Saini et al. (2003b) and scanning micro x-ray diffraction Campi et al. (2015) have a similar temperature region of existence.

II Group $G\left(\alpha\right)$ of stripe pairs

The research was inspired by the study presented in the work, Zhang et al. (2018) where hidden symmetries were investigated in the stripe structure of high-temperature superconductors. Here, we focus on the tilting effects as a required attribute of observed stripes. Bianconi et al. (1996b) Based on the large isotope effect in high-Tc superconductors with the stripe nanostructure, Sasagawa et al. (2005) we tried to study the phonon nature of the stripe structure in high-Tc cuprates, and to identify the stripes according to the type of their tilting effects. For example, the $U$ and $D$ stripes in Fig.1 show the $\left({a^{-}a^{-}a^{0}}\right)$ and $\left({a^{-}a^{0}a^{0}}\right)$ tilts in Glazer’s notations. Glazer (2011) Unfortunately, in these notations it is impossible to classify the stripes rotated about the axis $z$ by different angles $\theta_{n}$. Therefore we introduce new notations in which an orientation of the tilting angle $\alpha$ can vary within a single Glazer’s notation. To do this, we replace $a$ by the angle $\pm\alpha$, and omit the top indices $+$ and $-$ as they do not change under the action of rotation about the axis $z$. Then, all the possible $U$ and $D$ stripes: $D\left(0\right)=\left({0,\alpha,0}\right)$, $U\left(45^{\circ}\right)=\left({\alpha,\alpha,0}\right)$, $D\left({90^{\circ}}\right)=\left({\alpha,0,0}\right)$, $U\left({135^{\circ}}\right)=\left({\alpha,-\alpha,0}\right)$, $D\left({180^{\circ}}\right)=\left({0,-\alpha,0}\right)$, $U\left({225^{\circ}}\right)=\left({-\alpha,-\alpha,0}\right)$, $D\left({270^{\circ}}\right)=\left({-\alpha,0,0}\right)$, $U\left({315^{\circ}}\right)=\left({-\alpha,\alpha,0}\right)$ form a group of a complex root of unity $\sqrt[g]{{{e^{\left({2\pi i}\right)}}}}$, where $g=8$. This is the symmetry group of an isolated CuO6 octahedron with respect to its rotation by an angle $\theta_{n}=n(\pi/4)$. In the LCO and LSCO layered materials, this group reduces to the two $U\left({{\theta_{n}}}\right)$ and $D\left({{\theta_{n}}}\right)$ groups because of different orientation of the CuO4 planar complex relative to the rocksalt spacer layers.

In accordance with the study, Pickett (1989) the undoped LCO has the orthorhombic Abma (D${}^{18}_{2h}$) structure at a temperature below 500K, where the tilts of the perovskite octahedra are ordered into a $(a^{-}a^{-}a^{0})$ sequence. The latter can formally be presented here as a superposition $\left({a^{-}a^{-}a^{0}}\right)=\left({a^{-}a^{0}a^{0}}\right)+\left({a^{0}a^{-}a^{0}}\right)$, and we further consider the mono $U$ stripe structure of the undoped LCO as the confinement of a pair of tetragonal structures rotated relative to each other by an angle $\Delta\theta_{n}={\pi\mathord{\left/{\vphantom{\pi 2}}\right.\kern-1.2pt}2}$ (Fig.2). As a consequence, the number of stripes, in the doped LSCO, is reduced to one species ($U$ stripes) with extensive overlapping areas ($D$ stripes) between them.

To study the nature of the spatial stripe distribution we take into account that the $U\left({{\theta_{n}}}\right)$ and $D\left({{\theta_{n}}}\right)$ structures have a common origin and differ only by orientation (i.e. $\theta_{n}$) of the tilting angle $\alpha$. Let’s collect the $U(\theta_{n})$ and $D(\theta_{n})$ stripes into a new group $G\left(\alpha\right)$ built from the stripes rotated relative to each other by an angle ${\theta_{n}}=n\left({\pi/4}\right)$

[TABLE]

and etc. Following the superposition for the $U\left({{\theta_{n}}}\right)$ and $D\left({{\theta_{n}}}\right)$ stripes, we derive the rules of group multiplication for the stripes: $U\left(45^{\circ}\right)$, $U\left({135^{\circ}}\right)$, $U\left({225^{\circ}}\right)$, $U\left({315^{\circ}}\right)$ and $D\left(0\right)$, $D\left({90^{\circ}}\right)$, $D\left({180^{\circ}}\right)$, $D\left({270^{\circ}}\right)$ (see Tab.1) , where the group unit $O$ refers to the untilted prototype in which there are no tilts at all. The group $G\left(\alpha\right)$ of the stripe pairs presented in Tab.1 appears if we assume the replacement $\left({2\alpha}\right)\leftrightarrow-\alpha$ in the squared group element $D{\left(0\right)^{2}}=\left({0,\alpha,0}\right)+\left({0,\alpha,0}\right)=\left({0,2\alpha,0}\right)=D\left({180^{\circ}}\right)$ , i.e. there can be only one specific angle $\alpha\sim const$ in the material. In this case, we obtain an abelian group $G\left(\alpha\right)$ of the ninth order, which is a direct product of any two cyclic subgroups:

[TABLE]

of the four possible:

[TABLE]

In contrast to the observed orthorhombic symmetry with the $(a^{-},a^{-},a^{0})$ tilting effects in the undoped LCO cuprates, Pickett (1989) described only by the subgroup ${U_{1}}$ (or ${U_{2}}$ ) while in the doped cuprates LSCO, the effects $(a^{-}a^{-}a^{0})$ and $(a^{-}a^{0}a^{0})$, $(a^{0}a^{-}a^{0})$ from the direct product ${U_{1}}\times{U_{2}}$ might be observed. Bianconi et al. (1996b)

The irreducible representations of the group $G(\alpha)$ are homomorphisms from this group into a group of three elements: 1, $e^{2\pi i/3}$, $e^{4\pi i/3}$, that is, a third degree root of unity. The homomorphism is completely defined by how it acts on the generators of a group. In our case, these are two elements which do not belong to the same subgroup, for example $D(0)$ and $U(0)$. If $D(0)$ is mapped to $e^{2\pi i/3}$, and $U(0)$ - to $e^{4\pi i/3}$, then $U(135)=D(0)U(0)$ is mapped to 1, and similarly, for all other elements of the group. The three variants of the homomorphism for each of the pair of generators of the group result in the nine different representations, and the latter are numbered by pairs of indices $\{i,j\}$, where $i(j)=1,2,3$.

Since the active CuO2 plane does not contain any untilted prototypes, there is no neighborhood of $U$ stripes rotated by $180^{\circ}$ relative to each other. All of them are separated by the $U\left({{\theta_{n}}}\right)$ stripes with ${\Delta\theta_{n}}=90^{\circ}$ (or ${\Delta\theta_{n}}=270^{\circ}$ ), which have an overlapping region ($D\left({{\theta_{n}}}\right)$ stripes). For example, $U\left(45^{\circ}\right)$/ $D\left({270^{\circ}}\right)$/ $U\left({135^{\circ}}\right)$/ $D\left(0\right)$/ $U\left({225^{\circ}}\right)$/ $D\left({90^{\circ}}\right)$/ $U\left({315^{\circ}}\right)$/ $D\left({180^{\circ}}\right)$/ $U\left(45^{\circ}\right)$, or $U\left(45^{\circ}\right)$/ $D\left({270^{\circ}}\right)$/ $U\left({135^{\circ}}\right)$/ $D\left({270^{\circ}}\right)$/ $U\left(45^{\circ}\right)$/$D\left({270^{\circ}}\right)$/ $U\left({135^{\circ}}\right)$ with a period of the stripe structure less than the order of the group $G(\alpha)$.

The spatial distribution of the stripes in Fig.3 and Fig.4 directly follows from the well-known four color theorem on the map. Biggs et al. (1986) In four colors, $U\left(45^{\circ}\right)$, $U\left({225^{\circ}}\right)$ -red ($R$), $U\left({135^{\circ}}\right)$, $U\left({315^{\circ}}\right)$ -blue ($B$), and also $D\left(0\right)$, $D\left({180^{\circ}}\right)$ -green ($G$), $D\left({90^{\circ}}\right)$, $D\left({270^{\circ}}\right)$ - yellow ($Y$) we can always color an arbitrary map on the plane . Biggs et al. (1986) The four colors: $R$, $G$, $B$, $Y$ are just four subgroups: ${D_{1}}$, ${D_{2}}$ and ${U_{1}}$, ${U_{2}}$, from which the untilted prototypes $O$ have been removed. For example, the unidirectional charge stripe order and a chessboard electronic crystal Okamoto and Furukawa (2012); Seibold et al. (2007) can correspond to one of the color maps: $R$/$Y$/$B$/$Y$/$R$, $R$/ $Y$/$B$/$G$/$R$/$Y$/$B$ in Fig.3 and Fig.4 respectively.

Any stripe structures, including a random structure are more convenient for representation using planar graphs with a chromatic number $\chi\leq 4$. Biggs et al. (1986) In accordance with the group $G\left(\alpha\right)$, all the maps are symmetric with respect to the replacement of $U$ by $D$ and vice versa.

Recently, the four color theorem has been used to interpret the domain topology of the magnetic material FexTaS2. Horibe et al. (2014)

III Conclusions

1. The overlapping between the stripes may be an alternative for Bloch and Neel type of domain walls in oxides Cherifi-Hertel et al. (2017) at the subatomic scale. The overlapping looks like the $D$ stripes, whose properties follow from the group multiplication in Tab.1. In this group, the overlapping areas of the $D$ and $U$ stripes are equal, and the untilted prototype $O$ with $\alpha=0$ is the group unit. In the undoped LCO cuprate, the structure of one large $U$ stripe is observed, and the material is at the state with $(a^{-}a^{-}a^{0})$ tilting effects. Under doping, the disappearance of the long-range stripe order is accompanied with the generation of the $D$ stripes in the region of the immediate boundary between the $U\left({{\theta_{n}}}\right)$ stripes oriented at different angles ($\Delta{\theta_{n}}={\pi/2}$) relative to each other. The spatial distribution of the $D\left({{\theta_{n}}}\right)$ and $U\left({{\theta_{n}}}\right)$ stripes is reduced to possible maps in the four color problem for the CuO4 plane, where the color represents the $D\left({{\theta_{n}}}\right)$ or $U\left({{\theta_{n}}}\right)$ stripes combined into four subgroups of the group $G(\alpha)$. Any observed spatial distributions, namely, the unidirectional charge-stripe order, chessboard electronic crystal or totally random structures can be represented by planar graphs with different chromatic numbers, where $2\leqslant\chi\leqslant 3$ and $\chi=4$, respectively. $\chi=1$ at the long-range stripe ordering in the undoped material LCO. By their origin, the boundaries between the $U$ and $D$ stripes should atomically be sharp. Moreover, in layered materials the reduced symmetry of the CuO4 square in the $D$ stripes agrees well with the symmetry of the local ${b_{1g}}$ vibrations that are active in the $\left({{}^{2}{a_{1}}+{}^{2}{b_{1}}}\right)\otimes\left({{b_{1g}}+{a_{1g}}}\right)$ JT effect. Bersuker et al. (1992); Bersuker and Polinger (2012). However, unlike the usual local ${b_{1g}}$ mode, there no contribution to the multimode JT effect from the tilting mode in the CuO6 octahedron in the isotropic environment. In the $U$ stripes, the local symmetry of all four plane oxygens is identical even in layered materials. We expect the inhomogeneous distribution of electron density over the $U$ and $D$ stripes since the JT energy depends on the electron concentration only in the $D$ stripes. To observe the colors using STM spectroscopy, it is necessary to recover variously oriented tilting effects from the STM charge replica of the CuO2 plane. Note, compatible types of cooperative JT and tilting effects within a single structure were investigated in the study, Carpenter and Howard (2009) and an interplay between the tilting and JT distortions was studied previously in LaMnO3, YVO3, and YTiO3. Mizokawa et al. (1999)

2. What is the physical meaning of the group $G(\alpha)$? It is not a symmetry group of the lattice Hamiltonian since the group elements $U\left({{\theta_{n}}}\right)$ and $D\left({{\theta_{n}}}\right)$ do not reproduce any initial stripe distribution. The group $G(\alpha)$ is just the symmetry of a pair of interacting stripes. By analogy with the Woodward-Hoffmann’s rules for orbital symmetry conservation in chemical reactions, Woodward and Hoffmann (1971) we can associate the initial and final reagents in a reaction with the stripes $U\left({{\theta_{n}}}\right)$ and $D\left({{\theta_{n}}}\right)$, respectively. In this reversible ”reaction”, the orbital symmetry changes, but the tilting angle $\alpha$ remains a constant. Due to the group, we have established the relationship between the annihilation/creation of new stripes and ordering/disordering process of the tilts in the CuO2 plane, where the characteristic shape of the stripes coincides with the regions ordered by the $\theta_{n}$ angle.

Acknowledgements.

One of the authors (VAG) is grateful to RFBR, the Government of the Krasnoyarsk Region and Krasnoyarsk Regional Fund of Science for the research grant No.18-42-240017