# The Superconducting Critical Temperature

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fermion Dynamical Symmetry and Superconductivity

_{BCS}symmetry. If Coulomb repulsion is strong and antiferromagnetism significant, onsite pairing is disfavored relative to bondwise pairing and antiferromagnetic operators become important in addition to pairing operators. This reduces SO(8) to a 15-generator subgroup SU(4), with generators representing AF, spin-singlet and spin-triplet bondwise pairs, spin, and charge operators; explicit forms for the operators and their commutation relations are given in Refs. [23,33]. Three dynamical symmetry chains have exact solutions and correspond (through their matrix elements) to physical states thought to be relevant for cuprate doped and undoped states:

- SU(4) ⊃ SO(4), which represents an antiferromagnetic (AF) Mott insulating state that is the low-temperature ground state for zero doping.
- SU(4) ⊃ SU(2), which represents a d-wave singlet superconducting (SC) state that can become the low-temperature ground state for non-zero doping.
- SU(4) ⊃ SO(5), which represents a critical dynamical symmetry interpolating between the SU(2) superconducting and SO(4) antiferromagnetic solutions.

## 3. SU(4) Dynamical Symmetry and Cuprate Phenomenology

**Origin of the Phase Diagram**: Universality of the cuprate phase diagram suggests a unifying principle independent of microscopic details. The SU(4) model implies that symmetry alone dictates many basic properties, and that these properties lead to a highly universal phase diagram, illustrated in Figure 4,that is described quantitatively by the SU(4) model. Only two significant parameters enter: the effective strength of singlet pairing ${G}_{0}$ and the effective strength of AF correlations $\chi $ (triplet pairing strength ${G}_{1}$ has minimal influence). The best fit is for the smooth dependence of ${G}_{0}$ and $\chi $ on the doping P shown in the inset of Figure 4, but the basic features survive if these parameters are held constant with doping (see Ref. [33]). Thus, the cuprate phase diagram is a consequence of SU(4) symmetry correlating emergent d-wave singlet pairing and antiferromagnetism; it depends only parametrically on microscopic details such as pairing formfactors.

**AF Mott Insulator States at Half Filling**: By counting, SU(4) symmetry requires no double occupancy of lattice sites by correlated fermions [25]. Hence, charge transport is suppressed at half band-filling and the undoped ground state is a Mott insulator. Moreover, this state has SU(4) ⊃ SO(4) dynamical symmetry and the matrix elements of an AF Néel state [23,24,33]. Thus, the undoped SU(4) ground state is an AF Mott insulator, just as observed for cuprates.

**Cooper Instability of the Doped Mott Insulator**: The same SU(4) symmetry requiring the undoped ground state to be an AF Mott insulator implies that this state is fundamentally unstable against condensing Cooper pairs when doped [31,33]. This results in a quantum phase transition (QPT) to be discussed more extensively below, and implies a rapid transition to a superconducting state upon doping, as observed for data in Figure 4.

**Optimal Doping for Superconductivity**: For doping larger than that near the peak of the superconducting dome (optimal doping) in Figure 4, SC properties were observed to become better defined. As discussed further below, this is a natural consequence of SU(4) symmetry, which implies a quantum phase transition near optimal doping exhibiting critical behavior [33]. At subcritical doping the superconducting wave function is perturbed by residual AF correlations. At the QFT the AF correlations vanish identically, leaving pure d-wave, BCS-like, singlet SC above critical doping. This is consistent with various cuprate experiments.

**Existence of a Pseudogap**: A pseudogap is a partial gap at the Fermi surface above ${T}_{c}$. From Figure 4, at lower doping it is the “normal state” from which SC can be produced by lowering the temperature through the doping-dependent critical temperature ${T}_{c}$. As explained further below, a PG is expected from AF–SC competition in the SU(4) Hilbert subspace [33].

**Quantum Critical Behavior**: A highest symmetry with multiple dynamical symmetry subchains leads naturally to quantum phase transitions as tuning parameters such as doping, magnetic field, or pressure shift the balance between competing dynamical symmetries. The SU(4) theory is microscopic so one can determine whether these transitions are associated with critical behavior and examine the corresponding physical consequences. Thus, SU(4) and its dynamical symmetry subchains are a laboratory for quantum critical behavior in HTSC. As we now discuss, the SU(4) model implies two fundamental instabilities, leading to quantum phase transitions that are central to understanding HTSC, and a critical dynamical symmetry that generalizes a quantum critical point to an entire quantum critical phase, which proves useful in understanding the underdoped region in general and the PG region in particular.

**The SU(4) Cooper Instability**: The SU(4) solution at $T=0$ for the pairing order parameter $\Delta $ given in Equation (24b) of Ref. [26] implies that [31]$${\left(\right)}_{\frac{\partial \Delta}{\partial x}}x=0=\infty ,$$

**The SU(4) AF Instability**: SU(4) symmetry implies a second fundamental instability. From the SU(4) solution for the AF order parameter Q given by Equations (24b, 14) of Ref. [26],$${\left(\right)}_{\frac{\partial Q}{\partial x}}x={x}_{q}=-\infty ,$$

**Dynamical Symmetries and Critical Behavior**: Quantum phase transitions and quantum critical points are a natural consequence of fermion dynamical symmetries, implying that quantum critical behavior is a corollary of unconventional superconductivity, not a cause. Furthermore, some dynamical symmetry solutions generalize quantum critical points to entire quantum critical phases exhibiting critical behavior [24,32,33]. The SU(4) ⊃ SO(5) dynamical symmetry is an example. This is seen most easily in generalized coherent state approximation [24], which represents symmetry-constrained Hartree–Fock–Bogoliubov solutions that permit SU(4) results to be expressed in terms of gap equations and quasiparticles, and that lead naturally to total energy surfaces connecting SU(4) solutions microscopically to Ginzburg–Landau theory. SU(4) coherent state energy surfaces are displayed in Figure 6a–c. The flat critical nature of the SO(5) surface is evident for low doping in Figure 6b.

**Complexity in the Underdoped Region**: As suggested in Figure 6d,f, the underdoped ∼SO(5) energy surface exhibits complexity because many potential ground states with very different order parameters have almost the same energy. Complexity implies susceptibility to fluctuations in AF and SC order induced by small perturbations and the phase defined by the SO(5) dynamical symmetry is a critical dynamical symmetry. Critical dynamical symmetries are a fundamental organizing principle for complexity in strongly correlated nuclear structure and condensed matter systems [32,33]. In Ref. [32] we have proposed that stripe and checkerboard patterns, amplification of proximity effects, and related phenomena common in underdoped compounds may be a consequence of complexity enabled by the critical nature of the energy surface there. This complexity can occur with or without associated spatial modulation of charge. Charge is not an SU(4) generator so it is not fundamental for HTSC, but it can play a secondary role by perturbing critical energy surfaces in underdoped compounds.

**Competing Order or Preformed Pairs**: In the competing-order model the PG is an energy scale for order competing with SC. In the rival preformed pairs model pairs form at higher energy with phase fluctuations that suppress long-range order until at a lower energy the pairs condense into a SC with long-range order. In SU(4) the PG scale is an AF correlation competing with SC but the AF operators are generators of SU(4) and the collective subspace is a superposition of pairs. Thus, the PG results from a superposition of SU(4) pairs that can condense into a strong superconductor only after AF fluctuations are suppressed by doping. Hence the SU(4) pseudogap state results from competing AF and SC order in a basis of fermion pairs and SU(4) unifies the competing order and preformed pair pictures.

**Fermi Arcs and Anisotropy of Pseudogaps**: The SU(4) cuprate model implies strong angular dependence in k-space, which leads to temperature and doping restrictions on regions of the Brillouin zone where ungapped Fermi surface can exist [30,33]. If this region is interpreted in terms of Fermi arcs, the SU(4) model gives a natural description of arc lengths as a function of temperature in quantitative accord with ARPES data [30,33]. If the Fermi surface is interpreted in terms of small pockets instead, SU(4) symmetry restricts their possible location and size.

**SU(4) and Conventional BCS Superconductivity**: The relationship of SU(4) to BCS SC was given in Figure 3. Conventional SC is the limit of SO(8) ⊃ SU(4) SC when Coulomb repulsion is small and AF correlation is negligible. Thus, SO(8) ⊃ SU(4) dynamical symmetry provides a unified view of conventional and unconventional superconductivity [33].

**Dynamical Symmetry and Universality of Emergent States**: Dynamical symmetries for a variety of emergent states suggest an even broader unification transcending fields and subfields Figure 7.

## 4. Transition Temperatures for Unconventional Superconductors

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bednorz, J.G.; Müller, K.A. Possible high T
_{c}superconductivity in the Ba-La-Cu-O system. Z. Phys.**1986**, B64, 189–193. [Google Scholar] [CrossRef] - Kamihara, Y.; Watanabe, T.; Hirano, M.; Hosono, H. Iron-Based Layered Superconductor La[O
_{1-x}F_{x}]FeAs (x = 0.05–0.12) with T_{c}= 26 K. J. Am. Chem. Soc.**2008**, 1330, 3296–3297. [Google Scholar] [CrossRef] [PubMed] - Guo, J.; Jin, S.; Wang, G.; Wang, S.; Zhu, K.; Zhou, T.; He, M.; Chen, X. Superconductivity in the iron selenide K
_{x}Fe_{2}Se_{2}(0 ≤ x ≤ 1.0). Phys. Rev.**2010**, B82, 180520. [Google Scholar] [CrossRef] [Green Version] - Norman, M.R. The Challenge of Unconventional Superconductivity. Science
**2011**, 332, 196–200. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jérome, D. Organic Superconductors: When correlations and magnetism walk in. arXiv
**2012**, arXiv:1201.5796. [Google Scholar] [CrossRef] [Green Version] - Bonn, D.A. Are high-temperature superconductors exotic? Nat. Phys.
**2006**, 2, 159–168. [Google Scholar] [CrossRef] - Norman, M.R.; Pépin, C. The electronic nature of high temperature cuprate superconductors. Rep. Prog. Phys.
**2003**, 66, 1547–1610. [Google Scholar] [CrossRef] - Johnston, D.C. The puzzle of high temperature superconductivity in layered iron pnictides and chalcogenides. Adv. Phys.
**2010**, 59, 803–1061. [Google Scholar] [CrossRef] [Green Version] - Paglione, J.; Greene, R.L. High-temperature superconductivity in iron-based materials. Nat. Phys.
**2010**, 6, 645–658. [Google Scholar] [CrossRef] [Green Version] - Oh, H.; Moon, J.; Shin, D.; Moon, C.; Choi, H.J. Brief review on iron-based superconductors: Are there clues for unconventional superconductivity? Prog. Supercond.
**2011**, 13, 65–84. [Google Scholar] - Bardeen, J.; Cooper, L.N.; Schrieffer, J.R. Theory of Superconductivity. Phys. Rev.
**1957**, 108, 1175–1204. [Google Scholar] [CrossRef] [Green Version] - Cooper, L.N. Bound Electron Pairs in a Degenerate Ferxai Gas. Phys. Rev.
**1956**, 104, 1189–1190. [Google Scholar] [CrossRef] - Anderson, P.W. Spin-Charge Separation is the Key to the High Tc Cuprates. Physica
**2000**, C341–C348, 9–10. [Google Scholar] [CrossRef] [Green Version] - Pines, D. Quantum Protectorates in the Cuprate Superconductors. Physica
**2000**, C341–C348, 59–62. [Google Scholar] [CrossRef] [Green Version] - Laughlin, R.B.; Pines, D. The Theory of Everything. Proc. Natl. Acad. Sci. USA
**2000**, 97, 28–31. [Google Scholar] [CrossRef] [Green Version] - Wu, C.L.; Feng, D.H.; Guidry, M.W. The fermion dynamical symmetry model. Adv. Nuc. Phys.
**1994**, 21, 227–443. [Google Scholar] - Iachello, F.; Arima, A. The Interacting Boson Model; Cambridge University Press: Cambridge, UK, 1987. [Google Scholar]
- Bijker, R.; Iachello, F.; Leviatan, A. Algebraic Models of Hadron Structure. Ann. Phys.
**1994**, 236, 69–116. [Google Scholar] [CrossRef] [Green Version] - Iachello, F.; Levine, R.D. Algebraic Theory of Molecules; Oxford University Press: Oxford, UK, 1995. [Google Scholar]
- Iachello, F.; Truini, P. Algebraic Model of Anharmonic Polymer Chains. Ann. Phys.
**1999**, 276, 120–143. [Google Scholar] [CrossRef] - Anderson, P.W. Random-Phase Approximation in the Theory of Superconductivity. Phys. Rev.
**1958**, 112, 1900–1916. [Google Scholar] [CrossRef] - Glauber, R.J. Photon Correlations. Phys. Rev. Lett.
**1963**, 10, 84–86. [Google Scholar] [CrossRef] [Green Version] - Guidry, M.W.; Wu, L.-A.; Sun, Y.; Wu, C.-L. SU(4) model of high-temperature superconductivity and antiferromagnetism. Phys. Rev.
**2001**, B63, 134516. [Google Scholar] [CrossRef] [Green Version] - Wu, L.-A.; Guidry, M.W.; Sun, Y.; Wu, C.-L. SO(5) as a critical dynamical symmetry in the SU(4)model of high-temperature superconductivity. Phys. Rev.
**2003**, B67, 014515. [Google Scholar] [CrossRef] [Green Version] - Guidry, M.W.; Sun, Y.; Wu, C.-L. Mott insulators, no double occupancy, and non-Abelian superconductivity. Phys. Rev.
**2004**, B70, 184501. [Google Scholar] [CrossRef] [Green Version] - Sun, Y.; Guidry, M.W.; Wu, C.-L. Temperature-dependent gap equations and their solutions in the SU(4) model of high-temperature superconductivity. Phys. Rev.
**2006**, B73, 134519. [Google Scholar] [CrossRef] [Green Version] - Sun, Y.; Guidry, M.W.; Wu, C.-L. Pairing gaps, pseudogaps, and phase diagrams for cuprate superconductors. Phys. Rev.
**2007**, B75, 134511. [Google Scholar] [CrossRef] [Green Version] - Sun, Y.; Guidry, M.W.; Wu, C.-L. k-dependent SU(4) model of high-temperature superconductivity and its coherent-state solutions. Phys. Rev.
**2008**, B78, 174524. [Google Scholar] [CrossRef] [Green Version] - Guidry, M.W.; Sun, Y.; Wu, C.-L. A unified description of cuprate and iron arsenide superconductors. Front. Phys. China
**2009**, 4, 433–446. [Google Scholar] [CrossRef] [Green Version] - Guidry, M.W.; Sun, Y.; Wu, C.-L. Strong anisotropy of cuprate pseudogap correlations: Implications for Fermi arcs and Fermi pockets. New J. Phys.
**2009**, 11, 123023. [Google Scholar] [CrossRef] - Guidry, M.W.; Sun, Y.; Wu, C.-L. Generalizing the Cooper-pair instability to doped Mott insulators. Front. Phys. China
**2010**, 5, 171–175. [Google Scholar] [CrossRef] - Guidry, M.W.; Sun, Y.; Wu, C.-L. Inhomogeneity, dynamical symmetry, and complexity in high-temperature superconductors: Reconciling a universal phase diagram with rich local disorder. Chin. Sci. Bull.
**2011**, 56, 367–371. [Google Scholar] [CrossRef] [Green Version] - Guidry, M.W.; Sun, Y.; Wu, L.-A.; Wu, C.-L. Fermion dynamical symmetry and strongly-correlated electrons: A comprehensive model of high-temperature superconductivity. Front. Phys.
**2020**, 15, 43301. [Google Scholar] [CrossRef] [Green Version] - Dai, P.; Mook, H.A.; Hayden, S.M.; Aeppli, G.; Perring, T.G.; Hunt, R.D.; Doğan, F. The Magnetic Excitation Spectrum and Thermodynamics of High-T
_{c}, Superconductors. Science**1999**, 284, 1344–1347. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Campuzano, J.C.; Ding, H.; Norman, M.R.; Fretwell, H.M.; Randeria, M.; Kaminski, A.; Mesot, J.; Takeuchi, T.; Sato, T.; Yokoya, T.; et al. Electronic Spectra and Their Relation to the (π,π) Collective Mode in High-T
_{c}Superconductors. Phys. Rev. Lett.**1999**, 83, 3709–3712. [Google Scholar] [CrossRef] [Green Version] - Armitage, N.P.; Fournier, P.; Greene, R.L. Progress and perspectives on electron-doped cuprates. Rev. Mod. Phys.
**2010**, 82, 2421–2487. [Google Scholar] [CrossRef] - Fang, L.; Luo, H.-Q.; Cheng, P.; Wang, Z.-S.; Jia, Y.; Mu, G.; Shen, B.; Mazin, I.I.; Shan, L.; Ren, C. Roles of multiband effects and electron-hole asymmetry in the superconductivity and normal-state properties of Ba(Fe
_{1-x}Co_{x})_{2}As_{2}. Phys. Rev.**2009**, B80, 140508. [Google Scholar] [CrossRef] [Green Version] - Knebel, G.; Aoki, D.; Floquet, J. Magnetism and Superconductivity in CeRhIn
_{5}. arXiv**2009**, arXiv:0911.5223. [Google Scholar] - Kang, N.; Salameh, B.; Auban-Senzier, P.; Jérome, D.; Pasquier, C.R.; Brazovskii, S. Domain walls at the spin-density-wave endpoint of the organic superconductor (TMTSF)
_{2}PF_{6}under pressure. Phys. Rev.**2010**, B81, 100509. [Google Scholar] [CrossRef] [Green Version] - Guidry, M.W.; Sun, Y. Superconductivity and superfluidity as universal emergent phenomena. Front. Phys.
**2015**, 10, 107404. [Google Scholar] [CrossRef] [Green Version] - Wu, L.-A.; Guidry, M.W. The Ground State of Monolayer Graphene in a Strong Magnetic Field. Sci. Rep.
**2016**, 6, 22423. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**a**) Emergent-symmetry truncation of Hilbert space to a collective subspace using principles of dynamical symmetry. (

**b**) Comparison of matrix elements among different theories and data. Wavefunctions and operators are not observables. Only matrix elements are directly related to observables.

**Figure 2.**Schematic difference between bondwise $(D,\pi )$ and onsite $(S,{S}^{*})$ pair energies. If onsite repulsion is weak, the pairing states are nearly degenerated, yielding an SO(8) symmetry. If it is strong onsite pairs are pushed up in energy, reducing the symmetry to an effective SU(4) low-energy symmetry.

**Figure 3.**The relationship between SO(4), SO(8), and BCS SU(2) symmetry for conventional and unconventional superconductors.

**Figure 4.**SU(4) cuprate temperature T and doping P phase diagram compared with data taken from Refs. [34,35]. Strengths of the AF and singlet pairing correlations were determined in Ref. [27] by global fits to cuprate data (inset plot). Pseudogap temperatures are indicated by ${T}^{*}$. The two PG curves correspond to whether momentum is resolved or not in the experiment. The inset shows the variation of the AF and pairing coupling with doping P.

**Figure 5.**Two fundamental SU(4) instabilities that govern the behavior of high temperature superconductors. The plots illustrate (

**a**) the generalized Cooper instability and (

**b**) The AF instability in terms of the values of the order parameters calculated within coherent state approximation.

**Figure 6.**(

**a**–

**c**) Coherent-state energy surfaces for symmetry limits of the SU(4) Hamiltonian [24]. The horizontal axis measures AF order. Curves are labeled by lattice occupation fractions with the value 1 corresponding to half filling. The parameter $\sigma $ is the ratio of AF coupling to the sum of AF and pairing coupling strengths. (

**d**–

**f**) Effect of altering the ratio $\sigma $ for three values of doping in the cuprates. In (

**d**,

**f**) the system is in the stable minima associated with AF and SC, respectively, and changing $\sigma $ by 10% hardly alters the location of the energy minima, but in (

**b**) the energy surface is critical and the perturbation can flip the nature of the ground state between SC and AF minima.

**Figure 7.**Universality of superconductivity and superfluidity. (

**a**) Phase diagram for hole- and electron-doped cuprates [36]. Superconducting (SC), antiferromagnetic (AF), and pseudogap (PG) regions are labeled, as are Néel (${T}_{N}$), SC critical (${T}_{c}$), and PG (${T}^{*}$) temperatures. (

**b**) Phase diagram for Fe-based SC [37]. (

**c**) Heavy-fermion phase diagram [38]. (

**d**) Phase diagram for an organic superconductor (SDW denotes spin density waves) [39]. (

**e**) Generic correlation-energy diagram for nuclear structure [40].

**Figure 8.**Similarity in the dynamical symmetry chains and the ground coherent state energy surfaces for (

**a**) dynamical symmetry in nuclear structure [16], (

**b**) high-temperature SC [23,24], and (

**c**) monolayer graphene in a strong magnetic field [41]. The plot contours show total energy as a function of an appropriate order parameter, with different curves corresponding to a particle number parameter.

**Figure 9.**(

**a**,

**b**) Formation of BCS superconductor by lowering the temperature of a Fermi liquid through ${T}_{c}$. Direction of vectors indicates relative strength of competing order (x) and SC (y); length indicates total SU(4) strength. The SC transition converts a high-entropy state (

**a**) into a highly ordered one (

**b**), implying a low ${T}_{c}$. (

**c**,

**d**) Formation of SC from a parent state having order that competes with SC but is related to SC by symmetry. This requires imposing SC order (

**d**) on a state (

**c**) already highly ordered, which can occur at a higher ${T}_{c}$ because it is a collective rotation in the group space between two low-entropy states. (

**e**) Collective rotation in SU(4) group space. (

**f**) SU(4) Cooper instability.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Guidry, M.; Sun, Y.; Wu, L.-A.
The Superconducting Critical Temperature. *Symmetry* **2021**, *13*, 911.
https://doi.org/10.3390/sym13050911

**AMA Style**

Guidry M, Sun Y, Wu L-A.
The Superconducting Critical Temperature. *Symmetry*. 2021; 13(5):911.
https://doi.org/10.3390/sym13050911

**Chicago/Turabian Style**

Guidry, Mike, Yang Sun, and Lian-Ao Wu.
2021. "The Superconducting Critical Temperature" *Symmetry* 13, no. 5: 911.
https://doi.org/10.3390/sym13050911