$\mathit{SO}\left(5\right)$ theory of antiferromagnetism and superconductivity

Antiferromagnetism and superconductivity are both fundamental and common states of matter. In many strongly correlated systems, including the high-$T_c$ cuprates, the heavy-fermion compounds, and the organic superconductors, they occur next to each other in the phase diagram and influence each other’s physical properties. The $\mathit{SO}\left(5\right)$ theory unifies these two basic states of matter by a symmetry principle and describes their rich phenomenology through a single low-energy effective model. In this paper, the authors review the framework of the $\mathit{SO}\left(5\right)$ theory and compare it with numerical and experimental results.

Figure 1

Phase diagram of the NCCO and the YBCO superconductors.

Figure 2

Spin-flop transition. (a) The spin-flop transition of the $XXZ$ Heisenberg model; (b) the Mott insulator-to-superfluid transition of the hard-core boson model or the $U<0$ Hubbard model. (c) Dscription of both transitions as the spin or the pseudospin flop transition in the $\mathit{SO}\left(3\right)$ nonlinear $\sigma$ model, induced either by the magnetic field or by the chemical potential.

Figure 3

Systems with competing CDW and superfluid phases: (a) Zero-temperature phase diagram of the $XXZ$ Heisenberg model, the hard-core boson model, or the negative-$U$ Hubbard model. Phase I is the Ising or the charge-density-wave (CDW) phase, phase II is the $XY$ or the superfluid phase, and phase III is the fully polarized or the normal phase. Class-$A$ transition is induced by the anisotropy parameter $g=J∕V$, while the class-$B$ transition is induced by the chemical potential or the magnetic field. (b) Finite-temperature phase diagram for the class-$A$ transition in $D=2$. Because of the $\mathit{SO}\left(3\right)$ symmetry at $J=V$ point, the transition temperature vanishes according to the Mermin-Wagner theorem. (c) Finite-temperature phase diagram for the class-$B$ transition in $D=3$. $T_{bc}$ denotes the $\mathit{SO}\left(3\right)$-symmetric bicritical point.

Figure 4

Hilbert space of the projected $\mathit{SO}\left(5\right)$ model: (a), (b), and (c) express the five bosonic states of the projected $\mathit{SO}\left(5\right)$ model in terms of the microscopic states on a plaquette. (d), (e), and (f) represent states with well-defined superspin directions, which can be obtained from the linear combinations of (a), (b), and (c). These states are analytically defined in Eq. (55) and Table .

Figure 5

Illustration of hopping processes of the magnons and the hole pairs on a ladder. The × denotes the center of a plaquette. An ellipse enclosing two sites denotes a spin singlet. (a) $J_s$ describes the magnon hopping; (b) $J_s^{\prime }$ describes the spontaneous creation and annihilation of a magnon pair; (c) $J_c$ describes the hopping of a hole pair; (d) $J_c^{\prime }$ describes the spontaneous creation and annihilation of a hole pair and a particle pair. In the full $\mathit{SO}\left(5\right)$ model, $J_s=J_s^{\prime }$ and $J_c=J_c^{\prime }$. In the projected $\mathit{SO}\left(5\right)$ model, the particle pair states are removed and $J_c^{\prime }=0$.

Figure 6

The chiral $\mathit{SO}\left(5\right)$ sphere. This sphere has an $\mathit{SO}\left(5\right)$-symmetric shape but allows only one sense of the rotation in the SC plane $(n_1,n_5)$. Small oscillations around the equator, or the $\pi$ triplet resonance, are unaffected by the chiral projection. However, small oscillations around the north pole, or the $\pi$ doublet mode, are strongly affected: only one of the two such modes is retained after the projection.

Figure 7

Phase diagram of the projected $\mathit{SO}\left(5\right)$ model (50) (for the case $J_c=2J_s\equiv J$) as a function of ${\delta }_s={\Delta }_S∕4J$ and ${\tilde{\delta }}_c={\tilde{\Delta }}_c∕4J$. Variation of the chemical potential changes ${\tilde{\Delta }}_c$ and traces out a one-dimensional trajectory as shown by the dotted line. $x={\sin }^2\theta$ and $y={\cos }^2\varphi$.

Figure 8

Spectra of the collective excitations of the projected $\mathit{SO}\left(5\right)$ model as a function of density. The region $0<\rho <{\rho }_c$ corresponds to the uniform mixed phase of superconductivity (SC) and antiferromagnetism (AF). Region $\rho >{\rho }_c$ corresponds to the SC phase. The left panel shows the spectra of the spin excitations. For $\rho <{\rho }_c$, there are two gapless spin-wave modes and one gapped spin-amplitude mode. For $\rho >{\rho }_c$, there is a spin-triplet $\pi$-resonance mode. The right panel shows the spectra of the gapless charge excitations (in the absence of long-range interactions). For $\rho <{\rho }_c$ the charge mode has quadratic dispersion. The dispersion relation changes from $\omega \propto k^2\text{to}\omega \propto k$ for the $\rho >{\rho }_c$ regime.

Figure 9

The three types of phase state in the $\mathit{SO}\left(5\right)$ model: The energy (a) and the free energy (b) can depend on the density of a uniform AF/SC mixed state with a negative curvature when $u_{12}>\sqrt{u_1{u}_2}$ (classified as type 1) or a positive curvature when $u_{12}<\sqrt{u_1{u}_2}$ (classified as type 2). The $\mathit{SO}\left(5\right)$-symmetric limiting case of zero curvature, classified as type 1.5, is realized when $u_{12}=\sqrt{u_1{u}_2}$. (c) The type-1 phase transition from the AF to SC state is a direct first-order transition. There are two second-order transitions from the AF to SC state in the type-2 case. $\mathit{SO}\left(5\right)$ symmetry is realized at the intermediate case of type 1.5.

Figure 10

The finite-temperature phase diagram in $D=3$ for the class-B1 transition shown in Fig. 13. (a) Direct first-order phase transition between AF and SC, as a function of the chemical potential; (b) first-order AF-to-SC transition as a function of doping, classified as the type-1 transition; (c) two second-order phase transitions with a uniform AF/SC mix phase in between, classified as a type-2 transition. The AF and SC transition temperatures $T_N$ and $T_c$ merge into either a bicritical $T_{bc}$ or a tetracritical point $T_{tc}$.

Figure 11

Renormalization-group flow in the $(u_1∕u_{12},u_2∕u_{12})$ plane. (In this figure, $u=u_1$, $\mathit{v}=u_2$ and $w=u_{12}$.) The renormalization-group flow is initially attracted towards the symmetric Heisenberg point labeled by $H$. The RG trajectories diverge near the Heisenberg model, with a very small exponent. From 211.

Figure 12

Some possible ground states of the projected $\mathit{SO}\left(5\right)$ model (see also Fig. 4). The × depicts a resonating-valence-bond (RVB) -like spin-singlet state on a plaquette, the arrow denotes the direction of the superspin, and the open circles depict hole pairs. (a) A plaquette RVB state, described by $\theta \left(x\right)=0$ on every plaquette; (b) an in-phase SC stripe with $\alpha \left(x\right)=0,\pi ∕2,0,\pi ∕2$, on each stripe; (c) a superspin spiral with $\alpha \left(x\right)=0,\pi ∕2,\pi ,3\pi ∕2$ on each stripe; (d) a hole-pair checkerboard state with $\alpha \left(x\right)=0$ everywhere, except on the hole-pair plaquettes, where $\theta =\pi ∕2$ and $\alpha =\pi ∕2$.

Figure 13

(Color) A typical global phase diagram of the extended $\mathit{SO}\left(5\right)$ model in the parameter space of chemical potential and the ratio of boson hopping energy over interaction energy (see 55 for details). This phase diagram shows self-similarity among the insulating states at half filling and other rational filling fractions. There are two types of superfluid-insulator transition. The quantum phase transition of class $A$ can be approached by varying the hopping energy, for example, by applying a pressure and magnetic field at constant doping. The quantum phase transition of class $B$ can be realized by changing the chemical potential or doping. This theoretical phase diagram can be compared with the global phase diagram of the high-$T_c$ cuprates. Different families of cuprates correspond to different traces of class $B$. For example, we believe YBCO is $B1$-like, BSCO may be close to $B2$-like, and LSCO is $B3$-like. The vertical dash-dotted line denotes a boundary in the overdoped region beyond which our pure bosonic model becomes less accurate. All the phase boundaries in this figure can be classified into direct first-order transitions (type 1), two second-order transitions (type 2), or a marginal case with enhanced symmetry (type 1.5). Type-2 transitions between CDW lobes and the superconducting state lead to intermediate supersolid phases.

Figure 14

Effect of the chemical potential and of temperature on hole density: (a) ∎, hole concentration $\delta =\rho ∕2=\frac{1}{2}⟨t_h^{†}{t}_h⟩$ and 엯, magnon density ${\Sigma }_{\alpha }\frac{1}{2}⟨t_{\alpha }^{†}{t}_{\alpha }⟩$ as a function of the chemical potential $\mu$ at $T∕J=0.03$. The small inset shows a detailed view of the $\mu$ region in which the hole-pair density jumps to a finite value; (b) hole densities of the coexisting phases on the first-order transition line from (almost) zero to finite hole density at $\mu ={\mu }_c$ as a function of temperature.

Figure 15

Quantum Monte Carlo simulations for the projected $\mathit{SO}\left(5\right)$ model in $D=2$: (a) Phase diagram [see Eq. (50) with $J_c=J_S$] in $D=2$: The squares between $S$ and the tricritical point $P$ trace the first-order line of phase separation. The solid line from $P$ to the right edge of the plot traces the Kosterlitz-Thouless transition between the SC and normal states. The dashed line separating $N_t$ (triplet-dominated region) and $N_h$ (hole-pair-dominated region) describes the line of equal AF and SC correlation lengths. The small inset shows the same phase diagram on a larger $\mu$ scale, covering the whole KT phase. The tricritical point $P$ appears as a result of the Mermin-Wagner theorem, which does not allow spin ordering in $D=2$ at finite temperature. (b) Energy of a single-magnon excitation in the projected $\mathit{SO}\left(5\right)$ model as a function of the chemical potential. This corresponds to the resonance energy of the $(\pi ,\pi )$ peak of the spin correlations in the fermionic model (magnons are defined to carry the momentum of the AF order). The numbers in parentheses indicate the peak weights, i.e., the area under the peak. ($20\times 20$ lattice at temperature $T∕J=0.1$.) From 142.

Figure 16

(Color) The phase diagram of the extended $\mathit{SO}\left(5\right)$ model obtained by quantum Monte Carlo simulation. The parameters used in simulation are ${\Delta }_s=4.8$, $V_c=4.1010$, $V_c^{\prime }=3.6329$, and $J_{\pi }=V_{\pi }=0$. The lines are guides to the eye only. The overall topology of the phase diagram agrees well with the global phase diagram presented in Fig. 13.

Figure 17

Scaling of the superfluid density near the $\mathit{SO}\left(5\right)$ bicritical point obtained by classical Monte Carlo simulations. The critical behaviors of the superfluid density for various $g$ fit into a single scaling curve, from which $\mathit{SO}\left(5\right)$ scaling exponents were obtained. From 130.

Figure 18

Quantum Monte Carlo simulations for the projected $\mathit{SO}\left(5\right)$ model in $D=3$: (a) $T\left(\mu \right)$ phase diagram with $J_s=J_c∕2$ and ${\Delta }_s={\Delta }_c=J$. $N_h$ is the hole-pair-dominated part, $N_t$ the triplet-dominated part of the high-temperature phase without long-range order. The separation line between $N_h$ and $N_t$ is the line of equal spatial correlation decay of hole-pairs and bosons. (b) $T\left(\delta \right)$ phase diagram as a function of hole doping $\delta =n_h∕2$. The first-order transition line from $S$ to $P$ in the $T\left(\mu \right)$ diagram becomes a forbidden region due to phase separation. These two phase diagrams are consistent with those presented in Figs. 10a, 10b based on general arguments.

Figure 19

(Color) Scaling of $T_N$ and $T_c$ near the $\mathit{SO}\left(5\right)$ bicritical point. Both $T_N$ and $T_c$ merge into the bicritical point tangentially, with the crossover exponent of $\varphi =1.43\pm 0.05$.

Figure 20

Quantum lattice model with exact $\mathit{SO}\left(5\right)$ symmetry: (a) Under the condition specified by Eq. (73), the 16 states on a rung are classified into six groups, each transforming irreducibly under the $\mathit{SO}\left(5\right)$ group. $\mid \Omega ⟩$ is an $\mathit{SO}\left(5\right)$ singlet state; $n_a\mid \Omega ⟩$ describes five states that transform as $\mathit{SO}\left(5\right)$ vectors; ${\Psi }_{\alpha }^{†}\mid \Omega ⟩$ are four states that form an $\mathit{SO}\left(5\right)$ spinor; four states ${\Psi }_{\alpha }\mid \Omega ⟩$ also correspond to a spinor; $R_{\alpha \beta }{\Psi }_{\alpha }{\Psi }_{\beta }\mid \Omega ⟩$ and $R_{\alpha \beta }{\Psi }_{\alpha }^{†}{\Psi }_{\beta }^{†}\mid \Omega ⟩$ are two $\mathit{SO}\left(5\right)$ singlet states. The figure also gives energies of all multiplets for the $\mathit{SO}\left(5\right)$-symmetric ladder model described by Eqs. (72, 73). (b) Strong-coupling phase diagram of the $\mathit{SO}\left(5\right)$-symmetric ladder model in the $(U,V)$ space. The $E_0$, $E_1$, and $E_3$ phases are regions in parameter space where the respective states have the lowest energy.

Figure 21

Phase diagram of the bilayer $\mathit{SO}\left(5\right)$ model plotted as $J_{\Vert }∕J$ vs $\mu$. The entire phase-transition line from the Mott phase into any of the ordered phases is a second-order quantum phase transition. The Mott insulating state has five massive collective modes. The $\mathit{SO}\left(5\right)$-symmetric AF/SC uniform mixed state at half filling has four gapless collective modes. The SC state has a spin-triplet $\pi$-resonance mode and one massless charge Goldstone mode.

Figure 22

Evolution of the quasiparticle states when doping is reduced: (a) pure $d$-wave SC gap with nodal quasiparticles; (b) the pure $d$-wave SC gap is rotated into an AF gap of the form $\mid \cos p_x-\cos p_y\mid$; (c) a large uniform component of the AF/Mott insulating gap is developed on top of the $\mid \cos p_x-\cos p_y\mid$ gap when doping is reduced close to zero.

Figure 23

Eigenstates of the spin $3∕2$ problem on a single site. The longer (shorter) arrows denote $S_z=3∕2$ $(1∕2)$ and the up (down) directions denote the + (−) sign. The $E1,4,6$ (singlet), $E2,5$ (quartet), and $E3$ (quintet) sets can also be classified as $\mathit{SO}\left(5\right)$ singlet, spinor, and vector representations.

Figure 24

Doping dependence of the ground-state energy (two upper curves) and staggered magnetization (lower curve) for the $t\text{−}J$ model with $J∕t=0.3$. The state with uniform AF and $d$-wave SC order has lower energy than the pure $d$-wave SC state for $0<\delta <10\%$; furthermore, the energy of the uniform AF/SC mixed state depends linearly on $\delta$, fitting into the $\mathit{SO}\left(5\right)$-symmetric type-1.5 transition classified in Fig. 9. From 121.

Figure 25

Exact diagonalization results for the dynamical correlation function of the $\pi$ operator on a 10-site Hubbard system with $U=8t$. A single $\delta$-function-like peak with pronounced weight is visible near $\omega =0$ for the $\pi$ operator, proving the eigenoperator relation (88) in the low-energy regime. This precession frequency ${\omega }_{\pi }$ decreases with decreasing doping. An alternatively constructed $s$-wave $\pi$ operator, with $g\left(p\right)$ in Eq. (28) given by $g\left(p\right)=\cos p_x+\cos p_y$, shown in the bottom graph exhibits only incoherent behavior and hardly any weight (note the difference in the $y$ scale). Here $⟨n⟩$ denotes average electron density, with $⟨n⟩=1$ being at half filling. From 195.

Figure 26

The magnon and hole-pair states obtained at level $\nu$, which is the total number of magnons and hole pairs, after expanding out the coherent state (55). These states are classified by their $(S_z,Q)$ quantum numbers in (a). The energy is independent of the $S_z$ quantum number because of the $\mathit{SO}\left(3\right)$ spin rotation symmetry. The energy can depend on $Q$ with three generic possibilities, as depicted in (b). (Compare with Fig. 9.) If the energy depends linearly on $Q$, there is no free-energy cost to rotate magnons and hole pairs into each other, and the potential energy is $\mathit{SO}\left(5\right)$ symmetric. This multiplet structure was tested in the $t\text{−}J$ model and is shown in Fig. 27.

Figure 27

The low-energy states within each total spin and charge sector $(S_z,Q)$ of the 18-site cluster $t\text{−}J$ model with $J∕t=0.5$. The states are grouped into different multiplets and are labeled by spin, charge, point-group symmetry, and total momentum. $A_1$ denotes the totally symmetric representation and $B_1$ the $d_{x^2-y^2}$-like representation of the $C_{4\mathit{v}}$ symmetry group. The quantum numbers of these states match those of the magnon and hole-pair states shown in Fig. 26. Furthermore, the energy depends approximately linearly on $Q$, demonstrating the $\mathit{SO}\left(5\right)$ symmetry of the interaction potential among the magnons and hole pairs.

Figure 28

Low-energy spectrum of the Hubbard model on a plaquette. Eigenstates are labeled by total spin $S$ and plaquette momentum $q_x$,$q_y=0,\pi$. Truncated high-energy states are shaded. The vacuum is defined as $\mid \Omega ⟩$, and quantized operators connect the vacuum to the lowest eigenstates as shown. (In this figure, $t^{†}$ denotes the magnon creation operator $t_{\alpha }^{†}$, and $b^{†}$ denotes the hole-pair creation operator $t_h^{†}$.)

Figure 29

Illustration of the basic idea of the contractor renormalization-group (CORE) method. To implement this method, one first decomposes the original lattice in plaquettes, and then truncates the spectrum of a given plaquette to the five lowest states, i.e., singlet, hole-pair, and three magnon states. An effective Hamiltonian for these bosons can then be calculated using the CORE method. Left: local bosons in the original lattice. Gray rectangle denotes the singlet RVB vacua, circles denote holes, and the sets of two parallel vertical arrows denote magnons. Right: local bosons on the lattice of the plaquette. Leaf-like pattern denotes a local $d$-wave hole pair on a plaquette. Canted arrows denote local magnons on the plaquette. The singlet RVB vacuum is denoted by an empty site.

Figure 30

This figure illustrates the construction of a “superblock” and its Hamiltonian $H_s$ out of two neighboring blocks, with intrablock Hamiltonian $H_0$ and interblock coupling $V$ (in the block basis: ${\left(H_0\right)}_{n,n^{\prime }}=⟨{\alpha }_n^{\mathrm{o}}\mid H\mid {\alpha }_n^{\mathrm{o}}⟩={ϵ}_n^{\mathrm{o}}{\delta }_{n,n^{\prime }}$ and ${\left(V_0\right)}_{nm,n^{\prime }{m}^{\prime }}=⟨{\alpha }_n^{\mathrm{o}}{\alpha }_m^{\mathrm{o}}\mid V\mid {\alpha }_{n^{\prime }}^{\mathrm{o}}{\alpha }_{m^{\prime }}^{\mathrm{o}}⟩$).

Figure 31

(Color) Boson hopping energies vs Hubbard $U$. The intersection region near $U=8$ is close to the projected $\mathit{SO}\left(5\right)$ symmetry point. All energies are in units of $t$.

Figure 32

The order-parameter space of the $\mathit{SO}\left(5\right)$ theory. $\pi$ operator performs a rotation between the AF and the $d$-wave SC states. This small fluctuation is the new Goldstone mode of the $\mathit{SO}\left(5\right)$ theory.

Figure 33

The two-particle continuum and the $\pi$ excitation for the tight-binding model. Note that the continuum of two-particle states collapses to a point when the center-of-mass momentum is $\Pi =(\pi ,\pi )$. The $\pi$ mode emerges as an anti bound state above the continuum.

Figure 34

Dyson’s equation for the $\pi$ resonance. The function $d\left(p\right)$ is defined in Eq. (26).

Figure 35

$\pi$-resonance contribution to the spin susceptibility in the SC state.

Figure 36

Spin susceptibility in the SC state for the model (106). The wave vector is along the 0 to $(\pi ,\pi )$ direction. Susceptibility was computed using the self-consistent linear response formalism in Fig. 35. The peak at $(\pi ,\pi )$ comes from the $\pi$ resonance. From 70.

Figure 37

Feynman diagram for the $\pi$ resonance below $T_c$ contrasted with the diagram above $T_c$. The + denotes the anomalous scattering in the SC state which converts a particle into a hole and vice versa.

Figure 38

Suppression of the resonance intensity by the magnetic field. From 64.

Figure 39

Doping dependence of the resonance energy and intensity measured in neutron-scattering experiments. From 93.

Figure 40

(Color) Temperature dependence of the resonance intensity compared to the specific heat. From 66.

Figure 41

SC pairing between electrons mediated by exciting a virtual magnon-$\pi$-mode pair.

Figure 42

Diagrammatic representation of SC pairing mediated by exciting a virtual-magnon $\pi$-mode pair: solid lines, electron propagators; dashed lines, interactions. The upper particle-hole ladder corresponds to the magnon and the lower particle-particle ladder corresponds to the $\pi$ mode.

Figure 43

(Color) SC vortex with AF core. Far from the center of the vortex core, the superspin vector lies in the SC plane and winds around the vortex core by $2\pi$. The superspin vector lifts up to the AF direction as it approaches the center of the vortex core. The arrows represent the direction of the superspin and the color scale represents the magnitude of the AF order parameter.

Figure 44

(Color) Field dependence of the AF moment for different parameters of the Landau-Ginzburg theory (as defined by 58). The parameters are ${\rho }_1={\rho }_2=a^2$, $r_1=-1$, $r_2=-0.85$, $u_1=u_2=1$, and $\chi =42.4$. Here the parameters are chosen such that the maximum SC order is 1 and the SC coherence length at zero field equals the lattice constant $a$ of the lattice model. (a) Field dependence for different values of $u_{12}$. The curvature strongly depends on $u_{12}$. (b) Fit to the neutron-scattering results of 146 of the ${\mathrm{Nd}}_{1.85}{\mathrm{Ce}}_{0.15}\mathrm{Cu}{\mathrm{O}}_4$ crystal with $u_{12}=0.95$. $B_{c2}$ is about $6.2\mathrm{T}$ in this sample. The experimental data are obtained by subtracting the magnetic-field response along the $c$ axis from the magnetic-field response in the $ab$ plane, so that the response from the Nd moment can be removed.

Figure 45

Illustration of the $d$-wave pair-density-wave state at $x=1∕8$. In this state, the $d$-wave hole pairs occupy every four nonoverlapping plaquettes on the original lattice. The charge unit cell is $4a\times 4a$. The $\mathit{SO}\left(5\right)$ model is defined on the center of the nonoverlapping plaquettes. Such a state could be realized around the vortex core, whose center is depicted by the × or realized as a generic competing state whenever SC order is reduced. In the actual realization of this state, the hole pair could be much more extended, and the AF ordering could be much reduced from the classical value.

Figure 46

The combined phase diagram as a function of lattice density $D$ in $\mathrm{Ce}{\mathrm{Cu}}_2{\left({\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x\right)}_2$ $(D<D_c)$ and in $\mathrm{Ce}{\mathrm{Cu}}_2{\mathrm{Si}}_2$ $(D_c⩽D)$ under pressure $P$. Note that $D\propto 1∕V$, where $V$ is the unit-cell volume, and $D=D_{\mathrm{Si}}[1(V_{\mathrm{Ge}}-V_{\mathrm{Si}})x∕V_{\mathrm{Ge}}]$ in the former case. From 155.

Figure 47

Doping dependence of the SC transition temperature and magnetic moment for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$. From 291.

Figure 48

(Color) Possible $\mathit{SO}\left(5\right)$ symmetry in BEDT salts: (a) Phase diagram. From 194. (b) Log-log plot of $T_1^{-1}$ vs $(T-T_c)∕T$ for (A) $\kappa -{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]\mathrm{Cl}$ (∎), and (B) deuterated $\kappa -{\mathrm{BEDT}\text{−}\mathrm{TTF})}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]\mathrm{Br}$ (◻). (Data from 150.)

Figure 49

Setting of the tunneling experiment for detecting the triplet particle-particle $\pi$ mode in the normal state.

Figure 50

Second-order tunneling diagram that gives rise to the resonant coupling of Cooper pairs and $\pi$ excitations in the junction shown in Fig. 49.

Figure 51

The superconductor-antiferromagnet-superconductor (SC/AF/SC) junction described by Eq. (117).

Figure 52

Predicted current-phase characteristics of a SC/AF/SC junction with different $d∕d_{c0}$.