---
address:
- 'Massachusetts Institute of Technology, Cambridge MA02139'
- ' ( 7 October 1997 ) '
-  
author:
- 'Jan Brinckmann and Patrick A. Lee'
title: 'Spin susceptibility and the $\pi$-excitation in underdoped cuprates.'
---

\#1[ to 10pt [ ]{} to 7pt [ ]{} ]{} The spin-triplet particle–particle excitation (‘$\pi$-excitation’) $$\displaystyle \widehat{\pi}^\dagger =
   \sum_k (\cos(k_x) - \cos(k_y) )
   c^\dagger_{-k\uparrow} c^\dagger_{k + q\,\uparrow}$$ at wave vector $q=(\pi,\pi)$ has been introduced in [@dem95] as a possible explanation for the ‘41meV resonance’ observed in neutron scattering on cuprates in the superconducting state (see e.g. [@fon95; @bou96]). It has been argued that $\widehat{\pi}^\dagger$ is an approximate collective eigenmode of the t–J or Hubbard model. The coupling of spin-triplet particle–particle excited states and spin-singlet particle–hole states in the superconducting phase then should lead to a resonance in the susceptibility $\chi_\pi''(\omega)$ at $q=(\pi,\pi)$ at the energy $\omega_0$ of this $\pi$-mode.

We compare the susceptibility and the propagator of the $\pi$-excitation in a slave-boson theory for a wide range of hole concentrations (doping). Both calculated spectra $\chi_\pi''$ and $\pi''$ show a pronounced resonance at the same energy $\omega_0$, which is roughly given by the chemical potential $\mu$ as $\omega_0 \approx
2|\mu|$. The outcome is in qualitative agreement with the aforementioned prediction and with recent numerical [@mei97pre; @eder97pre] and diagrammatic calculations [@zha97private]. However, our interpretation differs from that originally envisioned in [@dem95]: The diagrammatic spin-triplet particle–particle channel does not contribute as a collective mode to $\chi_\pi''$ or $\pi''$. In underdoped systems not far from the transition to the N[é]{}el state, the resonance is solely caused by the ‘RPA channel’, which describes spin fluctuations mediated through spin-singlet particle–hole excitations of fermions.

The ‘$\pi$-propagator’ is given as $$\label{equ-piprop}
 \displaystyle \pi(\omega)= 
   \langle {\cal T}_\tau \, 
      \widehat{\pi}(\tau) 
      \widehat{\pi}^\dagger(\tau')
   \rangle^\omega \;\,.$$ We start from the t–J-model and consider a Gutzwiller-projected $\pi$-propagator, Eq. (\[equ-piprop\]) with $\widehat{\pi} \to P_G \widehat{\pi} P_G$. The calculations for $\pi(\omega)$ as well as the susceptibility $\chi_\pi(\omega)$ are performed within the standard slave-boson scheme. Diagrammatic expressions are based on a self-consistent perturbation theory with self energies taken at Hartree-Fock (mean-field) level. We consider the superconducting state at very low temperature, i.e., the d-wave pairing phase of fermions and fully condensed bosons. The t-matrix approximation for $\chi_\pi(\omega)$ and $\pi(\omega)$ are indicated in Figs. \[fig-chi\] and \[fig-pipi\] respectively.

The susceptibility is given by the vertex-renormalized mean-field bubbles displayed in Fig.\[fig-chi\]a, which are to be inserted into $$\label{equ-rpa}
 \chi_\pi(\omega) = 
  \widehat{\chi}_\pi(\omega) /
  [ 1 - 2J \widehat{\chi}_\pi(\omega) ] \;\,.$$ Eq.(\[equ-rpa\]) represents the particle–hole RPA channel (random phase approximation).

The single and double arrowed lines in Fig. \[fig-chi\] stand for the normal and pairing Green’s functions of auxiliary fermions. The dashed line in Fig.\[fig-chi\]b is the t–J-model’s spin and density interaction for fermions on two nearest neighbor lattice sites $i,j$, $$\label{equ-inter}
  J \sum_{<i,j>}[ {\boldmath S}_i {\boldmath S}_j - \frac{1}{4}n_i n_j ]$$ with $n_i = \sum_\sigma f^\dagger_\sigma f_\sigma$.

The vertex corrections entering $\widehat{\chi}_\pi$ consist of the spin-singlet particle-hole ($ph$) ladder diagrams shown in Fig.\[fig-chi\]b. The double arrowed (anomalous) Green’s function introduces the $ph$ channel in both time directions. In general it also allows for a coupling of the spin-triplet particle-particle ($pp$) channel into the singlet $ph$ correlation function $\chi_\pi$ by transforming e.g. a spin-up fermion into a spin-down hole and vice versa. However, the $pp$ channel would appear in $\chi_\pi$ as a vertex-function, involving at least one interaction vertex Eq. (\[equ-inter\]) with equal spin on both sites, which is zero [@basand97pre]. This reflects the fact that Pauli’s principle blocks any exchange process $\sim J=4 t^2 / U$ for particles with equal spin in the Hubbard model. Thus the $pp$ channel contributes no spectral weight to $\chi_\pi$. This also holds if the $pp$ channel is ‘artificially’ switched on by replacing $n_i n_j$ in Eq.(\[equ-inter\]) with $g\,n_i n_j$ and turning $g=1 \to g=0$: Recent numerical cluster calculations for the t–J-model [@eder97pre] show that $\chi''$ is not affected by varying the coefficient of the density–density interaction. In the following we stick to the case $g=1$.

Numerical calculations in the t-matrix approximation are performed with mean-field parameters set to reflect a fermion bandwidth of $4 J$ and a superconducting gap $\Delta_0 = 40\mbox{\,meV} \approx 0.3
J$. As has already been observed in earlier RPA calculations [@fuk95], an instability to the N[é]{}el state occurs at an unphysically high hole concentration (doping) $x_c$. Since the vertex corrections of the t-matrix approximation turn out to have no significant effect, we assume a further renormalization of $J \to \alpha J$ in Eq.(\[equ-rpa\]). We have chosen $\alpha = 0.5$ such that $x_c$ is reduced to $\approx 0.02$.

Results for $\chi_\pi''(\omega)$ are shown in Fig. \[fig-tmat\](top) as continuous curves for several hole densities in the underdoped regime. The dominant feature is apparently a sharp and strongly doping dependent resonance. Its position $\omega_0$ shifts from $\approx 0$ at the magnetic instability ($x= x_c= 0.02$) to higher energies with increased doping, crossing the anticipated value $40\mbox{\,meV} \approx
0.3J$ around $x= x_m = 0.12$. $\omega_0$ is for $x>x_c$ roughly given by the chemical potential $\mu$ as $\approx 2|\mu|$. This shift of the resonance with hole concentration is found in neutron scattering experiments on optimal [@fon95; @bou96] and underdoped YBCO-compounds [@dai96; @fon97]. The ‘optimal’ doping $x_m\approx 0.12$ found here also compares to experimental values. However, the spectral weight $\int d^2q\,\chi''(q,\omega) /
\int d^2q$ comes out too small with respect to the experiment [@bou97pre].

The resonance is caused by the spin-fluctuation RPA channel Eq.(\[equ-rpa\]): For comparison, dashed curves in Fig. \[fig-tmat\] (top) show the results for $\widehat{\chi}_\pi''(\omega)$, i.e., the renormalized bubble diagrams in Fig. \[fig-chi\]a. The position of the doping dependent peak here is bound from below by $\approx 2\Delta_0 = 0.6J$, even for lowest doping. The contribution from the $ph$ vertex corrections is quite small, $\widehat{\chi}_\pi''$ differs only slightly from the well known mean-field susceptibility.

The results for the susceptibility may be compared to the $\pi$-propagator Eq. (\[equ-piprop\]). In slave-particle formulation, with bosons completely condensed at very low temperature, it reads $\displaystyle \pi(\omega) = x^2\, \pi_f(\omega)$. The prefactor $x^2$ is the (mean-field) probability of finding two empty lattice sites when adding a spin-triplet pair of particles. The $\pi$-propagator for fermions $\pi_f(\omega)$ appearing here is formally identical to Eq. (\[equ-piprop\]), its t-matrix approximation is displayed in Fig. \[fig-pipi\]. According to the discussion given above, the triplet $pp$ channel may contribute only as the mean-field bubble (1st diagram). The singlet $ph$ channel appears in vertex renormalizations in the 2nd and 3rd diagram. The 3rd diagram contains the contribution from the RPA channel via $\widetilde{ J}(q,\omega) / 2 = 
   \frac{1}{2} J(q) + \frac{1}{4}J(q) \chi(q,\omega) J(q)$, indicated as a double dashed line in Fig. \[fig-pipi\].

The resulting spectrum $\pi_f''(\omega)$ in the underdoped regime is shown in Fig. \[fig-tmat\] (bottom) for the same set of parameters and hole densities $x$ as the susceptibility. Continuous lines correspond to the t-matrix approximation, dashed lines are calculated with the 3rd diagram in Fig. \[fig-pipi\] (the coupling to the RPA channel) ignored. Again, the effect of the vertex corrections is negligible, the dashed curves differ only slightly from the mean-field theory (given by the 1st diagram in Fig. \[fig-pipi\]). Apparently $\pi''$ shows a pronounced peak which occurs at exactly the same position as the resonance in $\chi_\pi''$, if the same approximation is used for both quantities. As has been pointed out, the spin-fluctuation (RPA) channel has to be taken into account in the underdoped regime $|\mu| <
\Delta_0$, where the system is not far from the instability to the N[é]{}el state, and the RPA dominates $\chi_\pi''$. In this case the peak in $\pi''$ is entirely caused by the coupling to spin fluctuations through $\widetilde{ J}$. Note that its spectral weight decreases with reduced $x$, and vanishes at the transition to the N[é]{}el state ($x=x_c\approx 0.02$).

The picture changes in a highly overdoped situation $|\mu|\gg\Delta_0$: Fig.\[fig-hdope\] shows curves for a large chemical potential $\mu = -J$ (the breakdown of superconductivity in favor of the fermi-liquid state $\Delta_0=0$ is ignored for the moment). The peaks in $\chi_\pi''$ and $\pi''$ still occur at the same position $\approx 2|\mu|$, but the RPA induces only a slight shift, besides an enhancement of $\chi_\pi''$. In the normal state $\Delta_0=0$ the resonance in $\chi_\pi''$ vanishes, whereas the peak in $\pi''$ remains as a delta function , $ \pi''(\omega) \sim x^2 \delta(\omega + 2\mu)$, as has also been observed in numerical calculations on highly doped clusters (referenced in [@zharesponse]). In contrary to the low doping region, the highly overdoped regime $|\mu|\gg\Delta_0$ is well described by mean-field theory.

[**Acknowledgments:**]{} Discussions with W. Hanke and S.C. Zhang are gratefully acknowledged. One of the authors (JB) acknowledges a fellowship from the Deutsche Forschungsgemeinschaft, Germany.

[10]{}

E. Demler and S. C. Zhang, Phys. Rev. Lett.  [**75**]{}, 4126 (1995).

H. F. Fong [*et al.*]{}, Phys. Rev. Lett.  [**75**]{}, 316 (1995).

P. Bourges, L. P. Regnault, L. Sidis, and C. Vettier, Phys. Rev. B [**53**]{}, 876 (1996).

S. Meixner, W. Hanke, E. Demler, and S. C. Zhang, (1997), preprint (e-print [cond-mat/9701217]{}).

R. Eder, W. Hanke, and S. C. Zhang, (1997), preprint (e-print [ cond-mat/9707233]{}).

S. C. Zhang, (1997), in preparation.

G. Baskaran and P. W. Anderson, (1997), preprint (e-print [ cond-mat/9706076]{}).

H. Fukuyama, H. Kohno, and T. Tanamoto, J. Low Temp. Phys.  [**95**]{}, 309 (1995).

P. Dai [*et al.*]{}, Phys. Rev. Lett.  [**77**]{}, 5425 (1996).

H. F. Fong, B. Keimer, D. L. Milius, and I. A. Aksay, Phys. Rev. Lett.  [ **78**]{}, 713 (1997).

P. Bourges [*et al.*]{}, (1997), preprint (e-print [cond-mat/9704073]{}).

E. Demler, S. C. Zhang, S. Meixner, and W. Hanke, (1997), preprint (e-print [cond-mat/9705191]{}).
