Intrinsic phase fluctuation and superfluid density in doped Mott insulators

TL;DR

This work develops a mutual Chern-Simons gauge framework to describe superconductivity in doped Mott insulators, coupling spin and charge via phase-string–induced gauge fields. The superconducting transition temperature $T_c$ is determined by the low-energy spin resonance energy $E_g$ through $T_c\simeq E_g/(6.44\,k_B)$, while the zero-temperature superfluid density $\rho_{\text{s}}$ is renormalized by spin fluctuations as $\rho_{\text{s}}=\rho_{\text{s}}^0\frac{\lambda E_g}{\rho_{\text{s}}^0+\lambda E_g}$, with $\rho_{\text{s}}^0=4\delta J$. This framework yields a generalized Uemura relationship in the underdoped regime and explains the suppression of both $T_c$ and $\rho_{\text{s}}$ in the overdoped regime by the collapse of antiferromagnetic correlations, linking spin dynamics to SC phase coherence via the emergent resonance energy $E_g$.

Abstract

The doping dependence of the superfluid density $ρ_{\text{s}}$ exhibits distinct behaviors in the underdoping and overdoping regimes of the cuprate, while the superconducting (SC) transition temperature $T_c$ generally scales with $ρ_{\text{s}}$. In this paper, we present a unified understanding of the superconducting transition temperature $T_c$ and $ρ_{\text{s}}$ across the entire doping range by incorporating the underlying mutual Chern-Simons gauge structure that couples the spin and charge degrees of freedom in the doped Mott insulator. Within this framework, the SC phase fluctuations are deeply intertwined with the spin dynamics, such that thermally excited neutral spins determine $T_c$, while quantum spin excitations effectively reduce the superfluid density at zero temperature. As a result, a Uemura-like scaling of $T_c$ vs. $ρ_{\text{s}}$ in the underdoped regime naturally emerges, while the suppression of both $T_c$ and $ρ_{\text{s}}$ at overdoping is attributed to a drastic reduction of antiferromagnetic spin correlations.

Figures (5)

• Figure 1: The phase diagram for a doped Mott insulator based on the mutual Chern-Simons (MCS) gauge description. The low-temperature white-colored dome indicates the superconducting phase, of which $T_c$ is drastically reduced from a bare $T_v^0$ by the MCS gauge fluctuations. Here the phase coherence is protected by spin local moments forming the bosonic RVB pairing with a finite spin-spin correlation length $\xi$. Self-consistently the antiferromagnetic long-range order (bounded by the Neel temperature $T_N$ with a divergent $\xi$ in the gray area) is quickly destroyed by doped holes and evolves into a short-range ordered phase bounded by $T_0$ (the blue area), which in turn connects to a purple-colored overdoped regime with the Curie-Weiss-like paramagnetic phase of vanishing $\xi$ at doping $\delta^*\simeq 0.26$Weng.Ma.2014.
• Figure 2: (a) Intrinsic phase fluctuations arise from the long-range spin-charge entanglement in the MCS gauge theory: A holon carries a $\pi$-flux $\Phi^h \equiv \pm \pi n^h$ as perceived by a spinon of spin index $\sigma =\pm 1$, and, vice versa, a spinon carries a flux $\Phi^s \equiv \pi \sum_\sigma\sigma n^b_\sigma$ seen by holons. (b) A finite density of holons reshapes the $S=1$ spin excitation spectrum into a resonance-like mode at energy $E_g$ near antiferromagnetic wavevector $(\pi, \pi)$ at the mean-field level (with $\delta=0.125$). (c) The superconducting critical temperature $T_c$ according to Eq. \ref{['Tc']}, with the experimental data replotted from Ref. mei_spin-roton_2010.
• Figure 3: (a) Bare superfluid density $\rho_{\text{s}}^0$, renormalized superfluid density $\rho_s$, and superconducting transition temperature $T_c$ versus doping concentration $\delta$, where $\delta^*\approx 0.26$ denotes the superconducting ending point. The underlying parameters are the same as used at the mean-field level in Ref. Weng.Ma.2014. (b) $T_c$ versus $\rho_{\text{s}}$, where the blue curve and gray curve represent the underdoped (UD, $\delta<0.14$) and overdoped (OD, $\delta>0.14$) regimes, respectively. The green dashed line indicates the fitting of $T_c=3.17\rho_s$, with $\lambda=1/(2\pi^2)$.
• Figure 4: Experimentally observed superfluid density (open circle) is suppressed in the overdoped regime, which fits well with the rescaled quantity $\rho_{\text{s}}/\rho_\text{s}^\text{max}$ plotted against the normalized doping level $\delta/\delta^*$, where $\rho_\text{s}^\text{max}$ denotes the maximal $\rho_{\text{s}}$ where $\rho_\text{s}^\text{max}$ denotes the maximal $\rho_{\text{s}}$ shown in Fig. \ref{['fig:TcNsUemura']} (a). The experimental data are taken from Božović et al. Bozovic2016.
• Figure 5: The low momentum, low frequency scaling of $\mathop{\mathrm{Re}}\chi^{zz}$. Numerically, we show that the scaling behavior is $q^2$.