Yamaji effect in models of underdoped cuprates

TL;DR

The paper addresses the topology of the Fermi surface in the pseudogap phase of underdoped cuprates and whether small Fermi pockets arise from spin-density-wave (SDW) reconstruction or a fractionalized Fermi liquid (FL*) state. It computes the c-axis magnetoresistance ρ_{zz}(θ,φ) within a semiclassical Boltzmann framework for both SDW and FL* models, using the Ancilla Layer Model (ALM) for FL*. The key finding is that the FL* scenario accurately reproduces the Yamaji peak positions observed experimentally, whereas SDW predictions are highly sensitive to interlayer ordering and can generate unobserved features (e.g., an extra peak near θ≈70° at φ=45°). The work concludingly argues that Yamaji-angle measurements provide a powerful discriminator between competing pseudogap theories and supports the FL* interpretation with pocket area $A_{ ext{FS}}=p/8$. It also highlights how the observed Fermi arcs can be reconciled with a small, consistent pocket topology across doping, bolstering the FL* picture and guiding future high-field transport experiments.

Abstract

Recent angle-dependent magnetoresistance measurements in underdoped cuprates have revealed compelling evidence for small hole pockets in the pseudogap regime, including observation of the Yamaji effect in HgBa$_2$CuO$_{4+δ}$ (Chan et al., Nature Physics 10.1038/s41567-025-03032-2 (2025)). A key distinction between theories is their predicted Fermi volumes, measured as fractions of the square lattice Brillouin zone: $p/4$ per pocket for spin density wave (SDW) versus $p/8$ for fractionalized Fermi liquid (FL*), where $p$ is the hole doping. We calculate the $c$-axis magnetoresistance $ρ_{zz}(θ, φ)$ within the semiclassical Boltzmann formalism for both states, and using the ancilla layer model (ALM) for FL* in a single-band Hamiltonian. The results from the $\text{FL}^*$ phase show good consistency with current experimental data. Conversely, the results for the SDW phase are highly sensitive to the ordering momentum along the $z$-direction. An ordering vector of $Q = (π, π, π)$ yields predictions that starkly disagree with the experiment. The only possibility for agreement within the SDW scenario is to assume an ordering momentum of $Q = (π, π, 0)$. However, even in this specific case, the SDW scenario predicts a marginally smaller Yamaji angle at $φ=0$ than the FL* theory, and a second Yamaji peak near in-plane angle $φ= 45^\circ$, which was not observed in the experiment. In reality, the Néel ordering vector is likely uncorrelated between adjacent layers, so that there is no coherent interlayer transport of hole-pocket quasiparticles in the SDW scenario, and consequently no Yamaji effect. Our results support the FL* interpretation of Fermi arcs in the pseudogap phase, and establish Yamaji angle measurements as a discriminatory tool between theoretical models.

Figures (12)

• Figure 1: (a) and (b) shows the Fermi pocket by (a) SDW and (b) FL* by ALM at doping $p=0.1$. In (b), the red color represents the contribution of the physical electron to the spectral weight on the Fermi surface, and purple color represents the contribution from the ancilla fermion $\psi_1$. The dark dashed line in (a) indicate the folded Brillion zone corresponding by SDW $(\pi,\pi)$. (c) An illustration of the ancilla layer model. In the one-band model considered in this work, $c$ and $\psi_1$ are the active electrically charged degrees of freedom, while the neutral $\psi_2$ fermions decouple and do not contribute to charge transport. (d) Schematic of the polar $\theta$ and azimuthal $\phi$ angles of the magnetic field $\bf B$. And an illustration of the caliper momentum $k_{\mathrm{cal}}$ along the in-plange magnetic field direction $\bf B_{\parallel}$.
• Figure 2: Calculated interlayer resistivity $\rho_{zz}(\theta, \phi)/\rho_{zz}(0, 0)$ as a function of the polar angle $\theta$ for azimuthal angles $\phi = 0^\circ$, $22.5^\circ$, and $45^\circ$. Panels (a) and (b) show the results obtained within FL* by ALM for $\omega_c\tau = 1.3$ and $\omega_c\tau = 3.0$, respectively, with the mixing term $a=0$. The Yamaji angle at $\phi=0^\circ$ is extracted as $\theta\approx 59.5^\circ$. Panels (c) and (d) present the corresponding results calculated within the SDW framework for $\omega_c\tau = 1.3$ and $\omega_c\tau = 3.0$. The Yamaji angle at $\phi=0^\circ$ is extracted as $\theta\approx 57^\circ$.
• Figure 3: Comparison of the resistivity curve $\rho_{zz}$ contributed by different pockets. $\omega_c\tau=3.0$ and $a=0$ is used for all the calculations. (a) and (b) show the resistivity contributed by the two pockets of the FL* theory. (c) and (d) show the resistivity contributed by the two pockets of the SDW theory. The corresponding pockets are shown in the insets of each figure. The arrows indicate the directions of th ein plane magnetic $\mathbf B_\parallel$, with the arrow lengths proportional to the corresponding caliper momentum $k_{\mathrm{cal}}$.
• Figure 4: Yamaji angle of ALM for different mixing angles $a$, with $a$ defined in Eq. \ref{['eqn:c0cp']} (a) and (b): the effective $t_{z,\mathrm{eff}}(\mathbf k)$ as a function of momentum $\mathbf k$ for $a=-0.1\pi$ and $a=0.1\pi$. The dashed white line indicate the Fermi pocket and the red start marks the momentum $k^*$ of the corner of Fermi pocket. (c) The Yamaji angle for $\phi=0^\circ$ as a function of $a$. Data points were obtained from the peak positions of the interlayer resistivity $\rho_{zz}(\theta,0)$ calculated at $\omega_c\tau=4.0$. (d) The effective $t_{z,\mathrm{eff}}(\mathbf k^*)$ at the corner of the Fermi pocket $\mathbf k^*$
• Figure 5: Yamaji angle of FL* for different hybridization strength $\Phi$, with $\Phi$ defined in Eq. \ref{['eqn:H_anc']}. (a) The Yamaji angle $\theta_{\mathrm{Yamaji}}$ as a function of $\Phi$ for $\phi=0^\circ$ and $a=0$, extracted from the date of $\omega_c\tau=4.0$. The red star indicate the parameter we used in the main text. (b) Calculated interlayer resistivity $\rho_{zz}(\theta, \phi)/\rho_{zz}(0, 0)$ as a function of the polar angle $\theta$ for azimuthal angles $\phi = 0^\circ$, $22.5^\circ$, and $45^\circ$, with $\omega_c\tau = 3.0$.