Realizing the Emery Model in Optical Lattices for Quantum Simulation of Cuprates and Nickelates

Abstract

The microscopic origin of high-temperature superconductivity in cuprates remains one of the central open questions in condensed matter physics. Growing experimental and theoretical evidence suggests that the bare single-band Fermi-Hubbard model may not fully capture properties of cuprates such as superconductivity, motivating us to revisit the canonical three-band model of the copper-oxide planes - the Emery model - from which the single-band counterpart was originally derived. Here, we propose and analyze a quantum simulation scheme for realizing the Emery model in regimes relevant to cuprates and infinite-layer nickelates with today's ultracold atom quantum simulation platforms, enabling the exploration of the three-band physics on system sizes that are challenging for current numerical methods. Specifically, we show that a two-dimensional optical lattice with a superimposed pattern of repulsive potentials can be designed to study low-temperature properties for variable parameter regimes of the Emery model relevant to cuprates as well as infinite-layer nickelates. Our results pave the way for real material simulations with ultracold atom quantum simulators and a better understanding of the physics of unconventional superconductors.

Figures (9)

• Figure 1: The three-band Emery model of cuprates in cold atom quantum simulators: Band structure of cuprates (a) and nickelates (b) with copper $d$ and oxygen $p$ bands. Performing a particle-hole transformation effectively corresponds to reversing the sign of the energy $E_\mathrm{p}\to E_\mathrm{h}$. c) The $d$ and $p_x, p_y$ orbitals also appear in a lattice description of the copper-oxide layers in cuprates, with hopping strengths $t_{pd}$, $t_{pp}^{(\prime)}$. In a description for holes, $p_x, p_y$ orbitals are offset by $\Delta_{pd}>0$. d) Experimental setup: A single laser beam traverses the atomic cloud (gray) sequentially along the $x$ and $y$ axes and is retro-reflected to form a 2D lattice. By inserting a half waveplate (HWP) between the first and second passes, the polarization of the lattice along the $y$-axis can be dynamically tuned from the initial input polarization ($V$, green) to a fully orthogonal state ($H$, orange). As illustrated in the top-right inset, the resulting passively phase-stable lattice features a programmable offset between nearest-neighbor sites. By using a Digital Micromirror Device (DMD) to project repulsive local potentials (violet arrays) through the objective -- blocking out every fourth site -- a dynamically controllable potential for the Emery model is generated (e).
• Figure 2: Accessible Emery-model parameters in the proposed optical lattice setup: $\Delta/t_{\rm pd}$ (a), $U_p/U_d$ (b), $t_{pp}/t_{pd}$ (c) and $t_{pp}^\prime/t_{pd}$ (d). The gray masks in the bottom-right corners denote the region that fourth band is too small to support an accurate three-band tight-binding description. The sharp feature appearing in b around $\theta\sim0.05rad$ originate from a change in the band structure where the $d$ and $p$ bands begin to decouple and coincides with the trends of the other parameters note_band. In b, c and d, the two white curves indicate $\Delta_{pd}/t_{pd}=3.5$ relevant to cuprates (solid) and $\Delta_{pd}/t_{pd}=9$ relevant to nickelates (dashed). Panels e and f show band dispersions for $V_0/E_r=15$ at $\theta=0.025rad$ and $\theta=0.049rad$ (marked by the white circle and square in panels c and d), respectively. For panel c (cuprates), $\Delta_{pd}/t_{pd}=3.5$ with $t_{pd}=0.31\,E_r$, $t_{pp}/t_{pd} = 0.03$ and $t_{pp}^\prime/t_{pd} = -0.2$; for panel d (nickelates), $\Delta_{pd}/t_{pd}=9$ with $t_{pd}=0.24\,E_r$, $t_{pp}/t_{pd} = 0.06$ and $t_{pp}^\prime/t_{pd} = -0.3$. Dashed lines show the exact dispersions from the experimental lattice potential, while the colored lines show the reconstructed dispersions from the tight-binding Emery model, with color indicating the $d$-orbital character.
• Figure 3: Numerical results for a cylinder with $12\times 2$ unit cells ($72$ sites) for two sets of parameters relevant to the cuprates ($\Delta/t_{pd}=3.5$) and infinite-layer nickelates ($\Delta/t_{pd}=9.0$) that are accessible for quantum simulation platforms and changing the doping $\delta=N_\mathrm{h}/(L_xL_y)$. a) Average nearest-neighbor spin correlations $C^S_{d\nu}$ divided by average densities $\tilde{n}_\nu$, Eq. \ref{['eq:SS']}, with $\nu=d (p)$ in blue (orange). b) Number of occupied $d$-$p$-$d$ bonds (see sketch). For Zhang-Rice singlets, we expect $N_\mathrm{bond}=N_\mathrm{h}/2$ (gray line), see SM SM. The inset shows the spin correlations $C^\mathrm{bond}_{d\nu}$ (blue and orange) and the total spin $M_\mathrm{bond}$ (green), both post-selected for occupied $d$-$p$-$d$ bonds. For all calculations, we set $U_d/t_{pd}=8.0$, $U_p/t_{pd}=3.0$, $t_{pp}/t_{pd}=0.1$ and $t_{pp}^\prime/t_{pd}=-0.2$. All observables are evaluated from $2000$ snapshots drawn from the MPS, errorbars denoting the error of the mean are smaller than the markers.
• Figure S1: a. Copper-oxide layer orbitals with the explicit phases from the copper $d_{x^2-y^2}$ and oxygen $p_{x,y}$ orbitals. b. Copper-oxide layer orbitals after the gauge transformation Eq. \ref{['eq:trafo']}. c. The transformation Eq. \ref{['eq:trafo']} transforms the signs of the $\hat{p}_\mathbf{j}$ operators at every second unit cell, resulting in a redefinition of the quasi-momenta from $\textbf{k}\to \textbf{k}+(\pi,\pi)$.
• Figure S2: Numerical results for a cylinder with $12\times 2$ unit cells ($72$ sites) for two sets of parameters relevant to the cuprates ($t_{pp}/t_{pd}=0.0(0.5)$, $t_{pp}^\prime/t_{pd}=0.0$) and the experimentally accessible set ($t_{pp}/t_{pd}=0.1$, $t_{pp}^\prime/t_{pd}=-0.2$), changing the doping $\delta=N_\mathrm{h}/(L_xL_y)$. a. Average nearest-neighbor spin correlations $C^S_{d\nu}$, Eq. \ref{['eq:SS']}, with $\nu=d (p)$ in blue (orange) and divided by the average $\nu$-site density $\tilde{n}_\nu$ (without doublon-hole resolution). b.: Number of occupied $d$-$p$-$d$ bonds (see sketch). For Zhang-Rice singlets, we expect $N_\mathrm{bond}=N_\mathrm{h}/2$ (gray line), see SM SM. The inset shows the spin correlations $C^\mathrm{bond}_{d\nu}$ (blue and orange) and the total spin $M_\mathrm{bond}$ (green), both post-selected for occupied $d$-$p$-$d$ bonds. For all calculations, we set $U_d/t_{pd}=8.0$, $U_p/t_{pd}=3.0$ and $\Delta_{pd}/t_{pd}=3.5$. All observables are evaluated from $2000$ snapshots drawn from the MPS, errorbars denoting the error of the mean are smaller than the markers.