Quantum phase transition driven by competing intralayer and interlayer hopping in bilayer nickelates

TL;DR

This work addresses how superconductivity arises in bilayer nickelates under non-thermal tuning by examining a minimal bilayer Hubbard model focused on the Ni-$d_{3z^2-r^2}$ orbital. Using DMRG on a bilayer lattice with doping $\delta=1/8$ and $U=8t$, the authors identify a quantum phase transition controlled by the ratio $t_1/t$ near $0.5$, separating a quasi-long-range SDW phase from a Luther-Emery phase where superconductivity is enhanced and CDW correlations emerge. The transition features the opening of a spin gap, a shift in central charge from $c\approx 2$ to $c\approx 1$, and a move from SDW-dominated order to uniform extended-$s$-wave superconductivity with interlayer singlet pairing; PDW tendencies are noted in the SDW regime. The results provide a microscopic link between lattice anisotropy and superconductivity, suggesting that pressure or strain tuning and possible hybridization with the $d_{x^2-y^2}$ orbital could further elevate the superconducting transition temperature in bilayer nickelates.

Abstract

Bilayer nickelates exhibit high-temperature superconductivity under proper hydrostatic pressure or epitaxial strain, signifying the emergence of quantum phase transitions whose physical mechanisms remain unclear. Using a minimal bilayer Hubbard model incorporating only the Ni-$d_{3z^2-r^2}$ orbitals, we demonstrate that a phase transition naturally arises from tuning the ratio of intralayer to interlayer hopping amplitudes. The transition point separates regimes with a rich interplay between superconducting and density-wave orders. In the regime of weaker intralayer hopping, the ground state is characterized by quasi-long-range spin-density-wave order. As the intralayer hopping increases, the system undergoes a transition marked by the opening of a finite spin gap and the disappearance of spin-density-wave order. Meanwhile, superconductivity is dramatically enhanced, accompanied by the emergence of quasi-long-range charge-density-wave order, indicating that the system enters Luther-Emery phase. This quantum phase transition, driven by the competition between intralayer and interlayer hopping, provides a plausible microscopic explanation for the experimentally observed correlation between the superconducting transition temperature and ratio of out-of-plane to in-plane lattice constants. Our findings reveal a possible link between the suppression of spin-density-wave order and the prominence of superconducting order, which may assist future efforts to optimize experimental conditions for further enhancing superconductivity in bilayer nickelates.

Figures (12)

• Figure 1: (a) Schematic of the bilayer Hubbard model. Circles of different colors indicate electron sites in different layers. The strong interlayer hopping $t$ is indicated by thick lines while the weak intralayer hopping $t_1$ is shown in thin lines. (b) Three types of dimers relevant to ground states in the weak $t_1$ regime. (c) Two distinct phases separated by a quantum phase transition at $t_1\approx 0.5t$, calculated for onsite Hubbard interaction $U = 8t$ and doping $\delta = 1/8$. The dashed line indicates the dimer limit ($t_1=0$), where the multiply degenerate ground states only involve the three types of dimers in (b). SDW is the shorthand for spin density wave, and LE denotes Luther-Emery liquid, which features competing superconductivity and charge density wave.
• Figure 2: (a) Charge-density profiles along the $x$ direction for $t_1=0.2t$, $0.4t$, and $0.6t$. (b) Oscillation amplitude $A$ as a function of $t_1/t$. (c) Spatial profiles of spin-spin correlation $\langle D_x\rangle$ within each dimer at $t_1=0.2t$, $0.4t$, and $0.6t$. (d) Evolution of the averaged $\langle D_x \rangle$ with $t_1/t$. The blue dashed line marks the average value in the dimer limit. These plots are obtained on a bilayer $96\times 2$ lattice.
• Figure 3: (a) and (b) Charge (c), spin (s), and pairing (sc) correlation functions plotted in the log-log scales. Dashed lines represent power-law or exponential fits to the envelope function $f(x)$. $K$ denotes the decay exponent of the power-law fitting, and $\xi$ is the decay length in the exponential case. (c) Dependence of decay exponents on $t_1/t$. The dashed line indicates $K_c^A$, extracted from the charge-density profile. (d) Oscillation coefficient $B$ in $P(x)$ as a function of $t_1/t$. $|B|>1$ indicates that the correlation functions change signs periodically. These results are obtained from simulations on a $96\times 2$ bilayer lattice.
• Figure 4: (a) Charge structure factor. A peak appears in the LE phase ($t_1=0.6t$) at $(\pi/2,0,0)$. (b) Spin structure factor. The peak shows only in the SDW phase ($t_1=0.3t$) at $(\pi/2,0,0)$. (c) Pair structure factor. There is a singular point in the SDW phase at $(k_x,k_y) = (\pi/2,0)$. In the LE phase, the singularity only occurs at $(0,0)$. (d) Momentum distribution function. The effective Fermi point $k_F$, where $\tilde{n}_{\sigma}$ becomes singular, lies at $k_x = 3\pi/4$.
• Figure 5: Scalings of the charge gap $\Delta_c$ and spin gap $\Delta_s$ with $1/N_x$. The insets show the variations of extrapolated gaps with $t_1/t$ in the thermodynamic limit. After the transition, the spin gap becomes finite while the charge gap remains zero. The truncation error is fixed at $\epsilon = 1.0\times 10^{-7}$.