Density matrix renormalization group study of quantum-geometry-facilitated pair density wave order

TL;DR

The study addresses the emergence of intrinsic pair density wave (PDW) order in a strongly interacting two-band lattice model by leveraging quantum-geometric effects. Using large-scale DMRG on a bilayer square-lattice with interorbital mixing, the authors map the phase diagram as a function of doping $\delta$ and interorbital pair hopping $U_{sd}$, revealing a robust $(\pi,0)$ PDW with $K_{sc} \sim 0.3$ and no competing spin or charge order, while a uniform SC phase appears at small $U_{sd}$ with a $\hat{\Delta}_{++}$ configuration. The PDW arises when geometric contributions suppress the uniform superfluid weight, providing compelling numerical evidence for quantum-geometry-facilitated PDW order in strongly correlated systems. The results, valid in a topologically trivial model, suggest a general pathway for PDW stabilization via quantum geometry and motivate exploration of related orders in multiorbital systems.

Abstract

Understanding the formation of novel pair density waves (PDWs) in strongly correlated electronic systems remains challenging. Recent mean-field studies suggest that PDW phases may arise in strong-coupling multiband superconductors by virtue of the quantum geometric properties of paired electrons. However, scrutiny through sophisticated many-body calculations has been lacking. Employing large-scale density matrix renormalization group calculations, we obtain in the strong-coupling regime the phase diagram as a function of doping concentration and a tuning interaction parameter for a simple two-orbital model that incorporates quantum geometric effects. The phase diagram reveals a robust PDW phase spanning a broad range of parameters, characterized by a Luttinger parameter $K_{sc} \sim 0.3$ and the absence of coexisting competing spin or charge density wave orders. The observed pairing field configuration aligns with the phenomenological understanding that quantum geometry can promote PDW formation. Our study provides the most compelling numerical evidence to date for quantum-geometry-facilitated intrinsic PDW order in strongly correlated systems, paving the way for further exploration of novel PDW orders and quantum geometric effects in such systems.

Figures (10)

• Figure 1: (a) A representative manifestation of quantum geometric effects in multiband superconductors. Thanks to the interband velocity, Cooper pairs can tunnel between different Bloch bands, resulting in effective Josephson coupling between their order parameters. (b) Sketch of bilayer lattice model used in our DMRG simulation, with $s$ and $d_{xy}$ orbitals residing on separate layers. Details are provided in the text.
• Figure 2: (a) The quantum phase diagram as a function of doping level $\delta$ and the inter-orbital repulsive pair hopping $U_{sd}$, obtained from $L_y = 4$ DMRG calculations. The intra-orbital interaction strength is fixed at $U_{s}=U_{d}=8$. Circles indicate calculations performed using canonical ensemble DMRG, while squares represent those gathered using grand canonical ensemble DMRG. (b) and (c), illustration of the uniform SC and $(\pi,0)$-PDW states obtained in our calculations. The symbol '$++$' in (b) indicates $\hat{\Delta}_{++}$ configuration on each lattice site; while the symbols '$+-$' and '$-+$' in (c) denote $\hat{\Delta}_{+-}$ configuration on each site, with an additional overall phase modulation in $x$-direction.
• Figure 3: SC Correlation functions in PDW phase. (a) SC correlations on the $s$-orbital along the horizontal direction and their power-law fitting. (b) SC correlations on the 4-leg lattice with doping $\delta = 1/8$. The reference is the $s$-orbital intra-orbital pairing on site ${\bf r_0} = (12, 1)$. The upper panel shows the intra-$s$-orbital correlation, and the lower panel displays the inter-orbital correlation between $s$- and $d$-orbitals. The area of the colored dots encode the magnitude of the correlation, and their color indicate the sign of the correlation, i.e., green (orange) is positive (negative).
• Figure 4: Charge density and different correlation functions in the PDW phase with doping $\delta=1/32$ (triangles) and $\delta=1/8$ (squares). (a) Charge density profile of the $s$-orbital in half of the system. The other half is related by a reflection. (b) Intra-$s$-orbital density-density correlation functions with power-law fitting. Filled points indicate negative values while empty points indicate positive values. (c) and (d) display the intra-$s$-orbital spin correlation functions and single-particle correlation functions, respectively.
• Figure 5: (a) Charge denisty and SC pair field in the uniform SC phase for $U_{sd}=0.3$, $\mu_s = \mu_d = -4.7$, corresponding to approximate doping level $\delta = 0.25$. (b) Correlation functions at half filling and $U_{sd}=8$. (c) Charge density pattern at half filling, $U_{sd} = 0$, showing a CDW with modulation wavevector $(\pi,\pi)$. The data are obtained from grand canonical ensemble DMRG with (a) $D=10000$ and (b, c) $D=20000$ states.