Incommensurate pair-density-wave correlations in two-leg ladder $t$--$J$--$J_\perp$ model

TL;DR

The paper identifies an incommensurate pair-density-wave (iC-PDW) phase in a two-leg ladder $t$--$J$--$J_ot$ model under leg-density polarization, combining DMRG with Abelian bosonization. It reveals a spin-gapped regime hosting two gapless charge modes ($c=2$) with interlayer FFLO-like PDW at momentum $q= riangledown k_F$ and intralayer PDW at $q=2k_{F,1}$, robust over a wide polarization range; a small interlayer hopping $t_ op$ can generate a pair-hopping process that, if relevant, may drive a transition to a $c=1$ charge-4e phase, though PDW correlations persist in practical system sizes. The study maps a polarization-driven sequence: a uniform Luther–Emery liquid at $P=0$, an interlayer iC-PDW for $0<P<1$, and a commensurate PDW at $P=1$, with potential relevance to bilayer nickelates such as La$_3$Ni$_2$O$_7$ and realizations in optical lattices. These findings offer a unified framework for engineering distinct pairing regimes via polarization and provide signatures accessible to cold-atom experiments and solid-state platforms.

Abstract

We report the discovery of a generalized Luther-Emery liquid phase characterized by incommensurate pair-density-wave (iC-PDW) correlations in the two-leg $t$-$J$-$J_\perp$ ladder model. By tuning the potential difference between the legs, we explore the regime of intermediate layer polarization $P$. Combining density-matrix renormalization group (DMRG) simulations with bosonization analysis, we identify a spin-gapped phase at finite $P$, where the interlayer and intralayer pair correlations both oscillate, but with distinct periodicities. The interlayer correlations exhibit FFLO-like oscillations, driven by pairing between layers with mismatched Fermi momenta, with a period determined by their momentum difference. In contrast, the intralayer pair correlations arise from the coupling between charges on one layer and spin fluctuations on the opposite layer, with a momentum equal to twice the Fermi momentum of the opposite layer. The iC-PDW state is robust across a wide range of doping and polarization, although finite interlayer hopping eventually destabilizes it toward a state with charge-$4e$ correlations. We conclude by discussing the experimental realization of this model in optical lattice platforms and its relevance to the bilayer nickelate La$_3$Ni$_2$O$_7$.

Figures (11)

• Figure 1: (a) Illustration of the two-leg ladder $t$–$J$–$J_\perp$ model. Particle numbers on the two legs are independently conserved, and we set $N_1 \ge N_2$. (b) Polarization-imbalance–driven phase transitions at $t \sim J \sim J_\perp$. For $0 < P < 1$, the system hosts an interlayer iC-PDW as the dominant correlation, with pairing momentum $q = \delta k_F = |k_{F,1} - k_{F,2}|$. We also find intralayer pairing on layer 2 as a subdominant correlation, with a distinct ordering wave vector $q = 2k_{F,1}$. At $P = 0$, the ground state is a Luther–Emery (LE) liquid, while at $P = 1$ the model yields a commensurate PDW with $q = \pi$.
• Figure 2: Finite DMRG simulation of the $t$--$J$--$J_\perp$ model. We use parameters $t=1$ and $J_\perp=2$ with (a) $n=7/8$ and (b) $n=38/47$. (a,b) Spin gap as a function of $J$ for various polarizations $P$. Simulations are performed on a system of length (a) $L=64$ and (b) $L=94$ with bond dimension $\chi=3000$. (a) While the spin gap closes for $J<J_c=0.78$, our main interest lies in the spin-gapped regime $J>J_c$, where the spin-gapped iC-PDW is realized.
• Figure 3: Pair correlations of interlayer pair and intralayer pair. We use parameters $t=1$, $J_\perp=2$ with (a,c,e) $n=7/8$ and (b,d,e) $n=38/47$. (a) Real space pair correlation shows power-law decaying at $P=2/4$ with $J=1$ (circle) and $J=1.4$ (square), (b) $P=3/11$ with $J=1$ (circle) and $J=1.5$ (square). $\alpha$ is the power-law exponent and we find $\alpha_\perp<\alpha_{\parallel,2}<\alpha_{\parallel,1}$ indicates the interlayer pair correlations are dominant long-range correlation. (c,d) The interlayer pairing $\Delta_{\perp}$ shows a peak at $q=|k_{F,1} - k_{F,2}|=(1-n)P\pi$, denoted as dashed line. (e,f) The intralayer pairing $\Delta_{\parallel,2}$ exhibits a peak at $q=2k_{F,1}=n_1 \pi$, the Fermi momentum of layer 1, denoted as dashed line. This peak is sharp at large $P$, where $n_1$ (carrier of spin-fluctuation) is large, but becomes broadened as $P$ decreases. In contrast, $\Delta_{\parallel,1}$ does not exhibit a sharp peak for any value of $P$ (see Fig. \ref{['fig:s3']}). In evaluating $\langle \Delta^\dagger(0)\Delta(r)\rangle$, we omitted $|r|=0,1$.
• Figure 4: Infinite DMRG results at filling $n=7/8$ with parameters $t=1$ and $J=J_\perp=2$. (a) Entanglement entropy $S$ versus correlation length $\xi$. The central charge is extracted from the relation $S = \frac{c}{6}\log \xi$. In the intermediate-polarization regime $0<P<1$, corresponding to the iC-PDW phase, we find $c=2$, whereas $c=1$ for both the $P=0$ Luther–Emery phase and the $P=1$ commensurate PDW (C-PDW) phase. (b) Correlation lengths of different operators, computed using the transfer-matrix technique. The corresponding charge sectors are specified in terms of $(\delta N_1,\delta N_2,2\delta S_z)$ as $(0,0,0)$, $(1,0,1)$, $(0,1,1)$, $(1,1,0)$, $(0,2,0)$, $(0,0,2)$, and $(1,-1,0)$ for $N$, $c_t$, $c_b$, $\Delta_\perp$, $\Delta_\parallel$, $S_+$, and $P_+$, respectively. Here, $P_+$ is an operator raising $N_1$ and lowering $N_2$ by 1.
• Figure 5: DMRG results of interlayer iC-PDW in the presence of $t_\perp$. We use $t=J_\perp=1$ and $J=2$ with $n=7/8$ and add potential difference between two layers $\epsilon=0.6$ for a finite polarization $\langle P \rangle$. (a) Real-space pair correlation functions for $t_\perp = 0.1$ and $0.25$ ($L_x=64, \chi=4500$). The pair correlations exhibit power-law decay, indicating robust quasi-long-range order upon finite $t_\perp$. Inset shows its Fourier transform, $\langle \Delta_\perp ^\dagger (q) \Delta_\perp (-q)\rangle$, which shows a peak at $q=\delta k_F =(1-n)P \pi$. (b) Correlation lengths of different operators at $t_\perp = 0.25$ ($L_x=16$, $\chi\leq 5500$). The corresponding charge sectors are specified in terms of $(\delta N_\mathrm{tot},2\delta S_z)$ as $(0,0)$, $(1,1)$, $(2,0)$, $(0,2)$, for $N$, $c$, $\Delta$, and $S_+$, respectively.