Pair Density Wave in the Doped $t\text{−}J$ Model with Ring Exchange on a Triangular Lattice

In our previous work [Phys. Rev. Lett. 121, 046401 (2018)], we found a quantum spin liquid phase with a spinon Fermi surface in the two dimensional spin-$1/2$ Heisenberg model with four-spin ring exchange on a triangular lattice. In this work we dope the spinon Fermi surface phase by studying the $t\text{−}J$ model with four-spin ring exchange. We perform density matrix renormalization group calculations on four-leg cylinders of a triangular lattice and find that the dominant pair correlation function is that of a pair density wave; i.e., it is oscillatory while decaying with distance with a power law. The doping dependence of the period is studied. This is the first example where a pair density wave is the dominant pairing in a generic strongly interacting system where the pair density wave cannot be explained as a composite order and no special symmetry is required.

Figure 1

Pairing correlation along the long direction of the ladder for $1/16$ hole doping with $K/J=0.8$, $t/J=2$ and four-leg ladder length $L_x=72$. (a) Log-log plot of the pairing correlation $P_{22}$ and $P_{12}$. Here, $P_{22}$ is the correlation between pairing order parameters defined both in ${\mathbf{a}}_2$ orientation, $P_{12}$ is the correlation between pairing order parameters defined in ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ orientation, as showed in the inset. The red thin diamond is for $P_{22}$, olive diamond is for $P_{12}$, and the data for $P_{12}$ have been shifted vertically for clarity. The solid symbol is for positive value, while the open symbol is for negative value and we only plot the magnitude. The power law function $f(x)$ in the plot is a fit through the magnitude of the data points. (b) Pairing correlation $P_{22}$ and $P_{12}$ normalized by the power law function $f(x)$, which directly reflects the oscillation of the pairing correlation. The out of phase oscillation of $P_{22}$ and $P_{12}$ indicates a $d$-wave type pairing. (c) Fourier transform of $P_{22}(x)/f(x)$ and $P_{12}(x)/f(x)$. The peak lies at $0.15(2\pi /a_0)$ both for $P_{22}$ and $P_{12}$, which gives the total momentum of the pairing.

Figure 2

Pairing correlation $P_{22}$ for different hole doping, $1/16$, $1/12$, and $1/8$, with parameters $K/J=0.8$, $t/J=2$, and four-leg ladder length $L_x=72$. (a) Power law fitting of the magnitude of pairing correlation $P_{22}$ for each doping. The data for $1/12$ and $1/8$ hole doping have been shifted vertically for clarity. (b) Normalized pairing correlation which shows the oscillation part of $P_{22}$ for doping $1/12$ and $1/8$, while $1/16$ doping is already showed in Fig. 1. (c) Fourier transformation of the oscillation part of $P_{22}$ for each doping, which gives pairing momentum about $0.15(2\pi /a_0)$, $0.17(2\pi /a_0)$, and $0.25(2\pi /a_0)$ for $1/16$, $1/12$, and $1/8$ hole doping, respectively.

Figure 3

Spin correlation $S$ and dimer correlation $D$ for different hole doping, $1/16$, $1/12$, and $1/8$, with parameters $K/J=0.8$, $t/J=2$, and four-leg ladder length $L_x=72$. (a) Spin correlation $S$. Red thin diamond, green square, and blue circle are for $1/16$, $1/12$, and $1/8$ hole doping, respectively. Solid symbol is for positive value, open symbol is for negative value. The magnitude is plotted, and the data points are fitted with a power law. Note the vertical shift of the data for different doping for clarity. (b) Same as (a) for dimer correlation $D$. Yellow left triangular, magenta up and cyan down are for $1/16$, $1/12$, and $1/8$ hole doping, respectively. Solid symbol is for positive value, open symbol is for negative value. (c) Normalization with the power law function of the magnitude, giving the oscillation part of spin and dimer correlations. (d) Fourier transformation of the oscillatory part of the spin and dimer correlations. The peak gives the ordering momentum; it is $0.25(2\pi /a_0)$ for the spin correlation and $0.5(2\pi /a_0)$ for the dimer correlation at $1/8$ hole doping. For $1/12$ and $1/16$ hole doping it is $0.5(2\pi /a_0)$ for both correlations.

Figure 4

Momentum space density $n(\mathbf{k})$ and its first derivative in the $k_1$ direction $\partial n(\mathbf{k})/\partial k_1$, with parameters $K/J=0.8$, $t/J=2$, and four-leg ladder length $L_x=72$. In the plot, $k_1$ and $k_2$ are the coordinates of $\mathbf{k}$ points in the unit of primitive vectors in reciprocal space $\mathbf{k}=k_1{\mathbf{b}}_1+k_2{\mathbf{b}}_2$. As there is inversion symmetry, only two cuts $k_2=-0.25$ and $k_1=0$ are shown. The peak and dip positions (denoted by black arrows) of $\partial n(\mathbf{k})/\partial k_1$ give the ${\mathbf{k}}_F\mathrm{s}$. We have ${\mathbf{k}}_F=\pm \frac{7}{24}{\mathbf{b}}_1$, $(\frac{3}{8}{\mathbf{b}}_1-\frac{1}{4}{\mathbf{b}}_2)$ and $(-\frac{1}{4}{\mathbf{b}}_1-\frac{1}{4}{\mathbf{b}}_2)$ for $1/8$ hole doping [(a),(b)], ${\mathbf{k}}_F=\pm \frac{7}{24}{\mathbf{b}}_1$, $(\frac{5}{12}{\mathbf{b}}_1-\frac{1}{4}{\mathbf{b}}_2)$, and $(-\frac{1}{4}{\mathbf{b}}_1-\frac{1}{4}{\mathbf{b}}_2)$ for $1/12$ [(c),(d)] and $1/16$ hole doping [(e),(f)]. The precision of these values is limited by the finite system size.