Ground-state phase diagram of the one-dimensional $t$-$J$ model

We examine the ground-state phase diagram of the $t$-$J$ model in one dimension by means of the density matrix renormalization group. This model is characterized by a rich phase diagram as a function of the exchange interaction $J$ and the density $n$, displaying Luttinger-liquid (LL) behavior both of repulsive and attractive (i.e., superconducting) natures, a spin-gap phase, and phase separation. The phase boundaries separating the repulsive from the attractive LL phase as $J$ is increased, and also the boundaries of the spin-gap region at low densities, and phase separation at even larger $J$, are determined on the basis of correlation functions and energy gaps. In particular, we shed light on a contradiction between variational and renormalization-group (RG) results about the extent of the spin-gap phase, which results larger than the variational but smaller than the RG one. Furthermore, we show that the spin gap can reach a sizable value ($~0.1t$) at low enough filling, such that preformed pairs should be observable at temperatures below these energy scales. No evidence for a phase with clustering of more than two particles is found on approaching phase separation.

Figure 1

Phase diagram of the 1D $tJ$ model from DMRG for densities $0.1⩽n⩽0.9$ and in the range $0<J⩽4$, where we set $t=1$. $n=N/L$ is the electronic density ($N$ is the total number of particles and $L$ the number of lattice sites). Four phases are present: a metallic phase (M) or repulsive LL, a gapless superconducting (SC) phase, a singlet-superconducting phase with spin gap ($\text{SG}+\text{SS}$), and phase separation (PS). The number given to each line stands for the value of the Luttinger parameter $K_{\rho }$.

Figure 2

Structure factor $N(k)$ of the density-density correlation function for $n=0.5$, $J=2.0$ and $L=40$, 80, 120, 160, and 200.

Figure 3

Extrapolation to the thermodynamic limit of $N(k=2\pi /40)$ for $n=0.5$ and $J=2.0$.

Figure 4

$K_{\rho }$ as function of $J$ for different densities $n$.

Figure 5

$K_{\rho }$ as function of the density $n$ for $J=2$ (supersymmetric point).

Figure 6

Regions with different Luttinger parameters $K_{\rho }(n,J)$. Each curve corresponds to a constant value of $K_{\rho }$. The red (dashed) line $(K_{\rho }(n,J)=1)$ denotes the boundary between the metallic phase and the superconducting phases. Note that the density of lines increases with $J$ showing the fast growth of $K_{\rho }$.

Figure 7

Energy comparison between systems containing four, six, and eight particles and a gas of singlet bound pairs. We observe a region $2.0<J<3.0$ where all energies are the same within an error of ${10}^{-5}$. This opens the possibility of the formation of a gas of singlet bound pairs and consequently of the existence of a spin gap at very low densities.

Figure 8

$\Delta E_S$ vs $1/L$ for $n=0.2$ and various values of $J$. For $J=2$, where the system is still in the metallic phase, the spin gap extrapolates to zero in the thermodynamic limit.

Figure 9

Spin gap $\Delta E_S$ in the thermodynamic limit as a function of $J$ and for different densities $n$. For $J<2$ (metallic phase) the spin gap is zero for all densities. For $J>2$ a finite spin gap emerges with a value that increases for diminishing densities. We can observe a small but finite spin gap up to $n=0.55$.

Figure 10

Spin gap $\Delta E_S$ in the thermodynamic limit as a function of $n$ for $J=2.1$–2.8.

Figure 11

Inverse of the compressibility ${\kappa }^{-1}$ as a function of $J$ for $n=0.1$–0.9 with $\Delta n=0.05$.

Figure 12

Zoom of Fig. 11. The points $(n,J_c)$ where ${\kappa }^{-1}=0$ define the boundary of the phase-separated phase (infinite compressibility).

Figure 13

Structure factor $N(k)$ for the density-density correlation function for $L=200$ and for different values of $n$ and $J$. The location of $2k_F$ and $4k_F$ correspond to the ones resulting from folding them back to the sector $k>0$ in the first Brillouin zone.

Figure 14

Structure factor $S(k)$ for the spin-spin correlation function for $L=200$ and for different values of $n$ and $J$.

Figure 15

Structure factor $P_S(k)$ for the singlet pair-pair correlation function for $L=200$ and for different values of $n$ and $J$.

Figure 16

Momentum distribution function $n(k)$ for $L=200$ and for different values of $n$ and $J$.

Figure 17

Singlet $P_S(x)$ and triplet $P_T(x)$ correlation functions for $n=0.5$, $J=2.6$, and $L=200$. For the singlet case we plot $|P_S(x)|$. The straight lines correspond to power laws determined by $K_{\rho }$. Both the singlet and triplet channel have the same exponent.

Figure 18

Singlet $P_S(x)$ and triplet $P_T(x)$ correlation functions for $n=0.5$, $J=2.6$, $L=200$, and $m=150,200,250$. We observe at long distances that on a double-logarithmic scale, $P_T(x)$ becomes more linear upon increasing $m$, i.e., the spurious exponential decay introduced by the DMRG cutoff decreases. Note, however, that $P_S(x)$ becomes less linear at long distances when increasing $m$. We associate this behavior to the open boundary conditions used.

Figure 19

Triplet and singlet pair-pair structure factors for $n=0.5$, $J=2.6$, and $L=200$.

Figure 20

Maxima of $P_S(x)$ and $-N(x)$ in real space for $L=200$, $n=0.2$, and $J=2.5$ on a log-log scale. The power laws are determined by $K_{\rho }$ through Eqs. (15) and (17).

Figure 21

Scaling of $P_S(k=0)$ in the thermodynamic limit for $n=0.2$ and $J=2.5$ (spin-gap phase).

Figure 22

Density in real space $\rho (x)$ for $n=0.1$ and (a) $L=80$ and (b) $L=160$. In the metallic regime ($J=1.0$) we have one peak per particle due to Friedel oscillations. When $J$ is increased, the particles start to form pairs. This happens in the spin-gap region ($J=2.5$). Increasing $J$ even more, the particles tend to be confined in a region smaller than the system size (phase separation).

Figure 23

Density in real space $\rho (x)$ for $n=0.1$, $L=160$, and different values of $J$. In (a) we observe the pairing of particles. In the other figures only particles at the boundaries of the particle-rich region merge into one loose cloud, but the particles in the middle do not cluster further.

Figure 24

Magnetic structure factor $S(k)$ in the phase-separated region for values of $J$ close to the boundary and at the value, where a peak at $k=\pi$ emerges. Insets: the corresponding density profiles.

Figure 25

Phase diagram of Fig. 1 including the electron solid phase (ES). The circles denote the position at which a peak in $S(k)$ appears at $k=\pi$, and at the same time, the phase-separated island reaches a density $n=1$. Triangles reproduce the results of Ref. 21 and the dashed-dotted line those of Ref. 19.