Critical role of the sign structure in the doped Mott insulator: Luther-Emery versus Fermi-liquid-like state in quasi-one-dimensional ladders

The mechanism of superconductivity in a purely interacting electron system has been one of the most challenging issues in condensed matter physics. In the BCS theory, the Landau's Fermi liquid is the ground state of weakly interacting fermions dictated by the fermion sign structure, and the superconducting instability only occurs when an additional pairing force is added. By contrast, as shown by density matrix renormalization group calculation, a quasi-one-dimensional superconducting state (specifically a Luther-Emery state) is an intrinsic ground state of the $t-J$ two-leg ladder and four-leg cylinder at finite doping. Such a Luther-Emery state with pairing can be directly turned into a Fermi-gas-like normal state without pairing by merely switching off the phase-string signs hidden in the model via two schemes, which reduce the sign structure to the trivial fermion signs associated with doped holes. It thus demonstrates a new pairing paradigm in a model-doped Mott insulator as due to the generic phase-string sign structure. Finally, as an example, the latter is explicitly shown to lead to a “stringlike” pairing force after adiabatically connecting to a strong anisotropic limit of the model.

Figure 1

Charge-density profile $n\left(x\right)$ of the isotropic $t-J$ and $\sigma t-J$ models at $\delta =0.125$ and $L_x=64$, where only the central-half region is used to extract $A_{\text{CDW}}$ using Eq. (6) labeled by the solid lines. (a) and (b) correspond to two-leg and four-leg ladders, respectively.

Figure 2

Finite-size scalings of (a) CDW amplitude $A_{\text{CDW}}$ and (b) superconducting correlations $\mathrm{\Phi }(L_x/2)$ on $L_y=2$ ladders, and (c) charge density-density $D(L_x/2)$ and (d) superconducting $\mathrm{\Phi }(L_x/2)$ correlations on $L_y=4$ cylinders, for the isotropic $t-J$ model with $K_c=0.90\left(1\right)$ and $K_{sc}=1.19\left(2\right)$, and for the $\sigma t-J$ model with $K_c=1.96\left(1\right)$ and $K_{sc}=2.05\left(9\right)$. Insets: Finite-size scaling of the spin-spin $F(L_x/2)$, single-particle correlations $G_{\sigma }\left(r\right)$ for $t-J$ and $G_{\sigma }(L_x/2)$ for $\sigma t-J$ models. Results for $L_y=2$ ladders are summarized in Table 1. For $L_y=4$ cylinders, ${\xi }_G=2.1\left(1\right)$ and ${\xi }_s=4.0\left(1\right)$ for the $t-J$ model, and $K_G=1.07\left(1\right)$ and $K_s=1.95\left(1\right)$ for the $\sigma -t-J$ model.

Figure 3

The stable LE state in an anisotropic $t-J$ ladder with $\gamma \equiv t_{\perp }/t_{\parallel }$ and $J_{\perp }=J_{\parallel }=J$: (a) Correlation lengths and (b) the exponents of the power-law correlations. Inset in (a) shows the central charge $c$.

Figure 4

Fermi-gas-like state in the strong-rung regime ($\alpha =0.4$) of the anisotropic $t-J$ ladder with $\alpha \equiv t_{\parallel }/t_{\perp }\equiv J_{\parallel }/J_{\perp }$ and $t_{\perp }=t$ and $J_{\perp }=J$. Finite-size scalings of (a) CDW amplitude and (b) superconducting pair-field, spin-spin, and single-particle correlators. Insets are the second-order derivatives of the ground state energy density at doping $\delta =0.125$ (and with one more hole which determines ${\alpha }_c\sim 0.68$ at $t/J=3$) (see text).

Figure 5

A schematic illustration of an effective hard-core chain obtained by compressing the two-leg $t-J$ ladder in the large $J_{\perp }/J_{\parallel }$ limit with interchain hopping $t_{\perp }=0$.

Figure 6

Phase diagram of the isotropic $t-J$ model on two-leg ladders. (a) Extracted Luttinger exponents $K_c,K_{sc}$ and their product $K_c{K}_{sc}$ as a function of doping level $\delta$. (b) Extracted central charge $c$ as a function of doping level $\delta$.