Revealing quantum phase string effect in doped Mott-insulator: a tensor network state approach

TL;DR

This study uses imaginary-time evolution of a two-dimensional fermionic tensor-network (fTPS/PEPS) to compare the standard $t$-$J$ model with a $oldsymbol{\sigma t}$-$J$ benchmark that suppresses the quantum phase string sign $\tau_C=(-1)^{N_{h}^{\downarrow}[C]+N_{ex}^{h}[C]}$. The key finding is that phase strings drive strong spin–charge entanglement, causing rapid suppression of antiferromagnetic order, enhanced charge compressibility, incoherent single-particle propagation, and robust $d$-wave pairing in the $t$-$J$ model, while the $oldsymbol{\sigma t}$-$J$ model behaves like a Fermi liquid with decoupled spin and charge. The results provide a microscopic mechanism for non-Fermi-liquid behavior in doped Mott insulators and reinforce the view that phase-string physics is crucial for realizing high-$T_c$ superconductivity. The work showcases the power of fermionic tensor networks to dissect sign-structure–driven phenomena and suggests avenues to probe PSG, sign problems, and potential PDW states in strongly correlated systems, with implications for cuprate physics.

Abstract

We apply the fermionic tensor network (TN) state method to understand the strongly correlated nature in a doped Mott insulator. We conduct a comparative study of the $σt$-$J$ model, in which the no-double-occupancy constraint remains unchanged but the quantum phase string effect associated with doped holes is precisely switched off. Thus, the ground state of the $σt$-$J$ model can serve as a well-controlled reference state of the standard $t$-$J$ model. In the absence of phase string, the spin long-range antiferromagnetic (AFM) order is found to be essentially decoupled from the doped holes, and the latter contribute to a Fermi-liquid-like compressibility and a coherent single-particle propagation with a markedly reduced pairing tendency. In contrast, our TN calculations of the $t$-$J$ model indicate that the AFM order decreases much faster with doping and the single-particle propagation of doped holes gets substantially suppressed, concurrently with a much stronger charge compressibility at small doping and a significantly amplified Cooper pairing tendencies. These findings demonstrate that quantum many-body interference from phase strings plays a pivotal role in the $t$-$J$ model, mediating long-range entanglement between spin and charge degrees of freedom.

Figures (11)

• Figure 1: A $2 \times 2$ unit cell of a fTPS on a square lattice, containing four independent fermionic tensors $A, B, C, D$ on lattice sites, and eight independent fermionic Schmidt weight matrices $\Lambda_0, ..., \Lambda_7$ on nearest neighbor bonds. Indices of a tensor corresponding to super vector spaces (or dual spaces) are represented by outgoing (or incoming) arrows.
• Figure 2: Ground state energies of the $\sigma{t}$-$J$ model. $m$ is the number of states kept in the DMRG sweeping. $t/J=3.0$.
• Figure 3: Compressibility $\kappa$ versus doping $\delta$ from the ground states of $t$-$J$ and $\sigma{t}$-$J$ models. Inset shows the doping $\delta$ versus the chemical potential $\mu$. $t/J=3.0$ and $D=12$.
• Figure 4: (a) Magnetizations in the ground states of $t$-$J$ and $\sigma{t}$-$J$ models. The dashed line shows the fully polarized magnetization $(1 - \delta)/2$. (b, c, d, e) show spin-spin correlation functions $C_{s}(r)$ for these two models with doping levels around $\delta\approx3\%, 5\%, 15\%, 20\%$, respectively. $t/J=3.0$. $D=12$.
• Figure 5: Superconductivity singlet pairing magnitudes in the (a) $t$-$J$ model (b) $\sigma{t}$-$J$ model. $t/J=3.0$.