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Physical Review Letters

Phase String Effect in a Doped Antiferromagnet

D. N. Sheng, Y. C. Chen, Zheng-Yu Weng

Published Dec 16, 1996• 107 citations
Publisher

Abstract

Based on the $t\ensuremath{-}J$ model, it is shown that a hole moving on an antiferromagnetic spin background always induces a phase-string effect. Such a previously unnoticed phase string is revealed by explicitly tracking the Marshall sign and can be rigorously shown to be nonrepairable at low energy. Its quantum interference can drastically modify the long-wavelength behavior of the doped hole, leading to a vanishing spectral weight $Z$ at the ground-state energy. Implication for finite doping is also discussed.

DOI
10.1103/physrevlett.77.5102

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Based on the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model, it is shown that a hole moving on an antiferromagnetic spin background always induces a phase-string effect. Such a previously unnoticed phase string is revealed by explicitly tracking the Marshall sign and can be rigorously shown to be nonrepairable at low energy. Its quantum interference can drastically modify the long-wavelength behavior of the doped hole, leading to a vanishing spectral weight ZZZ at the ground-state energy. Implication for finite doping is also discussed.
We reexamine the problem of a hole moving in an antiferromagnetic spin background and find that the injected hole will always pick up a sequence of nontrivial phases from the spin degrees of freedom. Previously unnoticed, such a stringlike phase originates from the hidden Marshall signs which are scrambled by the hopping of the hole. We can rigorously show that this phase string is nonrepairable at low energy and give a general proof that the spectral weight Z must vanish at the ground-state energy due to the phase-string effect. Thus, the quasiparticle description fails here and the quantum interference effect of the phase string dramatically affects the long-distance behavior of the injected hole. We introduce a so-called phase-string formulation of the t-J model for a general number of holes in which the phase-string effect can be explicitly tracked. As an example, by applying this new mathematical formulation in one dimension, we reproduce the well-known Luttinger-liquid behaviors of the asymptotic single-electron Green's function and the spin-spin correlation function. We can also use the present phase-string theory to justify previously developed spin-charge separation theory in two dimensions, which offers a systematic explanation for the transport and magnetic anomalies in the high-Tc{\mathrm{T}}_{\mathrm{c}}Tc​ cuprates.
By explicitly tracking the Marshall sign, a phase string induced by hopping is revealed in the one-hole-doped t−Jt-Jt−J model. It is rigorously shown that such a phase string cannot be eliminated through low-lying spin excitations, and it either causes the spectral weight Z=0Z=0Z=0, or leads to the localization of the doped hole. Implications for finite doping are also discussed.
Marshall sign and the doped t−Jt-Jt−J model
Marshall sign as a sole source of sign problem hidden in an antiferromagnet is explored under doping. By tracking the Marshall sign, a zero spectral weight ZZZ is revealed in the doped antiferromagnetic system. Z=0Z=0Z=0 is caused by a phase string induced by the ``bare'' hole. By eliminating such a phase string through nonlocal transformations, a non-perturbative scheme is obtained. It is argued that this formalism provides a unique way to get access to the real ground state of the doped t−Jt-Jt−J model for both one- and two-dimensions.
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 $\ensuremath{\sigma}t\text{\ensuremath{-}}J$ 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 $\ensuremath{\sigma}t\text{\ensuremath{-}}J$ model can serve as a well-controlled reference state of the standard $t\text{\ensuremath{-}}J$ model. In the absence of the 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 $t\text{\ensuremath{-}}J$ 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 significantly amplified Cooper pairing tendencies. These findings demonstrate that quantum many-body interference from phase strings plays a pivotal role in the $t\text{\ensuremath{-}}J$ model, mediating long-range entanglement between spin and charge degrees of freedom, which might hold the key mechanism for high-Tc{T}_{c}Tc​ cuprates.
The motion of a single hole in a Mott antiferromagnet is investigated based on the $t\ensuremath{-}J$ model. An exact expression of the energy spectrum is obtained, in which the irreparable phase string effect [Phys. Rev. Lett. 77, 5102 (1996)] is explicitly present. By identifying the phase string effect with spin backflow, we point out that spin-charge separation must exist in such a system: the doped hole has to decay into a neutral spinon and a spinless holon, together with the phase string. We show that while the spinon remains coherent, the holon motion is deterred by the phase string, resulting in its localization in space. We calculate the electron spectral function which explains the line shape of the spectral function as well as the ``quasiparticle'' spectrum observed in angle-resolved photoemission experiments. Other analytic and numerical approaches are discussed based on the present framework.
Understanding the doped Mott insulator is a central challenge in condensed matter physics. In this work, we first explicitly identify a new sign structure in the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model on the square lattice that replaces the conventional Fermi statistics for weakly interacting electrons. Then we show that the singular, i.e., the phase-string part of the sign structure in the partition function can be precisely turned off in a modified model. The density matrix renormalization group method is then employed to study these two models comparatively on finite-size systems, which is designed to unveil the consequences of the phase-string component. We find that the hole pairing is present not only in the quasi-long-range superconducting phase but also in the stripe phase of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model. However, once the phase-string is switched off, both the superconducting and stripe orders together with the underlying hole pairing disappear. The corresponding ground state reduces to a trivial Fermi-liquid-like state with small hole Fermi pockets that is decoupled from the antiferromagnetic spin background. It is in sharp contrast to the original $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model where large Fermi surfaces can be restored in the stripe phase found at ${t}^{\ensuremath{'}}/t<0$ or the superconducting phase at ${t}^{\ensuremath{'}}/t>0$ in the six-leg ladder calculation. Our study clearly demonstrates that the strong correlation effect in doped Mott insulator should be mainly attributed to the long-range quantum entanglement between the spin and charge, which is, nonperturbatively, beyond a simple spin-charge separation under the no double occupancy constraint.