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arXiv:cond-mat/9612040 · Physical review. B, Condensed matter

Phase string effect in the t-J model: General theory

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

Published Feb 1, 1997• 123 citations

Abstract

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-${\mathrm{T}}_{\mathrm{c}}$ cuprates.

DOI: 10.1103/physrevb.55.3894

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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-

{\mathrm{T}}_{\mathrm{c}}

cuprates.

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.

By explicitly tracking the Marshall sign, a phase string induced by hopping is revealed in the one-hole-doped

t-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=0

, or leads to the localization of the doped hole. Implications for finite doping are also discussed.

Marshall sign and the doped

t-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

Z

is revealed in the doped antiferromagnetic system.

Z=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-J

model for both one- and two-dimensions.

Injecting a single hole into a one-dimensional Heisenberg spin chain is probably the simplest case of doping a Mott insulator. The motion of such a single hole will generally induce a many-body phase shift, which can be identified by an exact sign structure of the model known as the phase string. We show that the sign structure is nontrivial even in this simplest problem, which is responsible for the essential properties of Mott physics. We find that the characteristic momentum structure, the Luttinger liquid behavior, and the quantum phase interference of the hole under a periodic boundary condition can all be attributed to it. We use the density matrix renormalization group (DMRG) numerical simulation to make a comparative study of the t–J chain and a model in which the sign structure is switched off. We further show that the key DMRG results can be reproduced by a variational wave function with incorporating the correct sign structure. Physical implications of the sign structure for doped Mott insulators in general are also discussed.

The quasiparticle weight of a single hole in an antiferromagnetic background is studied in the semiclassical approximation. We start from the $t\ensuremath{-}J$ model, generalize it to arbitrary spin S by employing an appropriate coherent state representation for the hole, and derive an effective action for the dynamics in the long-wavelength low-energy limit. In the same limit, we find an expression for the single-hole Green's function which we evaluate in an

1/S

expansion. Our approach has the advantage of being applicable in one and in two dimensions. We find two qualitatively different results in these two cases: while in one dimension our results are compatible with a vanishing quasiparticle weight, this weight is found to be finite in two dimensions, indicating normal quasiparticle behavior of the hole in this last case.

The stripe phase, an intertwined order observed in high-temperature superconductors, is regarded as playing a key role in elucidating the underlying mechanism of superconductivity, especially in cuprates. Following Zaanen's early scenario [J. Zaanen, Phys. Rev. Lett. 84, 753 (2000); J. Zaanen et al., Philos. Mag. B 81, 1485 (2001)], the filled charge stripe, with one hole per unit cell of the charge order, can be taken as the interactive elastic quantum strings of holes, stabilized by $\ensuremath{\pi}$-phase shifts between neighboring magnetic domains. However, this scenario is challenging to explain, particularly in terms of electron pairing, which necessitates hole pairs. In this paper, we propose an effective model for describing the stripe phase in the hole-doped $t\ensuremath{-}{J}_{z}$ model. With respect to the antiferromagnetic background, the model comprises three types of color-labeled point defects coupling to an effective spin field, so-called colored string. Comparing with numerical results from large-scale density matrix renormalization group simulations, we find semiquantitative agreement in local hole density, magnetic moment, and the proposed spectrum features of the effective spin field. By systematically analyzing the hole-density distribution and the scaling of ground-state energy at different system sizes, we determine the effective core radius and the effective hopping amplitude of the quantum string. Furthermore, the local pinning field can be finely adjusted to drag the quantum string, offering a potential method for detecting it in optical lattices. Lastly, we further demonstrate the partially filled stripe with less than one hole per unit cell of the charge order can also be well described by the effective theory.