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We study the problem of a single hole in an Ising antiferromagnet and, usingthe magnon expansion and analytical methods, determine the expansioncoefficients of its wave function in the magnon basis. In the 1D case, the holeis "weakly" confined in a potential well and the magnon coefficients decayexponentially in the absence of a string potential. This behavior is in sharpcontrast to the 2D square plane where the hole is "strongly" confined by astring potential and the magnon coefficients decay superexponentially. Thelatter is identified here to be a fingerprint of the strings in dopedantiferromagnets that can be recognized in the numerical or cold atomsimulations of the 2D doped Hubbard model. Finally, we attribute thedifferences between the 1D and 2D cases to the magnon-magnon interactions beingcrucially important in a 1D spin system.

From "Weak" to "Strong" Hole Confinement in a Mott Insulator