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arXiv:cond-mat/9712316 · Physical review. B, Condensed matter

Charged domain walls as quantum strings on a lattice

Henk J. Eskes, Osman Y. Osman, R. Grimberg, Wim van Saarloos, Jan Zaanen

Published Sep 15, 1998• 42 citations
arXivHTMLPDF

Abstract

Recently, experimental evidence has been accumulating that the doped holes in the high-${T}_{c}$ cuprate superconductors form domain walls separating antiferromagnetic domains. These so-called stripes are linelike objects and if these persist in the superconducting state, high-${T}_{c}$ superconductivity is related to a quantum string liquid. In this paper the problem of a single quantum meandering string on a lattice is considered. A kink model is introduced for the string dynamics, which allows us to analyze lattice commensuration aspects. Building on earlier work by den Nijs and Rommelse [Phys. Rev. B 40, 4709 (1989)], this lattice string model can be related both to restricted solid-on-solid models, describing the world-sheet of the string in Euclidean space time, and to one-dimensional quantum spin chains. At zero temperature a strong tendency towards orientational order is found and the remaining directed string problem can be treated in detail. Quantum delocalized strings are found whose long-wavelength wandering fluctuation is described by free field theory and it is argued that the fact that the critical phase of delocalized lattice strings corresponds to a free Gaussian theory is a very general consequence of the presence of a lattice. In addition, the mapping on the surface problem is exploited to show the existence of different types of localized string phases; some of these are characterized by a proliferation of kinks, but the kink flavors are condensed so that the long-wavelength fluctuations of these strings are suppressed. The simplest phase of this kind is equivalent to the incompressible (Haldane) phase of the $S=1$ spin chain and corresponds to a bond centered string: The average string position is centered on bonds. We also find localized phases of this type that take arbitrary orientations relative to the underlying lattice. The possible relevance of these lattice strings for the stripes in cuprates is discussed.

DOI
10.1103/physrevb.58.6963

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Recently, experimental evidence has been accumulating that the doped holes in the high-Tc{T}_{c}Tc​ cuprate superconductors form domain walls separating antiferromagnetic domains. These so-called stripes are linelike objects and if these persist in the superconducting state, high-Tc{T}_{c}Tc​ superconductivity is related to a quantum string liquid. In this paper the problem of a single quantum meandering string on a lattice is considered. A kink model is introduced for the string dynamics, which allows us to analyze lattice commensuration aspects. Building on earlier work by den Nijs and Rommelse [Phys. Rev. B 40, 4709 (1989)], this lattice string model can be related both to restricted solid-on-solid models, describing the world-sheet of the string in Euclidean space time, and to one-dimensional quantum spin chains. At zero temperature a strong tendency towards orientational order is found and the remaining directed string problem can be treated in detail. Quantum delocalized strings are found whose long-wavelength wandering fluctuation is described by free field theory and it is argued that the fact that the critical phase of delocalized lattice strings corresponds to a free Gaussian theory is a very general consequence of the presence of a lattice. In addition, the mapping on the surface problem is exploited to show the existence of different types of localized string phases; some of these are characterized by a proliferation of kinks, but the kink flavors are condensed so that the long-wavelength fluctuations of these strings are suppressed. The simplest phase of this kind is equivalent to the incompressible (Haldane) phase of the S=1S=1S=1 spin chain and corresponds to a bond centered string: The average string position is centered on bonds. We also find localized phases of this type that take arbitrary orientations relative to the underlying lattice. The possible relevance of these lattice strings for the stripes in cuprates is discussed.
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.
We attempt to access the regime of strong coupling between charge carriers and transverse dynamics of an isolated conducting ``stripe,'' such as those found in cuprate superconductors. A stripe is modeled as a partially doped domain wall in an antiferromagnet (AF),(\mathrm{AF}),(AF), introduced in the context of two different models: the $t\ensuremath{-}J$ model with strong Ising anisotropy, and the Hubbard model in the Hartree-Fock approximation. The domain walls with a given linear charge density are supported artificially by boundary conditions. In both models we find a regime of parameters where doped holes lose their spin and become holons (charge Q=1,Q=1,Q=1, spin S3=0),{S}_{3}=0),S3​=0), which can move along the stripe without frustrating AF environment. One aspect in which the holons on the AF domain wall differ from those in an ordinary one-dimensional electron gas is their transverse degree of freedom: a mobile holon always resides on a transverse kink (or antikink) of the domain wall. This gives rise to two holon flavors and to a strong coupling between doped charges and transverse fluctuations of a stripe.
Recent experiments have shown the existence of an ordered stripe phase in the cuprates, suggesting that the nearby metallic phase is best described as a domain-wall fluid. As a minimal model to describe the melting of charged domain walls we introduce quantum lattice strings. This model is related to a generalized quantum spin chain. The string exhibits a rich phase diagram, containing several rough phases with low-lying excitations as well as gapped ordered phases. In this scenario the occurrence of the stripe phase is related to a transition from the quantum-rough (metal) to the classical flat phase.
A quantum string, which was brought into discussion recently as a model for the stripe phase in doped cuprates, is simulated by means of the density-matrix-renormalization-group method. String collides with adjacent neighbors, as it wanders, owing to quantum zero-point fluctuations. The energy cost due to the collisions is our main concern. Embedding a quantum string between rigid walls with separation d, we found that for sufficiently large d, collision-induced energy cost obeys the formula $\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}{\mathrm{Ad}}^{\ensuremath{\alpha}})$ with $\ensuremath{\alpha}=0.808(1),$ and the string's mean fluctuation width grows logarithmically $\ensuremath{\sim}\mathrm{log}d.$ Those results are not understood in terms of a conventional picture that the string is ``disordered,'' and only short-wavelength fluctuations contribute to collisions. Rather, our results support a recent proposal that owing to collisions, short-wavelength fluctuations are suppressed, but instead, long-wavelength fluctuations become significant. This mechanism would be responsible for stabilizing the stripe phase.
This is part one in a series of two papers dedicated to the notion that the destruction of the topological order associated with stripe phases is about the simplest theory controlled by local symmetry: Ising gauge theory. This first part is intended to be a tutorial- we will exploit the simple physics of the stripes to vividly display the mathematical beauty of the gauge theory. Stripes, as they occur in the cuprates, are clearly `topological' in the sense that the lines of charges are at the same time domain walls in the antiferromagnet. Imagine that the stripes quantum melt so that all what seems to be around is a singlet superconductor. What if this domain wall-ness is still around in a delocalized form? This turns out to be exactly the kind of `matter' which is described by the Ising gauge theory. The highlight of the theory is the confinement phenomenon, meaning that when the domain wall-ness gives up it will do so in a meat-and-potato phase transition. We suggest that this transition might be the one responsible for the quantum criticality in the cuprates. In part two, we will become more practical, arguing that another phase is possible according to the theory. It might be that this quantum spin-nematic has already been observed in strongly underdoped LSCO.