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arXiv:cond-mat/0102123 · Physical review. B, Condensed matter

Quantum-fluctuation-induced repulsive interaction of a quantum string between walls

Yoshihiro Nishiyama

Published Jul 19, 2001• 10 citations
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Abstract

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.

DOI
10.1103/physrevb.64.064510

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Previously we have indicated the relationship between quantum groups [Phys. Lett A272, (2000)] and strings via WZWN models in the context of applications to cuprates and related materials.The connection between quantum groups and strings is one way of seeing the validity of our previous conjecture [i.e. that a theory for cuprates may be constructed on the basis of quantum groups]. The cuprates seems to exhibit statistics, dimensionality and phase transitions in novel ways. The nature of excitations [i.e. quasiparticle or collective] must be understood. The Hubbard model captures some of the behaviour of the phase transitions in these materials. On the other hand the phases such as stripes in these materials bear relationship to quantum group or string-like solutions. One thus expects that the relevant solutions of Hubbard model may thus be written in terms of stringy solutions. In short this approach may lead to the non-perturbative formualtion of Hubbard and other condensed matter Hamiltonians. The question arises that how a 1-d based symmetry such as quantum groups can be relevant in describing a 3-d [spatial dimensions] system such as cuprates. The answer lies in the key observation that strings which are 1-d objects can be used to describe physics in $d$ dimensions. For example gravity [which is a 3-d [spatial] plus time] phenomenon can be understood in terms of 1-d strings. Thus we expect that 1-d quantum group object induces physics in 2-d and 3-d which may be relevant to the cuprates. We present support for our contention using [numerical] variational Monte-Carlo [MC] applied to 2d d-p model. We also briefly discuss others ways to formulate a string picture for cuprates, namely by exploiting connection between gauge theories and strings and t'Hooft picture of quark confinement.

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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∼exp(\ensuremath−Ad\ensuremathα)\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}{\mathrm{Ad}}^{\ensuremath{\alpha}})\ensuremath∼exp(\ensuremath−Ad\ensuremathα) with \ensuremathα=0.808(1),\ensuremath{\alpha}=0.808(1),\ensuremathα=0.808(1), and the string's mean fluctuation width grows logarithmically \ensuremath∼logd.\ensuremath{\sim}\mathrm{log}d.\ensuremath∼logd. 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.
An elastic string embedded between rigid walls is simulated by means of the density-matrix renormalization group. The string collides against the walls owing to the quantum-mechanical zero-point fluctuations. Such a ``quantum entropic'' interaction has come under thorough theoretical investigation in the context of the stripe phase observed experimentally in doped cuprates. We found that the excitation gap opens in the form of an exponential singularity $\ensuremath{\Delta}E\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}{\mathrm{Ad}}^{\ensuremath{\sigma}})$ (d=wallspacing)(d=\mathrm{wall}\mathrm{}\mathrm{spacing})(d=wallspacing) with the exponent $\ensuremath{\sigma}=0.6(3),$ which is substantially smaller than the mean-field value $\ensuremath{\sigma}=2.$ That is, the excitation gap is much larger than that anticipated from mean field, suggesting that the string is subjected to a robust pinning potential due to the quantum collisions. This feature supports Zaanen's ``order out of disorder'' mechanism, which would be responsible for the stabilization of the stripe phase.
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.
Motivated by the stripes of the high-Tc{T}_{c}Tc​ cuprates the problem was introduced recently of a system of free elastic quantum strings interacting via a hard-core condition embedded in 2+12+12+1 dimensions. At zero temperature this system is always a solid due to ``quantum-entropic'' (or ``kinetic'') interactions which dominate at long wavelengths. The high-temperature limit of this problem corresponds with thermally meandering elastic lines in two dimensions and this system is well known to be dominated by literal entropic interactions. Here we analyze in detail what happens in between zero and high temperature. We identify a ``renormalized classical'' intermediate regime where the on-string fluctuations have become predominantly quantum mechanical. Surprisingly, the entropic interactions keep their high-temperature nature in this regime. At a low, but finite temperature the quantum-mechanical kinetic interactions take over rather suddenly. Despite their origin in long-wavelength quantum fluctuations these are not affected by thermal fluctuations when temperature is low enough.
Previously we have indicated the relationship between quantum groups [Phys. Lett A272, (2000)] and strings via WZWN models in the context of applications to cuprates and related materials.The connection between quantum groups and strings is one way of seeing the validity of our previous conjecture [i.e. that a theory for cuprates may be constructed on the basis of quantum groups]. The cuprates seems to exhibit statistics, dimensionality and phase transitions in novel ways. The nature of excitations [i.e. quasiparticle or collective] must be understood. The Hubbard model captures some of the behaviour of the phase transitions in these materials. On the other hand the phases such as stripes in these materials bear relationship to quantum group or string-like solutions. One thus expects that the relevant solutions of Hubbard model may thus be written in terms of stringy solutions. In short this approach may lead to the non-perturbative formualtion of Hubbard and other condensed matter Hamiltonians. The question arises that how a 1-d based symmetry such as quantum groups can be relevant in describing a 3-d [spatial dimensions] system such as cuprates. The answer lies in the key observation that strings which are 1-d objects can be used to describe physics in ddd dimensions. For example gravity [which is a 3-d [spatial] plus time] phenomenon can be understood in terms of 1-d strings. Thus we expect that 1-d quantum group object induces physics in 2-d and 3-d which may be relevant to the cuprates. We present support for our contention using [numerical] variational Monte-Carlo [MC] applied to 2d d-p model. We also briefly discuss others ways to formulate a string picture for cuprates, namely by exploiting connection between gauge theories and strings and t'Hooft picture of quark confinement.