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arXiv:cond-mat/9909106 · Physical Review Letters

Order Out of Disorder in a Gas of Elastic Quantum Strings in 2+1 Dimensions

Jan Zaanen

Published Jan 24, 2000• 29 citations
arXivHTMLPDF

Abstract

A limiting case of a dynamical stripe state which is of potential significance to cuprate superconductors is considered: a gas of elastic quantum strings in 2+1 dimensions, interacting merely via a hard-core condition. It is demonstrated that this gas always solidifies, by a mechanism which is the quantum analog of the entropic interactions known from soft condensed matter physics.

DOI
10.1103/physrevlett.84.753

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arXiv:cond-mat/9712316•Physical review. B, Condensed matter•Sep 15, 1998•42 citations
A limiting case of a dynamical stripe state which is of potential significance to cuprate superconductors is considered: a gas of elastic quantum strings in 2+1 dimensions, interacting merely via a hard-core condition. It is demonstrated that this gas always solidifies, by a mechanism which is the quantum analog of the entropic interactions known from soft condensed matter physics.
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.
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.
In this thesis we deal with the specific collective phenomena in condensed matter - striped-structures formation. Such structures are observed in different branches of condensed matter physics, like surface physics or physics of high-temperature superconductors. These quasi-one-dimensional objects appear in theoretical analyses as well as in computer simulations of different theoretical models. Here, the main topic of interest is the stability of striped structures in certain quantum models, where a tendency towards crystallization competes with a tendency towards phase separation, and some basic properties of these structures.
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.
Superfluids and superconductors have been around for a long time and their explanation in terms of the Bogoliubov theory for bosons and the BCS theory for fermions belong to the highlights of twentieth century physics. However, it appears that these theories are too primitive to address the high- T c superconductivity found in copper oxides. These electron systems seem to behave more like a dense, strongly correlated liquid contrasting markedly with the conventional quantum gasses: these show strong dynamical correlations on mesoscopic length and time scales associated with stripes, a particular form of electron crystallization. Resting on the gauge theory of topological quantum melting in 2+1 dimensions relevant for the cuprates, we describe the limit which is exactly opposite to the gas limit: the superconductor with the maximum possible amount of transient translational order. We predict that in this “orderly limit” an extra collective mode appears, and this “massive shear photon” can be regarded as a universal fingerprint of the fluctuating stripes. This mode is visible in the electrodynamic response and the ramification of our theory is that electron energy loss spectroscopy can be employed to prove or disprove the existence of dynamical stripes in cuprate superconductors.
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.