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arXiv:cond-mat/0103253 · Physical review. B, Condensed matter

Gas of elastic quantum strings in 2+1 dimensions: Finite temperatures

Sergei I. Mukhin, Wim van Saarloos, Jan Zaanen

Published Aug 27, 2001• 6 citations
arXivHTMLPDF

Abstract

Motivated by the stripes of the high-${T}_{c}$ cuprates the problem was introduced recently of a system of free elastic quantum strings interacting via a hard-core condition embedded in $2+1$ 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.

DOI
10.1103/physrevb.64.115105

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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.
Disordered elastic systems are studied in the quantum, classical, and dynamic regime in the present thesis. The thesis consists of two main parts: In the first part we study the low temperature phase diagram of one-dimensional weakly disordered quantum systems like charge or spin density waves and Luttinger liquids by a full finite temperature renormalization group (RG) calculation. In the classical region, for vanishing quantum fluctuations those results are supplemented by an exact solution of the model in the case of strong disorder, described by the ground state and the correlation function. Furthermore, by a mapping of the problem onto a Burgers equation with noise, in the case of weak disorder, we can derive an expression for the correlation length. At zero temperature we reproduce the (quantum) phase transition between a pinned (localized) and an unpinned (delocalized) phase for weak and strong quantum fluctuations, respectively, as found previously by Fukuyama or Giamarchi and Schulz. At finite temperatures the localization transition is suppressed: the random potential is wiped out by thermal fluctuations on length scales larger than the thermal de Broglie wave length of the phason excitations. The existence of a zero temperature transition is reflected in a rich cross-over phase diagram determined by the correlation functions. In particular we find four different scaling regions: a classical disordered, a quantum disordered, a quantum critical, and a thermal region. The results can be transferred directly to the discussion of the influence of disorder in superfluids. Finally we extend the RG calculation to the treatment of a commensurate lattice potential, and for the case of density waves we discuss the influence of quantum phase-slips, which might lead to a new scenario for the unpinning (delocalization) transition at zero temperature. Additionally, we analyze the current noise in a classical one-dimensional charge density wave system in the weak pinning regime by solving the overdamped equation of motion numerically. At low temperatures and just above the zero temperature depinning threshold, the power spectrum of the current noise reveals the existence of 1/f or flicker noise. Our results are in agreement with experimental measurements in quasi-one-dimensional charge density wave systems. In part two, we examine the viscous motion of an interface driven by a periodically oscillating external field in a random medium. The velocity exhibits a smeared depinning transition showing a pronounced hysteresis, which is absent in the adiabatic case. Its width is determined by a new length scale, introduced by the frequency of the external drive and - in the low frequency regime - by the critical exponents of the zero frequency depinning transition. Scaling arguments and an approximate renormalization group treatment predict a power law behavior of the velocity as a function of frequency and field amplitude at the dc depinning threshold, which is confirmed numerically. Thermal fluctuations lead to an additional smearing of the transition. If the amplitude of the external driving force is smaller than the depinning field, the motion of the interface by avalanches has to be taken into account in order to describe the velocity hysteresis. To check the validity of our model with respect to experimental results, we calculate the complex susceptibilities using an adiabatic and non-adiabatic approach numerically for finite systems and compare it to measurements of a superferromagnetic granular multilayer system and find that the model of an ac driven interface in a disordered environment can describe the main features of the experimental results. Additionally, the influence of a strong surface potential on the critical depinning of an elastic system, driven in a random medium is considered. If the surface potential prevents depinning completely, the curvature of the displacement profile at zero temperature exhibits a pronounced rhombic hysteresis curve with width of two times the (bulk) depinning field. The hysteresis disappears at non-zero temperatures if the driving force is changed adiabatically. If the surface depinns by the applied force or thermal creep, the curvature is reduced with increasing velocity. The results apply, e.g., to driven magnetic domain walls, flux-line lattices, and charge-density waves. As addendum we examine low dimensional interacting, but clean Bose systems at low temperatures. The interaction leads to scattering events of the over-condensate particles for which we calculate the scattering times using Fermi's golden rule. With that we derive the thermal conductivity and the related weak localization corrections. This is done for short and long-range interactions.
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.
In this thesis we study the physics of quantum many-body systems confined to one-dimensional geometries. The work was motivated by the recent success of experimentalists in developing atom traps, capable of restricting the motion of the individual atoms to a single spatial dimension. Specifically, we look at aspects of the one-dimensional Bose gas including; excitation spectrum, correlation functions, and dynamical behaviour. In Chapter \ref{ch:excitation1D} we consider the Lieb-Liniger model of interacting bosons in one-dimension. We numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss novel features of the solutions, including deviations from the well-known string solutions due to finite size effects. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Our results are compared to those obtained via exact diagonalization of the Hamiltonian in a truncated basis. In Chapter \ref{ch:g2} we analytically calculate the spatial nonlocal pair correlation function for an interacting uniform one dimensional Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects. In Chapter \ref{ch:casimirdrag} we study the drag force below the critical velocity for obstacles moving in a superfluid. The absence of drag is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a non-zero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytic solutions to the Bogoliubov-de Gennes equations for a gray soliton. In Chapter \ref{ch:solitons} we study the effect of quantum noise on the stability of a soliton. We find the soliton solutions exactly define the reflectionless potentials of the Bogoliubov-de Gennes equations. This results in complete stability of the solitons in a purely one dimensional system. We look at the modifications to the density profile of a black soliton due to quantum fluctuations.
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