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Progress of Theoretical Physics Supplement

Quantum-Entropic Interaction of an Elastic String between Walls

Yoshihiro Nishiyama

Published Jan 1, 2002• 0 citations
Publisher

Abstract

Recently, there arose a problem of `quantum string,' which is a linelike object meandering quantum-mechanically and undergoes entropy-reducing collisions with adjacent neighbors. In this paper, we overview underlying physics of the quantum string, and report our simulation result on the collision-induced entropy loss of a quantum string between walls by means of the density-matrix renormalization group.

DOI
10.1143/ptps.145.229

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Recently, there arose a problem of `quantum string,' which is a linelike object meandering quantum-mechanically and undergoes entropy-reducing collisions with adjacent neighbors. In this paper, we overview underlying physics of the quantum string, and report our simulation result on the collision-induced entropy loss of a quantum string between walls by means of the density-matrix renormalization group.
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.
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.
The renormalization group is a tool that allows one to obtain a reduced description of systems with many degrees of freedom while preserving the relevant features. In the case of quantum systems, in particular, one-dimensional systems defined on a chain, an optimal formulation is given by White's ‘density matrix renormalization group’. This formulation can be shown to rely on concepts of the developing theory of quantum information. Furthermore, White's algorithm can be connected with a peculiar type of quantization, namely, angular quantization . This type of quantization arose in connection with quantum gravity problems, in particular, the Unruh effect in the problem of black-hole entropy and Hawking radiation. This connection highlights the importance of quantum system boundaries, regarding the concentration of quantum states on them, and helps us to understand the optimal nature of White's algorithm.
Quantum entanglement entropy has a geometric character. This is illustrated by the interpretation of Rindler space or black hole entropy as entanglement entropy. In general, one can define a "geometric entropy", associated with an event horizon as a boundary that concentrates a large number of quantum states. This allows one to connect with the "density matrix renormalization group" and to unveil its connection with the theory of quantum information. This renormalization group has been introduced in condensed matter physics in a heuristic manner, but it can be conceived as a method of compression of quantum information in the presence of a horizon. We propose generalizations to problems of interest in cosmology.
What is the meaning of entanglement in a theory of extended objects such as strings? To address this question we consider the spatial entanglement between two intervals in the Gross-Taylor model, the string theory dual to two-dimensional Yang-Mills theory at large NNN. The string diagrams that contribute to the entanglement entropy describe open strings with endpoints anchored to the entangling surface, as first argued by Susskind. We develop a canonical theory of these open strings, and describe how closed strings are divided into open strings at the level of the Hilbert space. We derive the Modular hamiltonian for the Hartle-Hawking state and show that the corresponding reduced density matrix describes a thermal ensemble of open strings ending on an object at the entangling surface that we call an E-brane.
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.