Abstract
We employ high-temperature series to investigate a two-parameter class of renormalization group transformations for the two-dimensional Ising model on the triangular lattice. For the static case we identify an optimal organization of the high-temperature expansion and an optimal transformation matrix and thus find, in second order, ν=0.96 and the magnetic eigenvaluey=2-η/2=1.76.
From recursion relations for flip rates we find the dynamic exponent to be the same for all transformations in our two-parameter class,z=2.32.
Our fixed-point flip rates do not describe a Markov process even though the corresponding master equation for the single-event probability displays no explicit memory effects. The non-Markovian nature shows up only in a violation of the Markovian detailed balance conditions.
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References
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Note thatR exp (Lt)\(\hat R\) is not, in general, a conditional probability. Actually, none of our arguments make use of (expLt)σ σ′ having that meaning
A simple analogy may serve to illustrate this point. Consider the everdamped harmonic oscillator,\(\dot q = p, \dot p = - \gamma p - q\), for γ≫1. The two eigenvalues are\( - \Gamma _ \pm = - (\gamma /2 \pm \sqrt {\gamma ^2 /4 - 1} )\). The ratio\(\dot q/q\) is somewhat similar in nature to the matrix quotientl(t) since the displacementq is the slow variable. The ratio\(\dot q/q\) assumes the time independent limit −ν − on the fast time scale, 1/ν +, and this limit is independent of the initial displacement as well as of the initial momentum. The asymptotic relation\(\dot q = - \Gamma _ - q\) may be understood as resulting from an adiabatic elimination of the momentum. Closer analogies exist with multidimensional stoachastic processes like the Ornstein-Uhlenbeck process [16] and the laser [17]
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Our summation convention is to count all equivalent pairs (and triples) of spins once
We could allot the configuration-independent flips of first-neighbor pairs of spins toL 1, i.e. takeψ 0 to be of first order. This modification is discussed in Sect. 9
For appropriate choices ofp andf these relations include the corresponding ones of [1, 5, 12]
See, e.g., Niemeijer, Th., Leeuwen, J.M.J., van: In: Physe transitions and critical phenomena. Domb, C., Green, M.S. (eds.). London: Academic Press 1976
It is because of the linearity of the magnetic recursion relations and because ofh * i =0 that we need not modify theh i
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Haake, F., Lewenstein, M. & Wilkens, M. Renormalization group and high-temperature series for the two-dimensional ising model. Z. Physik B - Condensed Matter 54, 333–350 (1984). https://doi.org/10.1007/BF01485831
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DOI: https://doi.org/10.1007/BF01485831