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Physical Review Letters

Measurement-Noise Maximum as a Signature of a Phase Transition

Zhi Chen, Clare C. Yu

Published Jan 30, 2007• 26 citations
Publisher

Abstract

We propose that a maximum in measurement noise can be used as a signature of a phase transition. As an example, we study the energy and magnetization noise spectra associated with first- and second-order phase transitions by using Monte Carlo simulations of the Ising model and 5-state Potts model in two dimensions. For a finite size system, the total noise power and the low frequency white noise S(f<f(knee)) increase as T(c) is approached. In the thermodynamic limit, S(f<f(knee)) diverges but f(knee)-->0 and the total noise power vanishes. f(-1)(knee) is approximately the equilibration time.

DOI
10.1103/physrevlett.98.057204

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We propose that a maximum in measurement noise can be used as a signature of a phase transition. As an example, we study the energy and magnetization noise spectra associated with first- and second-order phase transitions by using Monte Carlo simulations of the Ising model and 5-state Potts model in two dimensions. For a finite size system, the total noise power and the low frequency white noise S(f<f(knee)) increase as T(c) is approached. In the thermodynamic limit, S(f<f(knee)) diverges but f(knee)-->0 and the total noise power vanishes. f(-1)(knee) is approximately the equilibration time.
We study the energy and magnetization noise spectra associated with first and second order phase transitions by using Monte Carlo simulations of the Ising model and 5-state Potts model in 2D. For a finite size system, the total noise power and the low frequency white noise S(f < fknee) increase as Tc is approached. In the thermodynamic limit S(f < fknee) diverges but fknee → 0 and the total noise power vanishes. f-1knee is approximately the equilibration time. At high frequencies S(f > fknee) ~ f-μ. For the Ising model, we relate μ to the critical exponents.
We provide a general scheme to predict and derive the contribution to the noise spectrum of a stochastic sequence of pulses from the distribution of pulse parameters. An example is the magnetization noise spectra of a two-dimensional Ising system near its phase transition. At $T\ensuremath{\le}{T}_{c}$, the low-frequency spectra is dominated by magnetization flips of nearly the entire system. We find that both the predicted and the analytically derived spectra fit those produced from simulations. Subtracting this contribution at Tc{T}_{c}Tc​ leaves the high-frequency spectra which follow a power law determined by the critical exponents.
Using thermodynamic fluctuation theory, we study the finite-size rounding of anomalies occurring at first-order phase transitions of the corresponding infinite system. Explicit expressions for thermodynamic functions are derived both for "symmetric transitions" (such as the jump of the spontaneous magnetization in the Ising model from +Msp+{M}_{\mathrm{sp}}+Msp​ to $\ensuremath{-}{M}_{\mathrm{sp}}$ as the field changes from 0+{0}^{+}0+ to ${0}^{\ensuremath{-}}$) as well as for asymmetric cases, but restricting attention to (hyper)cubic system shapes. As an explicit example for the usefulness of these considerations in Monte Carlo simulations where it may be a problem to (i) locate a phase transition and (ii) distinguish first-order from second-order transitions, we present numerical results for the two-dimensional nearest-neighbor Ising ferromagnet in a field, both below the critical temperature Tc{T}_{c}Tc​ and at Tc{T}_{c}Tc​. The numerical results are found to be in very good agreement with the phenomenological theory and it is shown that one may extract the magnitudes of jumps occurring at first-order phase transitions in a well-defined and accurate way.
First-order phase transitions are ubiquitous in nature but their presence is often uncertain because of the effects which finite size has on all transitions. In this article we consider a general treatment of size effects on lattice systems with discrete degrees of freedom and which undergo a first-order transition in the thermodynamic limit. We review recent work involving studies of the distribution functions of the magnetization and energy at a first-order transition in a finite sample of size N connected to a bath of size N′. Two cases: N′ = ∞ and N′ = finite are considered. In the former (canonical ensemble) case, the distributions are approximated by a superposition of Gaussians corresponding to the different phases; all finite-size effects then vary as N or 1/N. The latter case involves the Gaussian ensemble where the entropy of the bath has a convenient form; for small N′, first-order transitions are characterized by van der Waals' loops in (for example) the energy vs. temperature curves. Results from extensive Monte Carlo simulations of Ising and Potts models in d = 2 are presented to confirm the predictions. Comparison is made with data from second-order transitions to show that the order of a transition can be distinguished through such studies, although problems still occur for first-order transitions very close to critical points.
Multifractal properties of the two-dimensional Ising model are studied by means of the $f(\ensuremath{\alpha})$ spectra of singularities of probability measures supported by energy spectra. These measures are determined through calculations of energy degeneracies for finite-size systems up to $12\ifmmode\times\else\texttimes\fi{}12$ spins in cases of square and triangular lattices and up to $22\ifmmode\times\else\texttimes\fi{}22$ spins in the case of the hexagonal lattice. The calculations are performed in an exact manner using the transfer-matrix method. It is argued that, in the thermodynamic limit, the scaling exponent ${\ensuremath{\alpha}}_{\mathrm{max}}$ associated with the most probable energy of the system takes at the critical temperature a minimum value and, consequently, it is argued that a given system reveals a phase transition at a finite temperature if ${\ensuremath{\alpha}}_{\mathrm{max}}$ possesses a minimum at the finite temperature. It is also shown that, in the thermodynamic limit, the spectrum $f(\ensuremath{\alpha})$ shrinks when the critical temperature is approached.
The total many-body correlations present in finite temperature classical spin systems are studied using the concept of mutual information. As opposed to zero-temperature quantum phase transitions, the total correlations are not maximal at the phase transition, but reach a maximum in the high-temperature paramagnetic phase. The Shannon mutual information and the Renyi mutual information in both Ising and Potts models in two dimensions are calculated numerically by combining matrix product state algorithms and Monte Carlo sampling techniques.