Questions tagged [critical-phenomena]

The physics of critical phenomena is the physics of systems close to a critical point, like the critical temperature in a ferromagnetic transition or the critical point of a gas-liquid transition. Examples of critical phenomena include dynamical slowing down, divergence of correlation length and ergodicity breaking.

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Andrew's experiment

In Thomas Andrew's experiment, consider the dome shaped saturation region. If we increase the pressure at constant volume until we reach the critical point, why does the density of vapours rise and ...
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What is the idea behind coarse-graining?

I don't think I fully understand the idea behind coarse-graining. I will elaborate. I was reading some lecture notes on statistical field theory and the text begins with some previous analyses on the $...
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65 views

Critical exponent mean field Ising model

I am given the following expression for the free energy: $$f = \frac{1}{2}r_0 m^2+um^4+vm^6,$$ where $r_0=k_B (T-T_c)$ with $T_c$ the critical temperature and $u=\frac{1}{12}k_B T$ and $v=\frac{1}{...
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Could the Odderon intercept be equal to $\alpha_\mathbb {O}(0)=0.813$? [closed]

The directed percolation dynamical universality class is characterized by just three independent critical exponents. These exponents are (in a 3d space): $$\beta=\beta'=0.813(9)$$ $$\nu_\perp=0.584(5)...
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Flow velocity ambiguity in transition region

I have calculated the following short table concerning the stationary flow of water in a 1 meter long pipe with a diameter of 16 mm and a completely smooth inner surface. The flow is driven by the ...
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78 views

Why the correlation function of 2D classical XY model is written so?

2D classical XY model $$H = -J\cos(\theta_{i}-\theta_{j})%$$ is famous for Berezinskii-Kosterlitz-Thouless phase transition. This is because of the difference of correlation function between hot and ...
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70 views

How do we understand the results of $1/N$ or $\epsilon$ expansion beyond leading orders?

When we do $1/N$ expansions in, say, 2+1$D$ $O(N)$ models and try to extract all kinds of critical exponents from it, we get the following results for the scaling dimensions of various operators up to ...
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130 views

Wolff cluster update in Monte Carlo simulation - at critical temperature [closed]

A general question to the Monte Carlo experts. When I use Wolff algorithm for global updates, say for Ising 2d, I always flip at least one spin (the initial random spin in the cluster). So, near the ...
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134 views

Ising Model with site-dependent magnetic field

Consider an Ising system in an external field, which is different at different sites. The Hamiltonian of the system is given by $H = -J\sum_{<i,j>}^{}s_i s_j - \sum_{i}^{} h_i s_i$ Here each ...
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53 views

Trouble with critical expotent β

In the context of the Landau Theory of phase transitions, applying the mean field theory in an attempt to describe transitions such as the Nematic - Isotropic, the Landau energy density is given by $...
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Reaching critical point in a fluid

I have carbon dioxide in a pressure reactor. I can control both temperature and pressure inside the container (or, equivalently, temperature and amount of fluid). I need to reach the critical point in ...
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What can happen on the “other side” of Berezinskii–Kosterlitz–Thouless (BKT) transition?

There is a generalized concept of Berezinskii–Kosterlitz–Thouless (BKT) transition in any dimension [not just in 2 dimensional classical system or 1+1 dimensional quantum system], such that the ...
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66 views

Compactification of Bosonic Closed Strings on $T^2$ and $T^3$

I am looking for a text to explain compactification of bosonic closed strings on $T^2$ and $T^3$ by focusing on its gauge groups enhancement. In fact, I want to know in each case ($T^2$ and $T^3$) how ...
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42 views

What's that theorem? All RG operators flow to directed percolation?

I can't for the life of me get this theorem straight. I can remember neither the name nor it's statement and would be grateful for anyone who wants to toss out some wisdom. It pertains to the ...
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Critical exponents from Ising free energy at zero magnetic field?

Consider the free energy of an Ising model $f(J,h,T)$, where $J$ is the coupling between neighboring sites, $h$ is the magnitude of a homogeneous external magnetic field and $T$ is the temperature. ...
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138 views

What is the anomalous dimension for two-dimensional multi-component spin systems?

My question is: What is the (predicted) anomalous dimension $\eta$ for the two-dimensional $n$-vector model (or $O(n)$ model)? Note: I am not looking for a derivation of $\eta$. A simple reference to ...
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State of the art results for locating critical points in 2D classical vertex models or 1D quantum spin chains

I'm looking for benchmarks for my own method, but I could not find an answer so easily. I've found so far: $\beta = 0.4407 \pm 0.0001$ for the 2D Ising model on a square lattice by Ghaemi, M., G. A. ...
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120 views

Bose-Einstein Condensation at higher critical temperature

The critical temperature $T_{c}$ of a Bose-Einstein Condensate is directly proportional to $n^\frac{2}{3}$, where $n$ is the density of the system which is to be condensed. The current $T_{c}$ for ...
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why when a circular superconductor cooled has magnetic filed?

i see in videos youtube that when a circular cooled about 200 kelvin it has some magnetic field around it,i think this Meissner effect is called , my question is how QM describe it??it is really ...
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147 views

What is the central charge of the disordered $q$-state Potts model, for large $q$?

The central charge of a model, is, heuristically related to the number of microscopic degrees of freedom. Is there a simple argument for the asymptotic behavior of the central charge for the ...
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58 views

Correlation function at zero distance

I'm confused about the definition of the correlation function (at equal time). I know it is defined from the thermal average of the scalar product of two random variables (for example the spins of a ...
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135 views

Does the critical dynamical exponent z of a 2D Ising model (simulated with Metropolis) vary with the temperature?

I have found in the literature that the critical dynamical exponent $z$ of an Ising model simulated with a local algorithm (such as Metropolis) is something around 2 near the critical temperature, ...
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188 views

Free energy second derivative at critical point of gas-liquid transition

In 2 lines this book says that the second derivative of the thermodynamic Helmholtz free energy density $a\left(\rho,T\right)$ with respect to density of a one-component fluid, $\rho$, when we ...
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Understanding this metaphor involving e-mails, chaos and phase transitions [closed]

I asked this question on the English Stack Exchange and people advised to try get the answer here. I can’t get the idea of metaphor in the last sentence of the following quote: Instead, email ...
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63 views

What is the length dimension in critical phenomena?

In this question it is said that: The best way to numerically work with continuous phase transitions is to study observables that have a vanishing length dimension (or mass dimension in the ...
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41 views

Critical Mass Exponents in $d=3$

I'm just a Bachelor student, so forgive me if my questions seem too silly. I want to show the convergence of the critical exponents in the Renormalization Group equations when $d=3$. When I construct ...
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49 views

$CP^N$ model in Peskin & Schroeder problem 13.3

In Peskin & Schroeder exercise 13.3 question d, it is asked to perform an expansion of the term $$iS =-N.tr\left[\log\left(-D^2-\lambda\right)\right]+\frac{i}{g^2}\int d^2x \lambda $$ where $D_{\...
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Susceptibility with a complex order parameter

I want to compute mean-field exponents in a theory that has a complex order parameter. So, let's say I have $$ F=\int d\vec x \left[ a|\psi|^2 - \frac{b}{2}|\psi|^4\right] \equiv \int d\vec x A[\psi,\...
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35 views

How to get mean field critical exponents for this Hamiltonian?

$$ \mathcal{H} = -J \sum_{\langle ij\rangle} \sum_{\alpha=1}^N s_i{}^\alpha s_j{}^\alpha -g \sum_{\langle ij\rangle} \sum_{\alpha\beta} (s_i{}^\alpha s_j{}^\alpha) (s_i{}^\beta s_j{}^\beta) $$ Above ...
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44 views

Derivation of the Ising free energy close to a critical point

In "Statistical physics of fields" Mehran Kardar states that the Ising free energy scales with, $$ f(t,h)\sim t^\alpha g_f\left(\frac{h}{t^\Delta}\right), $$ wherein $t=\vert T-T_c\vert/T_c$ ...
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What is the critical temperature for a BKT transition in the 2D quantum XY model with $S=1$ (not $S=1/2$)?

What is the critical temperature for a BKT transition in the 2D quantum XY model with $S=1$ (not $S=1/2$)? For instance, the classical XY model has KTc/J = 0.898 and the quantum XY model with S=1/2 ...
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Ferromagnet $\leftrightarrow$ paramagnet at Curie temperature

I think it's like this: $\, m=\tanh\left(\frac{Bμ}{k_bT}\right)$. If now the temperature decreases, then $\mu$ increases, until it flattens out ($\tanh$ function). Is the a point where $m$ flats out, ...
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How to deduce the formula of the Correlation Length on a periodic lattice?

Sometimes in Monte Carlo simulations we need to compute the correlation length, but this is a hard task without a formula. However, for instance, in an periodic cubic lattice of $L^3$ spins, some ...
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What’s the topology of critical region?

Duhem said the aim of physics is natural classification. I think topology and geometry are a wonderful way to link analogous parts among different phenomena. Thus we can classify and predict facts. ...
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35 views

Landau free energy expansion

On Huang page 417 he talks about Landau Theory and he says that in the neighborhood of a critical point where m(x) is small, we can expand the landau free energy $$\psi=\psi(m(x),H(x))$$ In powers ...
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34 views

What are phase transiton in different contexts?

I have come across the concept of phase transitions in various contexts. From simple phase transition between different states of matter like water to ice and so on, to phase transition in magnetic ...
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Two methods to find critical exponents from renormalization-group equations

Consider a renormalization-group flow for a set of quantities $(x_1, ..., x_N) \equiv \bf x$, which can be written in the form ${\bf x}_{t+1} = {\bf F}[{\bf x}_t, T]$,where $T$ is the temperature. At ...
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Aluminum critical current vs temperature fit

I have some data at different temperatures of Al's critical current (from 600 mK to 1.5K). Tc of Al is ~ 1.3. I am now trying to fit this data to a model to extract the theoretical critical current ...
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If a salt solution heated under a pressure slightly lower than critical pressure will it boil?

8.8% NaCl solution for example have a critical point of 450°C under 423 bar pressure. If we heat such solution to same temperature under slightly lower pressure i.e. 420 bar should it will boil or it ...
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System-size dependent phase transitions

I noticed that some physical phenomena require a system of size above some critical value to be observed. Two examples I know are: For a single-atom gold wire, there is a critical number of atoms to ...
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Question about Landau theory of Phase Transitions

The landau theory makes a mean-field approximation on the order parameter, which assumes that there are no fluctuations in the value of the order parameter at different sites (neglects the effects of ...
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What is the difference between these three definitions of specific liquid heat capacity?

This is an excerpt from the page 6.18 of book "Properties of gases and liquids, 5th ed". I can figure out the difference between the first one C_pL with other two, but cannot distinguish latter two, ...
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spontaneous symmetry breaking within critical phases

There are many examples of the spontaneous symmetry breaking in discrete symmetries which result in the gapped phases, such as dimerization phase of the quantum antiferromagnetic spin-1/2 chain which ...
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Landau theory; irrelevence of the lattice strcture?

In Ginzburg-Landau theory the derivative terms in the free energy depend on the structure of the lattice1. That said when looking at e.g. the O(3)-model the only derivative term kept is $$(\vec \nabla ...
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Trouble with critical exponents

I want to show that $$\frac{\langle S_iS_j\rangle}{\langle S_i\rangle^2}\rightarrow 0$$ in the ferromagnetic phase for dimension $d\geq 4$. My problem is the following: I know that $$\frac{\...
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86 views

Can thermal fluctuations be a source spatial variation in local value of the order parameter?

Usually, textbooks point out that such spatial variations of the order parameter (or order parameter "density") can arise due to inhomogeneous external fields e.g., the local magnetization $m(\textbf{...
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Does Heisenberg ferromagnet has inifinite number of phases below the critical temperature?

This is an upshot of the question here. The up-aligned and the down-aligned spin configurations are assumed to be two distinct phases in case of an Ising ferromagnet. But for Heisenberg ferromagnet, ...
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146 views

Mean field critical exponent

My notes (Gould and Tobochnik) come to the conclusion that the mean field critical exponent $\beta$ is 0.5 which is larger than the experimental value (in 3 dimensions) 0.3 so that: $$m_{mean} = \...
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119 views

Calculating Dynamic Critical Exponent in Continuum Field Theories

For a critical point controlled by $\delta$ described by a correlation length $\xi$ and a correlation time $\xi_\tau$, the critical exponent $\nu$ and the dynamic critical exponent $z$ is defined as $\...
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Criticality and the number of paths on a lattice

In the review "Scaling, universality, and renormalization: Three pillars of modern critical phenomena" by Stanley, he makes the following claim towards the end of the paper, which is neither derived ...