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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134 views

Why does Critical Points have fluctuations on all scales (Infinite correlation length?

I have been studying statistical field theory for a while and I still haven't found a physical explanation for this question. Every answer seems to be kind of circular. Basically something like this: "...
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148 views

Long Range order in 2D Ising model

We know from the exact solutions for 2D Ising model on square lattice the long range order appears bellow critical temperature, but how does this agree with the Mermin-Wagner theorem, from which we ...
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54 views

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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1answer
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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1answer
93 views

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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1answer
218 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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249 views

$pp$ and $p\bar p$ scattering energy scaling exponents and 3d directed percolation model critical exponents similarity/equality, why?

$pp$ and $p\bar p$ scattering can be approximately described (in the Regge limit, that is, when $s \gg m \gt |t|$) by the exchange of Reggeons defined by the following Regge trajectory (low $s$): $$\...
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Why does the RG group flow's linearization provide an eigenbasis at fixed points?

I'm reading Conformal Field Theory by David Sénéchal, Philippe Di Francesco, and Pierre Mathieu. Let $T$ be the map that generates the renormalization (semi-)group by taking couplings $J$ to $J'$ (...
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130 views

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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88 views

What happens to the dynamical critical exponent in the quantum-classical mapping?

It is well-known that one can, e.g., map the classical 2D Ising model to the 1D quantum Ising chain. Moreover, their critical points are related. Hence, if one is interested in critical exponents ...
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368 views

Why is the upper critical dimension of the Ising model 4?

I have read in various sources, that the critical exponents of the Ising Model are identical to the meanfield ones for dimensions $d \geq 4$. In trying to understand this better I came across the ...
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286 views

Why does a vanishing energy gap indicate a phase transition?

More concretely: When looking at the Ising model in the description of Bogoliubov fermions, we get an explicit expression for the energy gap, that vanishes for a particular value of the magnetic field....
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58 views

Is there any positive temperature from which superconductivity ceases?

From what I understand about superconductivity, it is due to a coupling between Cooper pairs and phonons. At the absolute 0, there is no phonon, so I assume superconductivity cannot exist at that ...
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84 views

Apparition of scale invariance

When did "scale invariance" started to be seen as an important concept in the theory of phase transition? Phase transition and critical points started to be investigated in earnest in the middle of ...
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Mean field critical exponents and the Gaussian approximation?

A while a go I asked this question on the difference between mean field theory and the Gaussian approximation. This question is related to that. The mean field critical exponents for the Ising model ...
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1answer
219 views

Correlation length at low temperatures?

The correlation length gives (approximately) the distance over which a spin flip has an effect. For systems with ordered phases, at low temperatures the correlation length is then small (since a ...
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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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77 views

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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42 views

Is it sufficient to consider a small part of a system (without potential energy sources which can be released) to determine if it's chaotic?

The world around us abounds with chaotic systems: dripping taps (when a certain dripping rate is reached the dripping becomes irregular, which can be seen in this old but very entertaining video, ...
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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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223 views

Is the Landau free energy scale-invariant at the critical point?

My question is different but based on the same quote from Wikipedia as here. According to Wikipedia, In statistical mechanics, scale invariance is a feature of phase transitions. The key ...
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1answer
74 views

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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425 views

Primary Operators in the Ising CFT

The 1D Ising model at criticality is given by the Hamiltonian $H=-\mathcal{N} \sum_i (\sigma^x_i \sigma^x_{i+1} + \sigma_i^z)$ in terms of Pauli operators and a normalization $\mathcal{N}$. In CFT ...
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107 views

Why is it interesting to study “quantum quench” at a critical point?

In the presentation, "Quantum Quenches in Extended Systems", by S. Sotiriadis, P. Calabrese and J. Cardy, it was pointed out that quatum quench through a critical point remains an open problem. Why ...
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67 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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Relationship between the validity of mean field treatment and the strength of coupling/interaction

Mean field is a quite common treatment in studying phase transition and critical phenomena, although it neglects fluctuations. Imagine we have a Hamiltonian consists of free part and interaction part:...
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Percolation theory: What is the critical amplitude for the “backbone” of a 2-D network?

Disclaimer: I am just learning about percolation theory for the first time, so I am not too familiar with some of the terminology. Suppose you have a 2-D square lattice with bonds connecting sites. ...
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Ain't “the straw that brakes the camel's back” an example of a critical phenomenon, instead of chaotic behavior?

In this old but very interesting video (the part where they show two concentric cylinders, Couette cells, in which the visible liquid shows very strange behavior if the velocity with which the ...
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387 views

Why is the critical exponent $\alpha$ negative at the Ising spin-glass transition?

The specific heat usually diverges at a phase transition - typically as a power-law, giving a critical exponent $\alpha > 0$. (Although in 2D, sometimes the divergence is only logarithmic, as with ...
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768 views

Why is the correlation function a power law at the critical point?

I’m taking my first exam in statistical field theory and critical phenomena. I’ve reached a point in which we use the fact that the pair correlation function decays as a power law at the critical ...
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42 views

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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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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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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484 views

Scale invariance at phase transitions

The Wikipedia entry for scale invariance states In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical ...
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328 views

What's about the critical exponents and RG flow in upper critical dimension $D=4$?

We know when $D>4$, i.e. $D$ larger than upper critical dimension, then critical exponents are exactly same as the ones of mean field . When $D<4$, critical exponents are not given correctly by ...
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108 views

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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Is there a phase coexistence line below the critical temperature $T_c$ in normal to superfluid transition?

In the case of water, there is a phase coexistence line (called the liquid-gas coexistence curve) which ends in a critical point. And this line separates the gas phase from the liquid phase in the P-T ...
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1answer
300 views

What is the universal definition of the order parameter that is valid irrespective of the nature of the phase transition?

Plausible definition Consider a phase transition from phase 1 to phase 2. The order parameter is zero in one of the phases 1 or 2 and nonzero in the other. For example, in normal (phase 1) to ...
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824 views

Are the first order phase transitions always associated with a latent heat?

Is the first order ferromagnetic transition below the critical temperature associated with latent heat? For example, the transition of ferromagnetic configuration with all its spins aligned up to a ...
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397 views

Isn't the transition at the critical point always a continuous phase transition?

At page 145 of Chaikin and Lubensky's Principles of Condensed matter physics, there are two figures 4.0.1(a) and 4.0.1(b). Figure (a) shows that at $T=T_c$, there is a continuous transition (the order ...
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300 views

What is the meaning of “Deconfined Quantum Critical Point”?

I tried to google it but couldn't found an intuitive explanation (not existing in Wikipedia). I have also tried to read the Science paper by T. Senthil et al, but couldn't fully understand the ...
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919 views

Free energy functions are analytic or non-analytic in phase transitions?

I already saw this Phys.SE post and it seems perfectly reasonable that the free energy describing a system must be a non-analytic function in order to display a phase transition. An analytic ...
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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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354 views

What does “no characteristic length or time scale” mean?

When looking into the topic of "self-organized criticality," (SOC) one often comes across descriptions of SOC as a state where "the system has no characteristic length or time scale." (Examples here ...
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150 views

Can thermodynamics be relied at the critical point?

Thermodynamics neglects fluctuations and deals with average macroscopic quantities (which is also important for the extensity of extensive thermodynamics coordinates e.g., the internal energy ${\rm U}$...
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1answer
862 views

How to evaluate the critical exponents of modified van der Waals equation?

The given modified van der Waals equation is $$(P+(a/v)^{n})(v-b)=RT$$ where $(n>1)$. What is the physical significance of the power $n$ in the above equation. How could one evaluate the critical ...
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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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1answer
452 views

Critical exponents and scaling dimensions from RG theory

In most books (like Cardy's) relations between critical exponents and scaling dimensions are given, for example $$ \alpha = 2-d/y_t, \;\;\nu = 1/y_t, \;\; \beta = \frac{d-y_h}{y_t}$$ and so on. Here $...
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1answer
129 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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Kleinert's Variational Perturbation Theory

I've been reading Kleinert's book and have been very intrigued by the chapter on variational perturbation theory. Namely, Kleinert derives a very good strong-coupling approximation to the ground state ...