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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What determines the specific value of the order parameter in spontaneous symmetry breaking?

Three examples in the spontaneous symmetry breaking that occurs at a phase transitions: A ferromagnet below the Curie temperature chooses an axis of quantisation along which all the spins align, ...
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Correlation length amplitudes in Ising 2D model

I am reading the article about Universal amplitude ratios in the 2D Ising model (https://arxiv.org/abs/hep-th/9710019) by G. Delfino. I have a question about page 3 of the paper. For a magnetic ...
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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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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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Phase diagram of the Z_N-invariant clock model

A well-known 1985 paper by Fateev and Zamolodchikov "Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in Z_N-symmetric statistical ...
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Example of critical (non-relativistic) quantum field theory in 1D?

Is there an example of a critical non-relativistic bosonic quantum field theory in 1D (no time)? So, the field theory can be describe by annihilation, $\psi(x)$, and creation operators, $\psi^\dagger(...
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References or resource recommendation for the mathematics concerning fission

I am working on a statistical problem that appears similar (in some respects...) to nuclear fission. I am interested in the properties of a system undergoing fission around, or near, delayed ...
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209 views

Assumptions behind Ornstein-Zernike correlation function

Let $S(\mathbf q)$ be come correlation function in Fourier space ($\mathbf q$ = wavevector). In the study of condensed matter systems, I have often encountered the statements that a reasonable form ...
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373 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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891 views

Infinite-range 1D Ising model

The Hamiltonian for this system is given by \begin{equation} \mathcal{H} \{S\} = -H\sum_i S_i - \frac{J_0}{2} \sum_{ij} S_i S_j, \end{equation} where $H$ is the external magnetic field and there is no ...
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Some Questions about the Critical Point

I'm currently trying to understand the physics of phase transitions and I'm having a hard time doing that. First of all, the discussions on the topic seem to be confusing and there is no methodical ...
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213 views

Why are RG flow fixed points associated with different phases?

Why are RG flow fixed points associated with different phases? I thought the RG makes only statements about behavior near to critical points... a definite phase is far away from the critical point, ...
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$pp$ 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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840 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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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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257 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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429 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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Defining renormalization factor

In textbooks it's stated that one should renormalize the field strengths in addition to coarse graining and rescaling by a factor of for example $z$. For a scalar $\phi^4$ theory they choose $z$ to be ...
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245 views

In renormalization group, What is the fixed point “control”?

I understand we have a fixed point in the couplings ("K") space (or in the scaling variable space). Then, there is a critical surface, which is attracted to it. This is a part of a system with some ...
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Substance below Critical Temperature but above Critial Pressure

I know that above the critical temperature and pressure, at the boundary between the liquid and gas phase disappears. But does this also hold when temperature is below critical temperature but ...
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AdS/CFT and Kondo problem/ Ginzburg-Landau theory

I was reading the review on Unconventional superconductivity by Mike Norman, towards the end (page 22) he comments two things about AdS/CMT: "In the condensed matter context in two dimensions, one ...
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356 views

Ising model Monte Carlo simulations in 4D and 5D

I'm going to be simulating the Ising Model in 4D and above to calculate spin-spin correlations and critical exponents and am wondering how to tackle this algorithmically. For example, in 1D, use an ...
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Relation between scaling dimension and critical exponents for harmonic peturbations in $O(N)$ Wilson-Fisher (WF) in an old paper

I am reading the paper "Harmonic perturbations of generalized Heisenberg spin systems" (D J Wallace and R K P Zia, 1975) - https://iopscience.iop.org/article/10.1088/0022-3719/8/6/014/meta . The ...
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Relation between mean field critical point and RG critical point

In the mean field / Landau picture a critical point is where the potential of the order parameter changes curvature. E.g. the mean field potential of a scalar $\phi^4$ theory is $$\mathcal{L} = a t \...
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How to quantify frustration for spin models with long range interactions?

Consider the following Hamiltonian: $$ H=-\sum_{i\neq j}J_{ij}S_iS_j-\sum_i H_iS_i $$ where $S_i\in\{-1,1\}$, and the summed pair $i,j$ can be any two distinct indices (not necessary adjacent spins)....
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Difference between domain size and correlation length in ferromagnetic materials?

I am getting confused about different length scales in magnetic materials. I understand that the correlation length for a ferromagnetic materials is defined as <(s(x)−<(s(x))>)(s(y)−<(s(y))>)>...
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At what critical Reynolds number does vortex shedding begin?

In: "Fluid Dynamics", Chapter 3 (Turbulence), Section 26, Landau and Lifchitz analyze the problem of the stability of a steady flow past a body of finite size. The fluid is assumed to be ...
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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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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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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 ...
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Can the terms in the microscopic model with nonzero conformal spin generate some new term(s) under RG (renormalization group) flow?

As in the book Bosonization and Strongly Correlated Systems at page 66, it says that "We see that the original perturbation with nonzero conformal spin generates the perturbation with zero conformal ...
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Books/resources for statistical field theory

I was wondering if anyone knows good, approachable textbook or other resources about statistical field theory (topics like in Kardar's Statistical physics of fields: lattice models, mean field theory, ...
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214 views

Neel order and O(3) model

The coarse grained fluctuations of the Neel order parameter in the half integer spin anti-ferromagnetic Heisenberg model is described by the O(3) non-linear sigma model with a strange berry phase term....
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121 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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573 views

Good layman definition of the critical point(phases) and supercriticality

I've heard of this point among others, but never really got what it meant. Wikipedia makes one's head spin. The only thing I picked up is that it occurs between liquid and gas, and displays ...
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What is the state of water at exactly 0°C?

Theoretically speaking, what is the state of water at bang on 0°C - not any lower or higher? Any lower would make it a solid whereas any higher would make it a liquid. But what about bang on 0°C? ...
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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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186 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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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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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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375 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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164 views

Is a system of free spinless fermions always critical?

Consider a system of free spinless fermions, whose Hamiltonian can be written as $$ H = \sum_{i,j}h_{ij}a_i^\dagger a_j-\lambda\sum_i a^\dagger_i a_i $$ with $h_{ij}=h_{ji}^*$ scalars and $a_i^\dagger$...
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202 views

Applicability of Cardy's “doubling trick” to the 2D Ising Model

In Section 11.2.2 of the book on Conformal Field Theory by di Francesco, Mathieu, and Senechal (page 417), the two point function on the Upper Half Plane is written as being equal to the four point ...
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324 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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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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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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364 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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171 views

Criticality in BCS Theory

Can someone provide me with a pedagogical introduction into the role of criticality in BCS theory? The QCD condensate is due to strong coupling. The BCS condensation involves only weak coupling - ...
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237 views

Is the Landau Free Energy U-TS or βH?

I'm having a hard time figuring out the physical meaning of the Landau Free Energy density: $$f(\phi,\nabla\phi,T) = \frac{1}{2}|\nabla\phi |^2 + \frac{a(T-T_c)}{2}|\phi |^2 + \frac{b}{4}|\phi |^4$$ ...
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345 views

Is water a gas at critical density, room temperature?

I am quoting Chaikin, Lubensky, Principles of Condensed Matter Physics, p. 4. Now suppose we have a closed container of water vapor at a density of 0.322 g/cc at room temperature. As the ...