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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Value of critical exponent $\alpha$ for 2D ising model

The Onsager solution for specific heat is $$C\approx -Nk\frac{2}{\pi}\bigg(\frac{2J}{kT_c}\bigg)^2\ln\Big|1-\frac{T}{T_c} \Big|\qquad (T \textrm{ near } T_c)$$ Critical exponent $\alpha\neq 0$. ...
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Nonzero spontaneous magnetization in two-dimensional Ising model

The two-dimensional Ising model with the nearest-neighbour interactions enjoys a $\mathbb{Z}_2$ symmetry under $S_i\to -S_i$; it displays sponatebous symmetry breaking at a finite temperature $T_C=2J[...
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Why does a divergent correlation function (not length) imply lack of the order in the system?

Consider a two-point correlation function defined as $$G_{ij}({\bf x},{\bf x}^\prime)\equiv \Big\langle\Big(\mathscr{O}_i({\bf x})-\big\langle\mathscr{O}_i({\bf x})\big\rangle\Big) \Big(\mathscr{O}_j({...
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What's the definition of Goldstone Mode?

My question is how to define a Goldstone Mode? Initially I thought that Goldstone Mode is a consequence of spontaneous symmetry breaking, but later I learned that in Kosterlitz–Thouless transition, ...
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Questions about Kosterlitz–Thouless (KT) transition

Why we extend $\theta$ from $(0,2\pi)$ to $(-\infty, \infty)$? I mean we cannot measure $\theta$ in experiment, can we? Secondly,the feature of vortex solution (at least in KT transition) can be ...
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Defining upper critical dimension

Considering the usual Landau functional of the form: $$ \beta L[\phi] = \int d^D r [\frac{1}{2} |\nabla \phi(r)|^2 + \frac{r_0}{2} |\phi(r)|^2 + \frac{u_0}{4} |\phi(r)|^4 ] $$ In searching for the ...
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Stuck with the derivation of correlation function from Huang's Statistical Mechanics

Context Section $16.2$ of Kerson Huang's Statistical Mechanics ($2$nd edition) deals with a derivation of two-point correlation function $\Gamma({\bf r})$, defined in terms of an order parameter ...
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For spontaneous order, how does the correlation length $\xi(T)$ change for $T<T_C$?

The correlation length, $\xi(T)$, defined from the correlation function $$C(r)\sim \frac{1}{r^{d-2+\eta}}e^{-r/\xi(T)},~~T>T_C\tag{1}$$ is exponentially small above the transition temperature ($T&...
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Renormalization of mass: does it change sign from high temperature to low temperature?

Consider a Landau Ginzburg theory for ferromagnets with Hamiltonian $$H=\int d^{D} x \frac{1}{2}(\nabla\phi(x))^{2} + \frac{1}{2} \mu^{2} \phi^{2}(x) + \frac{\lambda}{4!}\phi^{4}(x)$$ I can compute ...
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Non-zero order parameters in the disordered phase

We often learn that the order parameter is a perfect tool for the study of a phase transition (assuming Landau-Ginzburg theory is applicable). The order parameter is finite in the ordered phase, and ...
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Non-local order parameter for Kosterlitz–Thouless transition

It is known, that there's no local order parameter in Kosterlitz-Thouless transition. Is the order parameter in Kosterlitz-Thouless transition non-local?
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Power law behaviour at phase transition point & presence of fluctuations of all length scales

In physics, exponentials such as $\exp(-r/\xi) $ typically come with a natural length scale $\xi$ while power laws such as $\sim 1/r^n$ don't (at least not readily visible or identifiable). At the ...
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Ferromagnetic/Paramagnetic Phase Transition in a Non-Zero External Magnetic Field

I'm new to condensed matter theory, especially spin-glass systems. I understand that the Ising model exhibits a Phase Transition when there is no external magnetic field (h=0). And that at the ...
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Critical exponent relation for neural avalanche dynamics

I am trying to understand the origin of equation (4) in this paper. Per the arguments of Touboul and Destexhe, a power law distribution for avalanche size and duration, as well as observed data ...
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How to find the critical exponent of some directional dependent correlation length?

I am working on a two dimensional anisotropic system with correlation length diverging with different critical exponent in different directions. And I am wondering if there is any theoretical ...
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Critical supercurrent

What is meant by the concept of critical current when talking about superconducting phase diagrams and transitions? How does this relate to critical field and temperature? Thanks
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Three point correlation function 2D Ising model

What is the expected behaviour of the three point function $<\sigma_i \sigma_j \sigma_k>$ of the Ising 2D model at the critical point where conformal symmetry is valid? Do they have a power-law ...
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Ising model 1D spontaneous magnetisation

What does it mean to 'compute the spontaneous magnetisation'? According to wikipedia: 'Spontaneous magnetization is the appearance of an ordered spin state (magnetization) at zero applied magnetic ...
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Various questions on renormalization in lattice systems

Forgive the long, multi questioned-question. The setting of this question is inspired by this answer. Consider some theory on a lattice, for example the 2D $0$-field Ising model $$H=-K\sum_{\langle i,...
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Renormalization Group Flows

In the Renormalization Group flows, why there are two fixed points: Gaussian and Wilson Fisher and does the Gaussian Fixed point describe the critical behaviour of the system?
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What is a good definition for what this article talking about when it refers to “Universal Physics”?

I was puzzled when I read "Precise measurements find a crack in universal physics by Ingrid Fadelli" (Phys.org, Jan. 15, 2020). The article has some vague statements in the opening paragraphs about "...
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Approximately how long after forming a critical mass of fissionable material does it explode?

Just making up some quantities of variables to reduce the “depends” answers/comments. Given, say: enriched uranium mass of 1.5 x critical mass spherical shape brought together “quickly” (a few ...
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Yang-Lee article on phase transition (1952): possible problem in lemma demonstration

I'm studying the first article of Yang-Lee on phase transition C.N.Yang and T.D.Lee, Physical Review, ”Statistical Theory of Equations of State and Phase Transitions. I. Theory of ...
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Book recommendation for statistical physics

Before I start a PhD in Quantum Information I would like to study a bit of statistical physics. In particular I am interested in superfluids, critical phenomena, topological phases of matter and all ...
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Order parameter to quantify clustering?

I have a 1D system containing $N$ particles having positions $\{x_1(t),x_2(t),\dots,x_N(t)\}$ in a box of size $L$ with periodic boundaries. The number of particles is conserved. The dynamics of the ...
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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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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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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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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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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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$CP^N$ 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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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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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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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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why do matrix product states work at critical point?

Matrix product states satisfy the entanglement area law, which should be a property of gapped states. But usually, MPS work well in 1D quantum phase transition problems. As far as I know, ...
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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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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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What are the excitations in the near critical 2D-Ising model in a magnetic field?

Apparently it is well known that the 2D Ising model with $T=T_C$ in a small magnetic field has a mass gap and correlation length $\xi \sim h^{- \frac{8}{15}} $. Further, in a paper in 1989 ...
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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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Renormalization Group - Scaling fields and physical critical exponents (1D Ising model)

This is related to this question: Critical exponents and scaling dimensions from RG theory. TLDR: How to compute physical critical exponents $\alpha, \beta, \gamma, etc$ from the RG exponents when ...
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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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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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Integrability of a non-integrable quantum spin model at critical point

Is it right, that non-integrable quantum spin models in one dimension become integrable at their critical points? Or do they stay nonintegrable at the critical point also? Are there any examples known?...
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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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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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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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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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Divergent Coulomb integrals in superfluid fluctuations

In Chapter 3 of Kardar's statistical physics of fields, in the context of lower critical dimension, he works out an example about superfluids where starting from the probablity of a particular ...