# 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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### 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: "...
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 ...
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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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 $... 1answer 93 views ### Reaching critical point in a fluid I have carbon dioxide in a pressure reactor. I can control both temperature and pressure inside the container (or, equivalently, temperature and amount of fluid). I need to reach the critical point in ... 1answer 218 views ### Free energy second derivative at critical point of gas-liquid transition In 2 lines this book says that the second derivative of the thermodynamic Helmholtz free energy density$a\left(\rho,T\right)$with respect to density of a one-component fluid,$\rho$, when we ... 1answer 249 views ###$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$): $$\... 0answers 37 views ### 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' (... 1answer 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 ... 0answers 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 ... 1answer 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 ... 2answers 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.... 1answer 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 ... 1answer 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 ... 0answers 78 views ### 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 ... 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 ... 0answers 100 views ### 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 ... 1answer 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, ... 1answer 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, ... 0answers 83 views ### 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 ... 2answers 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 ... 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 ... 1answer 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 ... 1answer 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 ... 1answer 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 ... 0answers 43 views ### 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:... 2answers 86 views ### 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. ... 1answer 67 views ### 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 ... 1answer 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 ... 4answers 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 ... 1answer 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{\... 0answers 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 ... 1answer 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{... 1answer 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 ... 2answers 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 ... 1answer 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, ... 2answers 73 views ### 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 ... 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 ... 3answers 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 ... 1answer 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 ... 1answer 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 ... 3answers 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 ... 0answers 147 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} = \... 3answers 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 ... 1answer 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}$... 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 ... 0answers 93 views ### 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. ... 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$...
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 \$\...