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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160 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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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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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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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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72 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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292 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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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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65 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

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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321 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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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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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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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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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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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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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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101 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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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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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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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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244 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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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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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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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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755 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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383 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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106 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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180 views

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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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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346 views

Beyond the Ginzburg-Landau-Wilson theory/renormalization group

In the famous seminal paper by K G Wilson and J Kogut in Physics Reports (Aug 1974) on The renormalization group and the ε expansion, they achieve a pinnacle of uniting the Ginzburg-Landau theory and ...
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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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407 views

Landau and Mean Field Theory

I have often heard that the Landau theory of phase transitions is a mean field theory. Why is this so? What is the connection between the two ideas? One stresses symmetry breaking and one averages the ...
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456 views

Failure of Hertz-Millis-Moriya theory for quantum phenomena

In the quantum critical phenomena of condensed matter, the earlier work by Hertz, Moriya and Millis develope the the Hertz-Millis-Moriya (HMM) theory of quantum phase transition. Naively, they ...
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179 views

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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Why do spin correlation functions in Ising Models decay exponentially below the critical temperature?

I'm trying to form a better understanding of the 2D Ising Model, in particular the behaviour of the correlation functions between spins of distance $r$. I've found a number of explanatory texts that ...
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147 views

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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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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Why correlation length diverges at critical point?

I want to ask about the behavior near critical point. Let me take an example of ferromagnet. At $T < T_c$, all spins are aligned to the same direction thus it is in the ordered state, scale ...
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110 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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666 views

The relation between critical surface and the (renormalization) fixed point

In the book, I read some remarks about the criticality: Iterations of the renormalization (group) map generate a sequence of points in the space of couplings, which we call a renormalization group ...
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Fluids with critical point at ordinary temperature and pressure

Are there any fluids with critical point near STP or that are supercritical at STP? If not would it be feasible to design a molecule for a substance with critical point near STP using theoretical/...
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158 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 $a_i^\dagger$...