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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Measure of Lee-Yang zeros
Consider a statistical mechanical system (say the 1D Ising model) on a finite lattice of size $N$, and call the corresponding partition function (as a function of, say, real temperature and real ...
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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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Fisher exponent and fractal structure
In the context of critical phenomena, there are several critical exponents commonly used to characterize the singular behaviour at the point of phase transition. The Fisher exponent $\eta$ is defined ...
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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 does renormalization relate to emergence?
In statistical mechanics renormalization is often related to coarse-graining which in turn allows to calculate some macroscopic states. The resulting macroscopic description is sometimes called ...
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In general, do critical points of continuous phase transitions have $\beta =0$?
Consider a phenomenological modelling of a continuous phase transition, where the Lagrangian of the system is given by
$$L=\frac{a}{2}\phi^2+\frac{\lambda}{4}\phi^4-h\phi.$$
Here $\phi$ is the order ...
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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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How to calculate the emergent symmetry group in deconfined quantum critical point problem like $SO(3) \times Z_4 \to SO(5)$?
In the deconfined quantum criticality literatures like https://doi.org/10.1103/PhysRevX.9.041037, some equations are usually given: $$SO(3)\times Z_{4}\to SO(5),\\U(1)\times Z_{4}\to O(4),\\SO(3)\...
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Analytic change of free energy after renormalization
Suppose we have some model in statistical physics with Hamiltonian $H$ and partition function
$$Z=\mathrm{Tr}\left(e^{-H}\right) $$
the free energy per site is defined as
$$ f =\frac1N\log Z$$
A ...
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Nature of phase transitions in Kitaev honeycomb model
Short version of my question is this : what is the nature of the phase transition in the Kitaev honeycomb model ?
Longer version:
Kitaev honeycomb model undergoes a phase transition from a gapped to ...
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Critical temperature difference between Ising and XY model
The following formula gives the critical coupling (more precisely the ratio of the spin-spin coupling over the temperature) for $O(n)$ models on a triangular lattice:
$$\text{e}^{-2K}=\frac{1}{\sqrt{...
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Why is the supercritical fluid region a perfect rectangle?
The melting, sublimation and evaporation curves are all non straight lines in a (p,T) phase diagram, while the curves that divide liquid/gas regions from supercritical fluid region are perfectly ...
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Why is the energy of a vortex in a superconductor finite?
I just had a glimpse of the Ginzburg-Landau theory of superconductivity. I am surprised that that the energy of a vortex is finite. This is surprising because as far as I know, in superfluids, the ...
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Connection Between Renormalization Group and Phase Transitions
I have a couple of questions on the relation of RG and phase transitions. I've heard in many sources that the theory of most transitions (excluding novel phase transitions like Quantum Critical ...
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How much Curie temperature is obtained according to the Stoner model?
In Stoner model for itinarant ferromagnetism, the spin ordering of electrons is caused by the Coulomb interaction $U$ for onsite repulsion leading to the reduction of total energy against to the ...
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Non-renormalizable theory and mean field theory
For $\phi^4$ theory, when $d>4$, the theory becomes nonrenormalizable, but when $d>4$, we can use mean field theory to calculate the exact critical exponents. The intuition behind mean field ...
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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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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
$$\mathcal{L} = a t \...
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Example of critical (non-relativistic) quantum field theory in 1D?
Is there an example of a critical non-relativistic bosonic quantum field theory in 1D (no time)? So, the field theory can be describe by annihilation, $\psi(x)$, and creation operators, $\psi^\dagger(...
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Zeros of multiplicative wave function renormalization
It is probably needless to recall here that the Reimann zeta function $$\zeta(s)=\sum_{n=1}^\infty n^{-s}$$ and its generalizations are among the central objects of study in mathematics.
The main open ...
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Hertz-Millis theory and quantum criticality
Hartz-Millis(HM) theory is a model which exhibits quantum phase transition. The HM action following Altland & Simons is given by
$$
S = \frac{1}{\beta}\sum_{\omega_{n}}\int \frac{d^d q}{(2\pi)^d}\...
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Is the supercurrent just below critical temperature only carried by quasi particles?
Since the Superconducting gap vanishes at the critical temperature, the thermal energy gets sufficient to break up almost all the cooper pairs. But, as far as I understood, the supercurrent is solely ...
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Correlation length and renormalization group
In Scaling and Renormalization in Statistical Physics there's following block of information:
I have some misunderstanding of some ideas.
1) How to define correlation length for arbitrary theory?
I ...
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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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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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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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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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Why does critical temperature exist?
This question has been previously asked over here and the comment and answer there has already answered my original question (the one that I had in my mind), but the following question arises:
Why ...
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Ferromagnetic Potts models in a field and the endpoint of their first-order lines
The $q=3$, $d=3$ ferromagnetic Potts model has a first-order transition on varying temperature. I recently learned that at small $h>0$, where $h$ is a field favoring one of the three colors, there ...
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Equivalent definitions of (dis)continuous phase transitions at criticality
Consider a classical lattice model on $\mathbb{Z}^d$ and suppose that the system undergoes a phase transition as you lower the temperature, i.e., increase $\beta$. The most general definition of a ...
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Binder cumulant method for non-Gaussian distributions
In the Ising model, we know that the order parameter $m$ has a Gaussian distribution for temperatures below the critical point. Measuring the exact point where this phase transition takes place was ...
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Did Democritus predict atoms using Sharp Phase Transitions? How? Couldn't a classical field theory also have Sharp Phase Transitions?
In the Wikipedia page for the Ising Model it is written without citations:
One of Democritus' arguments in support of atomism was that atoms naturally explain the sharp phase boundaries observed in ...
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Analyzing a Kawasaki-evolving Ising model? (conserved-order-parameter Ising model)
Focusing on 2D in further text
I am struggling to understand how the conserved-order-parameter Ising model (also known/reached through Kawasaki algorithm) shows criticality and also how it can be ...
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How to derive the long range behavior of XY model?
In a lecture note (Lec 23) by Sachdev (https://canvas.harvard.edu/courses/76589/files/folder/Lectures?), he considers a model
$$Z=\int D\theta(x)\,exp(-\frac{K}{2\pi}\int d^{2}x\,(\nabla_x\theta)^2),$$...
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Goldstone Bosons and Criticality
In the presence of spontaneous symmetry breaking of a continuous symmetry, there are massless goldstone bosons. However, they are not treated in the discussions of critical phenomena. Naively I would ...
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Critical exponents and scaling dimension
It is often stated that the scaling exponents, e.g. $\alpha$ and $\beta$, of the critical 2D Ising model are related to the scaling dimensions $\Delta_{\sigma}$ and $\Delta_{\epsilon}$ of the ...
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How can I find the critical dimension for the Blum-Capel model near the tricritical point in mean field theory?
I believe that I have found the critical dimension for the critical temperatures on the critical line (that is, where the second order phase transition occurs), which is $D=4$. This is because the ...
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Simulation time for Ising model of large systems
I have tried to run simulation for Ising model of large-size square lattices at the critical point. Mostly I use Python optimized with numba decorator for $L=256$ it takes approx 2.5 min with ...
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Are quasiperiodic critical points described by non-unitary CFTs?
In a continuous phase transition driven by quenched disorder, the conformal field theory (CFT) describing the critical point often seems to be non-unitary. An example would be the Anderson ...
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Is there existing research suggesting that critical systems' power spectra are correlated with their eigenspectra?
I am interested in how the spatial and temporal spectral exponents – in other words the exponents of the power spectrum and eigenspectrum – of a high dimensional system at criticality are related. It ...
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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 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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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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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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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, ...