21
votes
Accepted
What leads to the existence of critical temperature?
Your description of critical temperature isn't quite right.
If you increase the temperature of a liquid beyond the critical point, the atoms are moving so quickly that persistent structure fails to ...
- 13.2k
15
votes
Accepted
Why do spin correlation functions in Ising Models decay exponentially below the critical temperature?
First note that, as you say, the 2-point function $\langle \sigma_i \sigma_j\rangle$ does not tend to zero as $\|j-i\|\to\infty$ when $T<T_{\rm c}$; namely,
$$
\lim_{\|j-i\|\to\infty} \langle \...
- 9,040
13
votes
Accepted
Is Wilson-Fisher fixed point unique?
As usual in interacting field theory, there is no rigorous answer to your question directly in three dimensions. However, there is strong numerical evidence from the conformal bootstrap approach for ...
- 1,979
11
votes
If boiling of water involves change in internal energy, then why does the temperature remain constant?
Temperature characterizes the average energy of atoms/molecules, as per Boltzmann distribution:
$$
w(\mathbf{p},\mathbf{q})\propto e^{-\frac{H(\mathbf{p},\mathbf{q})}{k_B T}},$$
where the potential ...
- 54.9k
11
votes
Is renormalization different to just ignoring infinite expressions?
Renormalization is more than just ignoring singularities, but rather studying how these signularities arise, how they behave, and finding relations between the properties of different particles/...
- 54.9k
10
votes
Accepted
Fluids with critical point at ordinary temperature and pressure
Whether an answer exists depends on your definition of "near" compared to STP.
There are a few fluids that have their critical point at a temperature close to STP, but higher pressure. For example, (...
- 118k
10
votes
Accepted
Argument on why spin correlation functions in Ising model decay exponentially with a correlation length?
Let me write the Hamiltonian
$$
H = -J \sum_i S_i^z S_{i+1}^z.
$$
This choice will avoid some annoying (and irrelevant) signs.
One way to formulate the statement in the OP precisely is as follows.
...
- 9,040
9
votes
Accepted
Why are there large fluctuations at the critical point and why does Landau theory work despite such large fluctuations?
As already mentioned in a comment by elifino, it is generally known that near a critical point, two (or several) different phases, with almost the same free energy, ...
- 2,071
9
votes
Accepted
Critical temperature and lattice size with the Wolff algorithm for 2d Ising model
The first thing to realize is that there are no "true" phase transitions (in the sense of non-analytic behaviour of thermodynamic potentials) in finite systems. This is the main difficulty one faces ...
- 9,040
9
votes
Accepted
What is the relation between non-local order parameters and topological phases?
When you refer to SPTO transitions as being 'nonlocal symmetry-breaking transitions', I guess you are referring to the idea of 'hidden symmetry-breaking'? This indeed the old way of looking at SPTOs (...
- 8,321
8
votes
Examples of important known universality classes besides Ising
Two systems belonging to the same universality class will have the same critical exponents.
There are many things that determine the universality class of a system, one being its dimension.
The 2D ...
- 15.9k
8
votes
What is the definition of correlation length for the Ising model?
The two-dimensional square-lattice Ising model, which is a simplified model of reality, exhibits a phase transition. Onsager showed that there is a specific temperature, called the Curie temperature ...
- 398
8
votes
Accepted
Nonzero spontaneous magnetization in two-dimensional Ising model
Your argument only applies to finite systems (otherwise the energy is ill-defined) and there are no phase transitions in finite systems. So, there is no contradiction there.
Moreover, your argument ...
- 9,040
7
votes
Why correlation length diverges at critical point?
I think your trouble is that a correlation length $\xi$ is not to be interpreted as correlation in the sense of statistics, e.g.
$ \frac{<(s(x)-\lt s(x) \gt)(s(y)-<s(y)>)>}{\sqrt{<\...
- 221
7
votes
Accepted
Free energy functions are analytic or non-analytic in phase transitions?
Landau free energy is just an approximation to the real free energy in the thermodynamic limit. For that reason, Landau free energy can be analytic, while the real one is not. Let me show how the ...
- 1,168
7
votes
Accepted
What is the meaning of "Deconfined Quantum Critical Point"?
Antiferromagnetic spin systems can have "spinon" excitations. Conceptually, you can make these by starting from a VBS ground state made of singlets, then "breaking" one singlet into a tensor product ...
- 45.5k
7
votes
Accepted
Scale invariance at phase transitions
In the types of system in which second-order phase transitions are studied, forces are generally short range. This means that the other scales you mention will be finite in size. However, at the ...
- 33.3k
7
votes
Why is the correlation function a power law at the critical point?
The exponent is not in general an integer, and therefore the divergence is not really polynomial. That being said, let's agree to call a structure of the form $x^a$ a polynomial, for any (real) $a$.
...
7
votes
Accepted
Primary Operators in the Ising CFT
Lattice-continuum correspondence for the primary fields of the Ising CFT:
Indeed, the identity field $1$ simply corresponds to the identity operator on the lattice. More generally, any lattice ...
- 8,321
7
votes
Accepted
Non-uniqueness of the Order Parameter and its Critical Exponent
I think the conventional wisdom is correct, the choice of order parameter does not matter (as long as the symmetries are preserved, and we redefine conjugate variables).
I think the main mistake in ...
- 18.2k
7
votes
Accepted
Is there any zero-order phase transition in nature?
A zeroth-order phase transition defined as a finite jump of some free energy or any other fundamental equation is incompatible with the requirement of convexity (concavity) for such state functions. ...
6
votes
Accepted
Good layman definition of the critical point(phases) and supercriticality
Characteristically, a critical point occurs somewhere anytime you have a continuous phase transistion. That is, if you have two phases of a substance that themselves share their intrinsic symmetries. ...
- 13.3k
6
votes
Fluids with critical point at ordinary temperature and pressure
The critical pressure is given by
$$P_c=\frac{a}{27b^2},$$
while the critical temperature is
$$T_c=\frac{8a}{27bR}=\frac{8bP_c}{R}.$$
The parameter $b$ is related to to the effective volume occupied ...
- 17k
6
votes
Accepted
Why correlation length diverges at critical point?
It is not the correlation length of the system that you should look at, but the correlation of the fluctuations. If T>>Tc the spins are randomly oriented and the lenghtscale of fluctuations is very ...
- 657
6
votes
Accepted
Divergent Coulomb integrals in superfluid fluctuations
When you say the integrals diverge i guess you are refering to the following:
$$ |C_d| \leq \int \frac{d^d\mathbf{q}}{(2\pi)^d} \frac{1}{\mathbf{q}^2} = \text{const.} \times
\int_0^\infty q^{d-3} dq $...
- 1,439
6
votes
Accepted
Three point correlation function 2D Ising model
Since there is a unique Gibbs state at $\beta_{\rm c}$, these correlation functions (actually all odd correlation functions) vanish by symmetry : $\langle \sigma_i \sigma_j \sigma_k \rangle_{\beta_{\...
- 9,040
6
votes
Accepted
Renormalization of mass: does it change sign from high temperature to low temperature?
The issue here is that the expectation value of the field, $\phi_0$, also gets a correction at one-loop. You should be careful to include this if you also include the loop corrections you've indicated....
- 1,979
6
votes
Accepted
Why are critical points of classical systems described by quantum conformal field theories?
The Feynman path integral gives an isomorphism between quantum field theories in $d$ dimensions and classical statistical mechanics in $d+1$ dimensions. Basically the time trace of the quantum time ...
- 48.2k
6
votes
Accepted
Mermin-Wagner and superconductivity
All the answers here and in the other question do not address the important difference between superconductivity and superfluidity: namely that the Nambu-Goldstone modes in superconductors are not ...
- 579
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