What does “strongly correlated” mean?

Question

For quantum many body problem, there is a common terminology “strong correlated systems” that appears in different context. However, it seems that the definition of it is ambiguous and sometimes inconsistent in different field of study. My question is that is there an unequivocal, universal definition of “strong correlated systems”?

Answer 1

Here I assume that we are focus on the condensed matter system which is composed of nuclei and electrons with the fundamental force: Coulomb Force. Furthermore, I just discuss the electron degree of freedom.

• A naive (wrong) answer: The interaction gets stronger when electron density is higher.

The Coulomb interaction is proportional to $\dfrac{1}{r}$, where $r$ is the separation between two electrons. Higher density means that the average the separation between electrons is smaller, so higher density means stronger interaction.

Take the electron gas as an example, the average distance $r_s$ between electrons can be estimated as $$\dfrac{N}{V} \equiv \dfrac{4}{3}\pi r_s^3 \tag{1}$$ where $\dfrac{V}{N}$ is the average volume occupied by one electron, and the density $n$ is defined as: $$n=\dfrac{N}{V} \tag{2}$$ Thus: $$r_s=(\dfrac{3}{4\pi n})^{1/3} \tag{3}$$

Furthermore, the average Coulomb interaction one electron feels can be estimated as: $$\dfrac{E_{int}}{N} \propto \dfrac{e^2}{r_s} \propto n^{1/3} \tag{4}$$

Therefore, the smaller $r_s$ (or high density $n$) means stronger interaction or strong correlation.

• However, the truth is the opposite. Because the electronic Hamiltonian is: $$H_{ele}=H_{k}+H_{int} \tag{5}$$

We need to compare interaction ($H_{int}$) with the kinetic energy ($H_k$) to determine whether we have a strongly correlated system or a weakly correlated system. The strong correlation means $H_{int} \gg H_k $ (electrons are correlated with each other) and the weak correlation mean $H_k \gg H_{int}$ (electrons are almost free in space).

We can also make a qualitative estimation for the total kinetic energy. The total kinetic energy $E_k$ in a Fermi gas is also a function of $r_s$ or density $n$:

$$E_{k}=2V\int_{\epsilon \ll \epsilon_F} \dfrac{d^3k}{(2\pi)^3} \dfrac{k^2}{2m}=2V \int_0^{k_F} \dfrac{k^4 dk}{4 \pi^2 m}=\dfrac{Vk_F^5}{10 \pi^2 m} \tag{6}$$

The total particle number $N$:

$$N=2V\int_{\epsilon \ll \epsilon_F} \dfrac{d^3k}{(2\pi)^3} = 2V \int_0^{k_F} \dfrac{k^2dk}{2 \pi^2 m} = \dfrac{Vk_F^3}{3 \pi^2} \tag{7}$$

Thus the total kinetic energy per electron is:

$$ \dfrac{E_k}{N} = \dfrac{\dfrac{Vk_F^5}{10 \pi^2 m}}{\dfrac{Vk_F^3}{3 \pi^2}} = \dfrac{3k_F^2}{10m} \propto \dfrac{1}{mr_s^2} \tag{8}$$

Note that from Eq.$(7)$ we can derive that: $$k_F=(3\pi^2 \dfrac{N}{V})^{1/3}=(3\pi^2 n)^{1/3} \tag{9}$$ which allows us to build a connection between $k_F$ and $r_s$.

Finally, the ratio between kinetic energy and interaction energy is

$$ \dfrac{E_{int}}{E_k} ~\propto \dfrac{e^2/r_s}{1/mr_s^2} = \dfrac{r_s}{a_0} \tag{10}$$

Here $a_0=\dfrac{1}{me^2}=0.529 \text{Angstrom}$.

• When $r_s \gg a_0$ (low density), we have a strongly correlated system with the energy of interactions greater than kinetic energy.

• When $r_s \ll a_0$ (high density), we have a weakly correlated system with kinetic energy greater than interaction.

In summary, we need to compare interaction ($H_{int}$) with the kinetic energy ($H_k$) to determine whether we have a strongly correlated system or a weakly correlated system.

• Ref1: https://www.youtube.com/watch?v=7SepOSfG-To&t=3915s
• Ref2: http://www-personal.umich.edu/~sunkai/teaching/Fall_2013/chapter2.pdf

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A mathematical definition from the viewpoint of statistical mechanics:

• The correlation function $\langle A B \rangle$ (here $\langle \cdots \rangle$ means statistical average) can be approximated as $\langle A \rangle \langle B \rangle$, then you obtain a weak correlation system;

• The correlation function $\langle A B \rangle$ can not be approximated as $\langle A \rangle \langle B \rangle$, then you obtain a strong correlation system;

Hope it helps.

Answer 2

Here is my summary of definition of strong correlated systems in different context.

For ab initio electronic structure Hamiltonian

The eigen value of the system can not be well approximated by single slater determinate approach (Hartree Fock or DFT). These system normally has large coefficient in there configurational interaction (CI) expansion. For some cases, if the system has close lying eigen states, a multi-reference scheme need to be applied.

For non-adiabatic nuclear dynamics

In these systems, Born-Oppenheimer approximation is invalid and the coupling between electronic structure and vibrational mode is known as non-adiabatic coupling. These coupling serves as off-diagonal matrix element in their model Hamiltonian represented in H.O. basis. When the dynamics behavior of these system deviate largely from the Born-Oppenheimer approximation, we treat it as strongly correlated systems.

For spin Hamiltonian

When the eigen state of the system can not be well approximated using mean field approach and interaction between neighbours spin need to be considered and the exchange integrals of these coupling terms are normally very large.

These are some examples for strongly correlated systems in different context. I think there are some common features here.

Answer 3

You got it completely correct when you said:

"it seems that the definition of it is ambiguous and sometimes inconsistent in different field of study."

As the other answers show, it's a bit of a "loose" term that can be used in slightly different ways depending on the sub-field of Matter Modeling in which the term is being used.

This might not be too surprising, because the word "strong" is a relative term too: to be "strong" or "weak" depends on who you're comparing yourself to. It's not like "longitudinal waves" and "transverse waves" where a wave is either one or the other, the "strongest" human can still be weaker than one of the "weakest" train engines.

This reminds me of a previous question that was asked here: What is static correlation, actually? We like to use the words "static correlation" and "dynamic correlation" to assist us in describing different "types" of correlation that come up in our calculations, and this sometimes creates confusion for people that are trying to calculate the same thing using completely different methods (e.g. direct solving of the Schroedinger equation rather than first doing an SCF calculation then a post-SCF calculation to add dynamic correlation).

To answer the question of what "strong correlation" means, we can start first with the definition of "correlation", which is a term that was originally used in statistics, before it was brought to quantum mechanics (though the statistics of measurements on an entangled system will be correlated in exactly the way statisticians use the term). For electron-correlation, you can try to categorize the correlations between electrons as being "weak" or "strong", but there will be systems with weaker or stronger electron correlation.

In the world of doing ab initio calculations by using "SCF + post-SCF" methods, systems with "strong correlation" tend to need many determinants in the SCF part of the calculation (i.e. a big active space in a CASSCF+PT2 calculation), but this is not the only way the term is being used. It's sometimes better to avoid terms that are "ambiguous" as you called it in your question, and instead be precise about what specifically you're conveying: for example, instead of saying "the system is strongly correlated", a more precise statement would be "the 1000 most dominant determinants all have the same weight in the FCI expansion".