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Mean field theory applied to spin-chains, where the ground state is assumed a product state and the energy is variationally minimized, is one of the tools in my toolkit. The examples I give in this question make me wonder if it behaves inappropriately under the addition of quartic terms and the like. I'm tempted that a resolution could be that one needs to do mean field theory on the effective theory picked by one's favorite method (symmetries a la Landau, path integral via resolutions of the identity or a Hubbard-Stratonovich transformation or the like, etc.), and that mean field theory applied via variational product states is not generally valid except for Hamiltonians with "low-order" terms in some sense. However, my main question is given below in terms of verifying or contradicting the predictions of this sort of mean field theory for a specific model.


I find it very enjoyable that mean field theory techniques, while not getting the critical behavior exactly in every dimension, still yield some important qualitative features of phase transitions. For example, the quantum transverse Ising model,

$$H=-J\left(\sum_{<ij>} \sigma^z_i \sigma^z_j + h/J \sum_{i} \sigma_i^x \right) =-J\left(\sum_{<ij>} \sigma^z_i \sigma^z_j + k \sum_{i} \sigma_i^x \right)$$ has a phase transition as one varies $k$ between a ferromagnetic phase with two nearly degenerate ground states followed by a paramagnetic phase at larger $k$ characterized by a single ground state. The ferromagnetic phase has a net magnetization in the $z$ direction, but the paramagnetic phase does not.

One variant of mean field theory that captures this is to assume the ground state is a translationally invariant product state without entanglement, $\otimes_i |\cos(\theta)\rangle_i$, where $\theta$ captures the directions of the spins between the $x$ and $z$ axis. Then one variationally minimizes with respect to $\theta$. Alternatively, one can characterize the product state by $m_z$ = $\langle \sigma^z\rangle$, as for spin-$1/2$ particles one has $m_z^2+m_y^2+m_z^2 = 1$, so that for this case with spins between the $x$ and $z$ axis, $m_x = \sqrt{1-m_z^2}$.

Then, taking $1d$ for ease (the mean field theory answer will only shift the location of the transition according to the coordination number of lattice), the expected energy per site is

$$\langle H \rangle/L = -J(m_z^2 + k m_x) =-J(m_z^2 + k \sqrt{1-m_z^2})$$

The magnetization $m_z$ that minimizes the energy above is the following, as a function of $k$:

Transverse ising MFT

The magnetization continuously goes to zero, with a clear nonanalyticity when it hits zero at $k=2$. While mean field theory gets the critical exponents and location of the phase transition wrong in $1d$, it still shows the qualitative behavior expected above of the $x$ field ultimately beating the $z$ interaction at strong enough fields.


Now, for my question. Let's add a quartic $z$ interaction: $$H=-J\left(\sum_{ijkl {\rm \: in\: a\: line}} \sigma^z_i \sigma^z_j \sigma^z_k \sigma^z_l + \sum_{<ij>} \sigma^z_i \sigma^z_j + k \sum_{i} \sigma_i^x \right)$$

Here the quartic term corresponds to four consecutive spins all in a row or a column.

Using the same mean field theory as above, the problem becomes to minimize (up to numerical prefactors like the coordination number, which will shift the location of the transition)

$$\langle H \rangle/L = -J(m_z^4 + m_z^2 + k \sqrt{1-m_z^2})$$

The resulting minimization problem has a discontinuity! See below: enter image description here

So mean field theory predicts a discontinuity for this model with an additional quartic term in all dimensions. My question is: For which dimensions, if any, does this model actually show a discontinuity in the magnetic field across the phase transition? (Also, does this model have a name?)

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    $\begingroup$ The $Z_2$ gauge theory has quartic and linear terms, but not the quadratic term (see equation (11) in Lattice Gauge Theories and Spin Models). The quartic term involves the four corners of a square, instead of a colinear sequence of sites. I know that's not quite the model you're asking about, but maybe it's another data-point in the space of similar models. $\endgroup$ Sep 19 at 22:46
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    $\begingroup$ @ChiralAnomaly Thanks, that is helpful. If I'm reading right, the $2d$ $Z_2$ gauge theory with quartic interaction, restricted to the "physical subspace", is dual to a transverse ising model with quadratic interaction, and so it should have a continuous phase transition like the quantum transverse ising model in $2d$ does. So I view this as a data-point leaning towards continuous phase transitions even in the presence of quartic interactions, though the high degree of local symmetries in the gauge theory perhaps distinguishes it from the model with quartic interactions in rows and columns. $\endgroup$
    – user196574
    Sep 20 at 3:40
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As you might already know, (Gutzwiller) mean-field theory becomes exact in the limit $z \to \infty$ (z is the coordination number). So, for a model like $$\frac{1}{N}\hat{H}=J_{4}\left[\frac{1}{N}\sum_{i=1}^{N}\sigma_{i}^{z}\right]^4+J_{2}\left[\frac{1}{N}\sum_{i=1}^{N}\sigma_{i}^{z}\right]^2+h\left[\frac{1}{N}\sum_{i=1}^{N}\sigma_{i}^{x}\right]$$ (defined on a mean-field/ all-to-all lattice) in the $N \to \infty$ limit, (Gutzwiller) mean-field theory becomes exact. Hence will exhibit the first-order transition with a jump in the $\mathbb{Z}_{2}$ symmetry breaking order-parameter.

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    $\begingroup$ This is another good data point in model space. I'll also check this with exact diagonalization for smallish system sizes, and maybe I'll see if there's a DMRG package that can go further just as a small sanity check for myself. As an aside, do you think it might be possible that despite MFT being exact as $z \to \infty$, there might yet not be a finite $z$ above which the transition is first order? Or if no, do you have any offhanded guesses of what the upper critical dimension would be? $\endgroup$
    – user196574
    Oct 6 at 20:34
  • $\begingroup$ Sorry for this late reply. I was relocating for a job. Its a good idea to check it for finite system sizes using exact diagonalization and see that indeed there is a first order transition in the thermodynamic limit through finite system size scaling.I guess one can do Hubbard-Stratanovich transformation on a finite dimensional lattice and see when the mean-field theory is justified by studying quantum fluctuations around mean field (ala Ginzburg criterion)? $\endgroup$
    – Sunyam
    Oct 11 at 13:03

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