# Talk:Gutzwiller wave function

## Reviewer A

The early work by Martin Gutzwiller on modifying many body wavefunctions to include local correlations has been very influential. This article focusses on exact analytic results in several special cases, due in most cases to Prof. Gebhard and collaborators, and on the application to ferromagnetic metals. It largely ignores the substantial body of work in recent years on strongly correlated fermions using Monte Carlo techniques to perform accurate numerical calculations. To include this latter work would require a major extension which the authors understandably may not wish to undertake.

The article provides a good summary of the results obtained on Gutzwiller wavefunctions. Unfortunately it does not adequately discuss the limitations of Gutzwiller wavefunctions as solutions to the Hubbard model. For example,it would be helpful to include a more complete comparison of the one dimensional Gutzwiller wavefunction to exact results e.g. Luttinger liquid theory and the Bethe Ansatz results at half filling. In higher dimensions the fact that it does give a Mott transition (except in infinite dimension) is obscured. A clear account of the important distinction between correlated and uncorrelated doubly occupied and empty site configurations and of the pros and cons of Gutzwiller wavefunctions would be very helpful to the uninitiated reader.

The historical note summarizes matters well except for one point. Hubbard was motivated by a desire to describe the Mott transition rather than the subject of ferromagnetism in metals which motivated Gutzwiller and Kanamori. This is clear from the Hubbard III paper. Subsequently most work on this Hamiltonian has focussed on this transition, which may explain why the model came to be referred to as the Hubbard model. It would be helpful if this point was included in the historical note.

## Authors: answer to reviewer A

(A-i) **The article lacks a discussion of numerical approaches to the Gutzwiller wave function.**

In our previous version, we mentioned the Variational Monte Carlo technique
at the end of the subsection *Variational energy for the single-band Hubbard model* with a reference to a recent review article by Edegger, Muthukumar
and Gros.

In order to make the reference more prominent we added
a sentence at the beginning of the chapter *Evaluation*
in which we refer to the Variational Monte Carlo (VMC) method:

*The numerical evaluation of expectation values is possible on finite lattices with the help of the Variational Monte Carlo (VMC) technique. (Edegger et al.).*

We agree that the discussion of Variational Monte Carlo and other numerical approaches to many-particle wave functions deserves a separate entry in scholarpedia.

(A-ii)
**The article should give a more complete comparison of the Gutzwiller wave function to exact results in one dimension.**

In our previous version, we discussed the shortcomings of the Gutzwiller wave
function for the Hubbard model in two paragraphs in the subsection
*Variational energy for the single-band Hubbard model*, see also figure 5.

Now we added the following two sentences at the end of the subsection
*Momentum distribution* in order to clarify its
relevance for the physics of one-dimensional metals:

*From the exact solution in one dimension via Bethe Ansatz it is known that the Hubbard model provides an example of a Luttinger liquid whose momentum distribution is continuous with a divergent slope *
at the Fermi wave number. The Gutzwiller wave function does not reproduce this generic behavior of one-dimensional metals.

(A-iii) **The article should give a clear account of the pros and cons of the Gutzwiller wave function.**

We think that we made a serious effort to do that in our previous version. For example, in one dimension we argued

**Pro:** The spin correlations are well described in one dimension. (Subsection *Spin correlations and Haldane-Shastry model*)

**Con:** The GWF lacks correlations between holes and doubly occupied sites for large interactions. (Subsection *Variational energy for the single-band Hubbard model*)

In the limit of high dimensions we argued

**Pro:** The Gutzwiller theory provides a reasonable description of liquid Helium-3 and nickel. (Subsection *Applications*)

**Con:** It is too crude approach for the description of the Mott transition. (Subsection *Gutzwiller Approximation and Brinkman-Rice transition*)

With the additional remarks on Luttinger-liquids vs. Fermi liquids, see (A-ii), we think that we provide a fair discussion of the merits and shortcomings of the Gutzwiller wave function.

(A-iv) **The fact that it does give a Mott transition is obscured.**

Of course, the Gutzwiller wave function has the potential of a Brinkman-Rice transition: all particle are localized at half band-filling when we set $g=0$. However, the minimization of the variational energy functional for the Hubbard model does not lead to $g=0$ for a *finite* value of the interaction strength in all finite dimensions.

In order to clarify the issue of the Brinkman-Rice transition,
we changed the wording in the subsection *Gutzwiller Approximation *
and Brinkman-Rice transition* as follows.*

*The Brinkman-Rice transition is an artifact of the limit of infinite*
dimensions, i.e., the variational result $g_{\rm opt}>0$
holds for the Hubbard model in finite dimensions for all finite
interaction strengths $U>0.''
(A-v) '''''In the historical note, the relevance of the model for the Mott transition rather than metallic ferromagnetism should be pointed out more clearly.'''''
Both in Hubbard''s original paper (Hubbard-I) and in a later note by Hubbard
[Citation Classic 22, 84 (1980)] he points out that his original motivation
was the study of ferromagnetism rather than the Mott transition.
It was only later that he realized the relevance of his ''solution''
to the problem of a metal-insulator transition due to electronic correlations.
This presentation of the historical development is in perfect agreement
with Martin Gutzwiller''s personal recollections.
== Reviewer B==
It is a good idea to write such an article. Still, I feel that there are a number of issues that can be improved:
-I did not like the discussion of the H molecule. The authors call the two site Hubbard model an H-molecule. One should at least comment on the fact that only 1s states are included and that the inter-atomic interactions have been fully ignored. After all the latter cause the binding of the molecule. A motivating example should be relevant. For the non-expert reader this discussion should be made clearer, otherwise he/she stop reading right away...
-I would love to see an explicit discussion of Eq.1 for a toy model (say the two site problem). Why not write down the states with their weights, analyze kinetic and potential energy, stress that one increases with g the other decreases and show that a compromise leads to a local minimum. Then a reader ''sees'' what is meant by Eq.(1). This is more important than the rather formalistic generalization to multiple orbitals etc. etc.
-On "Exact results in one spatial dimension". Please add a comment why the solution in d=1 is possible and how it compares with exactly known results (in addition to plotting the energies). Is the discontinuity at kF a sensible result or not? What do we learn? The same holds for infinite dimensions.
-On "Evaluation": The Variational Monte Carlo approach is not even mentioned here. This is the most powerful method to evaluate the GW wave function (without the GW-approx.). It allows to simulate a quantum many body system on the computer as if it was a classical system. Also, it was hugely important in the context of the Resonating Valence Bond theory of frustrated magnets and cuprates. A review should address this.
-On the Historical note: The current version is much improved over what I saw about 2 weeks ago. The specific model was independently proposed by Gutzwiller, Hubbard, and Kanamori. It has had a huge impact in the field. One of the three ''inventors'' of the model is a coauthor of this review. If he wants to share anecdotes that led to the model (and/or to his wave functions) with us, here is an opportunity. Which ideas didn''t work, but inspired this pioneering paper? What motivated the GW approximation? etc.
==Authors: answer to reviewer B ==
(B-i) '''''The discussion of the H molecule can be improved.'''''
We now emphasize the toy-model character of our analysis
(new title of the Subsection, beginning of the second sentence
''In this toy-model description, ...''). Naturally, the basis set
is far too small for a quantitative description of the hydrogen molecule.
(B-ii) '''''An explicit discussion of Eq. 1 for the two-site problem would be helpful.'''''
In the new version, we added the Hamiltonian for the two-site Hubbard model,
gave the explicit form of the wave functions and provided the expressions
for the variational parameter and the ground-state energy.
We very much hope that this is not too technical so that the reader
keeps on reading!
(B-iii) '''''Comments on ''exact results in one spatial dimension.'''''
- At the beginning of the Section ''Evaluation'' we explicitly stated
in the previous version that the expansion in $(g^2-1)$ plus particle-hole
symmetry permits the exact evaluation of expectation values. In the Section
''Exact results in the limit of infinite spatial dimensions'', the
Subsection ''Simplifications'' serves that purpose.
- For additional remarks, see (A-ii) and the new version.
(B-iv) '''''Variational Mote Carlo'''''
See (A-i) and the new version.
(B-v) '''''Historical note''''':
See (A-v) and the remarks by Reviewer~C.
== Reviewer C==
I took a quick look at the article. It seems OK to me.
The only point which could be contentious is the attribution of credit
as to who invented or introduced the Hubbard model, in the historical
note, (other physicists have a claim on that model ) but I would let a
historian of science rather than a physicists such as myself comment on
that.
== Reviewer D ==
(This review was submitted by reviewer D to the editor of Scholarpedia).
The article does not do justice to the considerable literature which has developed out of Gutzwiller’s original idea, primarily as a consequence of two papers which changed the focus of the method from being primarily a way of obtaining a variational approximation to a wave-function for a strongly-interacting system, to being a way to describe new physical phases and possibly to facilitate new kinds of controlled solutions.
The two crucial ideas were, first, the discovery—or rediscovery—by T M Rice and co-workers [1] of the perturbative canonical transformation which carries the effective Hamiltonian of a Hubbard-type model into a Gutzwiller-projected Hamiltonian or “t-J model” Hamiltonian which operates only within the Gutzwiller-projected subspace. This allows one, for instance, to describe atomic spins and metallic electrons within the same formalism, and thus to understand “mixed valence” systems. The second was Anderson’s idea [2] of Gutzwiller projecting B C S superconducting wave functions, thus providing a formal description of the old idea of Resonating Valence Bond or “spin liquid” phases, and showing a connection between antiferromagnetism and superconductivity which has been experimentally fertile. Already in 1988 a group around Rice [3] discovered a mean-field solution of the CuO2 plane t-J model which is still the mechanism of choice for high-Tc superconductivity.
Perhaps a third important group of generalizations has been the work of Laughlin, Bernevig and co-workers [4] in studying superconductivity in the presence of the original, partial Gutzwiller projection, leading to the idea of a “gossamer”superconductor.
[1] C Gros, T M Rice, and R Joynt, Phys Rev B36, 381 (1987). A good summary of this development is in T M Rice’s Varenna lectures for 1987, Scuola Internationale “Enrico Fermi”, Course 104, p 171-203, R Broglia and J R Schrieffer eds, Italian Physical Society, Bologna, 1988
[2] P W Anderson, Science 235, 1196 (1987)
[3] F C Zhang, C Gros T M Rice, H Shiba, J Supercond Sci & Tech 1, 36 (1988)
[4] B A Bernevig, R B Laughlin, D Santiago, Phys Rev Lett 91, 147003 (2003)
==Authors: answer to reviewer D==
(D) '''''The article does not do justice to other work.'''''
Our article is not a review of models for strongly correlated materials which
may be describable by the Hubbard model in the large-$U$ limit.
In this limit, wave functions necessarily have to obtain the constraint of
zero double occupancy and, thus, fall into the class of Gutzwiller-projected
wave functions. In our article we restrict ourselves to *exact* results for
Gutzwiller-correlated wave functions.

In our previous version, see Section `Further applications', we did mention Anderson's work of 1987 (Reference [2] of Reviewer D) and we made a reference to the latest, long review by Edegger, Muthukumar and Gros who quote the references [1] and [3] of Reviewer D. In the present version we cite these papers which certainly provide an example for the importance of the Gutzwiller wave function. For paper [1], we added a reference in the first paragraph of the chapter `Evaluation'. In doing so, we found it appropriate to cite the work by Yokoyama and Shiba, too.

For paper[3], we changed the last sentence in Subsection `Correlated superconductors' into

`This idea was worked out in more detail F.C. Zhang et al., 1988; for a review, see B. Edegger, V.N. Muthukumar et al., 2007.'

In order to cite the Reviewer's reference [4], we added the following sentence in the Subsection `Correlated superconductors':

`The concept of `gossamer superconductivity' put forward by Bernevig, Laughlin and Santiago is also based on Gutzwiller-correlated BCS wave functions.'

## Reviewer B on the references asked by reviewer D

As for the references put forward by Referee D, I would feel much better if the authors would refer to C Gros, T M Rice, and R Joynt, Phys Rev B36, 381 (1987). This paper has had great impact and was in many ways ahead of its time. I don't think this applies equally to Refs.[3,4] of Referee D (both are terrific papers and I have no objections to referring to them). Thus, my recommendation would be to include at least Ref.[1].