uc davis reu a systematic study of the mean field

2D lattice model strongly interacting electrons • August 2014 • UC Davis REU

A Systematic Study of the Mean FieldApproximation for the Hubbard Model

Denisa R. Goia∗

University of Wisconsin Whitewater

Abstract

We performed systematic investigations of mean field ground states of interacting two dimensional(2D) lattice models. Specifically, we address the effect of supercell ("cluster") size in the types of solutionsthat arise, treating doping levels that are either commensurate with, or incommensurate with, the supercellsize, for interaction strengths ranging from the perturbative range to the strong interactions regime.We consider the basic Hubbard model on a 2D square lattice, with hopping amplitude -t, providing thekinetic energy and the on-site repulsion U (Hubbard U) providing the energy of interaction. The meanfield solutions allow us to assess global characteristics such as charge and spin order, tendencies towardphase separation, and distribution of single particle eigenvalues (density of states [DOS]). The degree oflocalization of individual eigenstates, as provided by the inverse participation ratio, provides a new andinnovating window into the character of the system.

I. Introduction

Model Hamiltonian approaches have providedthe core of research on correlated electron sys-tems, displaying the basic phenomena and pro-viding understanding of the fundamental pro-cesses that are involved. In the absence ofexact solutions except for some very particularcases, approximations are required to enableprogress. The earliest approach was "Hartree-Fock" (HF) - a mean field approach that approx-imates electron-electron interactions, which arethe hangup, with interaction with an averagepotential (the mean field) that is determinedself-consistently to minimize the energy, or atfinite temperature the thermodynamic poten-tial. The HF method has provided invaluableguidance in interacting systems.

In spite of all of the increasing sophisti-cation and elegance of state of the art meth-ods, the mean field approach remains of greatimportance and may be experiencing a resur-gence. With simultaneous treatment of all ofthe charge, spin, orbital, and lattice degreesof freedom as well as variable electron den-

sity, there are many variables (parameters) tomanipulate, and mean field enables investiga-tions that are not viable with more realistic butmuch more time consuming methods. Meanfield studies remain an essential tool in thetheorist’s arsenal.

II. Theory Background

The tight-binding model is an approach tothe calculation of electronic band structureusing an approximate set of eigenfunctionsbased upon superposition of orbitals for iso-lated atoms located at each atomic site. Thetight binding approximation deals with thecase in which the overlap of the wave functionsis enough to require corrections to the pictureof isolated atoms, but not so much as to renderthe atomic description completely irrelevant.This model is incomplete for our needs. Ituses only kinetic energy and no potential en-ergy. Therefore we need a more specific model.Since I have used the name of electronic bandstructure in defining the TB model here is someinformation about it.

∗Funded by NSF grant PHY-1263201

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The electronic band structure of a solid de-scribes the ranges of energy that an electron inthe solid might or might not have. The rangesof energy in a solid where no electron statescan exist is called a band gap or an energy gap.In insulators and semiconductors an importantband gap is situated between the valence bandand the conduction band. The valence bandis situated below the band gap, if it is present.The electrons here are bound to the atom, theycannot move freely within the atomic lattice.They are present at absolute zero temperature.The valence band is the highest range of elec-tron energies. The conduction band sits on topof the band gap (if there is one). The electronis no longer bound to the atom and is thereforefree to move as it pleases within the atomiclattice. In insulators and conductors the levelsare respected, while in metals, the bands crosseach other.

Figure 1: The electronic band structure of a solidThere are many models used in physics, but

for our purpose, the Hubbard Model seemedto be a suitable choice. The Hubbard modelis an approximate model used to describe thetransition between conduction and insulatingsystems. It is the simplest model of interactingparticles in a lattice, with only two terms inthe Hamiltonian: a kinetic term allowing fortunneling (hopping) of particles between sitesof the lattice and a potential term consisting ofan on-site interaction. In our case, the HubbardModel adds an important element, which is thepotential energy of interaction.

But when talking about these models we re-ally need to explain something about the MeanField Theory. MFT is studying the behavior oflarge and complex stochastic (a system whose

state is non-deterministic, random) models bysimplifying them to a simple system. Whentalking about these random systems we arereferring to the electrons inside the periodiclattice which are free to hop randomly insidethe atomic lattice.

A many body system is very hard to un-derstand and solve exactly because there aretoo many interactions happening. That is whyall this interactions are approximated by anaverage, which becomes the external field. Theproblem goes now from an n-body system to aone body system, a problem that can be solved.That doesn’t mean that the problem is simple.There is still a part that is a little problematic.When summing over all states, the interactionterms in the Hamiltonian give some combina-torics (study of finite discrete structures) prob-lems, which are a little difficult to solve.

Since there are no exact solutions (witha few exception) we need to use approxima-tion methods to help in our study. The HFmethod is one of these methods and has pro-vided invaluable guidance in interacting sys-tems.The Hartree-Fock method is an approxi-mation method used in the determination ofthe wave function and the energy of a quantummany-body system in a stationary state (statewith a single definite energy). The Hartreeequation is the approximate solution of theSchrodinger equation given that after comput-ing the final field from the charge distribution,this field must be self-consistent with the initialfield.

III. Setting up the problem

When using the Hubbard Model we have tomake sure that some assumptions are met.The atoms are arranged in periodic patternsover long distances and are stable for a longtime. We have used lattices that are periodic,the 8x8, 16x16, 24x24, 32x32 and even some64x64 lattices. We assumed that atoms have asimple energy level the electrons can occupy,the electrons are interacting through the on-site Coulomb interaction, the kinetic energy ischaracterized by hopping of electrons between

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atomic sites and since the atomic orbitals dieoff exponentially, the electrons are allowed tohop only to the nearest neighbor.

We set up a Hamiltonian starting from ran-dom initial density. The Hubbard Hamiltonianfor fermions is written as:

H = −t ∑<α,β>σ

c†α,σ cβ,σ + U ∑α

nα↑nα↓ (1)

where cβσ destroys the electron with spin σat site β, and c†ασ creates an electron with thesame spin.

Since there can be up to 2 electrons per sitethe total number of electrons in the system canbe 2N (where N is the number of sites). Weset up the system, initially, to have only oneelectron per site, which is called half filling.

The hopping parameter is set to -1, whichallows the electrons to hop to neighboring sites.For the half-filling case we are looking at thereis 1 atom for cell and 1 orbital.

IV. Results

During our research we looked at a few dif-ferent things. We checked to see what thelowest energy (ground state energy) looks likein different lattices. We studied the depen-dence on supercell (cluster) size, comparedcommensurate vs incommensurate states (#ofelectrons/spin ←→ # of atomic sites), we as-sessed global characteristics like the distribu-tion of single particle eigenvalues (density ofstates), and in the end we looked at the in-verse participation ratio (IPR) vs the energyof the state, which may not have been donepreviously.

The amix is set to 0.3 or 0.1, with a fewexceptions for u=2, where both the 24x24 lat-tice and the 32x32 were not converging in 2500iterations. The amix represents the percentageof the output of the previous solution that ismixed into the input for the new one. Whatwe observed was that the larger the lattice theharder it was to find a percentage that would al-low for convergence. Therefore for some caseswe had to use an amix of 0.6, 0.7 and even 0.8.

If for the previous lattices, 30 percent of the so-lutions were diverging (with the best amix forthat case), when we went for a higher lattice,40x40, we noticed that most of the solutionswere diverging. It is true that we tried only afew cases, because due to the size of the ma-trix (1600x1600) it was taking too long to getsolutions. For example, it took three days toget 10 solutions for parameters, u=2, polariza-tion=1/8. This case is also the most difficultone. When the fraction p, of number of spinsup minus number of spins down over the totalnumber of spins is 1/8, it is very hard to getresults. It is a transition region from strong toweak onsite interactions, which caused prob-lems for us but also gave the most interestingresults. For the 24x24 and 32x32 lattices theinterval with problems was for polarization be-tween 1/4 and 1/16, and it was getting muchlarger for 40x40 lattices. For the small numbersof runs done for 64x64 lattices, it became a realproblem to get convergent solutions.

A. Varying the interaction strength

Looking at the ground state total energy (Fig-ure 2, first graph) it becomes clear very quicklythat for u=1, the ground state energy stays al-most the same with changes after the fourthdecimal. The behaviour is the same for all lat-tice sizes. What is interesting is that for polar-izations closing on 0, there are some changes atthe third and fourth decimal for the 16x16 lat-tice and the same changes happen in the 32x32lattice for larger polarizations. The 24x24 and32x32 lattices have the same ground state en-ergy, while when closing on polarization 1, the16x16 and 24x24 lattices have the same solution.For u=4 (Figure 2, last graph), due to the strongregime of the on-site energy the interactionsare localized, therefore we don’t get a globalorganized picture. We do not necessarily findthe ground state energy, unless we look at thecase with only spins up, where no matter whatsize of lattice we look at the energy stays thesame. For any other case the differences inenergy, in the same lattice, are varying from0.02 to 0.06 and this fluctuation happens for

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all lattices. But when taking an average of theenergy, for all lattices the energy tends to con-verge towards the same value. The u=2 case(Figure 2, second graph) seems to be on thetransition line between the higher on-site inter-actions that are localized and the lower onesthat are global. Therefore the solutions for theground state energy are different for differentlattice sizes, especially the 8x8 lattice. The 8x8lattice doesn’t allow us to see the big picture.The differences go up to the first decimal. Itseems that the larger the lattice becomes thesmaller the difference gets, where that happensonly after the fourth decimal. As in the case foru=4, when there are only spins up, the energyis identical for all 4 lattices.

Figure 2a: 32x32 lattices with p=1/2 and, u=1; thex-axis represents the run number while the Y-axisrepresents the total energy of that specific run

Figure 2b: 32x32 lattices with p=1/2 and u=2; thex-axis represents the run number while the Y-axisrepresents the total energy of that specific run

Figure 2c: 32x32 lattices with p=1/2 and u=3; thex-axis represents the run number while the Y-axisrepresents the total energy of that specific run

Figure 2d: 32x32 lattices with p=1/2 and u=4; thex-axis represents the run number while the Y-axisrepresents the total energy of that specific run

B. Varying the lattice size

A different way to look at the results is byanalyzing the cluster size. Due to the stronglocal on-site interactions happening for u=4,there isn’t much to see in the solution (Figure3).There is little correlation from site to site,the global interactions are very weak leavingthe magnetization to look just disorganized.We get the same type of solution for any lat-tice size and any combination of spins up andspins down.

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Figure 3: 32x32 lattice, u=4, p=1/2; each littlesquare represents a site and the colors go from spinup-dark red to spin down-dark blue

Due to the strong local interactions the so-lution for the ground state gives either the anti-ferromagnetic (the spins of the electrons alignin a regular pattern with the neighboring site -one up, one down) solution or some type of acluster or stripes solution. For small polariza-tions (up to p=1/16) the solution seems to bea variation of stripes, while for larger polariza-tion we get clusters. (Figure 4)

Figure 4a: The antiferomagnetic solution for u=1

Figure 4b: Stripped solution for small polarizationsand u=1

Figure 4c: Stripped solution for small polarizationsand u=1

Figure 4d: Cluster solution for larger polarizationsand u=1

There is an interesting result for p=1/9which is incommensurate with the chosen lat-tice sizes. See Fig.5. The 16x16 lattice has astripped solution while the 24x24 and 32x32lattices have cluster solutions. Otherwise, allsolution are consistent.

Figure 5a: For u=1; p=1/9; lattice size 16x16

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Figure 5b: For u=1; p=1/9; lattice size 24x24

Figure 5c: For u=1; p=1/9; lattice size 32x32

Another interesting result happens forp=1/2, where the bigger the lattice gets,the more disorganized the solution ap-pears.(Figure 6) There is a strong dependenceon lattice size.

Figure 6a: Polarization 1/2, u=1, 16x16 lattice

Figure 6b: Polarization 1/2, u=1, 24x24 lattice

Figure 6c: Polarization 1/2, u=1, 32x32 latticesBut the most interesting and diverse solu-

tions are seen in the transition case which isfor onsite interaction u=2. The only constantresults are that from the 8x8 lattice you can’tconclude much, and the larger lattices tend togive the same type of solution, whatever thatsolution might be. But here it ends because,the solutions are everything from stripes orclusters to discs, rings, waves and other shapesand forms with no names. A few examples ofphase separation can be seen in Figure 7

Figure 7a: Solution for u=2, lattice size 16x16

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Figure 7b: Solution for u=2, lattice size 24x24

Figure 7c: Solution for u=2, lattice size 32x32The HF solutions allow us to assess global

characteristics like the distribution of singleparticle eigenvalues. So, another way to lookat the results is by analyzing the density ofstates. DOS describes the number of states perinterval of energy at each energy level that areavailable to be occupied by electrons. A highDOS at a specific energy level means that thereare many states available for occupation. Plotsof the DOS can be seen on Figure 8. On the topimage, the interactions between electrons arenot strong when the onsite energy is 1 givingthis perfect solution with the peak right in themiddle and a shift between the spin up anddown, due to imposed spin polarization. In theright picture, due to the strong regime of theonsite energy (u=4), the number of states perinterval of energy are organized in a symmetri-cal picture, with a flip of the spin. The up spingoes down on the other side of the band gapand the down spin goes up. There can also beseen the peaks where the density of states is atits highest.

Figure 8: DOS for u=1 with shifted peaks due tothe diference in spins

Figure 8: DOS for u=4 with peaks for the highestdensity and gaps

C. A new type of characterization

A different way to look at the results is by ana-lyzing the solutions of the inverse participationratio versus the energy of states. The IPR pro-vides a measure of the extension over sites ofthe eigenstates. We did a set of runs only on24x24 and 32x32 lattices, for onsite repulsionu of 1, 2, and 4 and polarization 1/16, 2/16,and 3/16. It can be seen how much the in-teractions affect the results. Weak interactionsform a straight line(u=1), while strong inter-action(u=4) give a visible gap. The solutionsfor u=1, give an expected result. The values ofeigenenergies for every spin are close together,the majority forming a line. When looking atdifferent polarizations the values form a dis-tinctive line for bigger polarization and get alittle spread for smaller polarization. It canalso be easily seen that number of spins up

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and spins down play a role in the forming of agap around the point of 0 eigenenergy.

Figure 9: IPR vs energy for u=1, p=1/16

Figure 9: IPR vs energy for u=1, p=2/16

Figure 9: IPR vs energy for u=1,p=3/16The tendency for u=2 of the values of spins

up and down is to converge towards the pointof 0 IPR near 0 eigenenergy. The graph showsa sharper slope for higher polarization.

Figure 10a: IPR vs energy for u=2, p=1/16

Figure 10b: IPR vs energy for u=2, p=2/16

Figure 10c: IPR vs energy for u=2, p=3/16Stronger interactions of the electrons of the

onsite energy u=4, give a totally different pic-ture when compared to the solution of a stronginteraction. We don’t even get the curvy solu-tion of the u=2. Here, in either side of the 0eigenenergy forms an arrow head with a quitea big gap in between either side, as seen inFigure 11.

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Figure 11a: IPR vs energy for u=4, p=1/16

Figure 11b: IPR vs energy for u=4,p=2/16

Figure 11c: IPR vs energy for u=4, p=3/16

V. Conclusion

We have studied Mean Field Theory on a 2D lat-tice to obtain some insight into how correlatedelectron systems behave. The MFT helped usanalyze the results and assess global character-istics of the system. Even though they are notvery complicated or the most realistic meth-ods, for many properties what makes them sovaluable is the amount of variables that we canwork with and manipulate. Work like this, hasbeen done before, but usually only on 8x8 lat-tices. Since we have worked with much largerlattices (the largest being 64x64), we have beenable to see how the system changes and whatare the tendencies of those changes when work-ing at a bigger scale. One direction would beto work on even larger lattices, but for nowwe are unable to do so, due to computationalproblems. Computers are not fast enough togive results in real time. We hope that ourresults, while giving some answers that the-oretical physicist have been looking for, willmake them investigate further and revisit sim-ple methods that can still give insight into theworld of correlated electron systems.

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References

[1] Ashcroft, Neil W., Mermin, N. David, Solid state physics, Harcourt College Publishers, 1976.

[2] Kittel, Charles., Introduction to solid state physics 7th edition, John Wiley and Sons, 1996

[3] Khatami, Ehsan, Notes on the Second Quantization and the Hubbard Model, 2013

[4] Lattimore, DP, A 2D Multi-band Hubbard Model in Mean Field Approximation - Master’sthesis, UC-Davis, 1999

[5] Pickett, Warren E., "Tight Binding" Method: Linear Combination of Atomic Orbitals, privatenotes, 2014

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uc davis reu a systematic study of the mean field

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