Antiferromagnetism and Metal Insulator Transition in the Frustrated

274 Progress of Theoretical Physics Supplement No. 160, 2005

Antiferromagnetism and Metal Insulator Transition in theFrustrated Hubbard Model

Thomas Pruschke

Institute for Theoretical Physics, University of Gottingen, Friedrich-Hund-Platz 1,37077 Gottingen, Germany

We present a review of recent results on properties of the antiferromagnetic phase ofthe Hubbard model obtained within the dynamical mean-field theory. The equations ofthe dynamical mean-field theory are solved with a suitably extended version of Wilson’snumerical renormalization group.

For the particle-hole symmetric model we present evidence that in the antiferromag-netic phase at half filling a crossover between a Slater type antiferromagnet to a Heisenbergantiferromagnet takes place. Away from half filling phase separation is found.

Including magnetic frustration leads to the expected suppression of the antiferromag-net, but in contrast to previous expectations no distinct antiferromagnetic metal is found.Instead, a strong first-order transition between the paramagnetic metal and the antiferro-magnetic insulator occurs.

§1. Introduction

Transition metal compounds show a large variety of physical phenomena. Theseinclude metallic behavior with possibly strongly renormalized Fermi liquid scales,metal-insulator transitions in the paramagnetic phase and different kinds of orderedphases encompassing orbital, ferro- and antiferromagnetic order.1) Well-known ex-amples are V2O3, LaTiO3, LiV2O4, LaMnO3 or the cuprates, which all show metal-insulator transitions, magnetic and orbital order and superconductivity dependingon composition, pressure or other control parameters.1) From the point of view oftheory, the microscopic description of these phenomena, in particular magnetismand the correlation driven metal-insulator transitions, constitutes one of the majorresearch activities in modern solid state theory.

While the basic physics underlying the paramagnetic Mott-Hubbard metal-insulator transition (MHMIT) is well understood now,2)–5) there remain several openquestions to be answered. First, most of the theoretical studies of the MHMIT weredone for models showing particle-hole symmetry.2)–4) In this case, one typically findsnesting of the Fermi surface in the noninteracting case, which strongly favors orderedinsulating ground states.

In real materials, however, particle-hole symmetry is the exception rather thanthe rule and a question that immediately occurs is whether this can suppress theordering sufficiently to uncover the MHMIT as it e.g. seems to happen in V2O3.2)

Furthermore, without nesting of the Fermi surface, there is a priori no need that anordered ground state is also insulating, which then opens the interesting possibilitiesof an additional MHMIT in the ordered state.

But even if the ordered phase completely hides the paramagnetic MHMIT, thereremains the interesting question whether there will be a qualitative change, possibly

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Antiferromagnetism and MIT in the Hubbard Model 275

via a phase transition, of the properties of the ordered insulator when increasingthe correlations: From a Slater insulator, i.e. an insulator driven by the reducedtranslational symmetry in the ordered state, at weak coupling6) to a state at strongcoupling which is Mott-Hubbard localized and shows order on top of that.7),8) Sucha crossover or transition will show up in e.g. the optical properties and can possiblyexplain experiments in V2O3.9),10)

In this overview I will address some of these questions using the one-band Hub-bard model as simplest model that shows both phenomena, viz a correlation-drivenMHMIT and an antiferromagnetically ordered ground state. Since the Hubbardmodel cannot be solved exactly except for one spatial dimension, where no orderedphases are present, I employ the dynamical mean-field theory2) (DMFT) to solvethe model. The DMFT is a non-perturbative technique that allows to study thephysical properties of correlated models like the Hubbard model qualitatively andthermodynamically consistent, covering the full parameter regime and both T = 0and T > 0.

§2. The Hubbard model

The one-band Hubbard model8)

H = −∑i,j,σ

tijc†iσcjσ +

U

2

∑iσ

niσniσ , (2.1)

was originally introduced to describe band-ferromagnetism in transition metals. Inthe standard representation used in (2.1), c

(†)iσ annihilates (creates) an electron at site

i with spin σ, niσ = c†iσciσ, tij denotes the hopping amplitude between sites i andj and U is a parameter representing the local Coulomb repulsion between electronsmeeting at a site i. In this simplest version of the model, the longer-range parts ofthe Coulomb repulsion are neglected. Usually, one also ignores long-range hoppingprocesses and concentrates on nearest-neighbor hopping only, i.e.

tij =

{ −t, if i and j are nearest neighbors ,

0 otherwise .(2.2)

Qualitatively, one can expect this model to show the desired features. If one fixesthe filling of the model to 〈n〉 = 1 (half filling) and increases the Coulomb repulsionU , it obviously becomes energetically more and more unfavorable for the particles tomove. Thus, one expectation is that the model (2.1) will undergo a MHMIT at somecritical Uc of the order of the bandwidth W of the system at U = 0. Furthermore, atU/t � 1 a perturbational analysis about the limit t = 0 shows that the low-energyproperties of the Hubbard model are captured by a Heisenberg model11)

Heff = −∑i,j

Jij�Si · �Sj (2.3)

with antiferromagnetic exchange coupling Jij = −2|tij |2/U . Furthermore, a simpleweak-coupling Hartree analysis shows, that for U/t → 0, too, antiferromagnetism

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276 Th. Pruschke

is the favored ground state.12) Thus, the MHMIT and antiferromagnetism are phe-nomena naturally occurring in the Hubbard model.

Apart from these qualitative expectations, reliable quantitative results for theHubbard model in more than one spatial dimension are extremely hard to obtain.One interesting conjecture can be deduced from Nagaoka’s theorem,13) which statesthat for U = ∞ and certain lattice types one hole or electron added to the half-filledstate leads to a fully polarized ferromagnet. It is an obvious and long-standing ques-tion, whether this ferromagnetic state remains stable, possibly not fully polarized,when a thermodynamically finite number of holes (or electrons) and also U < ∞ isconsidered.14)

Considerable progress in understanding the physics of this model has beenachieved in the last decade through the development of the dynamical mean-fieldtheory.2),15),16) A brief introduction to this method will be given in §3. Here I willreview the major findings for the Hubbard model on a simple cubic lattice withnearest-neighbor hopping. With the latter condition, this bipartite lattice showsperfect nesting at half filling. From a weak-coupling analysis at U/t → 0 and thestrong-coupling analysis based on the mapping to model (2.3) as U/t → ∞ one caninfer that the ground state at half filling is always antiferromagnetically ordered andinsulating due to the doubling of the unit cell. These expectations are confirmed by

doping U/(W+U)

AFI

FM

PM

AFI

PM

T/W

10.8

0.60.4

0.20

0.30.2

0.10

0.1

0.05

0

Fig. 1. Schematic DMFT phase diagram of the Hubbard model with nearest neighbor hopping on a

simple cubic lattice. PM denotes the paramagnetic metal, AFI the antiferromagnetic insulator,

FM the ferromagnetic metal.

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the DMFT,2),16),17) the results of which are collected in the schematic phase diagramas function of doping δ = 1 − 〈n〉 and the local Coulomb repulsion U in Fig. 1. Tocapture the whole range from U = 0 to U = ∞, the corresponding axis was rescaledto display U/(W +U), where W denotes the bandwidth of the dispersion for U = 0.At half filling the physics is dominated by an antiferromagnetic insulating phase(AFI) for all U > 0. For finite doping, the antiferromagnetic phase persists up to acritical doping δc.17),18) Eventually, for very large values of U , the antiferromagneticphase is replaced by a small region of Nagaoka type ferromagnetism.13),18),19)

One will wonder where the MHMIT mentioned previously can be found in thephase diagram in Fig. 1. This phase transition can be made visible at half filling ifone artificially ignores the antiferromagnetic phase. Then, a transition from a para-magnetic metal (PM) to a paramagnetic insulator (PI) is found. At T = 0, it occursat a value of the Coulomb parameter Uc ≈ 1.5W .2),3),17) Interestingly, the transitionis of first order2),4) for T > 0 with a second order end point at a Tc ≈ 0.017W andUc ≈ 1.2W . Note that Tc < Tmax

N . Both phase transitions as function of U/W areshown together in the phase diagram in Fig. 2. Obviously, any attempt to describereal materials, for example the phase diagram of V2O3

20),21) within the one-bandHubbard model has to account for that particular fact. A possible solution wasaccordingly suggested early on, namely that introduction of next-nearest-neighborhopping, leading to magnetic frustration, will reduce TN sufficiently to suppress itbelow Tc.2) Since the phase transition to the antiferromagnet is, at least for theparticle-hole symmetric model, of second order, such a crossing with the first or-der transition of the MHMIT must also change the order of the transition to theNeel state, and a new interesting question arises: How do both transitions becomecompatible as TN ↘ Tc?

0.06

0.05

0.04

0.03

0.02

0.01

0.000 0.5 1

U/W1.5 2 2.5 3

Metal

T/W

Insulator

Antiferromagnetic insulator

Fig. 2. DMFT phase diagram of the Hubbard model with nearest-neighbor hopping on a simple

cubic lattice at half filling.

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278 Th. Pruschke

Last, but not least, even in the situation TN > Tc, it might be argued that withinthe Neel state, too, an additional transition can occur. As already discussed, at smallU a weak coupling theory is expected to give accurate results, leading to a band orSlater insulator6) due to the doubled unit cell in the Neel state. At large U , on theother hand, the Hubbard model reduces to an effective Heisenberg model11) (2.3)with localized moments from the onset and the question naturally arises whetherthese two limits are linked continuously or via a phase transition at some finite valueof the Coulomb interaction U .

§3. DMFT and NRG

The dynamical mean-field theory (DMFT) to exactly solve the Hubbard modelin the limit D → ∞ — or, equivalently, large coordination number of the lattice — isbased on the work by Metzner and Vollhardt15) and is by now well-established.2),16)

The basic ingredient is that for D → ∞ the proper single-particle self-energy Σ(�k, z)becomes purely local or momentum independent, i.e. Σ(�k, z) D→∞−→ Σ(z).15),22) Thiscan be used to map the Hubbard model (2.1) onto an equivalent quantum impu-rity problem (effective single impurity Anderson model) supplemented by a self-consistency condition.2),16) The remaining problem (the solution of a quantum impu-rity model) is, however, still highly nontrivial. Several approximate and numericallyexact techniques are currently available2),23) to accomplish this demanding task.

Most of these methods cannot access T → 0 or are restricted to the weak-coupling regime of the Hubbard model. The most reliable technique to solve thequantum impurity problem for all interaction strengths U and fillings n at T = 0and low T is the numerical renormalization group (NRG).24) Originally, this methodwas set up to treat the paramagnetic problem only, but recent extensions have shownthat calculations with a symmetry breaking field are possible with a similar level ofaccuracy, too.25) Hence, with the NRG we are able to study magnetically orderedphases directly at T ≥ 0.

In contrast to the standard NRG, a more refined approach has to be used tocalculate dynamical quantities in the presence of a magnetic field. This has first beennoted by Hofstetter, who observed discrepancies in the magnetization calculated fromthe spectral functions and the ground state occupation numbers.25) To resolve thisproblem, he proposed a modification of the standard method26) to calculate thespectral function.

§4. Technical details of the calculation in the Neel state

4.1. DMFT in the symmetry-broken state

There are in principle two ways to determine the phase boundary between theparamagnetic and a magnetically ordered state. First, one can calculate the suscep-tibility corresponding to the anticipated order and look for a divergence. Second, onecan allow for a proper symmetry breaking in the one-particle Green function andsearch for the region in parameter space where a solution with broken symmetry

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becomes stable.Since we are, apart from determining the phase boundary, also interested in the

physics in the symmetry-broken phase, we use the second approach as our methodof choice. This prohibits the search for incommensurate phases, because only brokensymmetries with a commensurate wave vector can be implemented that way. How-ever, since we are interested only in the standard Neel type antiferromagnetic order,this deficiency introduces no fundamental problem here. Within our approach, theproper way to look for the Neel phase is to introduce an AB-lattice structure asdepicted in Fig. 3 and allow for different sublattice magnetizations.

With this AB tiling of the lattice in the Neel state, the DMFT equations have tobe modified as well to account for two inequivalent sublattices A and B in Fig. 3 withself-energies ΣA = ΣB .2),27) To this end, we introduce operators a

(†)iσ and b

(†)iσ which

act on sublattices A and B respectively. In the case of nearest-neighbor hoppingonly, the kinetic part of the Hamiltonian (2.1) can then be written as

Ht = −t∑〈i,j〉

∑σ

(a†iσbjσ + b†jσaiσ

). (4.1)

After Fourier transforming this expression we obtain

Ht =∑σ

∑k

′Ψ †

kσ

(0 εk

εk 0

)Ψkσ , (4.2)

where we introduced the spinors

Ψ †kσ =

(a†kσ , b†kσ

), Ψkσ =

(akσ

bkσ

)(4.3)

and εk is the dispersion on the bipartite lattice. The prime on the sum indicatesthat the summation is over all values of k in the magnetic Brillouin zone (MBZ) (see

A

Bkx

ky

Fig. 3. Left: AB tiling of the cubic lattice necessary to do calculations for a Neel ordered state.

Right: First Brillouin zone for the simple cubic lattice (dashed lines). With the AB tiling from

the left part of the figure, the first Brillouin zone of the antiferromagnetic state (MBZ) is given

by the full lines. Note that this is exactly the Fermi surface of the noninteracting system at half

filling (perfect nesting).

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280 Th. Pruschke

Fig. 3, right panel). Within this notation, the Green function within the DMFT andfor nearest-neighbor hopping becomes a matrix in the two sublattices,2)

Gkσ(z) =(

ζAσ −εk

−εk ζBσ

)−1

, (4.4)

where ζA/Bσ = z + µ − Σ

A/Bσ . From now on we employ the symmetry ζA

σ = ζBσ ≡ ζσ

of the Neel state and drop the indices A/B. For the calculation this means that wedo not have to solve independent quantum impurity models for the two sublattices,but only one for say sublattice A.

4.2. Ground state energy and filling

To find the correct ground state, we need to calculate the ground state energy

E

N=

1N

〈H〉 =1N

〈Ht〉 +U

N

∑i

〈ni↑ni↓〉 , (4.5)

where Ht is the kinetic part of the Hamiltonian (2.1). The expectation value 〈ni↑ni↓〉can be determined within the NRG directly. The quantity 〈Ht〉, on the other hand,depends on the phase we are looking at. For para- and ferromagnetic phases it issimply given by2)

1N

〈Ht〉 =∑σ

∞∫−∞

dε ε ρ(0)(ε)

∞∫−∞

dωf(ω)Aσ(ε, ω) , (4.6)

with ρ(0)(ε) the density of states (DOS) for the non-interacting system, f(ω) theFermi function and

Aσ(ε, ω) = − 1π�m

1ω + µ − ε − Σσ(ω + i0+)

the spectral function of the Hubbard model in the DMFT, i.e. with �k-independentone-particle self-energy.

For an antiferromagnetic state with Neel order one has to take into account theAB-lattice structure and the formula becomes2)

1N

〈Ht〉 = 2

∞∫−∞

dε ε ρ(0)(ε)

∞∫−∞

dωf(ω)B(ε, ω) (4.7)

instead, with

B(ε, ω) = − 1π�m

1√ζσ(ω)ζσ(ω) − ε

and ζσ(ω) = ω + µ − Σσ(ω + i0+). Obviously, expression (4.7) reduces to (4.6)without magnetic order, i.e. ζσ(ω) = ζσ(ω).

Note that a similar calculation for finite T would involve the calculation of thelattice entropy S, too. However, within the NRG, this is currently not feasible.

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4.3. Optical conductivity

An appealing property of the DMFT is the possibility to calculate transportquantities in a very simple fashion. Due to the local nature of the theory, ver-tex corrections to the leading particle-hole bubble of the current-current correlationfunction vanish identically,16),28) i.e. one needs to calculate the bare bubble only.This has been extensively used to study the optical conductivity and various othertransport properties in the paramagnetic phase of the Hubbard model.2),16),29) Onthe other hand, up to now a comparable investigation of the optical properties ofsymmetry broken phases, in particular the Neel state at half filling, has not beenperformed. However, such an investigation is interesting for the reasons discussed inthe Introduction.

Using the formalism developed in the previous subsection, the current operatoris given by

j = e∑

σ

∑k

′Ψ †

kσ

(0 vk

vk 0

)Ψkσ (4.8)

with vk = ∇kεk as usual. If we consider a lattice for which the conductivity tensoris diagonal, the elements σii ≡ σ can be calculated from (D is the spatial dimensionof the lattice)

D · σ(ω) = e1iω

D∑l=1

〈〈jl; jl〉〉ω+iδ (4.9)

with the current-current correlation function

〈〈jl; jl〉〉iν = e2∑σσ′

∑kk′

′vl

kvlk′

× 〈〈a†kσbkσ + b†kσakσ; a†k′σ′bk′σ′ + b†

k′σ′ak′σ′〉〉iν .

Again, due to the symmetry of the lattice, the index l can be dropped. The mostimportant simplification arises from the locality of two-particle self-energies withinthe DMFT.16),27),28) In analogy to the paramagnetic case this allows us to carry outthe k sums in diagrams containing two-particle self-energy insertions independentlyat each vertex. Since the single particle propagators only depend on k through theeven function εk and the vk are of odd parity, the sum over their product vanishes. Asa result, we obtain the exact expression for the current-current correlation functionin the DMFT,

〈〈j; j〉〉iν = −e2

β

∑ωn

∑σ

∑k

′v2

k

× [GAAkσ (iωn + iν)GBB

kσ (iωn)

+ GBBkσ (iωn + iν)GAA

kσ (iωn)+ GBA

kσ (iωn + iν)GBAkσ (iωn)

+ GABkσ (iωn + iν)GAB

kσ (iωn)]

,

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whereGAA

kσ (z) =ζσ

ζσζσ − ε2k

, GBBkσ (z) =

ζσ

ζσζσ − ε2k

andGBA

kσ (z) = GABkσ (z) =

εk

ζσζσ − ε2k

.

Next, we convert the k sum into an energy integral by introducing the averagesquared velocity,

〈v2〉ε :=1

D · N∑

k

′v2

kδ(ε − εk) . (4.10)

Making furthermore use of the spectral representation of the Green functions, thefrequency sum can be evaluated in a straightforward way and finally we obtain forthe conductivity

σ(ω) = c∑σ

0∫−∞

dε 〈v2〉ε∞∫

−∞dω′ f(ω′) − f(ω′+ ω)

ω

× [ρAAσ (ε, ω′)ρAA

σ (ε, ω′+ ω) + ρABσ (ε, ω′)ρAB

σ (ε, ω′+ ω)]

(4.11)

withρAA

σ (ε, ω) = − 1π�m GAA

σ (ε, ω + iδ)

andρAB

σ (ε, ω) = − 1π�m GAB

σ (ε, ω + iδ) .

Here f(ω) is the Fermi function and c collects various constants. Note that the form(4.11) is reminiscent of the results found in the case of superconductivity, which isdiscussed at length e.g. in the book by Mahan.30) Consequently, one can expect toobtain similar features from the evaluation of (4.11).

§5. Results

In the following subsections we will discuss properties of the Hubbard model(2.1) for a bipartite lattice and dispersion (2.2). The effects of long-range hoppingwill be addressed in §5.4.

5.1. Weak-coupling results

Let us briefly review some weak-coupling results obtained from Hartree approx-imation. At T = 0, one finds a transition into the Neel state for any U > 0 belowa critical doping δH

c (U). For small U → 0 the magnetization m as well as thecritical doping depend non-analytically on U , i.e. m, δH

c ∝ exp(−1/(Uρ(0)(0))

)/U

independent of the dimension.A quantity of particular interest in the DMFT is the single-particle Green func-

tion. The general structure of the Green function in the Neel phase for both Hartreetheory and DMFT is given by expression (4.4), where in the Hartree approximation

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Σσ(z) reduces to ΣHσ (z) = Unσ = 1

2U(n−σm) with n the filling and m the magneti-zation. The local Green function is obtained from (4.4) by summing over �k ∈ MBZ,which yields for example for spin up

G↑(ω) =ζ↓(ω)√

ζ↑(ω)ζ↓(ω)G(0)

(√ζ↑(ω)ζ↓(ω)

)(5.1)

with ζσ(ω) = ω + i0+ + µ − U2 n + σU

2 m and

G(0)(z) =

∞∫−∞

dερ(0)(ε)z − ε

. (5.2)

For the further discussion let us define

ω− = U2 n − µ − U

2 m ,

ω+ = U2 n − µ + U

2 m .

Then, as long as ω ≤ ω− or ω ≥ ω+, the radicant in (5.1) is positive and the resultingDOS can be expressed as

ρ↑(ω) =ζ↓(ω)√

ζ↑(ω)ζ↓(ω)ρ(0)

(√ζ↑(ω)ζ↓(ω)

).

For ωi < ω < ω+, on the other hand, the radicant in (5.1) is negative, i.e.√

ζ↑(ω)ζ↓(ω)= i√|ζ↑(ω)ζ↓(ω)|. Since for particle-hole symmetric DOS the Green function G(0)(z)

defined in (5.2) for purely imaginary arguments is purely imaginary, too, one finds

ρ↑(ω) = 0 ,

i.e. the DOS has a gap between ω− and ω+. As one approaches ω− from below orω+ from above, it is easy to confirm that

ρ↑(ω) ≈

√Um

|ω − ω−|ρ(0)(0) ω ↗ ω− ,

√|ω − ω+|

Umρ(0)(0) ω ↘ ω + .

(5.3)

The corresponding DOS for σ =↓ has a similar behavior. Here, however, the DOSdiverges like 1/

√|ω − ω+| at the upper gap edge, and vanishes like√|ω − ω−| at

the lower one.In order to determine the thermodynamically stable phase one has to calculate

the ground state energy as function of the doping δ = 1−n. The result up to secondorder in U is31)

E(δ) − E(0) = −U

2δ + αH · ΦH(δ/δ1) , (5.4)

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284 Th. Pruschke

ω−

ω+ ω

0

ρ ↑(ω

)

∼ 1

√ω − ω−

∼ √ω − ω+

U |m|

Fig. 4. Behavior of the DOS for the majority spins on a particular sublattice in Hartree approxi-

mation close to the gap edges.

where

ΦH(x) =

12x

(1 − 1

4x

)x < 1 ,

14

(1 +

12x2

)x > 1 ,

(5.5)

and δ1 is the critical doping for antiferromagnetism in Hartree approximation. Thecoefficient αH is given by αH = 2δ2

1/ρ(0)(0). The function ΦH(δ/δ1) appearing inexpression (5.4) leads to the full line in Fig. 5. Apparently, this function is notconvex for small δ, i.e. the resulting phase is thermodynamically unstable towardsphase separation for dopings less than δc =

√2δ1. The resulting ground state energy

is then obtained from a Maxwell construction, given by the straight dashed line inFig. 5.

5.2. Half filling

5.2.1. Single-particle propertiesLet us start with the single-particle properties at half filling, n = 1.18),33) Here,

the Neel phase is energetically stable. The variation of the DOS for increasing Ufrom U = W/2 (full curve) to U = W (dashed curve) is shown in Fig. 6. As expected,the DOS for small U resembles the form (5.3). The characteristic features howevervanish rapidly with increasing U , and already for U = 3W/4 the DOS mainly consistsof the Hubbard peaks at ω = +U/2 and ω = −U/2 for σ =↓ and σ =↑, respectively;reminiscent of the behavior expected for the Mott-Hubbard insulator, where only theincoherent charge excitation peaks at high energies are present.2),3),32) Note thatneither from the spectra in Fig. 6 nor from the behavior of the magnetic moment inthe inset of Fig. 6 one can infer that at U ≈ W the Mott-Hubbard metal-insulator

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0 0,5 1 1,5 2δ/δ1

0

0,2

0,4

0,6

ΦH(δ

/δ1)

Fig. 5. The function ΦH(δ/δ1) from Eq. (5.5). Note the concave curvature between δ = 0 and

δ = δ1. The dashed line shows the actual behavior of the ground state energy following from a

Maxwell construction.

-1 -0.5 0 0.5 1ω/W

0

1

2

3

4

5

W⋅A

↓(ω)

0

1

2

3

4

5

W⋅A

↑(ω)

U/W=0.25U/W=0.5U/W=0.75U/W=1

0 0.5 1 1.5 2U/W

0

0.5

1

n↑-n

↓

Fig. 6. DOS for spin up and down at half filling in the antiferromagnetic phase as function of U .

While for small values of U the weak-coupling form (5.3) is approximately reproduced, the DOS

for large U is basically that of the Mott-Hubbard insulator. The inset shows the magnetization

as function of U .

transition occurs in the paramagnetic state.3),32)

5.2.2. Optical conductivity and optical gapThe optical conductivity resulting from the spectra in Fig. 6 is shown in Fig. 7.

Apparently, the overall behavior seen in the DOS has its counter part in σ(ω). For

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286 Th. Pruschke

0 0,5 1ω/W

0

2

4

σ(ω

) [ar

b. u

nits

]

U/W=0.25

U/W=0.5

U/W=0.75 U/W=1

Fig. 7. Optical conductivity of the half-filled Hubbard model in the Neel phase for T = 0 as a

function of U . The full lines represent the calculated data, the dashed lines a fit with the

function ω · σ(ω) = �m˘eiφ (ω − ω0 + iγ)−α

¯(see text).

0 0.5 1 1.5U/W

0

0.5

1

1.5ω0/ms

U

α

φ/π

Fig. 8. Dependence of the fit parameters ω0, α and φ in (5.6) on U . The lines are meant as guides to

the eye. Note the rather well defined change in (α, φ) from (α, φ) = (1/2, π/2) to (α, φ) = (1, 0)

around U/W = 0.75.

small values of U , one finds a threshold behavior with a singularity, whereas forlarge U the optical conductivity closely resembles the one found in the paramagneticinsulator.17)

To proceed, let us note that in the Hartree limit, i.e. without an imaginary part

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of the self-energy, an approximate evaluation of (4.11) yields30),33)

ω · σ(ω) ∝ Θ(ω − 2∆0)√ω − 2∆0

with ∆0 = Ums/2 and ms = 〈n↑−n↓〉. Since this behavior is governed by square rootsingularities in the integrand in (4.11), it is reasonable to assume that for a finiteimaginary part of the self-energy the above singularity will become an algebraicfunction

ω · σ(ω) ∝ �m

{eiφ

(ω − ω0 + iγ)α

}(5.6)

with a general exponent α. The quantity γ approximately cares for the finite imagi-nary part introduced by the one-particle self-energy and φ allows for a more complexmixing of real and imaginary parts in the integral (4.11). The function (5.6) describesthe behavior of σ(ω) in the vicinity of the maximum very nicely for all values of U(see dashed lines in Fig. 7).

Let us now turn to the behavior of the parameters ω0, α and φ shown in Fig. 8.As U → 0, we expect that ω0 = 2∆0 = Ums, α = 1/2 and φ = π/2, i.e. ω · σ(ω) ∝ e (ω − ω0 + iδ)−1/2 = Θ(ω −ω0)/

√ω − ω0. We indeed find the anticipated square-

root singularity; however, even for small U/W , the value of ω0 significantly deviatesfrom the Hartree value, being systematically smaller but obviously approaching itas U → 0.

For values U > W , the behavior of ω · σ(ω) is best described by a Lorentian,which becomes apparent from the values of α and φ obtained in this region, vizα ≈ 1 and φ = 0, meaning ω · σ(ω) ∝ �m (ω − ω0 + iγ)−1 ∝ 1/

((ω − ω0)2 + γ2

). In

addition, the results for ω0 together with ms ≈ 1 indicate that ω0 ≈ U , in agreementwith the predictions of the Mott-Hubbard picture.2)

We find, however, no evidence that the Slater limit at U/W → 0 and the Mott-Heisenberg limit at U/W → ∞ are separated by some kind of phase transition. Allresults rather indicate that a smooth crossover takes place for U/W ≈ 3/4.

5.3. Doping δ > 0

Keeping U fixed at U = 3W/4 and increasing δ leads to the spectra shown inFig. 9. Quite interestingly, the typical weak-coupling characteristics reappear in thespectra for small doping and are still recognizable for δ = 13%. Note also that uponvariation of doping and hence of the magnetization the spectra are not shifted in thesame way as in Hartree theory. Instead, the dominant effect is a strong redistributionof spectral weight from the Hubbard bands to the Fermi level. Eventually, in theparamagnetic phase one recovers the well-known three peak structure of the dopedHubbard model in the DMFT.2)

The evolution of the spectra both at and off half filling can be understood withina simple picture. In Fig. 10 we show a sketch of the Hartree bandstructure of theHubbard model in the Neel state, which has two branches in the MBZ and a gapof width ∝ U |m| between them. If, on the other hand, we inspect the paramagneticsolution, one for example finds at half filling and for small values of U a Fermi liquidwith quasiparticles defined on an energy scale larger than U |m|. This situation is

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-1 -0.5 0 0.5 1ω/W

0

0.5

W⋅A

↓(ω) [

arb.

uni

ts]

0

0.5

W⋅A

↑(ω) [

arb.

uni

ts]

δ=7%δ=13%δ=16%δ=20%

Fig. 9. DOS for spin up and down for U = 3W/4 and different dopings δ = 7%, δ = 13%, δ = 16%

and δ = 20%. The system at δ = 20% is already in the paramagnetic phase.

−1/2 0 1/2

k/π

∝ U |m|

(a)

(b)

(c)

Fig. 10. Schematic picture of the Hartree bandstructure of the Hubbard model in the Neel state.

The arrows on the left-hand side of the figure represent the energy scales of the corresponding

paramagnetic Fermi liquid for half filling and weak coupling (a), half filling and intermediate

coupling (b) and finite doping and intermediate coupling (c).

indicated by the arrow labeled (a) on the left side of Fig. 10. Here we expect, andindeed find for the antiferromagnetic solution (see full curve in Fig. 6), a DOS thatshows the characteristic van-Hove singularities of Fig. 4. Increasing U eventuallyleads to a situation, where the energy scale for the quasiparticles in the paramagneticstate is finite but much smaller than U |m| (arrow (b) in Fig 10). The self-energy inthe energy region of the van-Hove singularities then has a large imaginary part andwill completely smear out the characteristic structures. Further increasing U into

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the Mott-Hubbard insulator will then not change the picture qualitatively, explainingthe similarity between the curves for U � UMIT and U > UMIT in Fig. 6. With finitedoping, we move the chemical potential into, e.g., the lower band; this means thateven for a relatively small quasiparticle energy scale one again sees the van-Hovesingularities at the band edge, which results in the well defined singularities in thespectra for small doping in Fig. 9.

From the occupation numbers nσ obtained after convergence of the DMFT cal-culation one can calculate the magnetization per electron, m = (n↑ − n↓)/(n↑ + n↓),as function of the doping δ. The results for U = W/2 and U = 3W/4 are shown inFigs. 11(a) and (b) together with a fit to a power law

m(δ) = m0

∣∣∣∣1 − δ

δAFc

∣∣∣∣ν

. (5.7)

The resulting fit parameters are summarized in Table I. As expected for a mean-fieldtheory, the value for the critical exponent is ν = 1/2.

Finally, with the converged DMFT self-energy Σσ(z) we can calculate the ex-pectation value 〈H〉/N according to Eqs. (4.5) and (4.6) respectively (4.7) for theparamagnetic and antiferromagnetic phase. The results for the characteristic func-tion

Φ(δ) = E(δ) +U

2δ − Emag(0)

are summarized in Figs. 12(a) and (b). In Fig. 12 the energies of the antiferro-magnetic phase are represented by the circles, those of the paramagnetic phase by

0 0,04 0,08

δ

0

0,2

0,4

0,6

0,8

1

m=

(n↑−

n↓)/

(n↑+

n↓)

(a)

0 0,1 0,2

δ

(b)

AFM

PM

AFM

PM

Fig. 11. Doping dependence of the magnetization per electron for U = W/2 (a) and U = 3W/4

(b). The full lines are fits with the function (5.7), the resulting fit parameters are summarized

in Table I.

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0 0,05 0,1

δ

0

0,005

0,01

0,015

Φ(δ

)=E

(δ)−

Em

ag(0

)+U

δ/2

δcAF

(a)

0 0,1 0,2

δ

0

0,05

0,1

0,15

0,2

δcAF

(b)

Fig. 12. Doping dependence of the energy of the paramagnetic phase (squares) and the Neel phase

(circles) for U = W/2 (a) and U = 3W/4 (b). The full lines are fits with the function (5.8), the

corresponding fit parameters are summarized in Table I. The dashed lines are the result of a

Maxwell construction for the ground state energy.

Table I. Results of the fits of m(δ) in Fig. 11 to expression (5.7) and E(δ) in Fig. 12 to (5.8).

Magnetization Energy

U m0 δAFc ν δPS

c δ1 α/αH γ

1 0.4 0.06 0.49 0.07 0.047 0.52 0

3 0.9 0.16 0.54 0.157 0.191 0.33 0.026

squares. The full lines interpolating the antiferromagnetic data are fits to the func-tion

Φ(δ) = αΦH(δ/δ1) + γ

(δ

δ1

)3

(5.8)

with ΦH(x) according to (5.5). The fit parameters are summarized in Table I. Theuse of the function ΦH(x) in (5.8) is motivated by the results of van Dongen.31) Thelines interpolating the paramagnetic data are meant as guides to the eye only. Thedotted vertical lines denote the value δAF

c as obtained from Fig. 11.The antiferromagnet obviously has the lower energy as compared to the para-

magnet in the region 0 ≤ δ ≤ δAFc . However, in both cases U = W/2 and U = 3W/4

we find a clear non-convex behavior in E(δ) in that region, i.e. the aforementionedsignature of an instability towards phase separation. The true ground state energyas function of δ is obtained again via a Maxwell construction, leading to the dashedlines in Fig. 12 and the values δPS

c given in Table I. Note that in both cases δAFc ≈ δPS

c

within the accuracy of the fitting procedure. Note that this finding differs from theprediction of the weak coupling result31) δPS

c =√

2δAFc .

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While for U = 1 the function Φ(δ) nicely follows the weak-coupling prediction(5.4) with renormalized constant α one finds a sizeable contribution ∼ δ3 for U = 3.This additional term results in a much weaker non-convex behavior of E(δ) for U = 3.

5.4. Including magnetic frustration

As a last subject let us address the question, what effect magnetic frustrationhas on the phase diagram at half-filling; in particular, whether a reasonable amountof longer-range hopping can lead to a sufficient suppression of TN to uncover theMHMIT. Unfortunately, next-nearest-neighbor hopping on a cubic lattice leads tonumerical complications, in particular when one tries to fix the filling to 〈n〉 = 1.We therefore adopt a variant of longer-range hopping first introduced by Muller-Hartmann22) and later adopted by Georges et al.2) to set up a simplified set ofDMFT equations2)

GAσ(z) =1

z + µ − ΣAσ(z) − t214 GBσ(z) − t22

4 GAσ(z),

GBσ(z) =1

z + µ − ΣBσ(z) − t214 GAσ(z) − t22

4 GBσ(z), (5.9)

for calculations in the Neel state. In the following, we fix t2/t1 = 1/√

3 ≈ 0.58.Figure 13 shows the NRG results for the DOS at T = 0 and spin up on an A

lattice site. Due to particle-hole symmetry the DOS for spin down on A sites (orspin up on B sites) can be obtained by ω → −ω. The full and dashed lines representthe AFI solution for U ↘ Uc and the PM solution for U ↗ Uc, respectively. Clearly,the magnetic solution is insulating with a well-developed gap at the Fermi energy.Quite generally, we were not able to find a stable AF metallic solution at T = 0.

The discontinuity in the staggered magnetization mS at the transition PM↔AFI

-1.5 -1 -0.5 0 0.5 1 1.5ω

0

0.5

1

1.5

A↑(ω

)

U/W=0.85, pmU/W=0.9, afi

Fig. 13. Density of states for spin up on an A lattice site as function of frequency.

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0.85 0.9 0.95 1U/W

0

0.2

0.4

0.6

0.8

1

1.2 m(U) from AFIA(0) from PMA(0) from AFIm(U) from PM

0 0.1 0.2 0.3(TN-T)/TN

0

0.5

ms2

Fig. 14. Staggered magnetization (solid lines) and total DOS at the Fermi energy (dashed lines)

as function of U in the vicinity of Uc for T = 0.0155W . The arrows indicate that the DMFT

solutions have been obtained by either increasing U (→) or decreasing U (←). To verify that

this hysteresis is not an artifact the inset shows the squared staggered magnetization mS as

function of T at large U = 2W , where the mapping to a Heisenberg model requires that m2S

vanishes continuously like TN − T as T ↗ TN .

implies a first order transition and the existence of a hysteresis region. Indeed,starting from the paramagnet at U � Uc and increasing U results in a magnetizationcurve different from the one obtained by starting at U � Uc and decreasing U . Thisis apparent from Fig. 14 (main panel, full lines) where a region of hysteresis can beobserved in the staggered magnetization (for temperature T = 0.0155W ). At thesame time the total DOS at the Fermi energy A(0) = A↑(0)+A↓(0) shows hysteresisbetween metallic and insulating behavior in exactly the same U region. Note, thatdue to the finite temperature the DOS at the Fermi level is not exactly zero in theNeel state, but strongly reduced as compared to the metal.4) Of course, at T = 0also the DOS at the Fermi level in the Neel state vanishes.36)

It is of course important to verify that the hysteresis found for small U is not somekind of artifact. This can most conveniently be shown by looking at the transitionat large U . Due to the mapping of the Hubbard model to a Heisenberg model in thisregime, one should expect the transition to be of second order, with the staggeredmagnetization vanishing continuously like mS ∝ √

TN − T when approaching TN

from below. That this is indeed the case is apparent from the inset to Fig. 14,where we show the squared staggered magnetization as function of T for U/W = 2.The transition is thus of second order with the expected mean-field exponent in thisregion of the phase diagram.

Collecting the results for the transitions and the hysteresis region for differenttemperatures leads to the phase diagram in Fig. 15. An enlarged view of the regionshowing coexistence of PM and AFI is given in the inset, where the full line repre-

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0 1 2U/W

0

0.05

0.1

T/W

0.85 0.9 0.95 10

0.01

0.02

0.03

PM

AFI

AFI

PM

Fig. 15. Magnetic phase diagram for the Hubbard model with frustration as defined by Eq. (5.9)

and t2/t1 = 1/√

3. The dotted lines inside the AFI denote the coexistence region for the

paramagnetic MIT. The inset shows an enlarged view of the region with coexistence of PM and

AFI.

sents the transition PM→AFI with increasing U and the dashed line the transitionAFI→PM with decreasing U . These two lines seem to merge at a value of U ≈ W forthis particular value of t2, with a critical temperature for this endpoint Tc ≈ 0.02W .Note that, even in the presence of such a sizeable t2, the antiferromagnetic phasestill completely encompasses the paramagnetic MIT (dotted lines in the main panelof Fig. 154)).

It is, of course, interesting to see how the magnetic phase evolves with increasingt2 and in particular how its boundary crosses the paramagnetic MIT. We find thatincreasing t2 does not change the form of the magnetic phase in Fig. 15 qualitatively,but mainly shifts the lower critical U and decreases the maximum TN . The calculatedestimates for those two quantities as function of t2 lead to the schematic evolution ofthe phase diagram presented in Figs. 16(a)–(c). Here, only the true phase boundariesare shown. A direct calculation of the free energy at finite temperatures is presentlynot possible with the NRG method, so we cannot calculate the actual transition lineseparating the paramagnetic and AF phases. The transition lines in Figs. 16(a)–(c)are therefore a guide to the eye only.

Figure 16(a) shows the qualitative phase diagram corresponding to Fig. 15 withthe line of first order transitions ending in a critical point. Upon further increasingthe value of t2, the first order transition lines from both the PM↔AFI and the Motttransition cross (Fig. 16(b)), thus exposing a finite region of the Mott insulator and atransition PI↔AFI. Finally, for even higher values of t2, the PM↔AFI transition atT = 0 approaches the Mott transition and TN is reduced significantly (Fig. 16(c)).Note that in the limiting case t2 = t1 the AFI phase completely vanishes due tothe structure of the DMFT equations (5.9). However, for t2 → t1 there is always a

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294 Th. Pruschke

T

t2/t1<0.75 (a)

(c)

(b)

T

t2/t1>0.75

U

T

t2/t1→1

AFI PI

PM PIAFI

PM PIAFI

Fig. 16. Schematic evolution of the magnetic phase diagram with increasing frustration. The dots

on the phase transition lines denote the critical endpoints of the first order transitions.

finite antiferromagnetic exchange J ∝ (t21 − t22)/U which is sufficient to stabilize anantiferromagnetic ground state for U > Uc of the Mott transition.

Acknowledgements

We would like to acknowledge intensive discussions and helpful comments by D.Vollhardt, R. Bulla, F. Gebhardt, F. Anders, G. Czycholl and G. Kotliar. This workwas supported by the DFG through the collaborative research centers SFB 484 andSFB 602. Supercomputer support was provided by the Leibniz Computer center, theComputer center of the Max-Planck-Gesellschaft in Garching the the Gesellschaft furwissenschaftliche Datenverarbeitung, Gottingen, Norddeutsche Verbund fur Hoch-und Hochstleistungsrechnen.

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Antiferromagnetism and Metal Insulator Transition in the Frustrated

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