The 2d Hubbard model on thehoneycomb lattice with Coulomb

interaction

Marcello PortaTutors: Prof. Giovanni Gallavotti, Prof. Vieri Mastropietro

Overview

I The model and its physical relevance,

I results and comparison with the literature,

I sketch of the proof: multiscale analysis, renormalization group andapproximate Ward identities,

I perspectives.

The model

We consider the following hamiltonian:

H = H0 + V

H0 = −∑

~x,~y∈ΛA∪ΛB~x,~y are n.n.

∑σ=↑↓

(a+~x,σb−~y ,σ + b+

~y ,σa−~x,σ

)

V =e2

2

∑~x,~y∈ΛA∪ΛB

(n~x − 1)ϕ(~x − ~y)(n~y − 1),

where:

I a±~x , b±~y are fermionic creation/destruction operators acting on two2d triangular sublattices ΛA, ΛB ;

I n~x is the density operator,

I ϕ(~x − ~y) is the 3d Coulomb potential.

The hamiltonian is stable and hole-particle simmetric, that is the systemis at half filling.

The honeycomb lattice

A B

Physical relevance of the model

The model has been thought as a schematization of graphene.Graphene was “discovered” in 2004 by Novoselov et al., and it can beconsidered a true bidimensional crystal; it consists in a single layer ofgraphite. Among its remarkable physical properties we mention:

I its stability,

I a very high electron mobility,

I the lowest known resistivity (∼ 10−6Ω · cm, less than silver),

I a minimum conductivity σ = e2

~ ,

I an anomalous quantum Hall effect,

I the insensitivity to localization effects usually induced by disorder,

I as to 2009, it is the strongest material ever tested.

The non interacting case

The hamiltonian in Fourier space is

H = H0 = − 1

|Λ|∑~k∈DL

∑σ=↑↓

v(~k)∗a+~k,σ

b−~k,σ + v(~k)b+~k,σ

a−~k,σ,

and the propagator is given by, setting k = (k0, ~k)

g(k) =1

k20 + |v(~k)|2

(ik0 −v∗~k−v(~k) ik0

),

where v(~k) = 1 + e−i 32 k1 cos

√3

2 k2; the electronic dispersion corresponds

to |v(~k)|.

Analogy with QED in 2 + 1 dimensions

The poles of the propagator correspond to the zeros of v(~k), that is tothe Fermi points

~p±F =

(2π

3,± 2π

3√

3

);

near ~p±F , v(~k ′ + ~pωF ) ' 32 (ik ′1 + ωk ′2), and

g(k′ + pωF ) ' −i1

γµk ′µ,

where pωF = (0, ~pωF ), and γµ, µ = 1, 2, 3 are the Dirac matrices.

Remark

The low energy excitations of the model “look like” massless Diracfermions in 2 + 1 dimensions; so far, graphene has been analyticallystudied in the relativistic approximation, neglecting the lattice andimposing “by hand” an u.v. cutoff (see Semenoff (1984), Guinea et al.(1993), Castro Neto et al. (2007) ...).

The conical dispersion of graphene

Short range interactions: results

By using constructive fermionic renormalization group methods it hasbeen rigorously proved by A. Giuliani and V. Mastropietro (2009) that athalf filling for sufficiently small coupling U

I the perturbative series defining the correlations of the ground stateis analytic in U,

I there is no mass gap, and no magnetic or superconductinginstabilities are present in the ground state,

I the Fourier transform of the two point Schwinger function is givenby, close to the singularities,

S(k′ + pωF ) ' 1

Z

(−ik0 −vF (ik ′1 + ωk ′2)

−vF (ik ′1 − ωk ′2) −ik0

)−1

,

where Z = 1 + O(U2), vF = 32 + O(U2),

I there is an anisotropy between the two components v1, v2 of thecharge velocity.

Remarks I

I The honeycomb lattice is not neglected, and no relativisticapproximation is made.

I The isotropy of the Fermi velocity, that is of the Dirac cone, underπ2 rotations is a highly non trivial fact, since the lattice explicitlybreaks this simmetry (see also Herbut (2009)); the same conclusioncannot be drawn from the relativistic approximation, which issymmetric by construction. Observations based on angle - resolvedphotoemission spectroscopy are compatible with this effect, seeZhou (2006), Bostwick (2007).

Remarks II

I The anisotropy of the charge velocity is due to irrelevant terms in arenormalization group sense, which would be neglected in therelativistic approximation; so far, this asimmetry has never beenexperimentally investigated.

I From the point of view of mathematical physics, this is one of thefew examples in which the ground state properties of interactingbidimensional fermions can be rigorously established.

Long range interactions: results

In a joint collaboration with A. Giuliani and V. Mastropietro we havefound that order by order, at half filling and for sufficiently small coupling

I the Coulomb interaction is unscreened;

I the Fourier transform of the two point Schwinger function is givenby, close to the singularities,

S(k′ + pωF ) ' 1

Z

(−ik0 −vF (k′) (ik ′1 + ωk ′2)

−vF (k′) (ik ′1 − ωk ′2) −ik0

)−1

where Z = 1 + O(e2) and vF (k′) ' 32 + a · e2 |log |k′||, a > 0;

I the dressed coupling constant e(k′) tends to e for k′ → 0;

I the free energy f and the specific heat Cv have the followingasymptotic behaviours:

f (T )− f (0) ∼ T 3

(1 + e2| log T |)2, Cv (T ) ∼ T 2

(1 + e2| log T |)2.

Remarks I

I As for the local case, the results are obtained without anyapproximation and keeping the lattice, and our statements are at allorders in perturbation theory. Convergence issues will be object offuture studies; at the present time we can only say that the n-thorder is bounded by n!en.

I The absence of screening for the Coulomb interaction has beenobtained at all orders by using a Ward identity, which implies that avery large cancellation is present in the perturbative series; the sameresult was previously known only from RPA approximations (seeGonzalez, Guinea, Vozmediano, (1993), (1998); Das Sarma (2007)),whose reliability is actively debated (see Mishchenko (2006), DasSarma et al. (2007)).

I Again, the Dirac cone is isotropic while the charge velocity is not.

Remarks II

I The presence of a single log in the Fermi velocity was previouslyestablished up to the fourth order in perturbation theory (seeGonzalez et al. (1998); Mishchenko (2007)). The eventualproliferation of logs at higher orders would imply the presence of an

anomalous dimension, since xε =∑

n≥0(ε log x)n

n! ; our result rules outthis possibility.

I The asymptotic infrared “undressing” of the effective couplingconstant is obtained through a Ward identity; as far as we know,this effect seems to have never been mentioned in literature.

I The asymptotic behaviours of f (T ), Cv (T ) are in agreement withthe ones obtained up to one loop in the current literature, seeSheehy (2007).

RG and multiscale analysis I

The methods that we are going to briefly discuss have been introduced inthe 80’s by G. Gallavotti in the context of scalar fields (renormalization ofϕ4 theory), and have been used to prove many rigorous result in one andtwo dimensional fermionic systems, like

I the vanishing of the beta function for 1d fermions (Benfatto,Mastropietro (2000); Mastropietro (2005)),

I the Fermi liquid behaviour of the 2d Hubbard model on squarelattice at small filling up to an exponentially small (BCS)temperature (Benfatto, Giuliani, Mastropietro (2005)).

RG and multiscale analysis II

First of all we write

Tre−β(H0+V ) =

∫P(dψ)eV (ψ),

where ψ±k,ρ are Grassmann variables and

V (ψ) = −e2

2

∑ρ,ρ′=1,2

∫dpϕ(~p)nρ(−p)e ip1δρ,2 nρ′(p)e−ip1δρ,2 ;

after a Hubbard - Stratonovich transformation,∫P(dψ)eV (ψ) =

∫P(dψ)P(dA)eV (ψ,A),

where Ak are complex variables, and

V (A, ψ) = ie∑ρ=1,2

∫dkdpApψ

+k+p,ρψ

−k,ρe−ip1δρ,2 ;

RG and multiscale analysis III

the covariances of the measures are given by

∫P(dψ)ψ+

k ψ−k = g(k) =

1

k20 + |v(~k)|2

(ik0 −v∗~k−v(~k) ik0

)∫

P(dA)AkA−k = gA(k) =|ρ(|~k |)|2

|~k |,

where ρ(|~k |) = 0 for |~k| large enough. Then, we introduce a scaledecomposition:

ψ±k = ψ±,(u.v .)k +

0∑j=−∞

ψ±,(j)k , Ak =

0∑j=−∞

A(j)k ,

where the fermionic/bosonic fields on scale j ≤ 0 live on momentadistant O(2j) from the Fermi points/zero.

RG and multiscale analysis IV

Correspondingly,

P(dA) =∏j≤0

P(dA(j)), P(dψ) = [∏j≤0

P(dψ(j))]P(dψ(u.v .));

after the integration of ψ(u.v .) and of the first |h| i.r. scales we find

∫P(dψ)P(dA)eV (ψ,A) =

∫P(dψ(≤h))P(dA(≤h))eV (h)(

√Zhψ

(≤h),A(≤h)),

where, if χh(k′) = 0 for |k′| ≥ 2h+1, χh(k′) = 1 for |k′| ≤ 2h,

V (h)(ψ(≤h),A(≤h)) = ieh

∑ρ

∫dkdpA(≤h)

p ψ+,(≤h)k+p,ρ ψ

−,(≤h)k,ρ + irr. terms

g (≤h)ω (k′) ' Z−1

h

(−ik0 −ch(ik ′1 + ωk ′2)

−ch(ik ′1 − ωk ′2) −ik0

)−1

χh(k′)

g (≤h)(~k) =1

|~k |+ ζhχh(~k).

RG and multiscale analysis V

Setting ~vh = (Zh, ch, ζh, eh) the flow of the β function has the form

~vh = ~vh+1 + ~β(~vh+1, ~vh+2, ..., ~v0), ~v0 =

(1,

3

2, 0, e

);

we have found that, at all orders in perturbation theory,

1. Zh − 1 = O(e2);

2. ch ' 32 + a · e2|h|, with a > 0;

3. ζh = O(

e2 2h

c2h

);

4. eh − e = O(e3).

Remark

Point 2 and 3 imply that

vF (k′) ≡ c(k′) ' 3

2+ a · e2| log |k′||, ζ(~k) = O

(c(~k)−1|~k|

),

since on scale h 2h ' |k′|, 2h

ch' |~k |.

RG and multiscale analysis VI

I Each step of the multiscale integration corresponds to a very largeresummation of Feynman graphs.

I All the results, except the bound for ζh, have been obtained inperturbation theory through an expansion of the running couplingconstants in renormalized Feynman graphs.

I The bound on the mass of the photon has been obtained via arigorous implementation of Ward identities in a regularized theory.

I The bound on the effective charge can be improved using Ward

identities; it follows that e(k′)− e ∼ e3

c(k′) .

Ward identities

The invariance of the theory under local gauge transformations impliesidentities between correlation functions; in particular, the invariance ofthe regularized generating functional under ψ±x → e±iαx0ψ±x gives

< np,ρnp,ρ >(h,N) |p=0 = limp0→0

R(h,N)(p0).

An explicit computation shows that |R(h,N)| ≤ O(

2h

c2h

); this proves the

desired result, since ζh = e2 < np,ρnp,ρ >(h,N) |p=0.

Remark

Neglecting the lattice and the cutoffs we would get R(−∞,∞)(p0) = 0,which is in contrast with the fact that < n0,ρn0,ρ >(−∞,∞) is non zero(and in particular is divergent).

Perspectives

1. Non istantaneous interaction; some preliminary results suggest thatvF (k′) < +∞, and that Z (k′) ∼ |k′|η with η = O(e2) (anomalousdimension).

2. If the coupling with the electromagnetic field is derived through theminimal substitution the photonic field will have three componentsAµ, µ = 0, 1, 2; it seems that there is mass generation for A1, A2,while A0 remains massless. In the relativistic approximation all thecomponents would be massless, by space - time symmetry.

3. Convergence of the perturbative series after the removal of thecutoffs: determinant bounds.

4. Borel summability of the theory; this would prove the unicity of ourresummation prescription.

Borel summability

I In QFT, the naıve perturbative series are often divergent; the n-thorder is typically bounded by n!αλn. Example: λϕ4 theory.

I Nevertheless, in some case it is possible to resum the perturbativeseries in such a way that the final result is finite. But is theresummation procedure unique?

I One way to prove uniqueness in a suitable space of functions is toverify that the resummation is the Borel sum of the naıveperturbative series; given

∑n≥1 anλ

n we say that f (λ) is its B. sumif it is analytic in a ball tangent to the origin of the λ - complexplane, and if

f (λ) ∼∑n≥1

anλn forλ→ 0,

∣∣∣∣∣f (λ)−M∑

n=1

anλn

∣∣∣∣∣ ≤ λM+1M!CM .

Then, if g(λ), f (λ) are both Borel sums of∑

n≥1 anλn

f (λ) = g(λ) =

∫ +∞

0

e−pλ

∑n≥1

an

n!pndp.

Borel summability of λϕ44 planar theory

In a joint collaboration with G. Gallavotti and S. Simonella we provedthat the perturbative series defining the planar part of the massive λϕ4

4

theory is Borel summable.Calling ~vk = (λk , αk , µk) the vector of the running coupling constantsand N the u.v. cutoff, the flow of the beta function has the form

~vk = ~vk+1 + ~β(~vk+1, ..., ~vN);

we proved that this equation admit a solution such that

λk ∼ λ

1 + λβ2k, β2 > 0

αk ∼ α

µk ∼ µ,

where (λ, α, µ) are the renormalized coupling constants. The solution isanalytic in α = µ = 0, and it is the Borel sum of the naıve perturbativeseries in λ = 0.

Remarks

I A similar result has been obtained by Rivasseau in the 80’s, choosingαN = µN = 0, by directly studying the perturbative series definingthe Schwinger functions; he showed that a huge cancellation ispresent in the λ - perturbative series, and proved that thiscancellation is responsible for asymptotic freedom.

I In our approach the problem is reduced to that of solving a recursiveequation, which “hides” the cancellation; the procedure seemsgeneral enough to be applied to fermionic systems.