Dirac Lattices

Dirac lattices: Down to High Energy!

Corneliu Sochichiu

SungKyunKwan Univ. (SKKU)

Chi¸in˘u, August 13, 2012
s a

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 1 / 36

Outline

1 Motivation & Philosophy

2 The model

3 Low energy limit

4 Emergent Dirac fermion

Based on: 1112.5937 (v2.0 to come soon), see also 1012.5354


Motivation Philosophy

QFT’s like Standard Model are relativistic theories, based on Lorentz
symmetry group
Lorentz symmetry is an exact symmetry, no was violation observed
apart from. . .
. . . But are they indeed exact symmetries? Why?
The Lorentz symmetry is not compact, and there are critics, claiming
the inconsistency of ﬁeld theories based on exact Lorentz symmetry
[Jizba-Sardigli2011]

An alternative is to consider the high energy QFT models as low
energy approximations to some non-relativistic model and Lorentz
symmetry as emergent approximate symmetry [. . . Hoˇava2009. . . ]
r


Motivation Philosophy

QFT’s like Standard Model are relativistic theories, based on Lorentz
symmetry group
Lorentz symmetry is an exact symmetry, no was violation observed
apart from. . . so far
. . . But are they indeed exact symmetries? Why?
The Lorentz symmetry is not compact, and there are critics, claiming
the inconsistency of ﬁeld theories based on exact Lorentz symmetry
[Jizba-Sardigli2011]

An alternative is to consider the high energy QFT models as low
energy approximations to some non-relativistic model and Lorentz
symmetry as emergent approximate symmetry [. . . Hoˇava2009. . . ]
r


Emergent Lorentz Gauge symmetry

Apart from what one can imagine, there are physical examples of emerging
Lorentz symmetry
Graphene: Since long time it is known that the electron wave function
in the low energy limit is described by relativistic Dirac fermion in
2+1 dimensions [Wallace1947]
The low energy theory has an emergent Lorentz and global
(nonabelian) gauge invariance. The global gauge invariance can be
promoted to local one by considering the low energy limit of lattice
defect ﬁelds [CS2011]
Tomonaga–Luttinger liquid. . .
Can the same scenario be applied to high energy particle physics in
four dimensions?
“Space diamond” lattice regularization [Creutz2007]


Fermi surface

Consider a Fermi system (Pauli exclusion principle)
In the low energy limit the dynamics is determined by the states near
the Fermi surface
Fermi surface can take the forms of various geometrical varieties:
points, lines, etc
Which of these shapes are stable?
ABS construction [Atiyah-Bott-Shapiro]: Varieties with non-trivial topological
(in fact, K-theory) charge [Hoˇava2005,Volovik2011]
r


Fermi point

“Mathematical Fact’’: Fluctuations around a Fermi point are
described by Weyl/Dirac/Majorana particle
Stable and non-stable Fermi points:
stability: no small deformations can lead to disappearance of the Fermi
point (no consistent mass term is possible)
non-stability: Small deformations can lift the Fermi point (one can
generate a consistent mass term)
In the case of a Fermi point, the stability can be provided by
nontrivial homotopy class of maps from the sphere surrounding the
point to the space of energy matrices [Volovik2011]


So, do we live on a Fermi point?

Fermi systems provide a convenient tool for the encoding of the
space-time geometry [Lin-Lunin-Maldacena]
Matrix models and gauge theories lead to Fermi systems or behave
like Fermi systems
The elementary particle spectrum can be seen as quasiparticle
excitations around Fermi surface [Volovik]
Gauge/gravity interactions can be generated dynamically [Sakharov1968]

So, a fermi system is all one needs to build a Universe like ours, but. . .
can we ﬁgure out a microscopic theory ﬂowing to the existent
particle models in the IR?



like Fermi systems

can we figure out a microscopic theory flowing to the existent
OK. . . However, first let’s look at a simpler problem!



like Fermi systems

can we ﬁgure out a the microscopic theory ﬂowing to the existent



like Fermi systems

can we ﬁgure out the microscopic theory ﬂowing to the existent


The Setup of the problem

Tight-binding Hamiltonian

H= txy ax ay = a† · T · a
†

xy

x, y are sites of a graph and T is its adjacency matrix

T = txy

txy are the transition amplitudes; they can be, in principle,
arbitrary depending only on the pair x, y , but we will restrict
ourselves to only those which admit a continuum low energy limit
Which structure of T leads to a Dirac fermion in this limit?


Graph structure

Consider physical restrictions on the adjacency matrix
The graph: a superposition of D-dimensional Bravais lattices with the
common base {ˆ}, i = 1, . . . , D
ı
unit cell consists of p sites labeled by the sublattice index α = 1, . . . , p
each site is parameterized by its Bravais lattice coordinates as well as
the sublattice index:
xαn = xα + ni ˆ,
ı
The sites inside the cell can be connected in an arbitrary way
Only “neighbor” cells are connected
Therefore the adjacency matrix has a block structure
it could be 2D, 3D, etc. blocks. . .
The block structure is needed in order to deﬁne the proper continuum limit


The Hamiltonian

The Hamiltonian can be rewritten as,

H= †
an+ˆΓi an + an Γ† an+ˆ +
ı
†
i ı
†
an Man
n,i n

Γi are the inter-cell adjacency matrix blocks and M is the intra-cell matrix
Now, consider the low energy limit for the theory described by this
Hamiltonian
We want to ﬁnd Γi and M leading to Dirac fermion


Fourier transform Brillouin zones

Due to the translational invariance of Bravais lattice we can do the Fourier
transform
dk
a(k) = an eik·n , an = a(k)e−ik·n
n B (2π)D

The (normalized) Brillouin zone B :

k = ki ˜,
ı −π ≤ ki π

{˜, i = 1, . . . , D} : the dual (basis to the) Bravais basis
ı

˜ ·  = δij
ı ˆ


Low energy limit

The Hamiltonian in the momentum space description:

dk
H= a† (k) Γi eiki + Γ† e−iki + M
i a(k)
B (2π)D
i

The low energy contribution is given by the modes near the lowest
energy states


Low energy limit


dk
i a(k)
B (2π)D
i

energy states
Recall: This is a fermionic system!


Low energy limit


dk
i a(k)
B (2π)D
i

energy states Fermi surface


Low energy limit


dk
i a(k)
B (2π)D
i

energy states Fermi surface
Assume also: symmetric energy spectrum and half-ﬁlling of the Fermi
sea: EF = 0


(Generalized) Fermi surfaces

Fermi surface is an interface between occupied and non-occupied
states in a system withe exclusion principle
Generic case: Fermi surface has (spacial) co-dimension one
In D = 3 it is a 2D Fermi surface
In D = 2 it is Tomonaga–Luttinger fermion
It is Fermi point and only Fermi point, which brings to the Dirac
Fermion in the low energy limit.
So, we are interested in systems which lead to point-like Fermi
surfaces.
Deformations/corrections can lead to the degeneracy of the Fermi
surface down to D − 2, D − 3, etc


Fermi point conditions

Fermi point conditions are

det[h(K ∗ )] = 0 (Fermi level)
∗ i ∗ 2
det[h(K + k)] = α (K )ki + O(k ) = 0 (point cond.)

The energy matrix h(K ) = Γi eiKi + Γ† e−iKi + M
i
i

The Hilbert space splits into energy bands: the groups of states
continuously connected by the variation of “momentum” k


Fermi point condition

Consider the subspace of states belonging to the gapless bands
The Fermi surface condition becomes h(K ∗ ) = 0
The Fermi point conditions imply that αi are generators of the
D-dimensional Cliﬀord algebra C D
Representation can be reducible
D
The ‘minimal’ irreducible representation has dimension 2[ 2 ]
D
(Therefore p = N 2[ 2 ] )
A way to obtain a reducible representation is by reduction of an
irreducible representation of C D for some D D

This algebra can be explicitly constructed. . .


ΣI -basis

The inter-cell adjacency matrices Γi and intra-cell adjacency M
should be elements of Cliﬀord algebra
They can be expanded in terms of a ‘holomorphic’ Cliﬀord algebra
basis consisting of matrices ΣI , I = 1, . . . , D /2, for some even
D ≥ D, Γi = ΓiI ΣI , Γ† = ΓiI Σ† ⇒ M = mI (ΣI + Σ† )
i I I
The ΣI matrices are satisfying the algebra

{ΣI , ΣJ } = {Σ† , Σ† } = 0,
I J {ΣI , Σ† } = δIJ I
J


Fermi point condition in ΣI -basis

In terms of the ΣI -basis the Fermi point equation

hI (K ) = 0, hI (K ) ≡ ΓiI eiKi + mI .
i

should admit only point-like solutions −→ Ki∗ , i = 1, 2, . . .
Such a solution will be described below. So far assume it exists. . .


Some properties of Fermi points

Reality condition: If K ∗ is a solution −K ∗ is a solution too
We can associate a topological charge to every point Ki∗ ,
D
Γ +1 ∧(D−1)
Ni = 2
D tr h−1 dh ,
D−1
pDπ 2 Sα

Due to compactness of the momentum space, the total charge should
vanish,
Ni = 0
i


Minimal case

Each pair ±Ki∗ will contribute one fermionic species leading to
degeneracy and enhancement of the global nonabelian internal
symmetry in the low energy limit. The minimal solution contains a
single pair ±K ∗
Therefore, the minimal nontrivial conﬁguration consists of two Fermi
points ±K∗ or o single point (the degenerate case) K∗ = 0
Let’s concentrate on the (non-degenerate) minimal case


Emergent Dirac fermion: Dirac matrices

In the vicinity of ±K ∗

αi (±K ∗ ) = iΓiI cos Ki∗ (ΣI − Σ† ) ± sin Ki∗ i(ΣI + Σ† ) ,
I I

and ki = KI KI∗
Introduce index associated with the sign of ±K ∗

αi = iΓiI cos Ki∗ (ΣI − Σ† ) ⊗ I + sin Ki∗ i(ΣI + Σ† ) ⊗ σ3
ˆ I I


Embedding into Cartesian basis

αi are linear combinations of matrices βa , a = 1, . . . , D forming the
ˆ
standard basis of the Cliﬀord algebra C D

β2I −1 = −(ΣI + Σ† ) ⊗ σ3 ,
I β2I = i(ΣI − Σ† ) ⊗ I
I

i.e. αi = ξia βa
ˆ
with ξi2I −1 = ΓiI sin Ki , ξi2I = ΓiI cos Ki .
With matrices βa we can associate a Cartesian coordinate system


Emerging geometry

The matrices βa satisfy Clifford algebra relations

{βa , βb } = 2δab I

ξia can be regarded as vielbein coefficients for the embedding of a
D-dimensional plane into RD , where the Clifford algebra is defined.

Induced metric: gij = ξia ξja

Introducing ‘Cartesian momentum’: q a = ξia ki , a = 1, . . . , D
The low energy Hamiltonian takes the form

dD q †
H=J ψ (q)βa q a ψ(q) J= det gij
(2π)D


The low energy action

1 Introduce γ 0 = iD /2 β β · · · β
1 2 2Np s.t. (γ 0 )2 = −1
2 ¯
Introduce the Dirac conjugate ψ = ψ † γ 0
3 Make a (continuous) inverse Fourier transform to real space and get
the low energy action

Sl.e. = −i ¯
dD+1 x ψγ µ ∂µ ψ, γa = γ0βa

That’s all!
We got:
A Dirac fermion
An induced geometry


The low energy action

1 Introduce γ 0 = iD /2 β β · · · β
1 2 2Np s.t. (γ 0 )2 = −1
2 ¯
Introduce the Dirac conjugate ψ = ψ † γ 0
3 Make a (continuous) inverse Fourier transform to real space and get
the low energy action

Sl.e. = −i ¯
dD+1 x ψγ µ ∂µ ψ, γa = γ0βa

That’s all!
We got:
A Dirac fermion
An induced geometry

Something is left behind, however. . .


Moduli space for ΓI

Recall, ΓI should satisfy the condition that equations

hI (K ) = 0, hI (K ) ≡ ΓiI eiKi + mI
i

have the only solutions for isolated ±K ∗ .
What are the ΓI ’s?


A Mechanical analogy
. . . we can ﬁnd a mechanical analogy for the Fermi point condition in
terms of arm-and-hinge mechanism
Consider [D /2] (D + 1)-gons with sides ΓiI . (I = 1, . . . , [D /2] counts
polygons, and i = 1, . . . .D counts sides within one polygon.) The orienta-
tion of the sides with number i in the complex plane is the same for every
polygon and is given by the factor eiKi .
Then,
The Fermi level condition corresponds to the closure of the polygon
The point-like nature corresponds to its rigidity
Think about [D /2] superposed hinge mechanisms in the two-dimensional
plane, each with D arms of lengths ΓiI and orientation Ki , as well as
one horizontal arm of length mI . The arms with the same number i in
diﬀerent mechanisms counted by I are kept parallel. The Fermi point
condition implies that the whole hinge mechanism is (i) closed, and (ii)
rigid


Arms and hinges

(a) K3 (b)
K3
K4 K2 K4 K2

|Γ2I | |Γ2I /mI |

|Γ1I /mI |
|Γ1I |

K5 K1 K5 K1

K6 |mI | K6 1

Figure : The (D + 1)-gon representing the equation i ΓiI eiKi + mI for mI = 0.
The lengths of the sides are given by |ΓiI |, the angles to the horizontal are Ki . In
the case of mI = 0 the (D + 1)-gon degenerates to a D-gon. (a) A heptagon for
the problem in D = 6. (b) A set of two heptagons with parallel sites, solving the
Fermi point problem can be regarded as ﬁnding a rigid hinge mechanism. In this
case more heptagons are needed to make the mechanism rigid.


D = 2 case

β
Γ12 eiK2 Γ11 eiK1

γ α
1

Figure : D = 2 situation. There is a unique triangle you can construct with given
three site lengths. The angles of the triangle are related to the momenta in the
following ways: α = π − K1 , β = π − K2 + K1 and γ = π + K2 .

A single hinge system is needed (a triangle is uniquely determined by the
lengths of its sites)
Γi eiKi + m = 0
i


D = 2 case: the solution
Triangle sine rule
Γ1 Γ2 m
= =
| sin K2 | | sin K1 | | sin(K2 − K1 )|

Leads to solution for Γi
∗
m sin K2 ∗
m sin K1
Γ1 = ∗ ∗ , Γ2 = ∗ ∗
sin(K1 − K2 ) sin(K2 − K1 )
∗ ∗
with Fermi points at ± (K1 , K2 )
More generally one can have congruent hinge mechanism based on the
above one:
ΓiI = ηI Γi , mI = ηI
The resulting system is equivalent (upon coordinate transformation) to
graphene


D = 3 case
3
ΓiI eiKi + mI = 0, I = 1, 2
i=1
means closure of a solid quadrilateral

Γ3 eiK3 Γ1 eiK1
iK2
Γ21 e Γ11 eiK1

Γ21 eiK2 Γ12 eiK1
Γ31 eiK3

Γ32 eiK3

1

Figure : The quadrilaterals corresponding to each polygon equation can be
obtained from a single master triangle with sides 1, Γ1 and Γ3 , by cutting the
upper angle by the side Γ2I eiK2 . The dotted segments correspond to η1,3 .


D = 3 case: the solution

The elementary geometry problem has the solution
∗ ∗ ∗
mI sin K3 + ξI sin(K2 − K3 )
Γ1I = mI Γ1 − η1I = ∗ − K ∗)
sin(K1 3
Γ2I = −η2I = −ξI
∗ ∗ ∗
−mI sin K1 + ξI sin(K1 − K2 )
Γ3I = mI Γ3 − η3I = ∗ − K ∗)
sin(K1 3


Arbitrary D: the Holomorphic Ansatz

For general polygon equation

ΓiI eiKi + mI = 0, I = 1, . . . , D /2,
i

consider the Ansatz
ΓiI = ΓI δi,2I −1 + ΓI δi,2I .
As a result of substitution, the polygon equations split into D /2
independent triangular equations,

ΓI eiK2I −1 + ΓI eiK2I + mI = 0.

Mimics the canonical form of the rotational matrix, which in an appropriate
basis is a composition of elementary rotations of two-dimensional planes


The solution

We know how to solve the triangular equation. . .
∗ ∗
mI sin K2I −1
mI sin K2I
ΓI = ∗ ∗ , ΓI = ∗ ∗
sin(K2I −1 − K2I ) sin(K2I − K2I −1 )

I = 1, . . . D /2
Embedding functions:
∗ ∗ ∗
2I −1 mI sin K2I sin K2I −1 mI sin2 K2I
ξ2I −1 = ∗ ∗ , ξ2I −1 =
2I
∗ ∗ ,
sin(K2I −1 − K2I ) sin(K2I −1 − K2I )
∗ ∗
mI sin K2I −1 cos K2I −1 ∗ ∗
mI sin K2I −1 cos K2I
2I 2I
ξ2I −1 = ∗ ∗ , ξ2I = ∗ ∗ ,
sin(K2I − K2I −1 ) sin(K2I − K2I −1 )


Induced metric
Induced metric is given by 2 × 2 blocks
g2I −1,2I −1 g2I −1,2I
g2I ,2I −1 g2I ,2I
where
mI2 sin2 (K1 ) (cos (2K2I −1 ) − cos (2K2I ) + 2)
g2I −1,2I −1 = ∗ ∗ ,
1 − cos[2(K2I −1 − K2I )]
∗ ∗ ∗
mI2 sin(K2I −1 ) 2 sin3 (K2I ) + cos (K2I ) sin 2K2I −1
g2I −1,2I = ∗ ∗ ,
1 − cos[2(K2I −1 − K2I )]
∗
mI2 sin K2I −1 ∗ ∗
2 sin3 (K2I ) + cos (K2I ) sin (2K2I −1 )
g2I ,2I −1 = ∗ ∗ ,
1 − cos[2(K2I −1 − K2I )]
g2I ,2I =
∗ ∗ ∗ ∗
mI2 (5 + cos(4K2I ) − 4 cos(2K2I ) cos2 (K2I −1 ) − 2 cos(2K2I −1 )
∗ ∗ .
4{1 − cos[2(K2I −1 − K2I )]}


Conclusion Outlook

We considered conditions under which a discrete model on a graph
produces a Dirac fermion in the low energy limit
These conditions translate to algebraic equations on the adjacency
matrix
We found the general solutions for the case of D = 2, 3 lattices and a
‘holomorphic’ solution in the general case. However, not clear
whether this solution is a general one.
As a ‘bonus’ we got induced geometry in the low energy theory
Next step would be to consider Fermi systems which generate
‘desired’ symmetries, e.g. that of Standard model.
Dynamical adjacency matrix. The gauge and gravity degrees of
freedom should be expected to emerge in the low energy limit
through the Sakharov mechanism


Backup

Expand the point condition h(K ∗ + k)|Zero subspace = αi (K ∗ )ki = 0

∗ ∗
αi (K ∗ ) = i Γi eiKi − Γ† e−iKi
i
Zero subspace


Example
Having the solution in 3D, we can easily write a (random) example of
lattice model, generating a 3 + 1 dimensional Dirac fermion in the low
energy limit
2 The Legend:
1 m+ξ
√
4 n + l1 3

2 3 2 − m+ξ
√
3
1 1
−ξ
4 n 4 n + l2
3 2 3 m
1
−m
4 n + l3
3

Figure : Symbolic representation of the lattice for the model described by
Hamiltonian. The model is local in the sense that interaction is limited to the
nearest unit cells. The hopping amplitudes are given by diﬀerent types of lines, as
explained in the legend. The last two line types in the legend correspond to
internal lines of the cell.

Dirac Lattices

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