This document discusses how a Dirac lattice model on a graph can give rise to emergent Lorentz symmetry and geometry in the low energy limit. The model involves a fermion particle moving on a graph with certain properties. Near Fermi points, the dynamics are described by a Dirac equation with an emergent vielbein and metric. By varying the graph parameters, virtually any constant background metric can be generated. However, full general relativity may require considering dynamical graphs and backreaction effects. The goal is to control both the emergent geometry and fermion spectrum through the underlying graph structure.

1. Local Lorentz symmetry from the graph.
Geometric aspects of Dirac lattices
Corneliu Sochichiu
SungKyunKwan Univ. (until the end of the month) →
Gwangju Institute of Science and Technology (from 2013/3/1)
& IAP, Chi¸in˘u
s a
AP School & Workshop on Gravitation and Cosmology
Jeju, 2013
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 1 / 12

2. Outline
1 Motivation
2 The Dirac lattice
3 Emergent geometry
Based on: 1112.5937 , see also 1012.5354 and work in progress...
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 2 / 12

3. Motivation
QFT models of HEP are relativistic theories, based on Lorentz
symmetry group
In gravity it is the local symmetry group
In the observable range of energies the Lorentz symmetry is an exact
symmetry, no violation observed so far
But is it indeed an exact symmetry?
Penrose ’71: The concept based on continuous space and Lorentz
symmetry is faulty and should be replaced in the theory including
quantum gravity. There are other models where the space-time and
Lorentz symmetry emerge in an effective description e.g. in low
energy theory
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 3 / 12

4. Fermi surface
An alternative is to consider a Fermi system (system with Pauli
exclusion principle)
Such a system generates “Fermi surfaces”. Generalized Fermi surface
can take the forms of various geometrical varieties: points, lines, etc
Fermi surface can encode geometrical data
Lorentz symmetry is a natural symmetry for the low energy dynamics
near Fermi points
Here we consider the model of a fermi particle on a graph
The structure of the graph leads to effective Lorentz symmetry in low
energy
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 4 / 12

5. The Dirac lattice
Consider the model with the 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
A class of adjacency matrices T lead to a Dirac fermion in the low
energy limit. . .
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 5 / 12

6. Low energy limit
Assuming D-dimensional translational invariance of the graph, the
Hamiltonian in the momentum space becomes:
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 Fermi
level, EF = 0
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 6 / 12

7. Fermi points
Remember, the Lorentz algebra emerges at Fermi points
The Fermi point conditions are
det[h(K ∗ )] = 0 (Fermi level)
det[h(K ∗ ) + αi (K ∗ )ki + O(k 2 )] = 0 (point cond.)
The energy matrix: h(K ) = Γi eiKi + Γ† e−iKi + M
i
i
We can expand the matrices in terms of holomorphic matrices ΣI ,
I = 1, . . . , D/2 of D-dimensional complex Clifford algebra basis
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 7 / 12

8. Dirac fermion
In the minimal case there are just two Dirac points at ±K ∗
The modes near ±K ∗ can be combined into the field ψ with the
action
S = dtdD k iψ † ψ − Jψ † ξa β a ki ψ
˙ i
where β a for a = 1, . . . , D are
β2I −1 = −(ΣI + Σ† ) ⊗ σ3 ,
I β2I = i(ΣI − Σ† ) ⊗ I
I
s.t. {βa , βb } = 2δab I, and
ξi2I −1 = ΓiI sin Ki , ξi2I = ΓiI cos Ki .
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 8 / 12

9. Dirac fermion
Defining
γa = γ0βa, γ i = ξa γ a
i
where
γ 0 = iD /2
β1 β2 · · · β2Np , such that, (γ 0 )2 = −1
Taking the inverse Fourier transform, the action becomes,
Slow energy = −i ¯
dD+1 x ψγ µ ∂µ ψ
with γ µ = (γ 0 , ξa γ a )
i
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 9 / 12

10. Emergent geometry
I recall that we started from a graph with translation symmetry but
no notion of metric
We ended up with Dirac particle in a frame given by the vielbein
i
vectors ξa
This defines the induced (inverse) metric
i j i
g µν = (g 00 = 1, g 0i = 0, g ij = ξa ξa ) with vielbeins ξa defined by
ξ2I −1 (K ) = ΓiI (K ) sin Ki ,
i
ξ2I (K ) = ΓiI (K ) cos Ki
i
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 10 / 12

11. Nontrivial geometry background
By choosing the representation of the Clifford algebra basis and the
location of the fermi point K , we can get virtually arbitrary
(constant) metric
Let us deform the graph, allowing an adiabatic variation of the Fermi
point location as well as of Clifford algebra representation, such that
the spacial metric becomes an arbitrary non-degenerate macroscopic
tensor field gij (x, t)
The time components of the metric are not generic: g 00 = 1,
g i0 = 0. This, however, does not restrict the geometry, given the
R × RD –structure used in most cosmological models
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 11 / 12

12. Outlook
We considered conditions under which a discrete model on a graph
produces a Dirac fermion and a metric space in the low energy limit
We can control the metric by adiabatically varying the parameters of
the microscopic model
Although we got generic geometry, the diff symmetry seems to be
partially broken. This happened because we started with “absolute”
time from very beginning. To get the general case, we should start
with a graph in which time extension is also generated (work in
progress)
We got non-dynamical background geometry because we consider
non-dynamical graph. Considering dynamical graph and fermionic
back reaction will produce dynamical part of gravity (work in project)
I did not discuss here the fermion particle spectrum. The spectrum
can also be controlled by the graph parameters. One can aim to get
the spectrum of SM (project). . .
C.Sochichiu (SKKU → GIST) Dirac Lattices Jeju 2013 12 / 12