# dirac lattices

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Dirac lattices: Down to High Energy!

Corneliu Sochichiu

SungKyunKwan Univ. (SKKU)

Chisinau, August 13, 2012

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

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 2 / 36

Motivation & Philosophy

QFTs like Standard Model are relativistic theories, based on Lorentzsymmetry group

Lorentz symmetry is an exact symmetry, no was violation observedapart 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 lowenergy approximations to some non-relativistic model and Lorentzsymmetry as emergent approximate symmetry [. . . Horava2009. . . ]

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 3 / 36

Motivation & Philosophy

QFTs like Standard Model are relativistic theories, based on Lorentzsymmetry group

Lorentz symmetry is an exact symmetry, no was violation observedapart 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 lowenergy approximations to some non-relativistic model and Lorentzsymmetry as emergent approximate symmetry [. . . Horava2009. . . ]

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 3 / 36

Motivation & Philosophy

QFTs like Standard Model are relativistic theories, based on Lorentzsymmetry group

Lorentz symmetry is an exact symmetry, no was violation observedapart 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 lowenergy approximations to some non-relativistic model and Lorentzsymmetry as emergent approximate symmetry [. . . Horava2009. . . ]

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 3 / 36

Emergent Lorentz & Gauge symmetry

Apart from what one can imagine, there are physical examples of emergingLorentz symmetry

Graphene: Since long time it is known that the electron wave functionin the low energy limit is described by relativistic Dirac fermion in2+1 dimensions [Wallace1947]

The low energy theory has an emergent Lorentz and global(nonabelian) gauge invariance. The global gauge invariance can bepromoted to local one by considering the low energy limit of latticedefect fields [CS2011]

TomonagaLuttinger liquid. . . Can the same scenario be applied to high energy particle physics in

four dimensions?

Space diamond lattice regularization [Creutz2007]

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 4 / 36

Emergent Lorentz & Gauge symmetry

Apart from what one can imagine, there are physical examples of emergingLorentz symmetry

Graphene: Since long time it is known that the electron wave functionin the low energy limit is described by relativistic Dirac fermion in2+1 dimensions [Wallace1947]

The low energy theory has an emergent Lorentz and global(nonabelian) gauge invariance. The global gauge invariance can bepromoted to local one by considering the low energy limit of latticedefect fields [CS2011]

TomonagaLuttinger liquid. . . Can the same scenario be applied to high energy particle physics in

four dimensions?

Space diamond lattice regularization [Creutz2007]

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 4 / 36

Emergent Lorentz & Gauge symmetry

Apart from what one can imagine, there are physical examples of emergingLorentz symmetry

Graphene: Since long time it is known that the electron wave functionin the low energy limit is described by relativistic Dirac fermion in2+1 dimensions [Wallace1947]

The low energy theory has an emergent Lorentz and global(nonabelian) gauge invariance. The global gauge invariance can bepromoted to local one by considering the low energy limit of latticedefect fields [CS2011]

TomonagaLuttinger liquid. . . Can the same scenario be applied to high energy particle physics in

four dimensions?

Space diamond lattice regularization [Creutz2007]

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 4 / 36

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 [Horava2005,Volovik2011]

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 5 / 36

Fermi point

Mathematical Fact: Fluctuations around a Fermi point aredescribed by Weyl/Dirac/Majorana particle

Stable and non-stable Fermi points:I stability: no small deformations can lead to disappearance of the Fermi

point (no consistent mass term is possible)I 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 bynontrivial homotopy class of maps from the sphere surrounding thepoint to the space of energy matrices [Volovik2011]

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 6 / 36

So, do we live on a Fermi point?

Fermi systems provide a convenient tool for the encoding of thespace-time geometry [Lin-Lunin-Maldacena]

Matrix models and gauge theories lead to Fermi systems or behavelike Fermi systems

The elementary particle spectrum can be seen as quasiparticleexcitations 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?

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 7 / 36

So, do we live on a Fermi point?

Fermi systems provide a convenient tool for the encoding of thespace-time geometry [Lin-Lunin-Maldacena]

Matrix models and gauge theories lead to Fermi systems or behavelike Fermi systems

The elementary particle spectrum can be seen as quasiparticleexcitations 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?

OK. . . However, first lets look at a simpler problem!

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 7 / 36

So, do we live on a Fermi point?

Fermi systems provide a convenient tool for the encoding of thespace-time geometry [Lin-Lunin-Maldacena]

Matrix models and gauge theories lead to Fermi systems or behavelike Fermi systems

The elementary particle spectrum can be seen as quasiparticleexcitations 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 the microscopic theory flowing to the existent

particle models in the IR?

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 7 / 36

So, do we live on a Fermi point?

Matrix models and gauge theories lead to Fermi systems or behavelike Fermi systems

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 the microscopic theory flowing to the existent

particle models in the IR?

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 7 / 36

The Setup of the problem

Tight-binding Hamiltonian

H =

taxay = a

T a

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

T = t

t are the transition amplitudes; they can be, in principle,arbitrary depending only on the pair < x , y >, but we will restrictourselves to only those which admit a continuum low energy limit

Which structure of T leads to a Dirac fermion in this limit?

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 8 / 36

Graph structure

Consider physical restrictions on the adjacency matrix

The graph: a superposition of D-dimensional Bravais lattices with thecommon base {}, i = 1, . . . ,Dunit cell consists of p sites labeled by the sublattice index = 1, . . . , p

each site is parameterized by its Bravais lattice coordinates as well asthe sublattice index:

xn = x + ni ,

The sites inside the cell can be connected in an arbitrary way

Only neighbor cells are connected

Therefore the adjacency matrix has a block structureI it could be 2D, 3D, etc. blocks. . .

The block structure is needed in order to define

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