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Discretizing the sphere with an irregular� lattice: Application to the �transverse-field Ising model

Grigorios Makris

Institute for Quantum Information, RWTH Aachen University

Supervisors:

Prof. Dr. Fabian Hassler Prof. Dr. Stefan Wessel

Institute for Theoretical Solid State Physics, RWTH Aachen University

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Outline of presentation

Regular and irregular lattices
The free particle on the sphere
The transverse-field Ising Hamiltonian on the sphere
Symmetry breaking of the Ising model
Scaling and conformal symmetry
Results on the sphere

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Regular lattices in general

Objective homogeneity quantification of a periodic surface using the Gini coefficient

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spectrum(100,1.3,0,5)

Lattices on the sphere

Regular lattices are only possible for specific number of points

Examples of Platonic solids

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Irregular lattices

Nearest neighbor couplings equal.

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Motivation for the sphere

The plane is conformally related to the cylinder

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Fibonacci lattice

Allocate points on spiral around the z-axis of the sphere.

golden angle,

Lattice implementations

golden ratio

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Free particle on the sphere

Decide on a lattice by solving the free particle problem.

Discretize on the tangent plane

of every point on the sphere

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Free particle on the sphere

degeneracies

Fibonacci lattice

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Transverse-field Ising Hamiltonian on the sphere

same as free particle hoppings!

Constant couplings

Free particle couplings

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symmetry

Symmetry breaking of the Ising model

No order without symmetry breaking

Symmetry may be broken when ground states of both sectors have same energy.

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Symmetry breaking of the Ising model

At infinite system size the first excited becomes truly degenerate with the ground state.

Phase transition happens at the critical point, more degeneracies.

Finite transverse field Ising chain

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for large separation

Quantum to classical mapping

At the critical point there are degeneracies, so that

Implies scale invariance

Original quantum model exists along chain

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Scale invariance

Two-point correlation functions of scaling fields have power behavior at the critical point

is the scaling dimension of the field

Ising model

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Conformal symmetry

Scale invariance is characteristic in second order phase transitions.

Dilations or rescalings are part of the more general conformal transformations.

Conformal transformations preserve angles.

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Conformal symmetry at the critical point:

J L Cardy 1985 J. Phys. A: Math. Gen. 18 L757

Conformal symmetry

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Finite-size scaling

Scaling assumption

The thermodynamic observables scale with system size as power laws of to ratios of scaling exponents.

Consider a system of size .

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Results with quantum monte carlo

Flat model:

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Thank you for your attention!

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Scaling

On the classical Ising model we can tune two variables, the temperature and the magnetic field.

The phase transition happens at a critical temperature at zero magnetic field.

Scaling of thermodynamic observables on infinite systems is a power law with universal scaling exponents.

implies scale invariance

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Maximizing the smallest nearest neighbor distance

Minimizing interacting potential

For arbitrary number of points irregular lattices are used

Refinement of the platonic solids. �Creation of the icosahedron-seed.

Lattice implementations

maximize

d

d

Spiral points – Fibonacci lattice

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0-generation

1-generation

2-generation

Icosahedron-seed

Lattice implementations for the sphere

Most points connect to 6 neighbors.

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Free particle on the sphere

degeneracies

Fibonacci lattice

Icosahedron-seed

We choose the Fibonacci lattice!

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Isotropy of the spherical Ising model

Our model locally approximates the flat 2D transverse field Ising model.

Isotropy established up to second order similarly to flat model.

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Free particle on the sphere

How good is the spectrum on the Fibonacci lattice?

Minimum angle of rotation:

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Free particle on the sphere

How do the behave?

Almost one-to-one correspondence with nearest neighbor distances .

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Free particle on the sphere

How do the behave?

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Outline of presentation

Constructing a discrete sphere

Lattice implementations
The free particle on the sphere
The transverse-field Ising Hamiltonian on the sphere

Transverse-field Ising model at criticality

Basics of phase transitions
Finite-size scaling and conformal symmetry
Results on the sphere

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Quantum to classical mapping

Insert completeness relations

Transfer matrix:

many spin state

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Quantum to classical mapping

due to gap in the spectrum for finite systems

for large separation

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Finite-size scaling

constant for all system sizes.

Binder Cumulant is constant for all system sizes.

Don’t need to know scaling exponents a priori.

Suppose we want to find the critical temperature .

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Conformal symmetry

The and the are conformally related,

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Results on the sphere

Examine also the classical Ising model by setting the transverse field to zero.

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Results on the sphere

Nevertheless, fitting with the exponents of the 3D Ising universality class may be applicable.

Collapse of Binder Cumulant

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Conclusions and Outlook

The crossing points are very sensitive to very small rescalings of the couplings.
Attempts to find couplings on more uniform lattices yield more uniform distribution of couplings and better spectra.

The Fibonacci geometry yields better spectrum.

TFIM has more degenerate energy levels when Laplacian-couplings are used.

Pronounced finite-size effects on the spin system.
Better results on classical system.

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