p. zizzi- quantum mind's collective excitations

8/3/2019 P. Zizzi- Quantum Mind's Collective Excitations

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QUANTUM MINDS COLLECTIVE

EXCITATIONS

P. Zizzi

III Quantumbionet Workshop

Pavia, 24/09/2010


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Quantum Mind = Collection of quantum processes occurring in the brain.

Q M

Definition based on: Quantum Brain Dynamics (QFT of the

brain) by Umezava-Ricciardi.

Developed by Vitiello (Dissipative QFT of the brain).

Philosophy: Ontologically materialist

Epistemologically monist

Psychology: Quantum Mind = unconscious

(we cannot grasp our unconscious thoughts as they aresuperposed quantum states)

Logic: The logic of the quantum mind is the logic of quantum

information.


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Biology: Penrose-Hameroff Orch-Or theory of quantum consciousness.

Superposed coherent tubulines states = unconscious

Decoherence consciousness

Quantum computing: Quantum Mind = QC

There are many Quantum Minds

Q M1

QM4

QM2

Q M3


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If we consider interactions among them.

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4

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Many Quantum Minds many-body quantum system

In solid state physics collective excitations solve the many-body problem as the

system can be considered as a whole.

Collective excitations: Phonons

Magnons

Plasmons

Laughin quasiparticles in Fractional Quantum Hall Effect

(FQHE) which have fractional charge, and are anyons

(neither bosons, nor fermions)


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Ex: Phonons (collective excitations of the atoms of a crystal)

Due to the connections between atoms, the displacement of one or more atoms from

their equilibrium positions will give rise to a set of vibration waves propagating

through the lattice.


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Quantum Hall Effect (QHE)

Quantum version of the Hall effect

2-dimensional electron system

low temperature

strong magnetic field B

In the quantum Hall effect, a two-dimensional electron gas (electron charge

e and density n) moves under the influence ofmagnetic fieldB normal to

the plane and an electric field E in the plane.

By the Lorentz force, a current J is induced perpendicular to E.


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Schematic of a two-dimensional electron gas with a current J induced

perpendicular to an electric field E and a strong magnetic field B.

The Hall conductivity takes on the quantized values:

= e2/h

e = elementary charge

h = Plancks constant = filling factor

For integer values of=1,2,3, Integer QHE(IQHE)

For rational fractions =1/3, 1/5, 5/2, FractionalQHE (FQHE)


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These plateaus of quantized conductivity indicate where the two-

dimensional electron gas acts as an incompressible fluid, meaning that allcharged excitations have a finite energy gap.

For integer , the gap can be understood without electron interactions

because each plateau corresponds to a completely filled Landau level..

The IQHE can be explained in terms of single particle orbitals in a

magnetic field (Landau quantization)

The energy levels of the quantized orbitals take on discrete values:

Landau levels: )2/1( += nE cn h

meBc

/= cyclotron frequency

For strong magnetic fieldeach Landau level ishighly degenerate(there are

many single particle states with the same energy En ).

For fractional filling, the energy gap can only be explained by including

interactions (Coulomb) i.e. the excitations are a collective phenomenon.


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The ground state degeneracy means that there are different vacuum states

with the same symmetry.there is no symmetry breaking in passing from aground state to another ground state.

Topological order

Topological order is a property possessed by some special quantum many-

body systems.

A system is topologically ordered if:

- Has a degenerate ground state separated by a gap from the rest of the

spectrum.

-Each state in the ground eigenspace looks locally the same as any other.


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Topological Order

- It is a new kind of order in a quantum state- It is beyond the Landau symmetry breaking theory- It cannot be described by local order parameters and long range

interactions

- It can be described by a new set of quantum numbers, like ground statedegeneracy, quasiparticles fractional statistic ecc.

- It is a pattern oflong-range entanglement in quantum states- The nature of entanglement is topological (because of the presence of

anyons, which have exotic statistics, and braiding)

- States with different topological orders can transform into each otherthrough a Quantum Phase Transition (QFT).


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Topological quantum systems are more familiar from the quantum Halleffect where a two-dimensional layer of electrons is subject to a strong

vertical magnetic field. The low-energy spectrum of these systems is

governed by a trivial Hamiltonian, H= 0. Nevertheless, they have an

interesting behaviour due to the non-trivial statistics of their excitations. It

has been proven that this behaviour is dictated by the presence of anyons.

Anyons

In space of three or more dimensions, particles are restricted to being fermions or

bosons, according to their statistical behaviour

Fermions respect the so-called FermiDirac statistics

while bosons respect the BoseEinstein statistics

In two-dimensional systems, however, quasiparticles can be observed which obey

statistics ranging continuously between FermiDirac and BoseEinstein statistics


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with i being the imaginary unit and a real number.

So in the case = we recover the FermiDirac statistics and in the case = 0 (or

= 2) the BoseEinstein statistics. In between we have something different. Frank

Wilczek coined the term "anyon" to describe such particles, since they can haveany phase when particles are interchanged.

Unlike bosons or fermions, anyons have a non-trivial evolution

when one circulates another.

A particle spans a loop around another one. In three dimensions, it is possible to

continuously deform the path 1 to the path 2, which is equivalent to a trivial path


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In two dimensions, the two paths 1 and 2 are topologically distinct. This gives the

possibility of having non-trivial phase factors appearing when one particle

circulates the other.

This can be visualized by having the particles carrying charge as well as magnetic

flux

The word lines of anyons cross over one another to form braids in a three-dimensional spacetime (i.e., one temporal plus two spatial dimensions).

Topological Degeneracy

System with degenerate ground states where:The degeneracy is protected by topology (genus) ex. g =2 in fig.

Degenerate states are not locally distinguishable


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Assume we can:

Create identifiable anyons e.g. measure them by interference experiments

Braid anyons

Fuse anyons

vacuum

time

The world lines of the anyons where the third dimension depicts time running

downwards. From the vacuum, two pairs of anyons and anti-anyons are generated,

depicted by a+, a-,b+, b-


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Then, anyons a- and b+ are braided by circulating one around the other. Finally,

the anyons are pairwise fused in c+ and c-, but they do not necessarily return to the

vacuum as the braiding process may have changed their internal state

Topological quantum computers (TQC)A topological quantum computer employs anyons.

The braids form the quantum logic gates that make up the quantumcomputer.

The advantage of a quantum computer based on quantum braids over using

trapped quantum particles is that the former is much more stable.

The smallest perturbations can cause a quantum particle to decohere andintroduce errors in the computation, such small perturbations do not

change the topological properties of the braids.

Recent experiments indicate the elements of a TQC can be created in the real worldusing semiconductors near absolute zero and subjected to strong magnetic fields.


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Quantum Phase Transitions (QPT)

All the phase transitions described by Landau symmetry breaking are classical

(although they occur in quantum systems) in the sense that they are driven by

classical (thermal) fluctuations which diverge near the critical point.

At zero temperature Tc=0, the classical fluctuations disappear.

However, there can still be other phases depending on other parameters, like

ground state degeneracy ecc.

A transition between phases at T= 0 is called quantum phase transition

One way to detect a QPT is to notice that the ground state drastically depends onslight changes in the parameters.

Example: Transitions between different FQH states is a QPT.

That is, different Topological Orders can be transformed into each other

by a QPT


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String-nets

In condensed matter physics, a string-net is an extended object whose collective

behavior has been proposed as a physical explanation for topological order by

Michael A. Levin and Xiao-Gang Wen. A particular string-net model may involveonly closed loops; or networks of oriented, labeled strings obeying branching rules

given by some gauge group or still more general networks.

Their model purports to show the derivation of photons, electrons, and U(1) gauge

charge, small (relative to the planck mass) but nonzero masses, and suggestionsthat the leptons, quarks, gluons, and graviton, can be modelled in the same way.

However, their model does not account for the chiral coupling between the

fermions and the SU(2) gauge bosons in the standard model.

For strings labeled by the positive integers, string-nets are the spin networksstudied in loop quantum gravity.


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Light is a fluctuation of closed strings of arbitrary sizes. Fermions are ends of

open strings.

Light and fermions come from the collective motions of string-like objects thatform nets and fill our vacuum.

Light and fermions exist because our vacuum is a quantum liquid of string-nets.


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Gauge interaction and Fermi statistics are just phenomena of quan-

tum interference in infinity dimension - many-body quantum entan-glements.No need to introduce gauge bosons and fermions by hand. They justemerge if our vacuum has a string-net condensation.

Constructed spin model on cubic lattice that reproduce QED andQCD .They are the U(1) and the SU(3) in the U(1) SU(2) SU(3) stan-dard model.But ... have trouble to get the chiral coupling of the SU(2).

Six fascinating properties of nature:Identical particles Gauge interactionFermi statistics Massless fermionsChiral fermions Gravity

The string-net condensation picture can explain four of them.

Emergence vs reductionism.!


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The analogy:

Quantum Minds Many-body quantum systems

Collective Unconscious (Jung) Collective quantum excitations

Archetypes deep unconscious(Jung) Degenerate ground state

Holographic QM (Pribram) 2-dimensional topological quantum system

Unconscious as a QC (Penrose-Hameroff) Topological QC

Emergence of individual QM individual unconscious (Jung) String-nets

p. zizzi- quantum mind's collective excitations

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