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PCE STAMP
Physics & Astronomy
UBC
Vancouver
Pacific Institute
for
Theoretical Physics
‘EMERGENCE’ vs ‘REDUCTIONISM’
The reductionist view is that all matter can be
understood in terms of its ‘basic constituents’.
It is an atomistic point of view. It is a
programme.
The ‘emergence’ point of view says that
structures of matter at higher levels, & in
more complex systems, CANNOT be
understood in terms of basic constituents- that they have ineluctably
‘complex’ properties, not understandable in terms of elementary
constituents, even in principle. This is also a programme.
NB1: Many if not most ‘emergence’ believers still
nevertheless assume that matter is composed of ‘bits’
(the ‘lego’ philosophy, or ‘soft’ emergence)
NB2: Some argue there is no end in sight to the long road
towards ‘elementary constituents’, in particle physics. Nature may just be
‘wheels within wheels..’ (ie. an unending series of effective Hamiltonians).
Hard & Soft EMERGENCE
The House is built of bricks, glass, metal,
etc.-with a sufficiently detailed blueprint,
one could rebuild it elsewhere. It is built
from sub-elements, whose nature &
definition do not depend on being part of
the House. However, one cannot
understand their disposition without
knowing the purpose(s) of the House.
(1) Soft Emergence: Without disputing
the existence of the fundamental “lego” building blocks (which may be reducible into ever-s
building blocks), one argues that a complete description of the House (including its ‘interna
requires more than a complete understanding of the lego blocks and their interactions. In ph
has led to extraordinary fruitful concepts (the order parameter and
its dynamics, pattern & structure formation, dissipative structures...
It is often argued that similar ideas can apply in other areas (eg.,
decoherence  classical world, or even  space time).
(2) Hard Emergence: A much stronger view- that the building
blocks cannot be understood or even properly defined without
reference to the larger whole (the House). For the House this is nonsensical. It is not obvious
in Quantum Mechanics. One should distinguish between the description of a system S (in te
set of constituent coordinates Qj) and the quantum state of S, which may entangle the consti
they do not have individual quantum states).
So- how do we look at this question in a formal way in physics?
Description using EFFECTIVE HAMILTONIANS
H
eff
Ec
Scale out
High-E
modes
Orthodox view
of Heff
“Renormalisation”
Wo
Heff (Ec )  Heff (Wo)
The RG mantra is: RG flow
fixed points
|yi> Hij(Ec) <yj|

|fa> Hab(Wo) <fb|
low-energy Heff
universality classes
Flow of Hamiltonian & Hilbert space with UV cutoff
MORE ORTHODOXY
i(Ec)
i(Wo)
Continuing in the orthodox vein, one
supposes that for a given system, there
will be a sequence of Hilbert spaces,
over which the effective Hamiltonian
and all the other relevant physical
operators (NB: these are effective
operators) are defined.
Then, we suppose, as one goes to low
energies we approach the ‘real vacuum’; the approach to the fixed
point tells us about the excitations about this vacuum. This is of
course a little simplistic- not only do the effective vacuum and the
excitations change with the energy scale (often discontinuously, at
phase transitions), but the effective Hamiltonian is in any case
almost never one which completely describes the full N-particle
states.
Nevertheless, most believe that the basic
structure is correct - that the effective
Hamiltonian (& note that ALL
Hamiltonians or Actions are
effective) captures all the basic physics
RG FLOW, CRITICAL PHENOMENA, & Condensed Matter
T.O.E.’s
One of the main obstacles to passing from a high-energy description of a system to a low-energy
one is the existence of phase transitions. Physicists have turned this into a virtue, by analysing the
way the effective Hamiltonian changes as one approaches a finite-T critical point. One gets “Universality
Classes” of effective Hamiltonian, describing many different systems, as they approach the critical
point- this means that they all have the same form of effective Hamiltonian, differing only in the
coefficients of the operators in the Hamiltonian.
One can also talk about T=0 phase transitions- as one changes some
parameter like pressure, a phase transtion can
be induced. However here there are no thermal
fluctuations, instead we have quantum critical
fluctuations.
It as been argued with increasing vigour in
recent years that this framework may allow us
to classify all possible low-E states, thereby
producing a kind of low-energy “Theory of
Everything” (Laughlin, Preskill). The interesting
recent theoretical connections found between
correlations in certain 1-d models and the
entanglement features is connected with this.
1ST CONUNDRUM- the ‘GLASS’
States in a glass- piled up at low E
The simple picture of excitations
perched above a vacuum gets a
rude shock when we consider
Glasses - systems with disorder &
‘frustrating interactions’. We are
surrounded by these! States pile
up at low energy, but these can’t
communicate with each other.
Frustrating interactions
‘Frustration’ means that at low energy, any
What this means is that no matter what
local change must re-organize simultaneously
energy or temperature one is working
a vast number of states. This forces the
at, the ground state of the spin glass
Hilbert space of the effective Hamiltonian to
effective Hamiltonian is meaningless.
have an ‘ultrametric’ geometry.
At finite T, the system can never reach
the
ground
state,
and the finite-T
Hilbert space is
disconnected from any ground state. At zero-T, the
system splits into subspaces that can never
communicate with each other. Thus the effective
vacuum & its structure are physically meaningless.
A glass can only be defined by its dynamic (non‘Ultrametric geometry’ of a glass Hilbert space
equilibrium) properties.
2ND CONUNDRUM- the HUBBARD MODEL
The ‘standard model’ of condensed matter
physics for a lattice system is the ‘Hubbard
model’, having effective Hamiltonian at
electronic energy scales given by


H  t  ci† c j  h.c.  U  ni ni
i, j
i
This apparently simple Hamiltonian has
some very bizarre properties. Suppose we
try to find a low energy effective
Hamiltonian, valid near the Fermi energyeg., when the system is near “half-filling”.
We therefore assume a UV Cutoff much
smaller than the splitting U between the
Mott-Hubbard sub-bands (we assume that
U > t).
The problem is that this appears to be impossible. Any attempt to write an
effective Hamiltonian around the Fermi energy must deal with ‘spectral weight
transfer’ from the other Hubbard sub-band- which is very far in energy from the
Fermi energy. Thus we cannot disentangle high- and low-energy states. This is
sometimes called UV/IR mixing.
3RD CONUNDRUMTOPOLOGICAL
FIELD THEORIES
RIGHT: A statistical
flux attaches itself
to an electron to
make an ‘anyon’here on a lattice
Most of the models discussed in string theory &
quantum field theory are topological in nature, with
topological excitations & complex vacua. Similar
models exist in condensed matter physics (eg.,
FQHE).
In string theory it is hard to get rid of tachyons, which
create the analogue of a lattice potential for the strings,
leading to the complexity of the dissipative WAH model.
A key feature of such theories,
and of any non-commutative
gauge theory, is the same
UV/IR mixing we saw in the
Hubbard model- ie., no
clearly-defined effective low-E
action or Hamiltonian. It is not
known how general this UV/IR
mixing is.
See, eg.,
M. van Raamsdonk, N Seiberg
hep-th/9912072, /0002186
Question: EMERGENCE from “DECOHERENCE” ?
E
When some quantum system with coordinate Q interacts
with any other system (with coordinate x) , the result is
typically that they form a combined state in which there is
some entanglement between the two systems.
Example: In a 2-slit expt., the particle coordinate Q couples to
photon coordinates, so that we have the following possibility:
Yo(Q)
Pq fqin

[a1 Y1(Q) Pq fq(1) +
a2 Y1(Q) Pq fq(2) ]
But now suppose we do not have any knowledge of, or control over, the photon states- we must then
average over these states, in a way consistent with the experimental constraints. In the extreme
case this means that we lose all information about the PHASES of the coefficients a1 & a2 (and in
particular the relative phase between them). This process is called DECOHERENCE
NB 1: In this interaction between the system and its “Environment” E (which is in effect performing a
measurement on the particle state), there is no requirement for energy to be exchanged between the
system and the environment- only a communication of phase information.
NB 2: Nor is it the case that the destruction of the phase interference between the 2 paths must be
associated with a noise coming from the environment- what matters is that the state of the
environment be CHANGED according to the what is the state of the system.
Question: How do we describe this for a ‘COMPLEX’ SYSTEM ?
DESCRIBING the QUANTUM-CLASSICAL BORDERLANDS
Quantum Dynamics
Classical Dynamics
H
eff
Suppose we want to describe the dynamics of some quantum system in the presence of de
As pointed out by Feynman and Vernon, if the coupling to all the enevironmental modes is w
can map the environment to an ‘oscillator bath, giving an effective Hamiltonian like:
A much more radical argument was given by Caldeira and Leggett- that for the purposes of TESTING
the predictions of QM, one can pass between the classical and quantum dynamics of a quantum
system in contact with the environment via Heff. Then, it is arguend, one can connect the classical
dissipative dynamics directly to the low-energy quantum dynamics, even in the regime where the
quantum system is showing phenomena like tunneling, interference, coherence,
Feynman & Vernon, Ann.
or entanglement; and even where it is MACROSCOPIC.
Phys. 24, 118 (1963)
This is a remarkable claim because it is very well known that the QM wavefunction is far richer than the classical state- and contains far more information.
Caldeira & Leggett, Ann.
Phys. 149, 374 (1983)
AJ Leggett et al, Rev Mod
Phys 59, 1 (1987
DELOCALISED
WHAT ARE THE LOWENERGY EXCITATIONS IN
A SOLID ?
LOCALISED
Phonons, photons, magnons, electrons, ………
.
Defects,
Dislocations,
Paramagnetic
impurities,
Nuclear Spins,
…….
At right- artist’s view
of energy distribution
at low T in a solid- at
low T most energy is in
localised states.
INSET: heat relaxation
in bulk Cu at low T
.
…………………..
.
`’~.,`.,’
..’`
.
~.
~
~.
.
~”
~.:
~`”:
~`/:
..: .
.’`
,’`.`,
.’`*
.’,
DECOHERENCE DYNAMICS from an EFFECTIVE
Consider the following
Heff :
H
P.C.E. Stamp, PRL 61, 2905
(1988)
NV Prokof’ev, PCE Stamp, J
Phys CM5, L663 (1993)
NV Prokof’ev, PCE Stamp,
Rep Prog Phys 63, 669 (2000)
H (Wo  { [Dt exp(-i Sk ak.k) + H.c.] + eotz (qubit)
+ tz wk.k + hk.k
(bath spins)
+ inter-spin interactions
At first glance a solution of this seems very forbidding. However it turns out one can s
for the reduced density matrix of the central spin in all interesting parameter regime
the decoherence mechanisms are easy to identify:
(i) Noise decoherence: Random phases added to different Feynman paths by
the noise field.
(ii) Precessional decoherence: the phase
accumulated by bath spins between qubit flips.
(iii) Topological Decoherence: The phase
induced in the bath spin dynamics by the
qubit flip itself
USUALLY PRECESSIONAL
DECOHERENCE DOMINATES
Noise decoherence source
Thus decoherence can be dominated
Precessional decoheren
by processes causing little or no
dissipation
Actually the above model describes some very interesting systems
SOME
RECENT
EXPTS
Expts on
Tunneling magnetic
molecules & Ho
ions
Wernsdorfer et al, PRL 82, 3903
(1999); and
PRL 84, 2965 (2000); and
Science 284, 133 (1999)
R. Giraud et al., PRL 87, 057203 (2001)
A. Morello et al.; PRL 93, 197202 (2004)
Expts on
The quantum
phase transition
in LiHoF4
BELOW: Expts on coherently
Tunneling SQUIDs
H.M. Ronnow et al., Science 308, 389 (2005)
RW Simmonds et al., PRL 93, 077003 (2004)
WARNING: 3rd PARTY DECOHERENCE
This is fairly simple- it is decoherence in the dynamics of a
system A (coordinate Q) caused by indirect entanglement
with an environment E- the entanglement is achieved via a
3rd party B (coordinate X).
Ex: Buckyball decoherence
Consider the 2-slit expt with
buckyballs. The COM
coordinate Q of the buckyball does not couple directly to the vibrational modes
{qk } of the buckyball- by definition. However BOTH couple to the slits in the
system, in a distinguishable way.
Note: the state of the 2 slits, described by a coordinate X, is irrelevant- it does
not need to change at all. We can think of it as a scattering potential, caused
by a system with infinite mass (although recall Bohr’s response to Einstein,
which includes the recoil of the 2 slit system). It is a PASSIVE 3rd party.
ACTIVE 3rd PARTY: Here the system state correlates with the 3rd party, which then goes on to change the
environment to correlate with Q. We can also think of the 3rd party X as PREPARING the states of both system
and environment. Alternatively we can think of the system and the environment as independently measuring the
state of X. In either case we see that system and environment end up being
correlated/entangled.
Note the final state of X is not necessarily relevant- it can be changed in an
arbitrary way after the 2nd interaction of X. Thus X need not be part of the
environment. Note we could also have more than one intermediary- ie., X, Y,
etc.- with correlations/entanglement are transmitted along a chain (& they
can wiped out before the process is finished).
REMARKS
R1: One could argue that despite all this, the idea that we can still think of matter as made
of ‘elementary constituents’ (the lego philosophy) is nevertheless intact.
If so, one would like to know how to formulate this in physical theory- at the present time
the fundamental formulation of the properties of any physical system is in terms of an
effective Hamiltonian or effective action- and this poses the problems discussed herein.
R2: These are not just condensed matter physics problems- they also arise in high energy physics
Notice that whereas the IR / UV mixing comes in in condensed matter systems
typically in the presence of a lattice, this is not necessary- eg., in non-commutative gauge
theory or open string theory there is no lattice. In any case- since all Hamiltonians are
effective, the problems we address seem to be generic to all ‘many-body’ quantum theories, in
condensed matter, particle theory, or quantum gravity.
R3: Some of the problems discussed so far exist in a classical theory. However features
like IR / UV mixing seem to be quantum mechanical. And of course, the ineluctable role of
entanglement is entirely a QM feature.
Note that some formulations of QM make the description of any quantum system dependent
on macroscopic objects, and their entanglement with them (eg., Copenhagen/Bohr).
R4: Quantum Mechanics is a much richer theory than classical mechanics- and Q states contain
much more information. This feature appears in the Quantum/Classical borderlands when one looks at
mechanisms and sources of decoherence- many of the most potent sources of decoherence do not have
a classical dissipative analogue, are not connected with the noise spectrum, & hardly affect the
classical dynamics.
TALK: see
http://physics.ubc.ca/~stamp