Transcript Two Kinds of Field Theory in CM Physics

PCE STAMP
TWO KINDS of FIELD THEORY in CM PHYSICS
7 PINES meeting, May 7, 2009
Physics & Astronomy
UBC
Vancouver
Pacific Institute
for
Theoretical Physics
SOME DIFFERENT KINDS of EFFECTIVE FIELD THEORY
in CM PHYSICS
CLOSED SYSTEM
DELOCALISED “qp”
STATES
(eg., Fermi liquid,
superfluid, FQHE, etc.)
SYSTEM COUPLED TO
‘BATH’ OF DELOCALISED
‘qp” STATES
(eg., ‘polaron’ coupled
to Fermi liq, phonons,
superfluid, etc; SQUID
coupled to electrons)
CLOSED SYSTEM
LOCALISED STATES
(eg., disordered spin
system, glass, etc.
SYSTEM COUPLED TO
‘BATH’ OF LOCALISED
STATES
(eg., qubit, or ‘polaron’
coupled to defects,
impurities, nuclear
spins, etc.
WE NEED TO REMEMBER THAT ALL OF THESE ARE IMPORTANT.
Ec
Orthodox view of
Heff
SYSTEM plus
ENVIRONMENT
Scale out
High-E
modes
“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 PHILOSOPHY vs QCP PHILOSOPHY;
T.O.E.’s
We can contrast 2 quite different views of the
RG flow in a typical condensed matter system.
At left is a depiction of a ‘hierarchy’ of fixed
points, cascading down to ever lower energies.
In this picture one determines a succession of
effective Hamiltonians and field theories by
gradually integrating out high energy modes.
In any complex system like a glass (or
practically any real solid) this cascade
continues down to extremely low energies –
perhaps ad infinitum in many systems in the
thermodynamic limit (if there is one!).
A different point of view starts from the ‘Quantum
critical point’ philosophy – that the structure of the
effective field theories is determined instead from
BELOW by a few zero-energy fixed points.
Some have even argued in recent years that this
QCP framework may allow us to classify all possible
low-E states, thereby producing a kind of
low-energy “Theory of Everything” (cf., eg., Preskill,
and perhaps Laughlin)).
I: EXTENDED SYSTEM OF DELOCALISED MODES
This is the sort of system that philosophers of science and most
particle physicists like to talk about when they think of statistical
mechanics or condensed matter physics. Typical examples:
Fermi liquids (he-3, metals, etc., without dirt)
Superfluids and superconductors (without dirt)
Semiconductors, Quantum Hall fluids, etc. (without dirt)
Magnetic metals and insulators (without dirt)
etc., etc.
Theory of this works pretty well at first.
However there are problems…….
1ST 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.
II: EXTENDED SYSTEMS OF LOCALISED MODES
(WITH DISORDER, INTERACTIONS, ETC.)
The VAST MAJORITY of REAL systems in the condensed matter world
have to be described by effective theories that look nothing like the
kinds of field theory used in other parts of physics
One can certainly make quantum field theories for these
systems, but they look very different.
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
.
…………………..
.
`’~.,`.,’
..’`
.
~.
~
~.
.
~”
~.:
~`”:
~`/:
..: .
.’`
,’`.`,
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.’,
2ND CONUNDRUM: REAL Solids at low T
Capacitance in pure SiO2
In most real solids, frustrating interactions, residual
long-range interactions, and boundaries give a complex
hierarchy of states. These have difficulty communicating- to relax,
many atoms, spins, etc. must simultaneously reorganize. This is
sometimes summarized in the ‘ultrametric’ picture of the states
(below right):
One model for the
low-E excitations
is the ‘interacting
TLS model’, with
effective
Hamiltonian:
THE PROBLEM: HOW
DOES THIS BEHAVE?
ABOVE: structure of
low-energy eigenstates
for interacting TLS model,
before relaxation
III: QUANTUM SYSTEM INTERACTING
WITH ITS SURROUNDINGS
This kind of problem includes systems as simple as a single ‘polaron’
interacting with surrounding phonons, electrons, etc., in systems
ranging from semiconductors to polymers; all the way to large objects
like SQUIDs or magnetic qubits interacting with environments of both
localised modes (defects, local phonons, nuclear spins, etc.) and
delocalised modes (phonns, electrons, magnons, etc.)
It is also important for understanding
problems like decoherence,
entanglement, quantum computing, &
the measurement process.
OSCILLATOR BATH ENVIRONMENTS: REDUCTION PROCEDURE
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 environmental modes is WE
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 argued, 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
CONDITIONS for DERIVATION of OSCILLATOR BATH MODELS
Starting from some system interacting with an environment, we want an effective
low-energy Hamiltonian of form
(1) PERTURBATION THEORY
Assume environmental states
and energies
The system-environment coupling is
Weak coupling:
where
In this weak coupling limit we get oscillator bath with
and couplings
(2) BORN-OPPENHEIMER (Adiabatic) APPROXIMATION
Suppose now the couplings are not weak, but the system dynamics is SLOW, ie., Q
changes with a characteristic low frequency scale Eo . We define slowly-varying
environmental functions as follows:
Quasi-adiabatic eigenstates:
Quasi-adiabatic energies:
‘Slow’ means
Then define a gauge potential
We can now map to an oscillator bath if
Here the bath oscillators have energies
and couplings
The oscillator bath models
are good for describing
delocalised modes; then usually
Fq(Q) ~ O(1/N1/2)
(normalisation factor)
All this is fine except when either : (i) oscillators couple to solitons
(ii) We have degenerate bath modes (iii) Environment contains localised modes
QUANTUM ENVIRONMENTS of LOCALISED MODES
Consider now the set of localised modes that
exist in all solids (and all condensed matter
systems except the He liquids). As we
saw before, a simple description of
these on their own is given by the
‘bare spin bath Hamiltonian’
where the ‘spins’ represent a set of
discrete modes (ie., having a restricted
Hilbert space). These must couple to
the central system with a coupling of
general form:
We are thus led to a general
description of a quantum
system coupled to a
‘spin bath’, of the form
shown at right. This is
not the most general possible
Hamiltonian, because the bath
modes may have more than 2
relevant levels.
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)
CONDITIONS for DERIVATION of SPIN BATH MODELS
We start again from a model of general form:
with interaction:
and bath
For this effective Hamiltonian to be valid we require that no other environmental levels
couple significantly to the localised bath levels. We also require that the bath modes
couple weakly to each other, satisfying the conditions:
(intra-bath mode-mode coupling weak compared
to the coupling to the central system); or failing
this, that:
The ‘external fields’ acting on the bath modes are
(ii)
much larger than the intra-bath couplings
There is no ‘Born-Oppenheimer’ requirement of ‘slow’ changes.
If the system changes
Then define:
The model is valid for all uk:
on a timescale T, so that:
(i)
INFLUENCE FUNCTIONAL
This is given by the standard form:
With the interaction action
However now we have
The bath action
contains a
topological term
DYNAMICS of DECOHERENCE from SPIN BATH
We are interested in the dynamics of the
density matrix, via:
A simple example is a qubit coupled
to a spin bath.
Interestingly, the main decoherence
mechanism does not involve any
Dissipation, only phase entanglement.
3RD CONUNDRUM:
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 2-slit expt with
buckyballs. The COM
Buckyball coordinate Q does not couple directly to the vibrational modes of the
buckyball. However BOTH couple to the slits, 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.
PCE Stamp, Stud. Hist Phil Mod Phys 37, 467 (2006)
ACTIVE 3rd PARTY: Here the system state correlates with the 3rd party, which then changes the environment to
correlate with Q. We can also think of the 3rd party X as PREPARING both system & environment; or, we can think o
system and the environment as independently measuring the state of X. System & environment are then entangled.
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. We could also have more than one intermediary- ie., X, Y, etc.- with
entanglement transmitted along a chain (which can wiped out before the process is finished).
PROBLEM: IN PRINCIPLE A SYSTEM CAN ENTANGLE
WITH ALMOST ANYTHING. SO WE HAVE TO ENLARGE
OUR EFFECTIVE HAMILTONIAN TO INCLUDE ALL THESE
OTHER SYSTEMS
SUMMARY
I have tried to open the following questions for discussion:
1: How should we look at the RG picture (nature of Hilbert space,
IR/UV mixing, QCP from below or RG from above)
2: What about lattices plus interactions (eg., Hubbard)?
3: What about ‘glasses’? What can we say about systems where we have
no self-averaging, no thermodynamic limit, no physically
meaningful ground state, etc.?
4: What about entanglement? What use is an effective Hamiltonian for
a closed system if it is never really closed?