PCE STAMP Physics & Astronomy UBC Vancouver Pacific Institute for Theoretical Physics TWO KINDS of FIELD THEORY in CM PHYSICS 7 PINES meeting, May 7, 2009

SOME DIFFERENT KINDS of EFFECTIVE FIELD THEORY in CM PHYSICS CLOSED SYSTEM DELOCALISED “qp” STATES ( eg., Fermi liquid, superfluid, FQHE, etc.) CLOSED SYSTEM LOCALISED STATES (eg., disordered spin system, glass, etc. SYSTEM COUPLED TO ‘BATH’ OF DELOCALISED ‘qp” STATES (eg., ‘polaron’ coupled to Fermi liq, phonons, superfluid, etc; SQUID coupled to electrons) 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.

“Renormalisation” Scale out High-E modes Orthodox view of H eff H eff (E c )  H eff (  o ) |  i > H ij (E c ) H  (  o )  <   | The RG mantra is: RG flow fixed points low-energy H eff universality classes EcEc oo Flow of Hamiltonian & Hilbert space with UV cutoff SYSTEM plus ENVIRONMENT

(Ec)(Ec) (o)(o) MORE ORTHODOXY 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 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)). 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!).

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…….

1 ST 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 This apparently simple Hamiltonian has some very bizarre properties. Suppose we try to find a low energy effective Hamiltonian, valid near the Fermi energy- eg., 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.

WHAT ARE THE LOW- ENERGY EXCITATIONS IN A SOLID ?......`,.’,,’`. DELOCALISED Phonons, photons, magnons, electrons, ……… LOCALISED 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

2 ND CONUNDRUM: REAL Solids at low T Capacitance in pure SiO 2 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: ABOVE: structure of low-energy eigenstates for interacting TLS model, before relaxation THE PROBLEM: HOW DOES THIS BEHAVE?

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 H eff Classical DynamicsQuantum Dynamics Feynman & Vernon, Ann. Phys. 24, 118 (1963) Caldeira & Leggett, Ann. Phys. 149, 374 (1983) AJ Leggett et al, Rev Mod Phys 59, 1 (1987 Suppose we want to describe the dynamics of some quantum system in the presence of decoherence. As pointed out by Feynman and Vernon, if the coupling to all the environmental modes is WEAK, 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 H eff. 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, or entanglement; and even where it is MACROSCOPIC. This is a remarkable claim because it is very well-known that the QM wave- function is far richer than the classical state- and contains far more information.

CONDITIONS for DERIVATION of OSCILLATOR BATH MODELS (1) PERTURBATION THEORY (2) BORN-OPPENHEIMER (Adiabatic) APPROXIMATION Assume environmental statesand energies The system-environment coupling is Weak coupling: where In this weak coupling limit we get oscillator bath with Suppose now the couplings are not weak, but the system dynamics is SLOW, ie., Q changes with a characteristic low frequency scale E o. 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 Starting from some system interacting with an environment, we want an effective low-energy Hamiltonian of form and couplings The oscillator bath models are good for describing delocalised modes; then usually F q (Q) ~ O(1/N 1/2 ) (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: (i) (intra-bath mode-mode coupling weak compared to the coupling to the central system); or failing this, that: (ii) The ‘external fields’ acting on the bath modes are much larger than the intra-bath couplings There is no ‘Born-Oppenheimer’ requirement of ‘slow’ changes. If the system changes on a timescale T, so that: Then define:The model is valid for all u k : 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.

3 RD CONUNDRUM: 3 rd PARTY DECOHERENCE 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 3 rd party. 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 3 rd party B (coordinate X). ACTIVE 3 rd PARTY: Here the system state correlates with the 3 rd party, which then changes the environment to correlate with Q. We can also think of the 3 rd party X as PREPARING both system & environment; or, we can think of the 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 2 nd 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). PCE Stamp, Stud. Hist Phil Mod Phys 37, 467 (2006) 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?