PCE STAMP

1. PCE STAMP Physics & Astronomy UBC Vancouver Pacific Institute for Theoretical Physics

2. ‘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).

3. 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. • Soft Emergence: Without disputing • the existence of the fundamental “lego” building blocks (which may be reducible into ever-smaller • building blocks), one argues that a complete description of the House (including its ‘internal dynamics’) • requires more than a complete understanding of the lego blocks and their interactions. In physics this • 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 obviously wrong in Quantum Mechanics. One should distinguish between the description of a system S (in terms of a set of constituent coordinates Qj) and the quantum state of S, which may entangle the constituents (so they do not have individual quantum states). So- how do we look at this question in a formal way in physics?

4. Description using EFFECTIVEHAMILTONIANS Heff

5. Orthodox view of Heff Ec Scale out High-E modes “Renormalisation” Wo Heff(Ec )  Heff (Wo) The RG mantra is: RG flow fixed points low-energy Heff universality classes |yi> Hij(Ec) <yj|  |fa> Hab(Wo)<fb| Flow of Hamiltonian & Hilbert space with UV cutoff

6. i(Ec) i(Wo) 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

7. 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.

8. States in a glass- piled up at low E 1ST CONUNDRUM- the ‘GLASS’ 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 local change must re-organize simultaneously a vast number of states. This forces the Hilbert space of the effective Hamiltonian to have an ‘ultrametric’ geometry. What this means is that no matter what energy or temperature one is working at, the ground state of the spin glass effective Hamiltonian is meaningless. 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-equilibrium) properties. ‘Ultrametricgeometry’ of a glass Hilbert space

9. 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 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.

10. 3RD CONUNDRUM- TOPOLOGICAL 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

11. Question:EMERGENCE from “DECOHERENCE” ? When some quantum systemwith 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. E 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  [a1Y1(Q)Pqfq(1) + a2 Y1(Q) Pqfq(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 ?

12. DESCRIBING the QUANTUM-CLASSICAL BORDERLANDS Quantum Dynamics Classical Dynamics Heff 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 enevironmental 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 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, 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. Feynman & Vernon, Ann. Phys. 24, 118 (1963) Caldeira & Leggett, Ann. Phys. 149, 374 (1983) AJ Leggett et al, Rev Mod Phys 59, 1 (1987

13. 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 ~`”: ~`/: . . ,’` ..: .’` .`, .’`* .’,

14. DECOHERENCE DYNAMICS from an EFFECTIVE H Consider the followingHeff : 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.sk) + H.c.] + eotz (qubit) + tz wk.sk + hk.sk (bath spins) + inter-spin interactions At first glance a solution of this seems very forbidding. However it turns out one can solve for the reduced density matrix of the central spin in all interesting parameter regimes- & 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 Thus decoherence can be dominated by processes causing little or no dissipation Precessional decoherence Noise decoherence source Actually the above model describes some very interesting systems

15. 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)

16. 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)

17. 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).

18. 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.

19. TALK: see http://physics.ubc.ca/~stamp