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PCE STAMP

PCE STAMP. Quantum Phase Transitions & the Spin Bath: DYNAMICS. PITP workshop – Q Info & the Many-Body Problem: Dec 1-3, 2007. Physics & Astronomy UBC Vancouver. Pacific Institute for Theoretical Physics. QUANTUM PHASE TRANSITIONS: The BASIC IDEA.

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PCE STAMP

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  1. PCE STAMP Quantum Phase Transitions & the Spin Bath: DYNAMICS PITP workshop – Q Info & the Many-Body Problem: Dec 1-3, 2007 Physics & Astronomy UBC Vancouver Pacific Institute for Theoretical Physics

  2. QUANTUM PHASE TRANSITIONS: The BASIC IDEA One imagines a system which has a phase transition at T=0, which is not 1st order in the T  0 limit (a condition not so easy to satisfy in practise). EXAMPLES: These can be classified into systems with extended modes, and those with localised modes Extended Modes MnSi ; TiBe2 ; ZrZn2 : conductors with FM/PM transition Localised Modes LiHoxY1-xF4 ; Mn-12 ; Fe-8 : insulators with order (FM; quantum spin glass) / disorder (PM, spin liquid?) transition Now – let’s suppose that we can change the critical parameter g(t) with time t, passing through gc What happens?

  3. A RELATED PROBLEM: Adiabatic Quantum Computation Assume a system Hamiltonian: H = (1- s) H1 + s H2 NB: Linear interpolation:s = vt Assume also that the ground state ofH1is easily accessible., and that the ground state ofH2encodes the solution to a hard computational problem. E. Farhi et al., Science 292, 472 (2001) One would like to analyse this in terms of the standard Landau-Zener formulae for the transitions. However in reality we know the system is more complex

  4. SIMPLE EXAMPLE Transverse field Q Ising model (the ‘TFQIM’): Simplify to 1-d nearest neighbour model: Asymptotic ground states Or else one of The critical parameter is: Energy gap is: gc

  5. SIMPLE EXAMPLE (cont.) Assume that: Then: Define: where Then we can do a standard Landau-Zener calculation to get the probability that there will be a non-adiabatic transition, corresponding to the nucleation of a ‘kink’ during the passage through the QPT. Then the result is that we get a density of defects, with a mean distance between defects given by One can argue that this resembles the Kibble-Zurek mechanism of defect nucleation. Zurek et al., PRL 95, 105701 (2005) PROBLEM: Near a QPT we expect a huge number of other states at low energy – this is particularly true when the interactions are ‘interesting’ (long-range, frustrating, etc.). What then?

  6. Toy Model: LANDAU-ZENER plus ENVIRONMENT Assume a model Central Spin Hamiltonian: with Can also use: COUPLINGS to ENVIRONMENT Oscillator Bath: Spin Bath: Can also make the simplification: (inert bath)

  7. RESULTS: OSCILLATOR BATH ENVIRONMENT If the coupling to the oscillator bath is diagonal, and we are at T=0, then we get no change – we just get the original LZ formula. If we have a non-diagonal coupling, or we are non-zero T, then the result is more complicated. At finite T the results are controversial. See, eg: M Wubs et al., PRL 97, 200404 (2006) RESULTS: SPIN BATH ENVIRONMENT If the interaction between bath spins is zero (inert bath), and the temperature T = 0, then we again find no change – we get the original LZ formula. ATS Wan et al., condmat /0703085 See, eg: These restrictions make the results of limited use – real systems are at finite T (very much so!). Moreover the neglected terms are usually there: and they are Important even if they are small. How much further can we go?

  8. LANDAU-ZENER + SPIN BATH – INCLUDING BATH FLUCTUATIONS Define and assume so that for no bath we have where With spin bath: and and the trace includes bath fluctuations averaged according to: with correlator so that the average of is just For most experiments the interesting limit is the high T fast sweep limit, with sweep rate s.t. Then we find that

  9. Candidates for Magnetic Qubits One of the candidates discussed for quantum computations is magnetic systems. Note that very large magnetic domain walls have already shown macroscopic tunneling, just like SQUID flux. Right now interest is focussed on magnetic molecules and ions which behave as 2-level systems- as ‘Qubits”. Ho ions in LiYF4 host

  10. The DIPOLAR SPIN NET - REALISATION The dipolar spin net is of great interest to solid-state theorists because it represents the behaviour of a large class of systems with “frustrating” interactions (spin glasses, ordinary dipolar glasses). It is also a fascinating toy model for quantum computation: H = Sj (Djtjx + ejtjz) + Sij Vijdiptiz tjz + HNN(Ik) + Hf(xq) + interactions CERTAIN NANOMAGNETIC SYSTEMS ALLOW US TO PROBE THIS SYSTEM IN GREAT DETAIL, & TO VARY THE PARAMETERS OVER A VERY WIDE RANGE Almost all experiments so far are done in the region where Do is small- whether the dynamics is dipolar-dominated or single molecule, it is incoherent. However one can give a theory of this regime. The next great challenge is to understand the dynamics in the quantum coherence regime, with or without important inter-molecule interactions

  11. Different Regimes for the Spin Net System Vo Dipole Interaction regime • The spin net offers a range of possibilities, • and our task is to find the behaviour in each • of the following regimes: • (i) DIPOLE INTERACTION-DOMINATED REGIME: If • one ignores the environment, this Quantum Ising • system simply localises into a glass if Vo>Do. • However the environment has a profound effect - • even extremely small xo will delocalise the spins, & • give quantum relaxation. If we increase the quantum • parameter so that Do>> xo (but still Do< Vo) then very • complex multi-spin entangled dynamics ensues. • DECOHERENT RELAXATION REGIME: Even with strong • decoherence/dissipation, the inter-spin correlations strongly affect • the relaxational dynamics. Again, the system is never frozen, even if Do<< xo , at finite T; but • strong decoherence can freeze it at T=0 • COHERENT QUANTUM REGIME: The most interesting but the most difficult to • understand – this is the full quantum computation problem, with N-spin entanglement. • The smallest environmental coupling eventually destroys coherent dynamics – higher spin • entanglement is the first to go. Many features of are not understood at all – this is a frontier • problem of great importance. It is commonly assumed in the quantum info literature that • for weak decoherence one can ignore all but uncorrelated errors (ie., single-spin • decoherence coming from interactions between individual qubits & the environment). • As we shall see below this is not in general correct. xo QPT Decoherent regime Quantum regime Do

  12. The Fe-8 MOLECULE Low-T Quantum regime- effective Hamiltonian (T < 0.36 K): Longitudinal bias: Eigenstates: Which also defines orthonormal states: Fe8S = 10 Feynman Paths on the spin sphere for a biaxial potential. Application of a field pulls the paths towards the field

  13. HYPERFINE COUPLING to spin bath (NUCLEAR SPINS) Hyperfine coupling: Define the set of fields: Static component is: Some of the couplings in Fe-8 (at H=0) Component which flips is: This gives a ‘central spin’ Hamiltonian:

  14. Structure of NUCLEAR MULTIPLET There are 215 nuclear sites in the molecule Transitions between states of different total polarisation (T1 process) are driven mainly by molecular tunneling) Total width of gaussian multiplet: (NB: This decreases with increasing applied field) For Fe-8 at H=0, Eo ~7 mK (depends on isotopic concentrations) For large D, the precessional decoherence rate is just gk~ D (Eo /D)2 Coupling to PHONONS Effective coupling to qubit: NV Prokof’ev PCE Stamp, J Low Temp Phys 104, 143 (1996) PCE Stamp IS Tupitsyn Phys Rev B69, 014401 (2004) A Morello, PCE Stamp, IS Tupitsn, PRL 97, 207206 (2006) where Giving a phonon decoherence rate:

  15. Quantum Phase Transition or COHERENCE experiments in Fe-8 A. Morello, P.C.E. Stamp and I.S. Tupitsyn, Phys Rev Lett 97, 207206 (2006)

  16. RESULTS for DECOHERENCE in the Fe-8 SPIN NET A very startling result emerges when one looks at the low-T decoherence in a dipolar spin net. Even for rather low T, the decoherence is dominated by correlated errors (ie., coming from pairs of qubits). Here we see results for the Fe-8 system. Note that at low T we can still get very high coherence: optimal coherent operation point at T = 50 mK Q  107 A. Morello, P.C.E. Stamp, I.S. Tupitsyn, Phys Rev Lett 97, 207206 (2006)

  17. QUANTUM RELAXATION REGIME: Derivation of Kinetic Eqtn. In both the dipolar-dominated regime and the environment-dominated regime, the dynamics is incoherent if Dois small. The we can use a classical kinetic equation. The kinetic eqtn for the magnetic qubit distribution Pa (x, r) is a BBGKY one, coupling it to the 2-qubit distribution P2. Here r is the position of the qubit, a = +,- is the polarisation of the qubit along the z-axis, and x is the longitudinal field at r. In this kinetic equation the interaction U(r-r’) is dipolar, and the relaxation rate t-1Nis the inelastic, nuclear spin-mediated, single qubit tunneling flip rate, as a function of the local bias field. As discussed before, this relaxation operates over a large bias range xowhere typically xo ~ Eo ( andEo is the width of the nuclear spin muliplet introduced before) The BBGKY hierarchy can be truncated with the kinetic equation above if the initial 2-qubit distribution factorizes. This happens if the system is either (i) initially polarized, or (ii) initially strongly annealed. Then we have: Thekineticequationcanthenbesolved,and givesthesquarerootshort-timebehaviour:

  18. Quantum Relaxation in a “Spin Net” of InteractingMAGBITS M(x) Vij >> Eo > D x At first glance the problem of a whole net of magbits, with long-range “frustrating” dipole interactions between them, looks insuperable. But actually the short-time dynamics can be solved analytically, in the quantum relaxation regime! This is because the dipole fields around the sample vary slowly in time compared to the fluctuating hyperfine fields. This leads to universal analytic predictions: (1) Only magbits near resonance make incoherent flips As tunneling occurs, the resonant surfaces move & disintegrate- then, for ANY sample shape dM(t) ~ [t/tQ]1/2tQ ~ (D2T2) Eo2N(x=eH)/W where Wis the width of the dipolar field distribution, and N(x) is the density of the distribution over bias. (2) Tunneling digs a “hole” in this distribution, with initial width Eo, and a characteristic spreading with time- so it depends again on the nuclear hyperfine couplings. IS Tupitsyn, PCE Stamp, NV Prokof’ev, Phys Rev B69, 132406 (2004) NV Prokof’ev, PCE Stamp, PRL 80, 5794 (1998)

  19. HOLE DIGGING up close We look at the time evolution of the INTERNAL DISTRIBUTION OF BIAS FIELDS M(x,t) (recall that x is the longitudinal bias field. A key feature of the theory is ‘Hole-digging’ in this distribution; the tunneling spins deplete the distribution. Only spins in resonance can tunnel, and this happens in a field range 2x0 (ie., controlled by the nuclear hyperfine interactions). The time evolution is non-trivial because the dipolar interactions scatter spins back into the hole (giving the square root time relaxation).

  20. FAST SWEEPING: APPLICATION TO Fe-8 SYSTEM Now we assume a full array of Fe-8 molecules with intermolecular dipole coupling & a fluctuating nuclear bath. Assume the sample has been annealed and then cooled to low T. The we can assume And the full kinetic equation gives: Near the nodes where and we can make the expansion Then near the n-th node one finds a decay rate given by: For pictures see next page…..

  21. For field along hard axis Nodal regions Field tilted away from nodes by a 1 degree angle

  22. FAST SWEEP EXPERIMENTS on Fe-8 The fast sweep of field gives a set of hysteresis curves as a function of sweep rate and temperature. For low T (no thermal activation but well above hyperfine energies) one gets the graphs at right. R Sessoli, W Wernsdorfer Science 284, 133 (1999) Interpretation of these expts according to naïve Landau-Zener gives the curves at right. The smoothing of the curves is not due to misalignment but instead to the internal fields

  23. NUCLEAR SPIN BATH in MAGNETIC SYSTEMS: The LiHoxY1-xF4 system The Ho ions interact with each other via dipolar interactions. This system is usually treated as the archetypal Quantum Ising system: The single spin has and a 1-spin crystal-field Hamiltonian with and In zero field there is a low-energy doublet, which we call This is separated from a 3rd state by a gap The g-factor is extremely anisotropic (factor of 20) giving an easy z-axis. Application of a transverse field gives a ‘tunneling splitting’ between the 2 doublet states (with transitions through the 3rd level) so that our low-E single spin effective Hamiltonian is just that above, with The dipolar interactions between the spins are just However including crystal field effects strongly modifies this to the above form, with the ‘zz’- interaction having strength with

  24. HYPERFINE COUPLING to the NUCLEAR SPIN BATH We have a simple interaction with The hyperfine splitting between nuclear levels is roughly 0.25K This interaction has a profound effect on the dynamics & on the effective Hamiltonian at low energy – electronic spins cannot flip unless multiple nuclear transitions also take place. Consider first what happens in low transverse field; we single out the 4 important states shown in the diagram. This problem is easily solved without the transverse hyperfine coupling; we get eigenstates etc where & with mixing coefficient & At low transverse field this just produces a classical Ising system: with and & renormalised spin The transverse hyperfine term only becomes effective when We then have a renormalised Hamiltonian:

  25. QUANTUM PHASE TRANSITIONS: HOW ARE THEY AFFECTED BY THE SPIN BATH?

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