PCE STAMP

Physics & AstronomyUBCVancouver

Pacific Institute for Theoretical Physics

Quantum Phase Transitions & the Spin Bath: DYNAMICS

PITP workshop – Q Info & the Many-Body Problem: Dec 1-3, 2007

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

MnSi ; TiBe2 ; ZrZn2 : conductors with FM/PM transition

Extended Modes

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?

E. Farhi et al., Science 292, 472 (2001)

Assume a system Hamiltonian:

H = (1- s) H1 + s H2

NB: Linear interpolation: s = vt

Assume also that the ground state of H1 is easily accessible., and that the ground state of H2 encodes the solution to a hard computational problem.

A RELATED PROBLEM: Adiabatic Quantum Computation

f

LZ

ts

eP

/1~~

:yprobabilit

Zener Landau

2/2

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

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

SIMPLE EXAMPLE

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.

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?

Zurek et al., PRL 95, 105701 (2005)

SIMPLE EXAMPLE (cont.)

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)

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.

See, eg: ATS Wan et al., condmat /0703085

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?

LANDAU-ZENER + SPIN BATH – INCLUDING BATH FLUCTUATIONS

Define and assume

so that for no bath we have

With spin bath: where

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

Single-molecule magnets (SMM)

Mn12 S = 10

Fe8 S = 10

V15 S = 1/2

Ni12 S = 12

Giant spins

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

The DIPOLAR SPIN NET - REALISATION

The dipolar spin net is of great interest to solid-state theoristsbecause it represents the behaviour of a large class of systemswith “frustrating” interactions (spin glasses, ordinary dipolarglasses). It is also a fascinating toy model for quantum computation:

H = j (j jx + j j

z) + ij Vijdip i

zjz

+ HNN(Ik) + H(xq) + interactions

Almost all experiments so far are done in the region where 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

CERTAIN NANOMAGNETIC SYSTEMS ALLOW US TO PROBE THIS SYSTEM IN GREAT DETAIL, & TO VARY THE PARAMETERS OVER A VERY WIDE RANGE

Vo

DipoleInteractionregime

Decoherent regime

Quantum regime

Different Regimes for the Spin Net System

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>. However the environment has a profound effect - even extremely small will delocalise the spins, & give quantum relaxation. If we increase the quantum parameter so that (but still Vo) then very complex multi-spin entangled dynamics ensues.

(ii) 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 , at finite T; but strong decoherence can freeze it at T=0

(iii) 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.

QPT

Fe8 S = 10

Feynman Paths on the spin sphere forFeynman Paths on the spin sphere fora biaxial potential. Application of a a biaxial potential. Application of a

field pulls the paths towards the fieldfield pulls the paths towards the field

The Fe-8 MOLECULE

Low-T Quantum regime- effective Hamiltonian (T < 0.36 K):

Longitudinal bias:Eigenstates:

Which also defines orthonormal states:

HYPERFINE COUPLING to spin bath (NUCLEAR SPINS)

Hyperfine coupling:

Define the set of fields:

Static component is:

Component which flips is:

This gives a ‘central spin’ Hamiltonian:

Some of the couplings in Fe-8 (at H=0)

Coupling to PHONONS

Structure of NUCLEAR MULTIPLETThere 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)

Effective coupling to qubit:

For large , the precessional decoherence rate is just ~Eo

where

Giving a phonon decoherence rate:

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)

A. Morello, P.C.E. Stamp and I.S. Tupitsyn, Phys Rev Lett 97, 207206 (2006)

Quantum Phase Transition or COHERENCE experiments in Fe-8

A. Morello, P.C.E. Stamp, I.S. Tupitsyn, Phys Rev Lett 97, 207206 (2006)

1.5 2.0 2.5 3.0 3.5 4.01E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

10

nuclear

phonon

dipolar 0.05 K 0.1 K 0.2 K 0.4 K

By (T)

optimal coherentoperation pointat T = 50 mK

Q 107

RESULTS for DECOHERENCE in the Fe-8 SPIN NETA 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:

QUANTUM RELAXATION REGIME: Derivation of Kinetic Eqtn.

The kinetic eqtn for the magnetic qubit distribution Pr) is a BBGKY one, coupling it to the

2-qubit distribution P2. Here r is the position of the qubit, is the polarisation of the

qubit along the z-axis, and is the longitudinal field at r.

In this kinetic equation the interaction U(r-r’) is dipolar, and the relaxation rate is

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 where typically Eo ( and Eo 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:

The kinetic equation can then be solved, and gives the square root short-time behaviour:

In both the dipolar-dominated regime and the environment-dominated regime, the dynamics is incoherent if is small. The we can use a classical kinetic equation.

Quantum Relaxation inQuantum Relaxation in aa “Spin Net” “Spin Net” ofof

InteractingInteracting MAGBITSMAGBITS

At first glance the problem of a whole net of magbits, withlong-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

M(t) ~ [t/Q]1/2 Q ~ (N(H)/W

where W is the width of the dipolar field distribution,

and Nis 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.

NV Prokof’ev, PCE Stamp, PRL80, 5794 (1998)

Vij >> Eo >

IS Tupitsyn, PCE Stamp, NV Prokof’ev, Phys Rev B69,

132406 (2004)

HOLE DIGGING up close

We look at the time evolution of the INTERNAL DISTRIBUTION OF BIAS FIELDS M(,t) (recall that 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 2 (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).

FAST SWEEPING: APPLICATION TO Fe-8 SYSTEMNow 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…..

For field along hard axis

Nodal regions

Field tilted away from nodes by a 1 degree angle

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.

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

R Sessoli, W Wernsdorfer Science 284, 133 (1999)

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

HYPERFINE COUPLING to the NUCLEAR SPIN BATH

We have a simple interaction with

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

The hyperfine splitting between nuclear levels is roughly 0.25K

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:

QUANTUM PHASE

TRANSITIONS:

HOW ARE THEY AFFECTED

BY THE SPIN BATH?