Chapter IV: MBL-Mobile: Many-body-localized engine
4.3 Numerical simulations
The engine can be implemented with a disordered Heisenberg model. A similar model’s MBL phase has been realized with cold atoms [4]. We numerically simu- lated a 1D mesoscale chain of N = 12 spin-12 degrees of freedom, neglecting dy- namical effects during strokes 1 and 3 (the Hamiltonian tunings). The chain evolves under the Hamiltonian
Hsim(t)= E Q(h(αt))
"N−1
X
j=1
σj·σj+1+h(αt)
N
X
j=1
hjσzj
#
. (4.35)
Equation (4.35) describes spins equivalent to interacting spinless fermions. Ener- gies are expressed in units ofE, the average per-site energy density. Forγ = x,y,z, theγthPauli operator that operates nontrivially on the jthsite is denoted byσγj. The Heisenberg interactionσj·σj+1encodes nearest-neighbor hopping and repulsion.
The tuning parameter αt ∈ [0,1] determines the phase occupied by Hsim(t). The site-j disorder potential depends on a random variable hj distributed uniformly across [−1,1].The disorder strengthh(αt) varies ash(αt) =αthGOE+(1−αt)hMBL. When αt = 0, the disorder is weak, h = hGOE, and the engine occupies the ETH phase. When αt = 1, the disorder is strong, h = hMBL hGOE, and the engine occupies the MBL phase.
The normalization factor Q(h(αt)) preserves the width of the density of states (DOS) and so hδi. Q(h(αt)) prevents the work extractable via change of band- width from polluting the work extracted with help from level statistics, (Sec. 4.2).
Q(h(αt))is defined and calculated in Suppl. Mat. C.5.
We simulated the spin chain using exact diagonalization, detailed in Suppl. Mat. C.5.
The ETH-side field had a magnitudeh(0) =2.0, and the MBL-side field had a mag- nitudeh(1)= 20.0. Theseh(αt) values fall squarely on opposite sides of the MBL transition ath ≈7.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 Wb/hδi
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
hWtoti/hδi
(a)
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Wb/hδi 0.4
0.5 0.6 0.7 0.8 0.9 1.0
ηMBL
(b)
Figure 4.5:Average per-cycle powerhWtoti(top) and efficiencyηMBL(bottom) as functions of the cold-bath bandwidthWb: Each red dot represents an average over 1,000 disorder realizations of the random-field Heisenberg Hamiltonian (4.35). The slanted blue lines represent the analytical predictions (4.12) and (4.14). WhenWb hδi(in the gray shaded region),hWtotiandηMBLvary linearly withWb, as predicted.
Adiabatic engine performance
We first simulated the evolution of each state in strokes 1 and 3 as though the Hamil- tonian were tuned adiabatically. We index the energiesEj(αt) from least to greatest at each instant: Ej(αt) < Ek(αt) ∀j < k. Let ρj denote the state’s weight on eigenstate j of the pre-tuning Hamiltonian H(αt = 0). The engine ends the stroke with weight ρj on eigenstate j of the post-tuning HamiltonianH(1).
The main results appear in Fig. 4.5. Figure 4.5a shows the average work extracted per cycle,hWtoti; and Fig. 4.5b shows the efficiency,ηMBL.
In these simulations, the baths had the extreme temperatures TH = ∞ andTC = 0. This limiting case elucidates the Wb-dependence of hWtoti and of ηMBL: Dis- regarding finite-temperature corrections, on a first pass, builds intuition. Finite- temperature numerics appear alongside finite-temperature analytical calculations in Suppl. Mat. C.1.
Figure 4.5 shows how the per-cycle power and the efficiency depend on the cold- bath bandwidthWb. As expected,hWtoti ≈Wb. The dependence’s linearity, and the unit proportionality factor, agree with Eq. (4.12). Also as expected, the efficiency declines as the cold-bath bandwidth rises:ηMBL ≈ 1− 2hδiWb .The linear dependence and the proportionality factor agree with Eq. (4.14).
The gray columns in Fig. 4.5 highlight the regime in which the analytics were performed, where Whδib 1. If the cold-bath bandwidth is small, Wb . hδi, the analytics-numerics agreement is close. But the numerics agree with the analytics
even outside this regime. IfWb & hδi, the analytics slightly underestimateηMBL: The simulated engine operates more efficiently than predicted. To predict the nu- merics’ overachievement, one would calculate higher-order corrections in Suppl.
Mat. C.1: One would Taylor-approximate to higher powers, modeling subleading physical processes. Such processes include the engine’s dropping across a chain of three small gapsδ01, δ02, δ03< Wbduring cold thermalization.
The error bars are smaller than the numerical-data points. Each error bar represents the error in the estimate of a mean (ofhWtotior ofηMBL := 1−hWhQtoti
ini) over 1,000 dis- order realizations. Each error bar extends a distance (sample standard deviation)/√
# realizations above and below that mean.
Diabatic engine performance
We then simulated the evolution of each state in strokes 1 and 3 as though the Hamiltonian were tuned at finite speed for 8 sites. (We do not simulate larger di- abatic engines: That our upper bounds on tuning speed for a mesoscopic engine go as powers of the level spacinghδi ∼ 2−L means that these simulations quickly become slow to run.) We simulate a stepwise tuning, taking
α(t) = (δt)bvt/(δt)c. (4.36) This protocol is considerably more violent than the protocols we treat analytically:
In our estimates, we leave v general, but we always assume that it is finite. In the numerics, we tune by a series of sudden jumps. (We do this for reasons of numerical convenience.) We work at βC = ∞and βH = 0, to capture the essential physics without the added confusion of finite-temperature corrections. In this case, we expect the engine to workwell enough—to output a finite fraction of its adiabatic work output—for
v (Wb)3 δ−
(4.37) [c.f. Eq. (4.30)].
In Fig. 4.6, we show work output as a function of speed. Despite the simulated protocol’s violence,Wtotis a finite fraction of its adiabatic value forv . (Wδb−)3 and even forv > (Wδb)3
− : Our engine is much less sensitive to tuning speed than our crude diabatic-corrections bounds suggest.
These numerics not only confirm the validity of our analytics, but also indicate the robustness of the MBL Otto engine to changes in the tuning protocol.
10−3 10−2 10−1 100 101 102 vδ−/Wb3
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007
WTOT
Figure 4.6:Average per-cycle work as a function of tuning speed for 995 disorder realizations of the random-field Heisenberg Hamiltonian (4.35) at system sizeL=8 (red dots), compared to the analytical estimate (4.12) for the adiabatic work output (blue line).
Each error bar represents the error in the estimate of the mean, computed as (sample standard deviation)/√
(# realizations).