SM.pdf

Supplementary Material

Dynamical anyon generation in Kitaev honeycomb non-Abelian spin liquids

Yue Liu,

Kevin Slagle,

1, 2

Kenneth S. Burch,

and Jason Alicea

1, 2

1

Department of Physics and Institute for Quantum Information and Matter,

California Institute of Technology, Pasadena, CA 91125, USA

2

Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA

3

Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA

(Dated: June 23, 2022)

Appendix A: Effective lattice model

Spin Liquid

γ

γ

γ

1

γ

2

γ

≡

γ

1

γ

≡

γ

2

γ

≡

γ

1

γ

≡

γ

2

2

2

FIG. 1. Mapping from the dumbbell setup to a helical Majorana chain. The dashed lines refer to edges of the dumbbell. We

imagine flattening the holes along the vertical arrows and obtain a helical chain whose two edge modes are represented by solid

lines.

In this appendix, we map the Majorana edge modes of the dumbbell to an effective spin model that can be simulated

more readily. The original system hosts chiral Majorana fermions

around two holes and two counter-propagating

Majorana fermions

L,R

along the edges of the bridge region. We imagine flattening the two holes along the vertical

arrows in Fig. 1. In the resulting deformed geometry, we identify

on the upper half of the flattened hole with a

left-mover

; on the lower half of the flattened hole we similarly identify

with a right-mover

(at the end of the

left hole, the fields obey boundary conditions

). An analogous identification maps

to left- and right-movers.

As Fig. 1 summarizes, at this point we have mapped the dumbbell onto a helical Majorana chain with fields

L/R

propagating over an effective length

eff

2 +

The Hamiltonian of the dumbbell is

∫

eff

[

−

ivγ

∂

ivγ

∂

]

−

∫

[

(

∂

) (

∂

)]

(A1)

We can recover the same low-energy Hamiltonian from the lattice model

eff

−

∑

−

∑

−

−

−

iδJ

(

−

)

(A2)

with

∝

δJ

∝

, and

{

,γ

}

= 2

i,j

. One can establish this correspondence by expanding

∼

+ (

−

, which yields

iγ

∼ −

iγ

∂

iγ

∂

. The

term describing interaction in the bridge region

can be similarly recovered by expanding four-fermion terms to leading order:

−

−

∼

(

∂

) (

∂

)

Finally, the

δJ

term counteracts the explicitly generated mass at endpoints of the bridge (due to the loss of translation

symmetry at the endpoints).

We can map the Kitaev chain model to an Ising spin chain using the usual Jordan-Wigner transformation:

−

−

∏

, γ

−

−

(A3)

where

and

are Pauli operators. The resulting spin Hamiltonian is a critical transverse-field Ising model

supplemented by three-spin interactions in the bridge and boundary transverse fields at the bridge endpoints:

−

eff

−

∑

−

eff

∑

int

−

∑

(

)

δH

−

δJ

(

)

(A4)

Above we separated the Hamiltonian into pieces involving

, and

δJ

, as used in the main text.

The above mapping establishes that the lattice model

and continuum Hamiltonian

manifestly contain the

same low-energy degrees of freedom. One may ask, however, whether the continuum edge Hamiltonian adequately

captures all of the low-energy physics of the dumbell in the spin liquid setup. Since the emergence of a chiral Ising edge

conformal field theory (CFT) is a universal property of the non-Abelian spin liquid (the edge mode can also be derived

from the Kitaev honeycomb model [1–3]), the question is equivalent to asking whether

misses additional low-energy

degrees of freedom that need to be included. One possibility is the dumbbell supports a

non-chiral

gapless edge mode

riding along with the chiral Ising CFT; such an outcome is technically compatible with the bulk topological order but

would require fine-tuning and so can be safely dismissed. A more plausible possibility is that ‘accidental’ low-energy

bulk fermions or visons reside at discrete locations near the edge (e.g., due to disorder, atomically sharp corners,

etc. [1, 4–9]) and hybridize with the chiral edge degrees of freedom that we include. The chiral edge—consisting of

gapless, extended states—would however ‘eat up’ these states, i.e., they would be absorbed by the edge and become

delocalized, at most changing the overall topological sector of the dumbbell. If we assume that the overall topological

charge around the dumbbell is trivial then our scheme continues to hold without modification. Finally, the boundaries

of our dumbbell should be viewed as ‘blurred’ on the scale of the bulk correlation length, which we assume here is

large compared to the atomic spacing. The dumbbell will thus not suffer from sharp corner effects in the sense that

the edge mode wavefunctions will be smeared out over many lattice spacings, spending little time near any atomically

sharp corners (which need not exist in any case in our setup due to the manner in which we envision manipulating

the spin liquid using magnetic tunnel junctions).

Instead of studying the effective lattice model, one could alternatively contemplate exploiting exact solvability of the

full 2D Kitaev honeycomb model to simulate the dynamics of the dumbbell setup more directly. We are interested,

however, in attacking the problem in an experimentally relevant regime that is potentially far from the exactly

solvable point. In this generic situation, gauge fluxes are no longer static, thus complicating an explicit solution to

the 2D problem, particularly for simulating nontrivial time dependence. Our anyon-generation scheme circumvents

this challenge by relying only on universal edge physics, which depends on a small number of parameters (e.g., the

edge velocity, the hole/bridge sizes, and the interaction between counter-propagating edge modes). The 1D effective

lattice model derived above directly incorporates these parameters while ‘filtering out’ unimportant bulk degrees of

freedom. Correspondingly, the effective lattice model enables efficient simulation of time dependence as carried out in

the main text. That said, in future work it would be interesting to develop efficient methods for studying dynamics

of the non-Abelian phase in the full 2D Kitaev honeycomb model—particularly in the presence of experimentally

relevant perturbations that spoil exact solvability.

Appendix B: Effective-model level structure and simulation details

Figure 2 plots the 10 smallest energy eigenvalues of the effective Hamiltonian [Eq. (A4)] versus

, obtained from

DMRG with

(

) = (320

120)

and

(320

56)

; note that the ground state energy has been fixed to zero for each

The many-body energies at

= 0

can be extracted from the theory of a single chiral Majorana fermion propagating

over length

= 2

+ 2

, as arises for the trivial-bridge spin liquid setup that the Hamiltonian emulates in this

limit. That is, the many-body energies follow from summing up single-fermion energies of the form

πv

(

+ 1

with integer

≥

. Dotted grey lines correspond to levels with odd fermion number, while solid black lines have even

fermion number. The former represent states that are not physically accessible in our spin-liquid problem (where we

assume trivial total topological charge for the dumbbell).

0.1

0.2

0.3

0.4

0.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

I

×

I

ψ

×

ψ

σ

×

σ

TCI

I

×

I

ψ

×

ψ

σ

×

σ

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.1

0.2

0.3

0.4

0.5

Energy

(

) = (320

120)

(

) = (320

56)

FIG. 2. Lowest 10 levels (with the ground state energy subtracted) obtained by DMRG for

(

) = (320

120)

and

(320

56)

Black curves indicate levels that have trivial total topological charge and thus represent physically accessible states in our spin-

liquid problem. Grey dotted curves corresponds to levels with an odd number of fermions—which we assume are inaccessible

in the spin liquid context.

The levels in Fig. 2 at first vary weakly with

, but undergo more dramatic evolution near the TCI point (

= 0

428

corresponding to the vertical dashed line). At

= 0

—where our protocol terminates—the lowest levels that are

accessible in the spin liquid problem correspond to the following anyon sectors:

(ground state),

(

excitation),

(

excitation), etc. See the labels on the right side of Fig. 2. Notice that the first excited state is

actually two-fold degenerate, with one of the degenerate levels representing an inaccessible, odd-fermion-parity state.

This pair of levels corresponds to the two possible fusion channels for Ising anyons (i.e.,

). In the spin

representation, one can identify the states in the

sector by the properties

〈

〉≈

and

〈

〉≈

for sites

emulating the bridge region; the

and

states instead have

〈

〉≈

and

〈

〉≈

in the bridge sites.

Our time-dependent protocol simulations initialize the system into the

= 0

ground state. Figure 2 indicates that

the ground state does not exhibit any level crossings en route to

= 0

; consequently, diabatic evolution is required

to access

(and other excited states). Throughout the simulation, the truncation error is set to be

−

and the

time step is set to 0.005 (in

= 1

units). We checked that the error due to truncation and the finite time step is

small by doubling the truncation error and the time step in a system with

= 320

= 56

, and

= 1200

, which

only affected the amplitudes for the

and

states by

Evolution from the ground state to the

fusion sector is forbidden even in the spin representation (due to

Ising symmetry), and even for diabatic evolution—modulo TEBD simulation errors from truncation and finite time

steps. When computing the probability of entering the

state at the end of the protocol, we use DMRG to

compute the pair of degenerate

eigenstates and then sum the probabilities for both. The probability of the

sector due to TEBD error is negligible (e.g.,

−

for

= 160

= 80

and

= 1200

In Fig. 3 we show an example of when the probability is transferred from

state to

state. We plot the

probabilities as a function of

(

)

for a

(

) = (80

80)

dumbbell with

= 400

. When

λ < λ

TCI

, the transition rate

is small. The major probability transition appears just after the TCI point where kinks of spontaneously generated

mass in the bridge carry Majorana zero modes and facilitate the generation of

state. When the bridge is deep

in the gapped phase (

λ > λ

TCI

), such kink excitations become gapped and difficult to generate, and the transition

rate correspondingly decreases to nearly zero again.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.2

0.4

0.6

0.8

P

×

|

(

)

|

′

(

)

|

(

)

|

TCI

(

) = (80

80)

FIG. 3. Transition probability from the

state (defined as a function of

via adiabatic continuity) for a system

with

(

) = (80

80)

and

= 400

. In the legend

′

(

)

≡

(

)

/dλ

denotes the derivative of the interaction term, and

|〈

(

)

′

(

)

(

)

〉|

measures the transition rate in the language of time-dependent perturbation theory.

σ

σ

∆

κ

∼

−

(

σ

)

7

/

4

−

δ

FIG. 4. Dumbbell setup incorporated in a thermal anyon interferometer.

Appendix C: Interferometric detection of Ising anyons in the dumbbell setup

Unitary time evolution in our protocol generically yields a superposition of

and

sectors for the holes in

our dumbbell. (For simplicity, here we neglect the

sector, which is suppressed in the desired operating regime.)

A projective measurement of the anyon charge on one of the holes is therefore required. The protocol repeats until

projective measurement returns the desired

sector.

Our dumbbell setup can be naturally incorporated into previously proposed interferometer designs to enable anyon

detection. In this appendix, we focus on the example of thermal interferometry (see Refs. 10 and 11), which can

be implemented in the device shown in Fig. 4. The five hatched regions are assumed to be dynamically tunable

between spin liquid and trivial phases using magnetic tunnel junctions that locally control the Zeeman field in the

Kitaev material. The hatched bridge connecting the holes in the dumbbell was discussed in detail in the main

text. Four additional hatched regions serve to bend the outermost spin-liquid edge states inward during the readout

stage—creating constrictions that allow tunneling of Ising anyons between the upper and lower edges. A temperature

difference

δT

between cold and hot reservoirs at the left and right ends, respectively, generates a heat current

; the

corresponding thermal conductivity is

δT

)

If the bulk thermal conductance is sufficiently small, the heat exchange is dominated by the chiral edge mode—

yielding quantized thermal Hall conductivity proportional to the central charge in the absence of constrictions. As

shown in Refs. 10 and 11, when the two constrictions are created, edge Ising anyons tunneling through the constrictions

introduce a correction to the heat conductance [see Eq. (39) in Ref. 11] that, crucially, is sensitive to the anyon charge

located in the enclosed hole in the dumbbell:

∆

κ/κ

−

−

{

L,σ

R,σ

+ [1 + (

−

]

L,σ

R,σ

}

(C1)

Here,

is the mean temperature;

(

)

,σ

is the tunneling amplitude of the edge Ising anyons at the left (right)

constriction; and

= 0

is the number of Ising anyons generated in the enclosed hole. In particular, the

L,σ

R,σ

contribution reflects an interference term—which

vanishes

when an Ising anyon resides in the enclosed hole. Physically,

the absence of interference in this case arises because an edge Ising anyon encircling a bulk Ising anyon changes the

fusion channel for the Ising anyon pair in the dumbbell, producing a state that is orthogonal to that arising when

edge Ising anyons do not encircle the hole. Therefore one can detect an Ising anyon in the hole by measuring the

thermal conductance of the above setup, providing a means of projective measurement required for our protocol to

deliver well-defined anyon charge in the dumbbell.

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