Dynamic properties of collective excitations in twisted bilayer Graphene

Gaopei Pan,1, 2 Xu Zhang,3 Heqiu Li,4 Kai Sun,4, ∗ and Zi Yang Meng3, †

1Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,Chinese Academy of Sciences, Beijing 100190, China

2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China3Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics,

The University of Hong Kong, Pokfulam Road, Hong Kong SAR, China4Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA

(Dated: August 31, 2021)

Employing the recently developed momentum-space quantum Monte Carlo scheme, we study thedynamic response of single-particle and collective excitations in realistic continuum models of twistedbilayer graphene. At charge neutrality, this unbiased numerical method reveals strong competitionbetween different symmetry breaking channels with a leading instability towards the intervalleycoherent state. Single-particle spectra indicate that repulsive interactions push the fermion spectralweight away from the Fermi energy and open up an insulating gap. The spectra of collectiveexcitations suggest an approximate valley SU(2) symmetry. At low-energy, long-lived valley wavesare observed, which resemble spin waves of Heisenberg ferromagnetism. At high-energy, these sharpemodes quickly become over-damped, when their energy reaches the fermion particle-hole continuum.

Introduction — To understand the rich physics intwisted bilayer graphene (TBG), as well as the mech-anism that governs this novel quantum system, a cru-cial step is to identify the ground state and to charac-terize the associated low-energy excitations [1–19]. Re-cently, many new theoretical insights have been obtainedusing real-space effective model analysis and large-scalenumerical simulations (e.g. quantum Monte Carlo andDMRG) [20–26], which indicate that even at integer fill-ings, correlation effects give rise to a very rich phase dia-gram with a variety of competing quantum phases. A keyadvantage of this approach is that these lattice modelscan be easily incorporated with well-established numer-ical techniques, but it remains a challenge to determinethe effective control parameters utilized in these mod-els from first principle. Another parallel approach uti-lizes continuum models with flat bands and fragile topol-ogy [27–29], where Coulomb interactions and first prin-ciple material parameters can be easily incorporated. Inthis approach, a key theoretical challenge is to handle thestrong Coulomb interactions. In certain special limit, ex-act solutions exist due to emergent high symmetry [30].For realistic material parameters away from these specialcases, Hartree-Fock mean-field and DMRG calculationssuggest that the ground state is likely to be an inter-valley coherent (IVC) state [31–34], which mixes elec-tron states from the two opposite valleys and breaks theUv(1) valley charge conservation. However, the competi-tion and intertwinedness with other symmetry-breakingground states, such as valley polarized (VP) states, arestill under investigation [32, 35–38]. To fully understandsuch a complex many-body system, unbiased numericalmethodology, which can solve such correlated problemsefficiently and accurately, is in great need.

In this Letter, we utilize the momentum-space quan-tum Monte Carlo (QMC) method [39–42] to achieve this

objective. The implementation of this method in contin-uum models of TBG has been developed recently [41, 42],but dynamic response, in particular the spectral infor-mation of the collective excitations, has not yet been ob-tained. In this work, we employ the momentum spaceQMC method, accompanied by the stochastic analytic(SAC) continuation scheme [43–49], to compute the spec-tra of both single-particle and particle-hole excitations.We find that, at the charge neutrality point (CNP), theIVC state is the leading instability, with strong competi-tion from the VP state. More interestingly, although thevalley SU(2) symmetry is broken explicitly when controlparameters take realistic values (with kinetic term), dy-namic response of particle-hole excitations still exhibitan approximate SU(2) symmetry. At low-energy, long-lived valley waves are observed in close analogy to spinwaves of a Heisenberg ferromagnet, and these modes be-come over-damped as their energy reaches the particle-hole continuum. These results reveal complex dynamicresponse in TBG and provide a foundation for the studyof other intriguing physics at and away from charge neu-trality, such as the mechanism of superconductivity andits possible topological origin [50, 51], the Pomeranchukeffect and beyond [52, 53].

Model and Method — In this study, we utilize the con-tinuum model of TBG flat band introduced in Refs. [1–6].In the plane wave basis, the single-particle Hamiltoniancan be written as:

Hτ0,k,k′ = δk,k′

(−~vF (k−Kτ

1) · σσστ U0U†0 −~vF (k−Kτ

2) · σσστ)

+(

0 Uτ1 δk,k′−τG1

Uτ†1 δk,k′+τG1 0

)+(

0 Uτ2 δk,k′−τ(G1+G2)Uτ†2 δk,k′+τ(G1+G2) 0

)(1)

arX

iv:2

108.

1255

9v1

[co

nd-m

at.s

tr-e

l] 2

8 A

ug 2

021

2

(a)

(b) (c)

FIG. 1. (a) The moire Brillouin zones (mBZ) at one valley. The red solid line marks the high-symmetry path Γ−M−K1(K2)−Γ.G1 and G2 are the reciprocal lattice vectors of the mBZ. Yellow dots mark possible momentum transfer in QMC simulations,q + G, and the blue dashed circle is the momentum space cut-off. Because the form factor decays exponentially with G [30],scatterings with momentum transfer larger than this cut-off are ignored. Here we show a 9 × 9 mesh in the mBZ, with 300allowed momentum transfers. In (b) and (c), blue lines are single particle spectra of L = 6, T = 0.067 meV, u0 = 33 meV and60 meV, respectively, obtained from the momentum space QMC with analytic continuation. The red stars and lines indicatethe bare dispersions of H0, which is the kinetic energy in our model in Eq.(3).

where vF is the Dirac velocity, τ = ± is the valley in-dex, and στ = (τσx, σy) defines the A,B sublattices ofthe monolayer graphene. Kτ

1,2 are the correspondingDirac points of the bottom and top layers, which aretwisted by angles ∓ θ2 respectively. As shown in Fig. 1 (a),G1 =

(− 2π√

3LM,− 2π

LM

)and G2 =

(− 4π√

3LM, 0)

are recip-rocal lattice vectors of the moire Brillouin zone (mBZ),with LM = a0/[2 sin(θ/2)] and a0 = 0.246 nm. Inter-

layer tunnelings are described by U0 =(u0 u1u1 u0

), Uτ1 =(

u0 u1e−τ 2π

3 i

u1eτ 2π

3 i u0

)and Uτ2 =

(u0 u1e

τ 2π3 i

u1e−τ 2π

3 i u0

)where u0 and u1 are the intra- and inter-sublattice in-terlayer tunneling amplitudes. In this Letter, we set~vF /a0 = 2377.45 meV, θ = 1.08 and u1 = 110 meV,which means the moire bands are completely flat at thechiral limit u0 = 0 [54–57].

We then project the charge-density operator at q + Gto the nearly flat bands relative to the filling of CNP:

δρq+G =∑

k∈mBZ,m1,m2,τ,s

λm1,m2,τ (k,k + q + G)

(d†k,m1,τ,s

dk+q,m2,τ,s −12δq,0δm1,m2

)= (δρ−q−G)†

(2)

where d†k,m.τ,s is the creation operator for a Bloch eigen-state, |uk,m,τ,s〉, with m, s, τ band, spin and valley in-dices. The form factor is defined as λm1,m2,τ (k,k + q +

G) ≡ 〈uk,m1,τ | uk+q+G,m2,τ 〉. As shown in Fig. 1 (a)q ∈ mBZ and q + G represents a vector in extendedmBZ, with G = n1G1 + n2G2, n1, n2 ∈ Z [55, 56].After projecting to the flat band, the Hamiltonian reads:

H = H0 +Hint

H0 =∑m=±1

∑kτs

εm,η(k)d†k,m,τ,sdk,m,τ,s

Hint = 12Ω

∑q,G,|q+G|6=0

V (q + G)δρq+Gδρ−q−G

(3)

where εm,η(k) is the eigenvalue of the continuum model inEq. (1). We define the long-ranged single gate Coulombpotential: V (q) = e2

4πε∫d2r

(1r −

1√r2+d2

)eiq·r =

e2

2ε1q

(1− e−qd

). d

2 is the distance between graphenelayers with d = 40 nm and ε = 7ε0. The volumeΩ = Nk

√3

2 L2M with Nk being the number of momentum

points in a mBZ (e.g., Nk = 81 for a 9× 9 mesh).The projected Coulomb interaction is solved via a dis-

crete Hubbard-Stratonovich transformation [24, 41, 58,59], eαO2 = 1

4∑l=±1,±2 γ(l)e

√αη(l)o + O

(α4), which in-

troduces the auxiliary field with the degree of freedomsl = ±1,±2 at each site of the 2 + 1 space-time configura-tional space with the total size∼ Nk×Nk×Lτ (details areshown in the Sec. I of Supplemental Material (SM) [60]).Up to corrections of order O(∆τ2), the partition functionwith the imaginary time discretization β = Lτ∆τ is:

3

Z =∑

l|q+G|,a,τ=±1,±2

Lτ∏τ=1

e−∆τH0 Trd

∏|q+G|6=0

γ(l|q+G|,1,τ

)γ(l|q+G|,2,τ

)16 eiη(l|q|1,t)Aq(δρ−q+δρq)eη(l|q|2,t)Aq(δρ−q−δρq)

where Aq+G =√

∆τ4V (q+G)

Ω and the trace over fermionoperators gives rise to the determinant for each auxiliaryconfiguration. The Markov chains of such configurationsare carried out with important sampling and physics ob-servables are computed via ensemble average [41, 60].

Exact ground states in the flat-band limit — When thekinetic energy is ignored (i.e., the flat-band limit), theTBG Hamiltonian at charge neutrality has an emergentU(4) symmetry and ground states can be obtained ex-actly [30, 42, 61]. To see the exact solution, one justneeds to realize that the valley polarized state, with allelectrons in one valley, is a zero-energy eigenstate of Hint.Because Hint is semi-positive definite, this must be aground state. In addition, any U(4) transformation ofthis ground state is also a degenerate ground state, in-cluding the VP, IVC and spin polarized states, as wellas infinite many other degenerate states. For simplicity,in this Letter, we will focus only on the VP and IVCstates, because the spin polarized states quickly becomeenergetically unfavored once kinetic energy is included.

We define the VP and IVC order parameters asOa(q, τ) ≡

∑k d†k+q(τ)Madk(τ), with Ma = τzη0 (η0

for band index) for VP and Ma = τxηy or τyηy for theIVC states [30, 36, 42]. It is straightforward to verify thatat q = 0, these three order parameters obey the commu-tation relations [Oa,Ob] = iεa,b,cOc and they all com-mute with the interaction Hamiltonian [Oa, Hint] = 0.Thus, they generate a SU(2) symmetry group, a sub-group of the full U(4) symmetry. In the ordered phase,the nonzero expectation value of these order parame-ters spontaneously breaks this SU(2) symmetry, result-ing in spin-wave-like gapless Goldstone modes, i.e. valleywaves. Same as ferromagnetism, such valley waves havea quadratic dispersion ω ∝ k2 at low-energy.

As for single-particle excitations, all these degenerateground states are insulators with a gap proportional tothe interaction strength. In the flat-band limit, single-particle Green’s function can be calculated exactly atT = 0 [30]. Despite of the strong Coulomb repulsion,electrons/holes exhibit free-fermion-like behavior, wherethe Green’s function shows four fermion bands with zerodamping: two conduction (valence) bands above (below)the Fermi energy.

In a real TBG, away from the flat-band limit, thisSU(2) symmetry is explicitly broken by the kinetic en-ergy down to Z2 (valley) and Uv(1) (valley charge con-servation), lifting the degeneracy between VP and IVCstates. Here, a IVC (VP) state breaks the continuous

U(1) (discrete Z2) symmetry, and dynamics fluctuationsin VP and IVC states shall exhibit different behaviors.However, if the kinetic energy term is small (i.e., smallband width), an approximate SU(2) symmetry may sur-vive, and qualitative features may still resemble the flat-band limit. The momentum space QMC technique offersa probe to directly visualize the breaking of the SU(2)symmetry as well as the remnant approximate symmetry.

Results and Analysis — In a previous work [41], wehave shown that Hint acquires a correlated insulatorground state at CNP. In this study, we added the ki-netic term H0 and carried out the simulations at u0 = 33meV and 60 meV with 6×6 and 9×9 momentum meshes.The single-particle spectra are shown in Fig. 1 (b) and(c). The bare (non-interacting) dispersions are depictedas red stars. At low-temperature, for both u0 = 33 meVand 60 meV, interactions push the fermion states awayfrom the Fermi energy, results in an interaction-drivenband gap of ∼ 20 meV, magnitudes larger than thatof the bare bandwidth. Although we are using realis-tic parameters away from the flat-band limit, as shownin Fig. 2 (c) and (d), the peak of single particle spec-tra agrees nicely with the exact solution of the flat-bandlimit [30], indicating that the system is not far from theexactly-solvable limit. As for the width of the peak, dueto the finite temperature and the presence of kinetic en-ergy, fermions here exhibit some damping of the order 10meV, which is significantly larger than T and the bandwidth of the bare dispersion. This is in contrast to theexactly-solvable limit at T = 0 where the damping van-ishes.

The next question is to reveal the symmetry-breakingchannels of this insulating state. The proposedsymmetry-breaking states at the CNP, based on Hartree-Fock mean-field analysis, are gradually pointing towardsthe IVC and VP states [31, 32, 35, 36]. Here, we measuretheir corresponding (dynamical) correlation

Sa(q, τ) ≡ 1N2k

〈Oa(−q, τ)Oa(q, 0)〉 (4)

where Oa is the order parameter of the VP or IVC statedefined early on. For static properties, we measure theequal-time correlation at imaginary time τ = 0. To ob-tain dynamic response, time-dependent Sa(q, τ) is mea-sured at τ ∈ [0, β], followed by the stochastic analyticcontinuation (SAC) [43–48] to obtain the real frequencyspectra [60].

The static order parameters are presented in Fig. 2 (a)

4

meV

meV

T(meV)

FIG. 2. (a) S(q = Γ, τ = 0), the squares of order parameters, for VP and IVC at u0 = 33 meV and L = 6, as a function oftemperature. (b) The same measurement at u0 = 60 meV with both L = 6 and 9. When kinetic energy is ignored, the two orderparameters are degenerate due to an emergent SU(2) [U(4)] symmetry. When the kinetic energy is taken into account (”withkin”), which breaks the symmetry, this degeneracy is lifted. At u0 = 33 meV, the splitting between VP and IVC is weak. Thissplitting becomes more pronounced at u0 = 60 meV, indicating that IVC is more favored at low temperatures in comparisonto VP, although the competition between these two symmetry-breaking channels remains. (c) and (d) single-particle spectraat T = 0.067 meV, u0 = 60 meV and L = 9, which shows an insulating gap ∼ 10 meV. The dashed lines are the analyticcomputation of the single-particle dispersion at the flat-band limit following Ref. [30]. (e) and (f) dynamical spectra of VP andIVC with the same parameters. Sharp and ferromagnetic-like valley waves are observed in both channels near q = Γ. At theenergy scale of twice the single-particle gap, ∼ 20 meV, valley waves are over-damped into the particle-hole continuum.

and (b), where we measure S(q = Γ, τ = 0), the squaresof the order parameter, for IVC and VP as a functionof temperature. Without the kinetic energy (H = Hint),IVC and VP share identical susceptibility, which reflectsthe SU(2) symmetry of the flat-band limit. Once thekinetic energy is included (”with kin” in the Fig. 2 (a)and (b)), this degeneracy is lifted. At u0 = 33 meV, asmall splitting between IVC and VP correlation functionsis observed. The splitting becomes more significant whenu0 reaches 60 meV, closer to the realistic case [62, 63],with IVC being the more favored ground state. It isworthwhile to note that from 6 × 6 to 9 × 9, the IVCS(q = Γ) does not change with system sizes, whereas theVP S(q = Γ) decreases as the system size increases. Oneshall also notice that although the degeneracy betweenIVC and VP is lifted, both correlation functions grows atlow T , indicating that the competition between IVC andVP remains strong and there is no a completely dominantsymmetry-breaking channel [36].

In addition to static correlations, we also compute thedynamic correlations of IVC and VP as defined in Eq. (4)and their spectra with the system size of 9 × 9 for therealistic case with kinetic energy at u0 = 60 meV atlow temperature T = 0.067 meV, much lower than thescale of the single-particle gap. The results are shownin Fig. 2 (e) and (f), with Fig. 2 (c) and (d) the associ-

ated single-particle spectra. The dashed lines mark theexact single-particle dispersion when the kinetic energyis ignored [30]. Measured from ω = 0, the single-particlegap is of size ∼ 10 meV and both the VP and IVC spec-tra develop a clear and sharp valley wave dispersion atlow-energy near Γ. Remarkably, although the static sus-ceptibility indicates that the SU(2) symmetry has beenexplicitly broken at u0 = 60 meV and the degeneracybetween IVC and VP is lifted [Fig. 2 (b)], the IVC andVP spectra are almost identical and are strikingly similarto the flat-band limit [30, 64]. These sharp Goldstone-like modes are in strong analog to SU(2) ferromagneticGoldstone modes with ω ∝ q2, indicating an approximateSU(2) symmetry survives in our model.

One other interesting feature of these valley waves isthat above the energy scale of ∼ 20 meV, the sharp collec-tive excitations become heavily damped. In general, thedamping of collective modes has two origins (1) scatter-ing between collective modes and (2) damping due to thefermion particle-hole continuum. The second dampingchannel arises for energy larger than twice of the fermiongap, and thus is responsible for the over-damped featuresat energy above 20 meV shown in Fig. 2 (e) and (f). Thisis in strong analogy to the damping of ferromagnetic spinexcitations in the graphene nanoribbons, where the flatband gives rise to the ferromagnetic long-range order but

5

the spin waves becomes over-damped in the particle-holecontinuum [64–66].

Discussion and outlook — Quantum dynamics of col-lective excitations holds the key to the understanding ofmany-body effects in twisted bilayer graphene and otherquantum moire systems. This study suggests that themomentum-space QMC method offers a powerful tool totackle this problem. In particular, the spectral functionobtained via this unbiased method offers a bridge wayto directly connect theoretical studies with experimentalmeasurements, especially spectroscopy methods, such asinelastic light- or neutron- scattering and tunneling spec-troscopy, making it possible to compare measurements inexperiments and large-scale quantum simulations at thequantitative level.

Acknowledgments — We are indebted to Yi Zhang forthe help in the form factor tables. We thank TianyuQiao, Jian Kang, Jianpeng Liu and Xi Dai for stimulat-ing discussions. G.P.P., X.Z. and Z.Y.M. acknowledgesupport from the RGC of Hong Kong SAR of China(Grant Nos. 17303019, 17301420 and AoE/P-701/20)and the Strategic Priority Research Program of the Chi-nese Academy of Sciences (Grant No. XDB33000000).H.L. and K.S. acknowledge support through NSF GrantNo. NSF-EFMA-1741618. We thank the ComputationalInitiative at the Faculty of Science and the Informa-tion Technology Services at the University of Hong Kongand the Tianhe platforms at the National SupercomputerCenter in Guangzhou for their technical support and gen-erous allocation of CPU time.

∗ sunkai@umich.edu† zymeng@hku.hk

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8

SUPPLEMENTAL MATERIAL FOR

DYNAMIC PROPERTIES OF COLLECTIVE EXCITATIONS IN TWISTED BILAYER GRAPHENE

Section I: Momentum space QMC methodology

Following the description in Ref. [41], in this section, we elucidate the momentum space quantum Monte Carlomethod in detail.

First, the partition function of the TBG Hamiltonian in Eq. (3) of the main text is given by:

Z = Tr[e−βH

]= Tr

[(e−∆τH)Lτ ]

= Tr[Lτ∏τ=1

e−∆τH0e−∆τHint

]+O(∆τ2)

(5)

For the interaction part Hint = 12Ω∑

q,G,|q+G|6=0 V (q + G)δρq+Gδρ−q−G, we have

∑q,G,|q+G|6=0

12ΩV (q + G)δρq+Gδρ−q−G =

∑|q+G|6=0

V (q + G)4Ω

[(δρ−q−G + δρq+G)2 − (δρ−q−G − δρq+G)2

](6)

then

e−∆τHint =∏

|q+G|6=0

e−∆V (q+G)4Ω [(δρ−q−G+δρq+G)2−(δρ−q−G−δρq+G)2]. (7)

The discrete Hubbard-Stratonovich transformation [24, 41, 58, 59] reads:

eαO2

= 14

∑l=±1,±2

γ(l)e√αη(l)o +O

(α4) (8)

where l = ±1,±2, and

γ(±1) = 1 +√

6/3, γ(±2) = 1−√

6/3

η(±1) = ±√

2(3−√

6), η(±2) = ±√

2(3 +√

6)(9)

This can be seen from the following simple derivation. Assuming,

γ(1) = γ(−1) = a, γ(2) = γ(−2) = b, η(1) =√c = −η(1), η(2) =

√d = −η(2) (10)

Taylor expands both sides of Eq. (8) to O(α4) and compare the coefficients, we obtain:

1 = 12(a+ b), 1 = 1

4(ac+ bd), 12 = 1

48(ac2 + bd2) , 1

6 = 11440

(ac3 + bd3) (11)

solve these equations, then we have:

a = 1 +√

6/3, b = 1−√

6/3c = 2(3−

√6), d = 2(3 +

√6)

(12)

as those in Eq. (8).For a fermion bilinear, i.e. free fermion system, its partition function can be expressed as a determinant,

Tr[e−∑

i,jc†iAi,jccj−

∑i,jc†iBi,jcj

]= Det

(1 + e−Ae−B) . (13)

9

Put Eqs. (8) and (13) together, the partition function of our interacting TBG system can be expressed as:

Z =∑

l|q+G|,a,τ=±1,±2

Lτ∏τ=1

e−∆τH0 Trc

∏|q+G|6=0

116γ

(l|q+G|,1,τ

)γ(l|q+G|,2,τ

)eiη(l|q|1,t)Aq(δρ−q+δρq)eη(l|q|2,t)Aq(δρ−q−δρq)

+O(∆τ2)

(14)where Aq+G =

√∆τ4V (q+G)

Ω , as shown in the main text. The free of sign-problem and the Monte Carlo samplingscheme are presented in Ref. [41].

Section II: Order Parameter

As discussed in the main text. For the correlation functions of VP order parameter, we define

SV P (q) ≡ 1N2 〈Oa(−q)Oa(q)〉

Oa(q) ≡∑k

d†k+qτzη0dk(15)

where η0 is for band index and τz is for valley index. Then its QMC implementation reads as,

SV P (q) = 1N2

∑k1,k2

∑n1,n2

∑τ1,τ2=±

(τ1τ2)⟨d†k1,n1,τ1

dk1+q,m1,τ1d†k2+q,n2,τ2

dk2,m2,τ2

⟩

= 1N2 〈

(∑k1

(∑n1

d†k1,n1,τdk1+q,n1,τ − dk1,n1−τ d

†k1+q,n1,−τ

))

·

(∑k2

(∑n2

d†k2+q,n2,τdk2,n2,τ − dk2+q,n2−τ d

†k2,n2−τ

))⟩

= 1N2

∑k1,k2

∑n1,n2

Gcn1n1,τ (k1, k1 + q)Gcn2m2,τ (k2 + q, k2)

+ G∗n1n1,τ (k1, k1 + q)G∗n2n2,τ (k2 + q, k2)+ Gcn1n2,τ (k1, k2)Gn1n2,τ (k1 + q, k2 + q)+ G∗n1n2,τ (k1, k2)Gc∗n1n2,τ (k1 + q, k2 + q)−Gcn1n1,τ (k1, k1 + q)G∗n2n2,τ (k2 + q, k2)−G∗n1n1,τ (k1, k1 + q)Gcn2n2,τ (k2 + q, k2)

(16)

where dk,m,−τ = m ∗ d†k,−m,−τ and d†k1,m,−τdk2,n,−τ = (mn)dk1,−m,−τ d†k2,−n,−τ = (mn)G∗−m,−n(k1, k2), note we

define the fermion Green’s function as Gij = 〈d†idj〉 and define Gcij = δij −Gji.For the correlation function of the IVC order parameter, we define

SIV C(q) ≡ 1N2 〈Oa(−q)Oa(q)〉

Oa(q) ≡∑k

d†k+qτxηydk(17)

10

and its QMC implementation reads as,

SIV C(q) = 1N2

∑k1,k2

∑n1,n2,

∑τ1,τ2=±

(n1n2)⟨d†k1,n1,τ1

dk1+q,−n1,−τ1d†k2+q,n2,τ2

dk2,−n2,−τ2

⟩= 1N2

∑k1,k2

∑n1,n2,

∑τ=±

(n1n2) Gcn1,−n2,τ (k1, k2)G−n1,n2,−τ (k1 + q, k2 + q)

= 1N2

∑k1,k2

∑n1,n2,

Gcn1,−n2,τ (k1, k2)Gc∗n1,−n2,τ (k1 + q, k2 + q)

+ G∗n1,−n2,τ (k1, k2)Gn1,−n2,τ (k1 + q, k2 + q)

(18)

Section III: Analytic continuation

From QMC simulations, we only obtain the imaginary time or imaginary frequency Green’s functions, we furtherperform the stochastic analytic continuation (SAC) method [41, 43–49, 67–71] to obtain the real frequency spectralfunction A(k, ω).

Here we give a brief description of the scheme.Firstly, we define : e−βΩ = Tr

(e−β(H−µN)) ,K ≡ H − µN . The imaginary time Green’s function is:

G(τ) =⟨Tτd (τ) d† (0)

⟩= Tr

[e−β(K−Ω)Tτe

τKd e−τKd†] (19)

where K|m〉 = Em|m〉. Then if we consider the Lehmann representation:

τ > 0 :G(τ) = eβΩ∑n,m

⟨n∣∣e−βKd(τ)

∣∣m⟩ ⟨m ∣∣d†(0)∣∣n⟩

G(τ) = eβΩ∑n,m

|〈n|d|m〉|2e−βEneτ(En−Em)(20)

Once again, imaginary frequency Green’s function is :

G (iωn) =∫ β

0dτeiωnτG(τ)

= −eβΩ∑n,m

|〈n|d|m〉|2e−βEn e(iωn+En−Em)τ |β0iωn + En − Em

= eβΩ∑n,m

|〈n|d|m〉|2 e−βEn ∓ e−βEmiωn + En − Em

(21)

here ∓ for boson and fermion. And we use eiωnβ = ±1.Then we carry out the analytic continuation: iωn → w+ iδ and obtain the retarded real frequency Green’s function

G (iωn) → Gret(ω), where Gret (ω) =∫∞−∞ eiωtGret (t) dt and Gret (t− t′) = −iθ (t− t′)

⟨[d(t)d† (t′) + d† (t′) d(t)

]⟩.

The spectral function is obtained by the retarded Green function: A(k, ω) = −(1/π) ImGret(k, ω)

A(k, ω) = −(1/π) ImGret(k, ω)

= eβΩ∑n,m

|〈n|d|m〉|2(e−βEn ∓ e−βEm

)δ(ω + En − Em) (22)

from Eqs. (21) and (22), we can get :

G(k, τ) =∫ ∞−∞

dω

[e−ωτ

1∓ e−βω

]A(k, ω) (23)

Note again ∓ for boson and fermion.

11

For boson Green function:

G(k, τ) =∫ +∞

0dωe

−τω + e−(β−τ)ω

1− e−βω A(k, ω). (24)

In the spectroscopy measurements such as the inelastic neutron scattering, the spectral function S(k, ω) =1

1−e−βω Imχ(k, ω), where χ(k, ω) is dynamical spin susceptibility. We can see Imχ(k, ω) is the spectral functionA(k, ω) mentioned above.

Now we discuss the details of stochastic analytic continuation. The idea is to give a very generic variational ansatzof the spectrum A(k, ω), and obtain corresponding Green’s function G(k, τ) following Eq. (24) . Then compare theGreen’s function with the Green’s function obtained from QMC by the quantity χ2

F/B . Definition of χ2F/B is

χ2F =

∑ij

(G (τi)−

∫ ∞−∞

dω

[e−ωτi

1 + e−βω

]A(ω)

)(C−1)

ij

(G (τj)−

∫ ∞−∞

dω

[e−ωτj

1 + e−βω

]A(ω)

)(25)

and

χ2B =

∑ij

(G (τi)−

∫ ∞0

dω

[e−ωτi + e−(β−τ)ω

1− e−βω

]A(ω)

)(C−1)

ij

(G (τj)−

∫ ∞0

dω

[e−ωτj + e−(β−τ)ω

1− e−βω

]A(ω)

)(26)

where

Cij = 1Nb (Nb − 1)

Nb∑b=1

(Gb (τi)− G (τi)

) (Gb (τj)− G (τj)

)(27)

and G (τi) is the Monte Calro average of Green’s functions of Nb bins.Then we perform the Monte Carlo sampling [43, 69] again to optimize the spectral function. We assume that the

spectral function has the following form: A(ω) =∑Nωi=1Aiδ (ω − ωi) and the weight of such Monte Carlo configuration

is: W ∼ exp(− χ2

2 ΘT

). Here ΘT is an analogy to temperature. Then we compute the average 〈χ2〉 at different ΘT ,

via the simulated annealing process, at the end of it, we can choose the converged ΘT to satisfy:

〈χ2〉 = χ2min + a

√χ2

min. (28)

Usually we set a = 2, and the ensemble average of the spectra at such optimized Θ is the final one to present in themain text.

We note that the QMC-SAC scheme for obtaining dynamical spetral function, is developed over the past decadesand has been verified in many works on quantum many-body systems and have been directly compared with theBethe ansatz, exact diagonalization, field theoretical analysis and spectropy experiments, such as the works on 1DHeisenberg chain [43], 2D Heisenberg model compared with neutron scattering and field theoretical analysis [44, 47],Z2 quantum spin liquid model with fractionalized spectra [45, 48], quantum Ising model with direct comparison withneutron scattering and NMR experiments [49, 70], the non-Fermi-liquid and metallic quantum critical point [71, 72]and the TBG system at flat-band limit [41].

Section IV: Exact Analytic Charge ±1 Excitations

Here we follow the the Ref. [30]. For ν = 0, ground state is:

Oq+G|Ψ〉 = 0 (29)

then: [Hint, d

†k,n,η,ss

]|Ψ〉 = 1

2Ωtot

∑m2

Rηm2n(k)d†k,m2,η,ss|Ψ〉 (30)

where

Rηm1n1(k) =

∑m,q,G,|q+G|6=0

V (q + G)λ∗m1,m,η(k,k + q + G)λn1,m,η(k,k + q + G) (31)

12

Diagonalize Rηm1n1 (k)2Ω and we obtain the charge ±1 excitations, as plotted as the dashed lines in the Fig. 2 (c) and

(d) of the main text with our model parameters.