Chiral superconductivity in the doped triangular-lattice Fermi-Hubbard model in two dimensions

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Chiral superconductivity in the doped triangular-lattice

Fermi-Hubbard model in two dimensions

Vinicius Zampronio1,2 and Tommaso Macrì3,2

1Institute for Theoretical Physics, Utrecht University, 3584CS Utrecht, Netherlands

2Departamento de Física Teórica e Experimental, Federal University of Rio Grande do Norte 59078-950 Natal-RN, Brazil

3ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA

The triangular-lattice Fermi-Hubbard

model has been extensively investigated in

the literature due to its connection to chi-

ral spin states and unconventional super-

conductivity. Previous simulations of the

ground state of the doped system rely on

quasi-one-dimensional lattices where true

long-range order is forbidden. Here we

simulate two-dimensional and quasi-one-

dimensional triangular lattices using state-

of-the-art Auxiliary-Field Quantum Monte

Carlo. Upon doping a non-magnetic chi-

ral spin state, we observe evidence of chi-

ral superconductivity supported by long-

range order in Cooper-pair correlation and

a ﬁnite value of the chiral order parameter.

With this aim, we ﬁrst locate the transi-

tion from the metallic to the non-magnetic

insulating phase and the onset of magnetic

order. Our results pave the way towards

a better understanding of strongly corre-

lated lattice systems with magnetic frus-

tration.

As a paradigmatic model for strongly cor-

related fermionic lattice systems, the Fermi-

Hubbard (FH) Hamiltonian still has many open

questions [1]. In two dimensions (2D), the FH

model captures the rich physics of metal to insu-

lator phase transitions (MIT) [2], itinerant mag-

netism [3] and spin liquids [4,5,6]. Quantum spin

liquids with non-abelian anyon excitations can

act as building blocks for topological quantum

computation and for the construction of fault-

tolerant quantum computers [7]. The idea be-

hind this non-magnetic insulator was proposed

in the seminal work by Anderson that describes

a resonating valence-bond state arising from geo-

metric frustration on the lattice [8,9]. The sim-

plest lattice structure containing this kind of frus-

tration is the triangular one, which is relevant

for the understanding of molecular materials of

the κ-ET family [10,11,12,13,14]. Besides

geometrical frustration, charge ﬂuctuations play

a role in the stabilization of quantum spin liq-

uids [15,16]. The simulation of charge ﬂuctua-

tions can be accomplished by introducing high-

order ring-exchange coupling to the eﬀective spin

Hamiltonian [17,18,19,20] or by considering the

Hubbard model itself. Additionally, the observa-

tion of Hubbard-model physics in triangular lat-

tices engineered in WeSe2/WeS2moiré superlat-

tices [21,22] and in quantum simulators [23,24]

has inspired scientiﬁc interest in such systems. In

this work, we focus on the triangular-lattice FH

model described by the Hamiltonian

H=−tX

⟨ij⟩,α c†

iαcjα +H.c.+UX

ni↑ni↓,(1)

where α=↑,↓is the electron spin, ⟨ij⟩indi-

cates a sum over nearest-neighbour sites, ciα and

c†

iα respectively annihilates and creates an elec-

tron with spin αat the i-th lattice site, niα =

c†

iαciα is the number operator, tis the hopping

strength, and Uis the intensity of the onsite in-

teraction. The non-interacting system is metallic

and, at half-ﬁlling (⟨niα⟩= 1/2), the strongly

interacting FH Hamiltonian can be mapped into

the antiferrognetic (AFM) Heisenberg one, whose

ground state contains 120◦long-range spin or-

der [25]. Away from these two cases (U/t = 0

and U/t ≫1), the precise nature of the system

is still under debate. For weak interactions, nu-

merical simulations assume an adiabatic connec-

tion with the non-interacting regime. Nonethe-

less, they might be overlooking a transition to

a phase with a small but non-vanishing gap [1].

In fact, renormalization-group calculations pre-

dict a d+id superconductor at U/t ≪1in

2D [26,27] and weak coupling analyses argue

that, at weak interactions, the quasi-1D system

Accepted in Quantum 2023-07-11, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.13551v2 [cond-mat.str-el] 12 Jul 2023

is a Luther-Emery liquid with time-reversal sym-

metry breaking [28]. For intermediate interac-

tions, the quest for spin liquids has been a sub-

ject of signiﬁcant interest. Still, there is not even

theoretical agreement on whether the spin liq-

uid state exists in the FH model. While calcu-

lations ranging from variational cluster approxi-

mation [29,30,31], path integral renormalization

group [32,33], strong coupling expansion [19],

dual fermion approach [34] and exact diagonaliza-

tion [35] to density matrix renormalization group

(DMRG) [36,37,38,39] and variational Monte

Carlo (VMC) [40,41] agreed in the existence of

a spin liquid state, dynamical cluster approxi-

mation studies [42] and earlier VMC computa-

tions [43,44] detected a direct transition from

a metallic state to a magnetic ordered phase.

Among the theories that support the existence

of a spin liquid, its nature remains controversial.

Inﬁnite-DMRG calculations predict a gapped chi-

ral spin liquid (CSL) [37,38], while VMC simu-

lations on full 2D systems and ﬁnite-DMRG [36]

support a gapless spin liquid that preserves time-

reversal symmetry. Another ﬁnite-DMRG study

also supports the gapped CSL [39]. On the

other hand, a multi-method approach ﬁnds that,

at intermediate interactions, there is a competi-

tion between chiral and two distinct magnetic or-

ders: collinear and 120◦order [45]. DMRG simu-

lations of the extended AFM-Heisenberg model

with four-spin interactions that arise naturally

from Mott-insulator physics corroborate the ex-

istence of a CSL in lattice geometries closer to

2D than the ones used in DMRG simulations of

the Hubbard model [20]. For the hole-doped sys-

tem, the quest for unconventional superconduc-

tivity in the Hubbard model is a matter of cur-

rent scientiﬁc interest due to its connection to

High-Tcsuperconductors [46,47,48,49]. A re-

cent DMRG study of the doped triangular FH

predicts a rich phase diagram with fractionalized

excitations, spin and charge deconﬁnement and

enhanced Cooper-pair correlations [50]. Another

DMRG study estimates the spectral function of

one single hole doped in the triangular-lattice

CSL and observes spinon dynamics [51]. Also,

DMRG simulations of the extended t−Jmodel

with three-spin chiral interactions in the triangu-

lar lattice predicted chiral superconductivity in

the system, evidenced by quasi-long-range-order

in the Cooper-pair correlations upon doping [52].

Shortly thereafter, emergent topological super-

conductivity was also reported in the simpler

t−Jmodel [53]. However, DMRG performed

in quasi-1D lattices does not display true long-

range order, and one has to rely on a slow decay of

Cooper-pair correlations. Numerical simulations

via the Linked-Cluster Expansion algorithm pro-

vide several benchmarks to the ﬁnite temperature

triangular FH at intermediate to strong interac-

tions [54], but a clear description of the weakly

interacting regime, the classiﬁcation of the spin

liquid state, and whether or not a superconduct-

ing phase would appear upon doping is still elu-

sive.

In this work, we numerically investigate the

ground state of the doped triangular FH in 2D.

Upon doping a non-magnetic chiral spin state

(CSS) we observe true long-range order in the

Cooper-pair correlations while the chiral order

parameter remains ﬁnite, i.e. a chiral supercon-

ductor. To simulate the CSS, we ﬁrst locate the

MIT and the transition to the AFM phase.

1 Methods

We report the implementation of state-of-

the-art Auxiliary-Field Quantum Monte Carlo

(AFQMC) to simulate the ground state of the full

2D triangular lattice FH model. By imaginary-

time projection to the ground state we intend

to reduce the bias from VMC calculations. A

constrained-path (CP) approximation is required

to restore polynomial convergence (otherwise

plagued by the sign problem [55]) and the bias

from the variational ansatz is not completely re-

moved. However, simulations made in the past

for square lattices away from half-ﬁlling have been

shown to be accurate and provided several bench-

marks [56,57,58,59,60]. See Appendix Afor fur-

ther details of the method and for a comparison

of the CP-AFQMC estimates of the triangular-

lattice ground-state energy with exact diagonal-

ization.

For the non-magnetic phases, we consider

the generalized Hartree-Fock ansatz (GHF) for

the imaginary-time projection. The mean-ﬁeld

Hamiltonian associated to the GHF state is ob-

tained considering a partial particle-hole trans-

formation on a BCS Hamiltonian,

HMF =−tX

⟨ij⟩α

c†

iαcjα +X

Mic†

i↑ci↓+H.c.,(2)

Accepted in Quantum 2023-07-11, click title to verify. Published under CC-BY 4.0. 2

where Mi=Ueﬀ⟨c†

i↑ci↓⟩. The GHF ground state

is obtained via a self-consistent diagonalization of

Eq. (2)[60]. As done with Unrestricted Hartree

Fock (UHF) wave functions [59], we consider

Ueﬀ = min(U, Umax

eﬀ ). We noticed that Umax

eﬀ /t =

4gives meaningful estimates of the non-magnetic

states of the system. Our CP-AFQMC simula-

tions with the GHF ansatz became unstable for

strong interactions, see Appendix Afor more ex-

planation concerning the instability. With this

ansatz, we were able to locate the MIT but not

the transition to the magnetic phase. To see

the transition to the 120◦AFM phase, we also

perform simulations starting from a full Hartree-

Fock (FHF) ansatz. The ansatz that provides

the smaller energy after imaginary-time evolution

is our best representative of the system ground

state, see Appendix Bfor details.

In our simulations, we mainly consider lattices

with Nx=Ny= 12 sites along the ˆex= (a, 0)

and ˆey= (a/2,√3a/2) directions respectively

(see FIG. 1for the lattice vectors and a visual

representation of the triangular FH model). Pe-

riodic boundary conditions are considered along

the horizontal and vertical directions. We also

run simulations with diﬀerent lattice sizes to anal-

yse the ﬁnite-size eﬀects. See the Appendix Cfor

the dependence of total energy and spin corre-

lations on the lattice size. Finally, to address

the eﬀect of the dimensionality, we investigate

quasi-1D lattices with Nx= 36,Ny= 3 and

4, PBC along ˆeyand open boundary condition

along ˆex. From now on, if not speciﬁed oth-

erwise, we are considering the 12 ×12 lattice.

We study spin-balanced systems at half ﬁlling

(n=N/M = 1, where Nis the number of elec-

trons and M=NxNy) and with hole doping

(n < 1).

2 Results at half ﬁlling

We start by deﬁning the charge structure factor

N(k) = 1

i,j

eik·rij (⟨ninj⟩−⟨ni⟩⟨nj⟩),(3)

with ni=ni↑+ni↓, around the origin k= 0

to determine whether the system is gapped or

not. Since the charge gap ∆cis proportional to

limk→0k2/N(k)[61,62], a linear behavior around

k= 0 indicates that the system is metallic (gap-

less) and a quadratic behavior indicates that the

Figure 1: Triangular FH model. We display the lat-

tice vectors ˆex= (a, 0) and ˆey= (a/2,√3a/2). The cir-

cle depicts the formation of a doublon (doubly-occupied

site) with energy cost U. We also represent a hopping

process with energy scale −t. Ellipses depict Cooper

pairs in the singlet (|s⟩= (| ↑↓⟩ − | ↓↑⟩)/√2) or triplet

(|t⟩= (| ↑↓⟩ +| ↓↑⟩)/√2) states formed on the ˆeyand

ˆexbounds of the triangular plaquettes.

system is insulating (gapped). More precisely, for

small kwe have N(k) = ak2+bk +O(k3), and

whenever b̸= 0,∆c= 0. We compute the coef-

ﬁcients aand bby performing a quadratic ﬁt to

our data. We considered the three allowed mo-

menta closer to the origin along k= (kx= 0, ky),

results are shown in FIG. 2where we located

the MIT around 7< Uc1/t ≤8. Our critical

interaction is in agreement with DMRG simula-

tions [36,37] and a multi-method study [45]. The

Linked-Cluster-Expansion calculations, after an

extrapolation to zero-temperature regime, pre-

dict a critical interaction Uc1/t ∼7[54], while

other DMRG simulations predict Uc1/t ∼9[39].

Figure 2: Metal-insulator transition. The coeﬃ-

cients of the expansion N(k)≈ak2+bk as a function

of the interaction U/t. A nonzero value of bimplies that

∆c= 0.

To further corroborate our ﬁndings, we com-

Accepted in Quantum 2023-07-11, click title to verify. Published under CC-BY 4.0. 3

pute the doublon density, d=Pi⟨ni↑ni↓⟩/M, for

which we expect diﬀerent behaviors in the metal-

lic and insulating phases [63,36]. In the metallic

phase the Brinkman-Rice picture predicts that d

decreases linearly with U[64], while for strong in-

teractions the doublon density shows Heisenberg

behavior [65], d= (2t2/U2Pδ(1/4− ⟨Si·Si+δ⟩),

where the sum runs over the nearest neighbours.

In Fig. 3we display results for das a function

of U/t. Our data for the doublon density show a

deviation from the linear behavior near the MIT.

Figure 3: Doublon density as function of the inter-

action strength U/t. The blue crosses represent our

data. The dotted line is a linear ﬁt of the doublon-

density data for U/t < 7. The shaded area delimits

the region where Uc1is located. The dashed curve is

the function d= (2t2/U2Pδ(1/4− ⟨Si·Si+δ⟩)with

⟨Si·Si+δ⟩=−0.1837(7), see Ref. [66]. Error bars are

smaller than the markers size.

Analogously, the presence of a spin gap can be

accessed by the spin structure factor

S(k) = 1

i,j

eik·rij ⟨Sz

iSz

j⟩,(4)

with Sz

i=ni↑−ni↓. We do not see the emergence

of quadratic behavior in S(k)(Fig. 4), which in-

dicates the absence of a spin gap. The excita-

tions of the 120◦Heisenberg antiferromagnet are

gapless magnons [67]. Therefore the presence of

a spin gap is not a good measure to locate the

AFM order in our system. On the other hand, the

presence of peaks in S(k)is a signature of long-

range magnetic order; for the 120◦AFM those

peaks appear on the Kpoints of the Brillouin

zone. In our simulations, we see the formation of

peaks on the Kpoints which indicates the tran-

sition to the 120 ◦phase with critical interaction

10 < Uc2/t ≤11 (see Appendix D). Our estimate

for the critical interaction is in agreement with

recent DMRG simulations [37], which locates the

transition at U/t = 10.6.

Figure 4: Spin structure factor S(k)as a function

of U/t for k= (kx= 0, ky). The linear behavior around

ky= 0 indicates gapless spin excitation.

We investigate the chiral order parameter

χ=



X

△⟨Si·(Sj×Sk)⟩



,(5)

where the sum is over every triangular plaquette

of the lattice with vertexes i,jand ktaken in the

clockwise direction and Sℓ=Pα,β=↑,↓c†

ℓασαβ cℓβ,

with σ= (σx, σy, σz)a vector of Pauli matrices.

Our results for χare show in Fig. 5along-

side the Smax of S(k). We observe a competi-

tion between chiral and magnetic orders as the

interaction increases as reported for quasi-1D lat-

tices [45]. We see a sharp increase in Smax in the

transition to the AFM phase.

To analyse the eﬀect of dimension in our re-

sults, we compute S(k)and χfor the quasi-1D

36 ×3lattice. We see that the results in 2D and

quasi-1D agree reasonably well, but in quasi-1D

we see the AFM insulator with the peaks of the

spin structure factor at the Mpoints, which is a

signature of a collinear AFM phase. We also anal-

yse the eﬀect of the width of the quasi-1D systems

on the results by the simulation of a 36 ×4lat-

tice, where we see competition between 120◦and

Accepted in Quantum 2023-07-11, click title to verify. Published under CC-BY 4.0. 4

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Chiralsuperconductivityinthedopedtriangular-latticeFermi-HubbardmodelintwodimensionsViniciusZampronio1,2andTommasoMacrì3,21InstituteforTheoreticalPhysics,UtrechtUniversity,3584CSUtrecht,Netherlands2DepartamentodeFísicaTeóricaeExperimental,FederalUniversityofRioGrandedoNorte59078-950Natal-RN,Brazil3I...

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Chiral superconductivity in the doped triangular-lattice Fermi-Hubbard model in two dimensions

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