Ferromagnetic domains in the large- UHubbard model with a few holes an FCIQMC study Sujun Yun

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Ferromagnetic domains in the large-UHubbard model with a few holes: an FCIQMC

study

Sujun Yun∗

Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany and

School of electronic engineering, Nanjing XiaoZhuang University, Hongjing Road, Nanjing 211171, China

Werner Dobrautz†

Department of Chemistry and Chemical Engineering,

Chalmers University of Technology, 41296 Gothenburg, Sweden

Hongjun Luo,‡Vamshi Katukuri, and Niklas Liebermann

Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany

Ali Alavi§

Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany and

Yusuf Hamied Department of Chemistry, University of Cambridge,

Lensﬁeld Road, Cambridge CB2 1EW, United Kingdom

Two-dimensional Hubbard lattices with two or three holes are investigated as a function of U

in the large-Ulimit. In the so-called Nagaoka limit (one-hole system at inﬁnite U), it is known

that the Hubbard model exhibits a ferromagnetic ground-state. Here, by means of exact FCIQMC

simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions

as a function of increasing U. The correlation functions clearly demonstrate the onset of ferromag-

netic domains, centred on individual holes. The overall total spin of the wavefunctions remain the

lowest possible (0, or 1

2, depending on the number of holes). The ferromagnetic domains appear at

interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in

ﬁnite systems with strictly one hole. The existence of such ferromagnetic domains is the signature

of Nagaoka physics in Hubbard systems with a small (but greater than one) number of holes.

I. INTRODUCTION

The Hubbard model is a simple yet important model in

the study of correlated electrons, as it captures complex

correlations between electrons on a lattice with a fairly

simple Hamiltonian [1]. Exact results of two-dimensional

Hubbard model are helpful for understanding a plethora

of phenomena in strongly correlated systems, includ-

ing pairing mechanisms in unconventional superconduc-

tors [2], the Mott metal-insulator transition [3], optical

conductivity [4, 5], and itinerant magnetism [6, 7]. For

the single-band 2D Hubbard model on a square lattice,

Nagaoka [8] proved analytically that the limit of inﬁnitely

strong interactions, in the presence of a single hole on top

of a Mott-insulating state with one electron per site, re-

sults in a ferromagnetic ground state. Intuitively, the

Nagaoka ferromagnetism can be understood as resulting

from an interference eﬀect between the diﬀerent paths

that the hole can take across the lattice. When the spins

are aligned, these paths interfere constructively and lower

the kinetic energy of the hole [9–12].

While Nagaoka ferromagnetism has been analytically

proven under extreme conditions, and has also been ob-

∗yunsujun@163.com

†dobrautz@chalmers.se

‡h.luo@fkf.mpg.de

§a.alavi@fkf.mpg.de

served in a quantum dot plaquette [13], the stability

of the Nagaoka ferromagnetic state at ﬁnite interaction

strengths on ﬁnite lattices has also been actively studied

[14–21]. However, open questions still exist, especially

concerning the thermodynamic stability of the ferromag-

netic state for systems with more than one hole. Extrap-

olations from the results on ﬁnite lattices have been used

to study properties in the thermodynamic limit. Thus

it is important to obtain exact results in systems with

two-, three-, and perhaps more-hole systems, on lattices

as large as possible.

In the two-hole system, the total spin of the ground-

state is zero (S= 0). This has been numerically

studied by exact diagonalization (ED) [22], the spin-

adapted full conﬁguration interaction quantum Monte

Carlo (FCIQMC) method [21] and analytical studies of

arbitrarily large systems [19]. However, the speciﬁc type

of magnetism is unknown, because anti-ferromagnetism,

para-magnetism, as well as low-spin-coupled ferromag-

netic domains, all correspond to S= 0. For the three-

hole system, on the other hand, ED results of an eﬀec-

tive Hamiltonian [22] show that the total spin of ground-

states on the 8- and 16-site lattices are 3

2and 7

2respec-

tively. Here the eﬀective Hamiltonian is constructed to

exclude double occupation and thus should be related to

the Hubbard model in the large Ulimit. It is interest-

ing to see whether the partial magnetization will still be

observed on larger lattices. In order to answer the above

open questions, in this work we investigate the two- and

arXiv:2210.12049v1 [cond-mat.str-el] 21 Oct 2022

three-hole systems with the full conﬁguration interaction

quantum Monte Carlo (FCIQMC) method.

FCIQMC is based on stochastic simulations of the

dynamic evolution of the wave function in imaginary

time. Diﬀerent to the diﬀusion quantum Monte Carlo

(DMC) [23] with the ﬁxed node approximation and the

auxiliary-ﬁeld quantum Monte Carlo (AFQMC) [24] with

the phaseless approximation, no systematic approxima-

tion is made in FCIQMC [25–27], and it thus serves as

a highly accurate method to approach the ground state

wave functions. The annihilation procedure of the algo-

rithm enables it to overcome the fermionic sign problem

exactly, as long as enough walkers are used. In practice,

for Hubbard systems at large U, this means, with the cur-

rently available hardware, systems up to 26 sites can be

studied [21]. (A 26-site lattice represents a useful increase

in size compared to exact-diagonalisation, for which 20-

sites is the largest lattice size so far reported [4]). In the

present paper, we extend this to the study of a system

with a few holes, as well as report on spin-spin correlation

functions for the obtained exact ground-states.

The software NECI, a state-of-the-art implementation

of the FCIQMC algorithm, utilizes a very powerful par-

allelization and scales eﬃciently to more than 24000 cen-

tral processing unit cores [27]. The FCIQMC method

in a Slater determinant (SD) basis has been extended to

calculate ground and excited state energies, spectral and

Green’s functions for ab initio and model systems, as well

as properties via the one-, two-, three- and four-body re-

duced density matrices(RDMs). To study magnetism, we

need to use the replica-sampled 2-RDMs [28–30] to ob-

tain the spatial spin distribution. The replica-sampling

technique removes the systematic error in the RDM, at

the expense of requiring a second walker distribution.

The premise is to ensure that these two walker distribu-

tions are entirely independent and propagated in parallel,

sampling the same (in this instance ground-state) distri-

bution. This ensures an unbiased sampling of the desired

RDM, by ensuring that each RDM contribution is derived

from the product of an uncorrelated amplitude from each

replica walker distribution. By using replica-sampled 2-

RDMs the spin-spin correlation function, hˆ

Si·ˆ

Sji, can

be calculated, where iand jare lattice site indices. This

spin-spin correlation function can then be used to iden-

tify the speciﬁc type of magnetism of the ground states.

FCIQMC in a spin-adapted basis is also used to study

the partial polarization in three-hole systems. Spin-

adapted FCIQMC uses SU(2) symmetry (arising from

the vanishing commutator [ ˆ

H, ˆ

S2] = 0) conservation.

SU (2) symmetry is imposed via the graphical unitary

group approach (GUGA) [31–33] which dynamically con-

strains the total spin Sof a multi-conﬁguration and

highly open-shell wave function in an eﬃcacious man-

ner. The spin-adapted version of the FCIQMC algorithm

based on GUGA has been developed in our group [34, 35],

with – among others – applications to ab initio sys-

tem [36, 37] and Nagaoka ferromagnetism in one-hole

system [21]. With the spin-adapted method, the mag-

netisation of the ground state can be determined in a

reliable way, especially for systems with small spin gaps.

The results of spin-adapted FCIQMC show the partial

spin-polarization only appears in small, three-hole sys-

tem (less than 18 sites) [22], which is the second impor-

tant result of this work.

The rest of this paper is organized as follows: In Sec.

II, we brieﬂy describe the methods, where we mainly

provide some more details on the measurements of the

spin-spin correlation function, hˆ

Si·ˆ

Sjifrom the replica-

sampled 2-RDMs in FCIQMC. In Sec. III, results about

the spatial spin distribution and partial spin-polarization

are discussed. Finally, we conclude in Sec. IV.

II. METHODS

The Hamiltonian of the Hubbard model in real space

takes the form

H=−tX

hijiσ

a†

i,σaj,σ +UX

ni↑ni↓(1)

where a†

iσ (aiσ) creates (annihilates) an electron with spin

σon site i, and niσ =a†

iσaiσ is the particle number oper-

ator. Urefers to the Coulomb interaction strength. We

consider only nearest neighbour hopping terms, where t

is positive and is used as the unit of the energy. When U

is inﬁnitely large, there will be no double occupancy and

the system can be treated with an eﬀective Hamiltonian

with constrained hopping terms [22]

Heﬀ =−tX

hijiσ

˜a†

i,σ ˜aj,σ,(2)

with ˜a†

i,σ =ai,σ(1 −ni,σ). In our current work, we want

to study the magnetic properties for ﬁnite Uand thus will

stay with the original Hamiltonian (1). Tough, we ﬁnd

that our results for the three-hole systems in the large U

limit (see Sec. III(B)), coincide with the result of Riera

et al. [22] for the eﬀective Hamiltonian, Eq. (2). In our

investigation we apply two diﬀerent FCIQMC methods,

which are based on full CI expansions in terms of SDs

and in terms of spin eigenfunctions (spin-adapted basis

states) respectively.

FCIQMC is a projector QMC method for obtaining

the ground state wave function |Ψ0i. By Monte Carlo

simulation of the imaginary-time evolution of the wave

function

|Ψ(τ)i=e−τ(ˆ

H−E0)|Ψ(0)i,(3)

the ground state wave function is approached in the long

time limit |Ψ(τ→ ∞)i∝|Ψ0i.

In a previous work [21], we have investigated the mag-

netism for one hole and two holes systems by using the

spin-adapted (SU (2) conserving) FCIQMC method. We

extend these investigations to three-hole systems in this

work. With the spin-adapted method, the magnetisation

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Ferromagneticdomainsinthelarge-UHubbardmodelwithafewholes:anFCIQMCstudySujunYunMaxPlanckInstituteforSolidStateResearch,Heisenbergstr.1,70569Stuttgart,GermanyandSchoolofelectronicengineering,NanjingXiaoZhuangUniversity,HongjingRoad,Nanjing211171,ChinaWernerDobrautzyDepartmentofChemistryandChemicalEn...

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Ferromagnetic domains in the large- UHubbard model with a few holes an FCIQMC study Sujun Yun

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