Hund bands in spectra of multiorbital systems M.Sroda1J. Mravlje2G. Alvarez3E. Dagotto4 5and J. Herbrych1 1Institute of Theoretical Physics Faculty of Fundamental Problems of Technology

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Hund bands in spectra of multiorbital systems

M. ´

Sroda,1J. Mravlje,2G. Alvarez,3E. Dagotto,4, 5 and J. Herbrych1

1Institute of Theoretical Physics, Faculty of Fundamental Problems of Technology,

Wrocław University of Science and Technology, 50-370 Wrocław, Poland

2Joˇ

zef Stefan Institute, SI-1000 Ljubljana, Slovenia

3Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

4Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA

5Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

(Dated: August 4, 2023)

Spectroscopy experiments are routinely used to characterize the behavior of strongly correlated systems. An

in-depth understanding of the different spectral features is thus essential. Here, we show that the spectrum of the

multiorbital Hubbard model exhibits unique Hund bands that occur at energies given only by the Hund coupling

JH, as distinct from the Hubbard satellites following the interaction U. We focus on experimentally relevant

single-particle and optical spectra that we calculate for a model related to iron chalcogenide ladders. The

calculations are performed via the density-matrix renormalization group and Lanczos methods. The generality

of the implications is veriﬁed by considering a generic multiorbital model within dynamical mean-ﬁeld theory.

Introduction. Strongly correlated systems are at the heart

of modern condensed matter physics. The celebrated single-

band Hubbard model, describing (doped) Mott insulators,

is still extensively studied in the context of Cu-based high-

temperature superconductivity [1–3]. Equally exciting case

is that of iron-based superconductors where the presence of

several active orbitals leads to novel effects beyond the “stan-

dard” Mott physics [4–6]. A nontrivial example is the orbital-

selective Mott phase (OSMP) [5,7–10], where Mott-localized

and itinerant electrons coexist.

A key probe of electronic excitations is the single-particle

spectral function A(k, ω), characterizing the excitations’ dis-

persion. It is experimentally accessible by angle-resolved

photoemission spectroscopy (ARPES) [11,12]. To under-

stand the origin of different spectral features, it is convenient

to consider idealized models that can be studied theoretically

and monitor how the signatures of correlations (e.g., the Hub-

bard bands) evolve with increasing Coulomb interaction U.

This is especially true for quantum systems of reduced dimen-

sionality, for which quasiexact numerical methods [13,14], or

even closed analytical solutions [15], provide unbiased infor-

mation on the elementary excitations. However, even in re-

duced dimensionality obtaining accurate results for the mul-

tiorbital Hubbard model remains challenging. The difﬁculty

lies in the exceptionally large Hilbert space. Because of that,

the spectral functions are often calculated using the dynami-

cal mean-ﬁeld theory (DMFT) [16–19]. This approach, that

strictly applies at large dimensionality, avoids the ﬁnite-size

limitation, but often relies on solvers in Matsubara frequen-

cies and hence the resulting spectral functions are blurred due

to analytical continuation (see Ref. [20] that discusses this and

introduces a method to alleviate the problem).

In this Letter, we numerically investigate the spectral func-

tions of several multiorbital models. Our main result is sum-

marized in Fig. 1(a). The electronic spectrum of a single-

orbital model (without the Hund coupling JH→0) consists

of the usual upper and lower Hubbard bands (UHB and LHB,

respectively) that develop with U. In multiorbital systems,

the ﬁnite JHgives rise to additional excitations. Some of

these states can appear at energies between UHB and LHB

that depend exclusively on JH(i.e., are independent of U),

paving the way to measure JHdirectly. Since such exci-

tations occur due to the Hund coupling and have a robust

dispersion [see Fig. 1(b,c) and [21] for the full spectrum of

A(k, ω)], we call them Hund bands. We recognize that the

Hund bands arise whenever single-particle removal/addition

processes yield a higher multiplet of the dominant valence

subspace. This can occur provided: (i) the higher multiplets

exist, (ii) these multiplets are allowed by the selection rules

upon adding/removing a particle, and (iii) the charge ﬂuc-

tuations are signiﬁcant. All these requirements are met for

Hund’s metals. Earlier work documented multiplet splittings

in the Hubbard bands [20,22,23], in the fully occupied or-

bital [24], found additional “holon-doublon” peaks [25–30],

and analyzed the energy-level structure, revealing multiplets

that violate the Hund’s rules [31]. Here, we stress that charge

excitations independent of Uare a generic consequence of the

multiorbital systems.

To reach these conclusions, we use the density-matrix

renormalization group method (DMRG) [32–37] and Lanczos

diagonalization [2,38]. To show that our ﬁndings are generic,

we study both the two- and three-orbital Hubbard model.

Furthermore, we supplement our analysis with the effective

model of the OSMP - the generalized Kondo-Heisenberg

Hamiltonian. Finally, we conﬁrm our ﬁndings with DMFT

calculations. Our results apply to many experiments investi-

gating the spectral properties of multiorbital materials, partic-

ularly iron-based compounds [39,40], ruthenates [24,41–43],

iridates [44,45], and nickel oxides [46–50].

arXiv:2210.11209v3 [cond-mat.str-el] 3 Aug 2023

(d)

Hund coupling 𝐽H

(a)

Hund

excitation

single-

orbital multi-

orbital Wave vector 𝑘

−𝜋−𝜋/20𝜋/2𝜋

(b) Spectral function 𝐴𝛾(𝑘, 𝜔)

in the orbital 𝛾=2

Wave vector 𝑘

−𝜋/20𝜋/2

in the orbital 𝛾=1

Frequency 𝜔−𝜇(eV)

Density of states 𝐴𝛾(𝜔)

localized (𝛾=2)

itinerant (𝛾=1)

gKH

0.00 0.05 0.10 0.15

−6

−5

−4

−3

−2

−1

Figure 1. (a) Sketch of the Hund band accompanying the standard Hubbard bands. (b), (c) Orbital- and momentum-resolved spectral function

Aγ(k, ω)in the two-orbital Hubbard model for n= 2.5,U/W = 1.3,JH/U = 0.25,L= 48 sites, and orbitals (b) γ= 2 and (c) γ= 1.

The horizontal line marks the chemical potential µ. (d) Orbital-resolved density-of-states Aγ(ω). Points depict the corresponding effective

generalized Kondo-Heisenberg model (gKH); see the text for details. The arrow points at the Hund band in the itinerant orbital. Results

obtained with DMRG using broadening η= 0.04.

Model. We focus on the SU(2)-symmetric multiorbital

Hubbard-Kanamori chain,

HH=−X

γγ′ℓσ

tγγ′c†

γℓσcγ′ℓ+1σ+ H.c.+X

γℓ

∆γnγℓ

+UX

γℓ

nγℓ↑nγℓ↓+ (U−5JH/2) X

γ<γ′

,ℓ

nγℓnγ′ℓ

−2JHX

γ<γ′

,ℓ

Sγℓ ·Sγ′ℓ+JHX

γ<γ′

,ℓ

P†

γℓPγ′ℓ+ H.c..

(1)

Here, c†

γℓσ creates an electron with spin σat orbital γof site

ℓ.tγγ′is the symmetric hopping matrix in orbital space. ∆γ

denotes the crystal-ﬁeld splitting. nγℓ =Pσnγℓσ repre-

sents the total density of electrons. Uis the standard repul-

sive Hubbard interaction. JHis the Hund coupling between

spins Sγℓ at different orbitals γ. The last term P†

γℓPγ′ℓde-

notes interorbital pair hopping, Pγℓ =cγℓ↑cγℓ↓. We assume

open boundary conditions, as required by DMRG. For the

two-orbital model, γ∈ {1,2}, we used (in eV): t11 =−0.5,

t22 =−0.15,t12 =t21 = 0,∆1= 0,∆2= 0.8; whereas

for the three-orbital model: γ∈ {0,1,2},t00 =t11 =−0.5,

t22 =−0.15,t02 =t12 = 0.1,t01 = 0,∆0=−0.1,∆1= 0,

∆2= 0.8. These values were previously used to study the

iron-based ladders of 123 family [9,10,51–54]. The band-

width of the two-orbital model, W= 2.1, is used as the en-

ergy unit [55]. All energy labels given throughout the text are

independent of the JH/U ratio.

We also study the minimal model of the OSMP: the gener-

alized Kondo-Heisenberg model (gKH). This model was de-

rived [10,53,54] to capture the static and dynamic properties

of BaFe2Se3iron-based ladder [56–58]. It describes interact-

ing itinerant electrons (with spin si) coupled via Hund cou-

pling to the localized spins Sl,

HK=−tiX

ℓσ

c†

ℓσcℓ+1σ+ H.c.+UX

ℓ

nℓ↑nℓ↓

+KX

ℓ

Slℓ·Slℓ+1 −2JHX

ℓ

siℓ·Slℓ.

(2)

For the gKH model: ti=−0.5,K= 4t2

l/U,tl=−0.15,

matching the OSMP of our two-orbital Hubbard model [10].

Hund bands. Let us study the orbital-resolved single-

particle spectral function Aγ(k, ω)and the density-of-

states (DOS) Aγ(ω)∝Pσ(⟨⟨c†

γ,L/2,σ;cγ,L/2,σ⟩⟩h

ω+

⟨⟨cγ,L/2,σ;c†

γ,L/2,σ⟩⟩e

ω)[21]. Here, kis the momentum, ωthe

energy, and ⟨⟨. . . ⟩⟩h,e

ωrepresent the hole and electron compo-

nents.

The origin of the Hund bands can be clearly illustrated in

an OSMP system. Figure 1(b)-(d) presents data for the two-

orbital Hubbard model (2oH) at electron ﬁlling n= 2.5and

interaction U≃W. Clearly, the narrow orbital γ= 2

[Fig. 1(b)] has a gap at the Fermi level µ, while the orbital

γ= 1 [Fig. 1(c)] is metallic with a ﬁnite DOS at µ(or a

narrow pseudogap-like feature originating in the magnetic or-

der [59]). This behavior is consistent with the OSMP [10],

the narrow orbital is Mott-localized with the electron density

equal to 1. However, instead of two excitation bands (UHB

and LHB), expected from the Mott physics, we observe a

prominent three-peak structure [see also the DOS in Fig. 1(d)].

This structure is also visible in the itinerant orbital (γ= 1),

Fig. 1(c), with an electron density equal to 1.5. Note that the

itinerant orbital’s spectrum is accurately reproduced by the ef-

fective gKH model.

Let us take a closer look at how the three-peak spectrum

develops with the interaction U. Figure 2(a) shows A1(ω)for

the gKH model at noninteger ﬁlling n= 1.5. In the U→0

limit, we recover the noninteracting behavior: a single metal-

lic band. However, already at U/W ≃0.8, i.e., close to the

(b) Atomic excitation spectrum

Frequency 𝜔−𝜇(eV)

Interaction 𝑈/𝑊

−8

−6

−4

−2

0.0 0.5 1.0 1.5 2.0 2.5

(a) Density of states 𝐴1(𝜔)at 𝑛=1.5

−2𝐽H

−(𝑈+𝐽H)

−2

0.5 1.0 1.5 2.0

𝑛=1

−8

−6

−4

−2

Frequency 𝜔−𝜇(eV)

−8

−6

−4

−2

(d) →

Interaction 𝑈/𝑊

0.0 0.5 1.0 1.5 2.0 2.5−8

−6

−4

−2

(e) →

Figure 2. (a) Interaction Udependence of the itinerant orbital’s

density-of-states (DOS) A1(ω)obtained for the gKH model with

L= 48 sites, JH/U = 0.25, and n= 1.5. Results obtained with

DMRG using η= 0.04 broadening. The solid lines represent the

atomic-limit transitions. Inset depicts the results for half electron

ﬁlling n= 1. (b) Atomic excitation spectrum. For clarity, we mark

only the hole-like (electron removal) excitations and show only one

spin projection. D, T, S, H labels stand for doublon, triplet, singlet,

and holon, respectively. (c)-(e) DOS A1(ω)projected on the speciﬁc

ﬁnal conﬁgurations: (c) parallel spins, (d) antiparallel spins, and (e)

on the holon; see the text for details. Results obtained with Lanczos

diagonalization of L= 8 lattice with broadening η= 0.05.

OSMP transition [9,10], the three-peak structure is visible in

A1(ω), and becomes clearer the larger the interaction Ube-

comes. Since the three-peak structure is most pronounced

for U≫W, it is instructive to examine the atomic limit

U, JH→ ∞ of the gKH model; see Fig. 2(b). The atom real-

izes the noninteger ﬁlling n= 1.5provided the ground states

(gs) of the 1- and 2-electron sectors are degenerate, which is

achieved at µ=U+JH/2. Then, the gs consists of a local in-

terorbital triplet, denoted as |T⟩, which is degenerate with an

itinerant doublon with localized spin, denoted as |D⟩. By re-

moving an electron from the triplet, one creates a holon in the

itinerant orbital (|T⟩→|H⟩), with the cost of energy U+JH.

Interestingly, from the doubly occupied state, one can remove

an electron in two different ways. Depending on the spin pro-

jection of the removed electron, one can arrive at a local triplet

or singlet, |D⟩→|T⟩or |D⟩→|S⟩, respectively. The former

is a zero-energy transition between degenerate states of the gs,

while the latter costs an energy 2JHas it breaks the Hund’s

rule. In Fig. 2(a), we plot the relevant energy scales of the

atomic limit (U+JHand 2JH) and ﬁnd good agreement with

the full many-body calculations of the gKH chain.

Projections on the atomic conﬁgurations. To make a

stronger case for the atomic-limit interpretation of the three-

peak spectrum, we decompose the spectral function of the

full many-body calculation into individual transitions [29].

To this end, we use the projector Ponto speciﬁc conﬁg-

urations of the on-site Ising basis |γ= 1, γ = 2⟩, i.e.,

⟨⟨c†

γ,L/2,σ;Pcγ,L/2,σ⟩⟩h

ω[21]. For clarity, we discuss only

the hole part (below µ), as the electron part can be de-

scribed analogously. Upon removing an electron from the

itinerant orbital, we distinguish three contributions. (i) In

Fig. 2(c), we project onto the parallel-spin conﬁguration,

P=|↑,↑⟩⟨↑,↑| +|↓,↓⟩⟨↓,↓|. The resulting weight forms

a band of excitations close to the Fermi level ω≃µ. This

transition is responsible for the metallic properties of the lat-

tice. (ii) In Fig. 2(d), we instead project onto the antiparal-

lel conﬁguration, P=|↑,↓⟩⟨↑,↓| +|↓,↑⟩⟨↓,↑|. We observe

large weight in the middle band and some smaller weight at

ω≃µ. The middle band represents the interorbital singlet

which breaks the Hund’s rule: this is the 2JHHund excita-

tion. The band at ω≃µrepresents the Sz= 0 component

of the triplet (|↑,↓⟩ +|↓,↑⟩), costing zero energy to excite.

(iii) Finally, in Fig. 2(e), we project onto the holon conﬁgu-

ration, P=|0,↑⟩⟨0,↑| +|0,↓⟩⟨0,↓|. This gives the energet-

ically lowest band of excitations, which we recognize as the

LHB, arising from triplet to holon transitions. The starting

state needs to be a triplet because singlets are excluded from

the gs by the Hund’s rule.

Noninteger vs integer ﬁlling. As shown above, for non-

integer ﬁlling (doped system), the atomic limit is enough to

explain the Hund bands. When the atomic gs of adjacent

particle-number subspaces, say Nand N−1, are degenerate,

there is no cost Ufor the transition from the gs of subspace

Nto the gs of subspace N−1. The excitation cost is zero;

it is compensated by µwhich is tuned to cause the degener-

acy. However, if the N−1subspace contains not only the

gs but also higher multiplets, these multiplets can be accessed

in the photoemission process N→N−1with just the en-

ergy ∝JH. Analogous reasoning applies to inverse transitions

N−1→N. Thus, remarkably, this results in U-independent

Hund bands.

Consider now this behavior in a more general system, host-

ing more atomic conﬁgurations with different n. In Fig. 3we

present the three-orbital Hubbard model (3oH) results [60] for

various electron ﬁllings. For n= 4.5, the atomic limit of our

setup [21] predicts one Hund excitation (between states with 5

and 4electrons) with energy 2JH[61], along with several U-

dependent Hubbard excitations. We pinpoint the Hund band

using the projector analysis, shown in Fig. 3(b). We differen-

tiate transitions arriving at |↑↓,↑,↑⟩ and |↑↓,↓,↑⟩. Similarly,

for the n= 3.5ﬁlling, the atomic limit implies Hund bands in

photoemission at 3JHand 5JH. They are shown in Fig. 3(c).

The 3JHband is a transition to a low-spin S= 1/2state [P

onto |↑,↓,↑⟩; see Fig. 3(a)]. The 5JHband originates in states

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HundbandsinspectraofmultiorbitalsystemsM.´Sroda,1J.Mravlje,2G.Alvarez,3E.Dagotto,4,5andJ.Herbrych11InstituteofTheoreticalPhysics,FacultyofFundamentalProblemsofTechnology,WrocławUniversityofScienceandTechnology,50-370Wrocław,Poland2JoˇzefStefanInstitute,SI-1000Ljubljana,Slovenia3ComputationalSciences...

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Hund bands in spectra of multiorbital systems M.Sroda1J. Mravlje2G. Alvarez3E. Dagotto4 5and J. Herbrych1 1Institute of Theoretical Physics Faculty of Fundamental Problems of Technology.pdf

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Hund bands in spectra of multiorbital systems M.Sroda1J. Mravlje2G. Alvarez3E. Dagotto4 5and J. Herbrych1 1Institute of Theoretical Physics Faculty of Fundamental Problems of Technology

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