Effect of many-body interaction on de Haas-van Alphen oscillations in insulators Gurpreet Singh and Hridis K. Pal Department of Physics Indian Institute of Technology Bombay Powai Mumbai 400076 India

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Eﬀect of many-body interaction on de Haas-van Alphen oscillations in insulators

Gurpreet Singh and Hridis K. Pal

Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

(Dated: November 10, 2023)

De Haas-van Alphen (dHvA) oscillations are oscillations in the magnetization as a function of the

inverse magnetic ﬁeld. These oscillations are usually considered to be a property of the Fermi surface

and, hence, a metallic property. Recently, however, such oscillations have been shown to arise, both

experimentally and theoretically, in certain insulators which have a narrow gap and an inverted band

structure. In this work, we develop a theory to study the eﬀect of many-body interaction on these

unconventional oscillations. We consider weak interaction, focusing on the eﬀect of renormalization

of the quasiparticle spectrum on these oscillations. We ﬁnd that interaction has an unusual eﬀect:

unlike in metals, in a certain regime the amplitude of oscillations may be enhanced substantially,

both at zero and nonzero temperatures, even when the interaction is perturbatively weak.

One of the striking consequences of Landau quantiza-

tion in a magnetic ﬁeld in metals is the appearance of

quantum oscillations. These are oscillations in physical

observables of metals, both thermodynamic and trans-

port, with a change in the magnetic ﬁeld. The underlying

mechanism is simple: as the magnetic ﬁeld increases, the

spacing between the Landau levels widens. This forces

the highest occupied level to spill out of the Fermi level

and become depopulated periodically, which manifests as

oscillations in various observables. Evidently, these os-

cillations are expected only in metals and are measured

routinely in experiments to map the Fermi surface of such

systems [1, 2].

In recent years, however, quantum oscillations have

been reported experimentally in various insulators. De

Haas-van Alphen (dHvA) oscillations, which refer to os-

cillations in the magnetization, have been observed in the

Kondo insulator SmB6[3–6], while Shubnikov-de Haas

(SdH) oscillations, which refer to oscillations in the resis-

tivity, have been observed in the Kondo insulator YbB12

[7], quantum well heterostructures [8, 9], WTe2[10], and

moir´e graphene [11]. Such unexpected ﬁndings prompted

intensive theoretical investigations which have now re-

vealed that contrary to the canonical picture, quantum

oscillations can indeed arise in insulators, provided the

insulators have a narrow gap and an inverted band struc-

ture [12–16].

Although the phenomenon is now well-understood at

the noninteracting level, the eﬀect of interactions on

these unconventional oscillations remains insuﬃciently

explored. Research in this direction has predominantly

concentrated on speciﬁc models of correlated insulators

where interaction causes the opening of the gap [17–20]

but not on generic band insulators with interaction. Ex-

ploring the latter is important since it oﬀers a controlled

approach to compare oscillations in insulators with those

in metals. Moreover, from an experimental perspective,

investigating this aspect is signiﬁcant since some of the

systems where unconventional oscillations have been ob-

served are of this nature [8, 9], and the abundance of

possibilities in this category suggests more explorations

in the future.

In this work, we develop a theory of quantum oscilla-

tions in interacting band insulators, focusing speciﬁcally

on dHvA oscillations. We consider weak interactions,

incorporating it within the Hartree-Fock approximation.

Notably, we ﬁnd that in a certain parameter regime, even

weak interactions can signiﬁcantly modify the amplitude

of oscillations unlike in metals. This arises because in-

teractions now competes with a new energy scale in the

form of a gap which is absent in a metal, thus inﬂuencing

oscillations in a qualitatively diﬀerent manner.

To put our results for the insulator in context, we

ﬁrst review the theory of dHvA oscillations in metals.

The oscillating part of the magnetization is given by

Mosc =−∂Ωosc

∂B , where Ωosc is the oscillating part of the

grand potential and Bis the external magnetic ﬁeld. In

a three-dimensional interacting metal with a parabolic

spectrum of spinless electrons, considering only the renor-

malization of the energy spectrum due to a static interac-

tion potential at the Hartree-Fock level, it is found that

[2, 21] (ℏ= 1)

Ωosc =∞

l=1

Al(T)cos 2πl ˜µ

˜ωc−πl+1

4,(1)

where ˜ωc=eB/ ˜mis the cyclotron frequency with eas

the absolute value of the charge and ˜mas the mass of

an electron, respectively, ˜µis the chemical potential, and

Al(T) is the temperature (T)-dependent amplitude of the

l-th harmonic of oscillations given by (kB= 1)

Al(T) = ˜ωc

(eB)3/2

8π4l5/2

2π2lT /˜ωc

sinh(2π2lT /˜ωc).(2)

Above, and henceforth, the presence (absence) of tilde de-

notes renormalized (bare) values of the respective quanti-

ties. The expressions (1) and (2) are, in fact, identical to

the ones that appear in the noninteracting case, except

for the appearance of renormalized parameters [22, 23]:

µ→˜µ=µ(1 + b) (3a)

ωc→˜ωc=eB

˜m=eB

m(1 + a),(3b)

where band acapture the degree of renormalization in

µand m, respectively. One can readily summarize the

following salient features:

arXiv:2210.10475v2 [cond-mat.mes-hall] 9 Nov 2023

FIG. 1. Schematic band structure for the Hamiltonian in (4)

with ϵ1kand ϵ2khaving curvatures of diﬀerent sign. Dotted

curves show bands before hybridization. The hybridization

results in a gap.

1. The phase does not change with interaction.

2. The change in frequency is negligible: Because

µ≫ωc, V , where Vis the strength of interaction,

˜µ/˜ωc≈µ/ωc.

3. At T= 0, the change in the amplitude is small in

proportion to the strength of the weak interaction

since ˜

Al(0) ∼˜ωc. The amplitude decreases mono-

tonically with rise in T.

We now proceed to investigate dHvA oscillations in an

insulator. Consider the following Hamiltonian:

H=X

i,k

εikc†

ikcik+X

kγkc†

1kc2k+ h.c.+Hint.(4)

Here, c†

ik(cik), i= 1,2, denotes the creation (destruc-

tion) operators for particles with momentum kin the

i-th band with dispersion εikin three dimensions. These

two bands are hybridized by γk. For simplicity, we choose

ε1k=k2

2m1−∆ and ε2k=−k2

2m2with m1,2,∆>0, and

assume γk=γto be independent of kwith |γ| ≪ ∆.

Also, all particles are assumed to be spinless. The ﬁrst

two terms in Eq. (4) describe the noninteracting part

and is easily diagonalized leading to an insulator with an

inverted band structure and a narrow gap—see Fig. 1.

The chemical potential µis chosen to lie inside the gap.

The last term in Eq. (4), Hint, introduces interparticle

interaction, whose exact form is not necessary for the

results to be derived—we only assume that the interac-

tion is static and weak so that its eﬀect can be included

perturbatively at the Hartree-Fock level which leads to a

renormalization of the energy levels but no broadening.

In the presence of a magnetic ﬁeld, discrete Landau

levels are produced that are aﬀected by the interaction.

We calculate the grand potential using the standard for-

mula [22]:

Ω = −TTr 

X

ζm

ln{−[˜

G−1(ζm)]}

.(5)

Here, ξm= (2m+ 1)πT , with m∈Z, are the Matsub-

ara frequencies, Tr stands for the trace over all energy

states, and ˜

Gis the Green’s function corresponding to

(4) in a magnetic ﬁeld given by ˜

G−1=G−1−Σ, where

Gis the noninteracting Green’s function and Σ is the

self-energy due to the interaction. In the band-basis, we

have G−1

11 =iζm−ωc1n+1

2−k2

2m1+∆+µ,G−1

22 =

iζm+ωc2n+1

2+k2

2m2+µ, and G−1

12 =G−1

21 =−γ,

where nis the Landau level index and ωc1,2=eB/m1,2.

In general, Σ is a function of both Band Tand requires

a substantial eﬀort to calculate. However, as far as dHvA

oscillations in three dimensions are concerned, it suﬃces

to consider Σ calculated at zero Band T—as in the case

of metals, the eﬀect of nonzero Band Tleads to sublead-

ing corrections in orders of ωc1,2/∆≪1 and T/∆≪1,

respectively [23–25]. Within this approximation, we eval-

uate the oscillating part of Eq. (14) and ﬁnd,

Ωosc =∞

l=1

Al(T)cos "2πl ˜

∆

˜ωc1+ ˜ωc2−πl+1

4#,(6)

where

Al(T) = (eB)3/2

π2l3/2TX

ζm>0

e−πl

˜ωc1˜ωc2√ζ2

m(˜ωc1+˜ωc2)2+4˜ωc1˜ωc2˜γ2

×cosh πlζm(˜ωc2−˜ωc1)

˜ωc1˜ωc2.(7)

Details of the derivation are provided in Supplemental

Material (SM) [26]. The above expressions are charac-

terized by the following renormalized parameters:

ωc1,2→˜ωc1,2=eB

˜m1,2

=eB

m1,2

(1 + a1,2),(8a)

∆→˜

∆ = ∆(1 + b),(8b)

γ→˜γ=γ(1 + t),(8c)

µ→˜µ=µ+δµ. (8d)

Note that, while ˜ωc1,2,˜

∆, and ˜γappear explicitly in

Eqs. (6) and (7), ˜µenters implicitly through the Mat-

subara sum. Thus, it aﬀects only the T-dependence of

the oscillations, and has no eﬀect on the T= 0 behav-

ior, as long as µand ˜µlie in the gap. For simplicity,

we have assumed ˜µ=−˜

∆ ˜m1

˜m1+ ˜m2, chosen such that it lies

at the intersection of the two renormalized bands prior

to hybridization [27]. Equation (6) along with Eqs. (7)

and (19) generalize Eqs. (1), (2), and (3) from an in-

teracting metallic system to an interacting gapped sys-

tem. Thus, for a given form of Hint in Eq. (4), one

simply needs to calculate the renormalization parameters

ai, b, t to study the eﬀect of interaction on dHvA oscil-

lations. We will come back to this calculation later; for

now, we discuss the qualitative features that arise from

these expressions. It is seen that the phase remains un-

altered and the frequency does not change appreciably

since 1 ≪˜

∆

˜ωc1+˜ωc2≈∆

ωc1+ωc2; thus, both these quanti-

ties remain qualitatively similar to those in metals. In

FIG. 2. The eﬀect of interaction on the amplitude at T= 0 according to Eqs. (9) and (19). ˜

A1(0) denotes the amplitude of

the ﬁrst harmonic at T= 0 in the presence of interaction. It is normalized by its noninteracting value A1(0). We show its

variation with a1and tkeeping a2= 0.2 ﬁxed in all the plots. The points Q, R, and Sare arbitrarily chosen in the interaction-

parameter-space deﬁned by (a1, a2, t) which are referred to in Fig. 3 later.

contrast, the amplitude is signiﬁcantly aﬀected by inter-

actions, in a way that is qualitatively diﬀerent from that

in metals.

First, we consider ˜

Al(0), the amplitude at T= 0.

Changing the summation to an integral over the fre-

quency in Eq. (7), we ﬁnd,

Al(0) = |˜γ|(eB)3/2

2π3l3/2K14πl|˜γ|

˜ωc1+ ˜ωc2,(9)

where Kαis the modiﬁed Bessel function of the second

kind. Equation (9), together with Eqs. (19), gives a quan-

titative description of how the amplitude is aﬀected by

interaction. It leads to an unusual feature that is unique

to the insulating case: even a weak interaction can lead

to a substantial change in the amplitude of oscillations

at zero temperature. Indeed, the fate is decided by a

delicate interplay between ˜γand ˜m1,2in Eq. (9). Us-

ing the parametrization of Eq. (19) in Eq. (9), we plot

the ﬁrst-harmonic-amplitude in Fig. 2 for diﬀerent val-

ues of m1/m2(determining the particle-hole asymmetry)

and γ/ωc1(determining the strength of the gap as com-

pared to the Landau level spacing). It is seen that when

γ/ωc1∼1, there is a pronounced enhancement in the

zero-temperature-amplitude, which can amount to even

an order of magnitude.

Next, we consider the dependence of the amplitude on

T. This is calculated numerically from Eq. (7) and is

presented in Fig. 3 for various choices of interaction pa-

rameters m1/m2and γ/ωc1as used in Fig. 2. In the limit

T≫γ, as expected, the T-dependence follows the usual

metallic behavior, contributed by the two participating

bands. It only depends on ˜m1,2and is independent of

˜γ. In the other limit, T<

∼γ, the T-dependence deviates

from the metallic behavior, and depends on both ˜m1,2

and ˜γ. The deviation is most striking when m1/m2≫1

and γ/ωc1∼1. In this regime [Fig. 3(d)] There is a size-

able enhancement in the amplitude in the form of a non-

monotonic upturn driven by Ton top of the enhancement

at T= 0 discussed earlier. An interesting observation is

that since the behavior of the amplitude at low Tin the

particle-hole asymmetric case is very sensitive to γ/ωc1,

for a given material (with a ﬁxed γand ˜γ) the eﬀect of

temperature depends crucially on the ﬁeld at which the

oscillations are being studied to extract the amplitude.

Indeed, Figs. 3(c) and (d) can be interpreted as the tem-

perature dependence of the amplitude of the same dHvA

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Effectofmany-bodyinteractionondeHaas-vanAlphenoscillationsininsulatorsGurpreetSinghandHridisK.PalDepartmentofPhysics,IndianInstituteofTechnologyBombay,Powai,Mumbai400076,India(Dated:November10,2023)DeHaas-vanAlphen(dHvA)oscillationsareoscillationsinthemagnetizationasafunctionoftheinversemagneticfiel...

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Effect of many-body interaction on de Haas-van Alphen oscillations in insulators Gurpreet Singh and Hridis K. Pal Department of Physics Indian Institute of Technology Bombay Powai Mumbai 400076 India

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