zotero/storage/IT4X4XKH/.zotero-ft-cache

The Sagnac eﬀect in a rotating ring with Dirac fermions
A.Yu. Fesh1 and S.G. Sharapov2, 1
1Kyiv Academic University, 03142 Kyiv, Ukraine 2Bogolyubov Institute for Theoretical Physics, National Academy of Science of Ukraine,
14-b Metrologichna Street, Kyiv, 03143, Ukraine (Dated: September 20, 2023)
The observation of the Sagnac eﬀect for massive material particles oﬀers a signiﬁcant enhancement in sensitivity when compared to optical interferometers with equal area and angular rotation velocity. As a result, there have been suggestions to employ solid-state interferometers that rely on semiconductors and graphene. However, in the case of monolayer graphene, its quasiparticles exhibit a linear dispersion, thus making the Sagnac eﬀect in graphene comparable to that of for light. We investigate the Sagnac eﬀect in the Dirac materials governed by the relativistic dispersion law and ﬁnd the value of the fringe shift. The analysis reveals that optimal sensitivity is achieved in materials featuring a reduced value of Fermi velocity. Notably, the sign of the fringe shift depends on the nature of the charge carriers – whether they are electrons or holes.

arXiv:2309.10497v1 [cond-mat.mes-hall] 19 Sep 2023

Introduction. Physical phenomena associated with rotation possess a captivating allure that spans multiple levels. One of the fundamental illustrations of this allure lies in the impossibility of establishing a standard clock synchronization procedure along a closed curve when the metric is non-static, as is the case of a rotating frame of reference (see the textbooks [1, 2] and [3]). The Sagnac eﬀect refers to the phenomenon where a phase shift is observed between two coherent beams that travel along opposite paths within an interferometer situated on a rotating disk (see Refs. 4–7 for the reviews). This phase shift, which was ﬁrst demonstrated for light by Sagnac [8] in 1913 is the intrinsically relativistic eﬀect. Thus, it can be essentially viewed as a consequence of the impossibility of synchronization of clocks along the circumference of the rotating disk. It is not restricted to light waves, but is observed for electron waves in vacuum [9], neutrons [10] and atoms [11] (see also the latest work [12]). Moreover, the observation of the Sagnac phase shift for massive particles in solids, speciﬁcally superconducting Cooper pairs, dates back to as early as 1965 [13].
While practical applications of the Sagnac eﬀect currently rely on light waves, there is a compelling physical explanation as to why massive particles are signiﬁcantly more advantageous for its realization. The Sagnac fringe shift, denoted as ΘS, with respect to the fringe position for the stationary interferometer, reads [6, 9, 14]

ΘS

=

4EAΩ c2

.

(1)

This formula is applicable to waves comprising both massless and massive particles, and E represents the total energy of a corresponding particle. The value A denotes the area enclosed by the light or particle beams in the interferometer, Ω is the angular velocity of the interferometer’s rotation within an inertial frame, is the reduced Planck constant, and c is the the free-space velocity of light. This equation is written neglecting a small relativistic correction and under the assumption that the plane of the interferometer is perpendicular to the axis of rotation. Substituting in Eq. (1) the energy E = ω,

where ω is either the frequency of the light or the frequency of the de Broglie wave of a material particle, we recover the standard formula for the Sagnac phase shift [4–7, 9] ΘS = 4ωAΩ/c2.
When considering light and using the dispersion relationship ω = 2πc/λ, where λ represents the vacuum wavelength of light, we arrive at another commonly used form for the Sagnac phase shift [4–6],

ΘS

=

8πAΩ cλ

.

(2)

In the case of slow massive particles (nonrelativisitic case), the energy E = mc2 is associated with their rest
mass m, and the phase fringe acquires the following form

ΘS = 4mAΩ .

(3)

Comparing the phase shift for the matter-wave and optical interferometers, one ﬁnds that for the equal area and angular velocity the phase shift is enhanced by a factor mc2/( ω) [15]. For atoms, the matter-wave interferometer is signiﬁcantly more sensitive to rotation, with a factor reaching the value of 1010. This constitutes the primary reason why the existing optical gyroscopes necessitate the utilization of either several kilometers of optical ﬁber or a substantial area to achieve the necessary sensitivity. Conversely, the sensitivity enhancement for matter-waves has led to proposals to utilize cold atom interferometers in the search for smaller signals beyond Earth rotation [12].
For free electrons, this factor also reaches the value 106 and, as mentioned earlier, the Sagnac eﬀect was observed using an electron interferometer in vacuum [9]. There is a possibility of realizing the Sagnac eﬀect in solid-state by employing a serial array of mesoscopic ring-shaped electron interferometers, which was discussed in Refs. 16– 19.
It has to be noted that the simulations for ring arrays discussed in [16–18] are conducted with the assumption that electrons in solids have an eﬀective mass, m∗. Consequently, the enhancement factor m∗c2/( ω) is slightly

2

reduced, estimated to be in the range of 105 to 106. Moreover, as highlighted in [18, 19], graphene emerges as a promising material for electron interferometry, attributed to its extraordinary electronic properties. Indeed, recent experiment [20] on Aharonov-Bohm oscillations in ringshaped gated bilayer graphene provides further conﬁrmation of this potential.
However, at least monolayer graphene belongs to the new class of Dirac materials with a zero eﬀective carrier mass m∗ and linear dispersion relation as for light. At ﬁrst glance, due to the assertion discussed e.g. in Ref. [6] that the phase shift remains independent of the phase velocity of the wave, the Sagnac eﬀect in graphene seems to be analogous to the case of light.
The objective of the present work is to examine this issue also taking into account that the density of carriers in graphene is ﬁnite. More broadly, the goal is to investigate the Sagnac eﬀect in relativistic Dirac materials and to clarify the distinctions from the previously known cases.
Model and formalism. We consider the Dirac materials characterized by the following dispersion relation:

ǫ(k) = ± 2v2k2 + ∆2 − µ,

(4)

where k represents the wave vector counted from the Dirac points, which can be either 2D or 3D. However, since we are examining a ring on the plane geometry, its z-component, denoted as kz, can be eﬀectively disregarded by setting it to zero. Also in Eq. (4) v is the Fermi velocity, µ is the chemical potential and ∆ is the gap in the quasiparticle spectrum. In the case of graphene, the value of µ (including the change of the character of carriers, either electrons or holes) is tunable by applying the gate voltage to the devices, and µ > 0 corresponds to electrons. The gap term ∆ is present, in particular, in the Hamiltonian derived by Wolf (see Ref. [21] for a review) for Bi and similar eﬀective Hamiltonians describing other 3D Dirac materials. It can also be induced in graphene monolayer by placing it on top of hexagonal boron nitride (G/hBN).
As already mentioned, the quasi-relativistic spectrum (4) follows, for example, from the Wolf Hamiltonian as well as other eﬀective low-energy either 2D or 3D Dirac Hamiltonians used to describe Dirac materials (for graphene see, e.g. the review [22]).
To focus on the Sagnac eﬀect for the quasiparticles with the relativistic-like dispersion we restrict ourselves by considering the squared Dirac Hamiltonians neglecting the coupling between pseudospin degree of freedom with the rotation of the frame [23]. Thus we assume that a free quasiparticle in the electron subsystem of the Dirac material in the inertial frame of reference obeys the following wave equation

′

+

∆
2

ψ(t′, r′) = 0,

′

≡

1 v2

∂2 ∂ t′2

−

△′.

(5)

Here ′ and △′ are the d‘Alembertian and Laplace operators, respectively, ψ(t′, r′) is the electron wave func-

tion in the inertial frame of reference denoted as primed. The chemical potential should also be included in Eq. (5) by the standard prescription for the relativistic systems, i ∂t → i ∂t + µ.
Seeking for a solution of Eq. (5) in the form ψ ∼ exp(−iǫt′/ + ikr′), one reproduces the spectrum (4). Clearly, relating the energy gap ∆ to the mass, m = ∆/v2, and setting v = c and µ = 0, the equation (5) reduces to the usual Klein-Gordon-Fock (KGF) equation.
We shall consider the Sagnac eﬀect for the quasisparticles characterized by the wave equation (5) from the point of view of a co-rotating observer employing the approach of Ref. [14]. It can be traced back to the explanation of the Sagnac eﬀect proposed by Langeven in the framework of general relativity in 1921 [24] (see also Refs. [4, 7] for the reviews and the textbook [1]). The invariant interval in polar coordinates (t′, r′, φ′) in the inertial rest frame is

ds′2 = c2dt′2 − dr′2 − r′2dφ′2,

(6)

where as mentioned above we restricted ourselves by the
planar geometry. The transformation to a new non-
primed frame of reference rotating about the z-axis with angular velocity Ω is done by t′ = t, r′ = r and φ′ = φ + Ωt, so the invariant interval reads

ds2 = c2

1

−

Ω2r2 c2

2

dt2

−

2r2Ω c

cdtdφ

−

dr2

−

r2dφ2.

(7)

The corresponding contravariant metric tensor (µ, ν =

0, 1, 2 = t, r, φ) is

1 0 gµν =  0 −1

−

Ω c



0 .

(8)

−

Ω c

0

−

1 r2

+

Ω2 c2

Note that g = det[gµν ] = r2 > 0 because 2 + 1 dimensional space is considered.
To elucidate the propagation of electron waves within a rotating coordinate frame, characterized by the metric (8), it becomes necessary to employ the following equation

+

∆
2

ψ(t, r) = 0,

(9)

with the generalized d‘Alembertian operator [25]

ψ

=

∇µ∇µψ

=

1 √g

∂µ

(√ggµν

∂ν

ψ).

(10)

Recall that by the deﬁnition the covariant derivative

∇µ ≡ ∂µ =

∂ ∂xµ

,

where

xµ

=

(vt, r, φ).

It easy to ob-

tain that the operator = 0 + Ω with

0

=

1 v2

∂2 ∂t2

−

1 r

∂ ∂r

r

∂ ∂r

−

1 r2

∂2 ∂φ2

,

Ω

=

−

2Ω vc

∂2 ∂t∂φ

+

Ω2 c2

∂2 ∂φ2

.

(11)

3

Obviously, for Ω = 0 Eq. (9) reduces to Eq. (5) written in the rest frame in the polar coordinates. Also for v = c and µ = 0 Eq. (9) reduces to the KGF equation in the rotating frame considered in Ref. [14] to investigate the Sagnac eﬀect and in Ref. [26] to study the relativistic superﬂuidity. There is, however, an essential diﬀerence in the approaches used in [14] and [26]. The KGF equation within the rotating frame was obtained in [14] by replacing ∂t → ∂t−V·∇r, where V = Ω×r is the local rotating velocity. Consequently, its extension to the case v = c is obscure because it is not enough to replace c by v in the KGF equation in the rest frame. On the other hand, the derivation in Ref. [26] utilizes the operator (11) as done in this work. The ﬁrst term of Ω with the mixed derivative, which is sensitive to the rotational direction, corresponds to the Coriolis force. Meanwhile, the second one is related to the centrifugal force [26].
The v = c case. Before going ahead with the analysis of Eq. (9) for the general v = c and ﬁnite µ case, it is instructive to recapitulate the derivation of Eqs. (1) and (3) made in Ref. [14]. The signals designated as ±, which propagate in the clockwise and counterclockwise directions around a circle of the radius R, are considered. These signals are described by the solutions of Eq. (9) which depend on t and the angular variable φ with ∆ = mc2. The solutions which have the frequency ω in the local Lorenz frame at the source are

ψ±(t, φ) = exp iγ

±k

+

ΩωR c2

Rφ − iωt/γ

(12)

with

γ=

1

−

Ω2R2 c2

−1/2
.

(13)

The wave number in Eq. (12) is determined by the KGF dispersion relationship ω2 = c2k2 + m2c4/ 2. Let’s assume that the two counter-propagating signals originate in phase from the same source at φ = 0. Subsequently, these signals are detected after completing a full roundtrip, with phases the counterclockwise signal is detected at φ = 2π, and the clockwise at φ = −2π. The phase diﬀerence between the two detected signals is therefore ΘS = 4πγωΩR2/c2. Considering that the circular interferometer has an area of A = πR2, one ﬁnds that the last expression reproduces Eq. (1) rewritten via the frequency ω up to the relativistic factor γ. This can be further elaborated upon by examining whether Eq. (1) is written for the rest or rotating frame (as discussed in the review by Post [4]). Nevertheless, this distinction is not crucial to our discussion, as ΩR ≪ c by several orders of magnitude, rendering the corrections to the fringe shift arising from γ (which are of the order Ω2R2/c2) practically indiscernible in experimental observations.
In the nonrelativistic limit, the phase fringe shift characterized by Eq. (3) emerges as the KGF equation reduces to the Schr¨odinger equation. This can be obtained by transforming away quickly oscillating rest energy dependence, ψ = χ exp(−imc2t/ ) which results in the

Schr¨odinger equation in the rotating frame:

i

∂χ ∂t

=

1 2m

(−i

∇

−

mV)2

χ

−

1 2

mV2

χ.

(14)

Its solution for the circularly propagating waves [14] results in the phase fringe (3).
It follows from Eq. (14) that a profound analogy exists between Aharonov-Bohm (AB) oscillations in mesoscopic rings and the Sagnac eﬀect in the nonrelativistic case, as described by Eq. (3). The rotation of a thin ring, and thus the Sagnac eﬀect, can be associated with the AB eﬀect occurring in a uniform (Larmor) magnetic ﬁeld, H = 2mΩ/e, or expressed in terms of the eﬀective AB ﬂux penetrating the ring as ΦΩ = 2mcΩA/e [14, 27] (for additional references and discussion see the reviews [5, 6]). Here −e < 0 is the electron charge.
One can also observe that Eq. (3) can be derived directly by substituting the nonrelativistic limit of the KGF dispersion relation, given by ω ≈ mc2/ + k2/(2m), into the solution (12). It is worthwhile to remind that the original consideration by de Broglie of the matter waves relied on the special relativity and contained the de Broglie frequency, ω = mc2/ associated with a resting particles. Although the Schr¨odinger theory does not contain explicitly such quantity, it appears when one calculates the phase diﬀerence between the two counterpropagating wave packages [28].
Since the de Broglie frequency is not a commonly used quantity, it is convenient to express the phase fringe for massive particles in the form (2). For example, one can reformulate Eq. (3) for electrons in the same manner as Eq. (2), but with the photon wavelength λ replaced by Compton wavelength, λC = 2π /(mec) ≈ 0.0243 ˚A, where me represents the electron mass. This representation simpliﬁes the comparison between interferometers using light and matter, both having the same area and angular velocity. For example, when comparing blue light with a frequency ωB and a wavelength of λB = 450 nm to electrons, one can estimate the enhancement factor as mec2/ ωB = λB/λC ≈ 1.85 × 105, in agreement with the aforementioned estimation.
The case of the Dirac materials, v < c. Now we may return to the consideration of a rotating ring made of the Dirac material. The Dirac electron subsystem in the rotating ring is described by Eq. (9) with a ﬁnite chemical potential introduced by the prescription given below Eq. (5). We neglect the eﬀect of deformation of the ring due to rotation and consider one dimensional rigid ring of the radius R, so the wave function ψ(t, φ) is independent of the radial coordinated as before. Then the solutions of Eq. (9) for the two counter-propagating electron waves [cp. Eq. (12)] reads

ψ±(t, φ) = exp iγ

±k

+

Ω(ǫ

+ µ)R cv

Rφ

−

i

ǫ

t γ

(15)

with the relativistic factor γ given by Eq. (13) and the

energy ǫ = ǫ(k) and wave number k obeying the dis-

persion law (4) for the Dirac quasiparticles. The phase

4

diﬀerence between the two counter-propagating electron waves is therefore ΘS = 4γ(ǫ + µ)ΩR2/( cv).
Let’s begin by examining the case where µ = ∆ = 0, which corresponds to a quasiparticle exhibiting a linear dispersion relationship given by ǫ = ± vk. Expressing the wave vector k via the wavelength λgr by the relation k = 2π/λgr, one again arrives at the expression (2) with λ replaced by λgr. As was anticipated, the phase shift remains independent of the phase velocity v of the wave. As pointed out in Ref. 6, the Sagnac eﬀect holds holds not only for quantum-mechanical particles (photons, electrons, etc.) but also for ordinary acoustic waves.
The physical meaning of Eq. (2) considered for light and material particles is further elucidated when one rewrites it as follows [5], ΘS = 4π(V /c)(P/λ) with P = 2πR. The fringe shift is proportional to the number of the corresponding wavelengths along the wave path. This interpretation of the phase shift ΘS is always valid, although the meaning of the wavelength λ has to be clariﬁed for each case.
Let us proceed with the cases of a ﬁnite µ or/and ∆. If the temperature T is much smaller than |µ|, only the quasiparticle excitations near the Fermi surface contribute to the transport. For the spectrum given by Eq. (4) the Fermi surface is determined by the condition ǫ(kF ) = 0, where kF represents the Fermi wave vector. Then neglecting the factor γ ≈ 1, the Sagnac fringe shift for the Dirac fermions reads

ΘS

=

4µAΩ vc

=

4 sgn (µ)mcAΩ

v c

,

mc

=

|µ| v2

.

(16)

Here in the second equality ΘS is written in terms of a

ﬁctitious “relativistic” mass, mc, which plays the role of the cyclotron mass in the Lifshits-Kosevich formula [29]

and also allows to rewrite Eq. (16) in the form resem-

bling the nonrelativitic expression (3). As was already

mentioned, in graphene µ and thus mc are easily tunable by the gate voltage. Interestingly, the value of ΘS turns out to be sensitive to the sign of µ or character of

the carriers. At ﬁrst glance, this seemingly paradoxical

result is actually sensible, as it arises from the fact that

the motion of hole carriers corresponds to the rotation of

electrons in opposite directions.

The representation of ΘS in terms of mc also turns out to be very convenient for an estimate of the phase fringe.

Indeed, one ﬁnds in Ref. 29 that for the carrier density

n ≈ 7 × 1012 cm−2 the mass mc ≈ 0.06me. Also consid-

ering that in graphene, c/v ≈ 300, one can estimate that

ΘeS/ΘgSr = comparing

cme/(vmc) ≈ 5 × 103. with blue light, ΘgSr/ΘBS

Alternatively, ≈ 37.

when

This enhancement factor is not signiﬁcant enough to

make graphene attractive for applications. However, a

more substantial value could be potentially achieved by

further increasing the carrier density. In contrast, there is a 3D topological insulator, speciﬁcally Bi2Te3, exhibiting a low Fermi velocity [30]. While the Fermi energy in Bi2Te3 is approximately 10 times lower than in graphene, the Fermi velocity is v ≈ 3260 m/s which is over 104 times smaller than in graphene. This signiﬁcant diﬀerence makes Bi2Te3 and similar materials attractive for electron interferometry.
Equation (16) is valid for both gapless and gapped cases. In the former, ∆ = 0 case, the Fermi wave vector, kF , is determined by the relationship vkF = |µ|. Once again, it is evident that Eq.(16) can be reformulated in the manner of Eq.(2), utilizing the Fermi wavelength, λF = 2π/kF . Using the relationship between the carrier imbalance n and chemical potential for graphene, |n| = (µ2−∆)/(π 2v2) (see e.g. Ref. [22]), one obtains for ∆ = 0 that λF = 2 π/|n|. Taking n = 7 × 1012 cm−2 one gets λF ≈ 13.5 nm. This corresponds to the enhancement factor λB/λF ≈ 33 which is quite close to the previous estimate made in terms of mc.
The opening of the gap ∆ in graphene can be taken into account by expressing the chemical potential via the carrier imbalance, µ2 = ∆2 + π 2v2|n|. This relation proved to be useful for analyzing the dependence µ(n) in G/hBN structures [31]. In particular, for n ≈ 0 one obtains that the fringe shift ΘS = 4 sgn (µ)∆AΩ/( vc). To produce the same phase fringe as light with a photon energy ω, the value of the gap is ∆ = ω(v/c). Accordingly, for the blue light with ωB ≈ 2.76 eV to reach the same sensitivity in graphene the gap has to be ∆ ≈ 9 meV, while in the experiment [31] on G/hBN ∆ ≈ 3.4 meV. However, the inclusion of the gap, along with a ﬁnite carrier density, and more notably, the reduction in the Fermi velocity, v, deﬁnitely enhances the sensitivity.
Conclusion. To conclude, we have obtained analytic expressions for the Sagnac fringe shift in the Dirac materials. The direction of the shift relies on the charge carriers’ nature – whether they are electrons or holes. When considering graphene, the enhancement factor is not as substantial as that achievable in conventional semiconducting materials with ﬁnite eﬀective carrier masses. Our analysis illustrates that the most signiﬁcant enhancement factor values are attainable in materials characterized by a reduced Fermi velocity.
We would like to thank the Armed Forces of Ukraine for providing security to perform this work. The authors acknowledge support by the National Research Foundation of Ukraine grant (2020.02/0051) “Topological phases of matter and excitations in Dirac materials, Josephson junctions and magnets”. We would like to thank Iu.A. Chernii, E.V. Gorbar, V.P. Gusynin, A.A. Kordyuk, E.G. Len, V.M. Loktev, and A.A. Varlamov for the numerous stimulating and enlightening discussions.

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