Generalized many-body exciton g-factors: magnetic hybridization and non-monotonic Rydberg series in monolayer WSe$_2$

Daniel Hernang\'omez-P\'erez; Jaroslav Fabian; Paulo E. Faria Junior; Sivan Refaely-Abramson; Tomer Amit

REVIEW 3 major objections 3 minor 2 cited by

Generalized many-body exciton g-factors: magnetic hybridization and non-monotonic Rydberg series in monolayer WSe$_2$

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A generalized GW-BSE formalism including off-diagonal spin and orbital angular momentum matrix elements reproduces the non-monotonic Rydberg series of exciton g-factors in monolayer WSe2.

desk verdict Generalized exciton g-factors are a real step forward, but the paper overstates that off-diagonal terms are necessary for the non-monotonic Rydberg series—the diagonal-only control is missing. read the letter →

arxiv 2505.18468 v2 pith:KFA6TWXM submitted 2025-05-24 cond-mat.mes-hall cond-mat.mtrl-sci

Paulo E. Faria Junior , Daniel Hernangómez-Pérez , Tomer Amit , Jaroslav Fabian , Sivan Refaely-Abramson This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.mtrl-sci

keywords excitong-factorsGW-Bethe-SalpeterequationmonolayerWSe2off-diagonalangularmomentumnon-monotonicRydbergseriesmagneticbrighteningspin-valleyphysicsmagneto-optics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that previous ab initio computations of exciton g-factors missed key physics because they kept only diagonal spin and orbital angular momentum matrix elements. By including off-diagonal elements in both the single-particle and exciton bases, the authors' generalized many-body GW-BSE framework reproduces known magneto-optical results for monolayer WSe2 and yields, without free parameters, the experimentally observed non-monotonic sequence of A-exciton g-factors from 1s to 4s. The same off-diagonal terms are claimed to drive the magnetic brightening of dark and grey excitons and the magnetic hybridization of higher-energy s- and p-like excitons. A sympathetic reader would care because this turns a long-standing experimental puzzle into a calculable many-body effect rather than an unexplained sample-dependent anomaly.

What carries the argument

The central object is the generalized exciton g-factor matrix element $g^\epsilon_{SS'} = \sum_{vck} (A^S_{vck})^* \big[\sum_{c'} A^{S'}_{vc'k} g^\epsilon_{cc'k} - \sum_{v'} A^{S'}_{v'ck} g^\epsilon_{vv'k}\big]$, where $A^S_{vck}$ is the GW-BSE exciton amplitude and $g^\epsilon_{\alpha\alpha' k} = \langle \alpha k | \hat{L}^\epsilon + \hat{\Sigma}^\epsilon | \alpha' k\rangle$ are single-particle g-factor matrix elements built from orbital and spin angular momentum operators in the Bloch basis. The off-diagonal components of these matrices, absent in previous diagonal-only treatments, allow an external magnetic field to couple different exciton states, including degenerate A-exciton partners and distinct s- and p-like Rydberg states. The orbital part is evaluated through momentum matrix elements and a finite band summation, while symmetry analysis based on the $D_{3h}$ point group supplies the allowed coupling patterns between the bright, grey, and dark exciton subspaces for arbitrary field orientation.

What would settle it

Compute the 1s-4s A-exciton g-factors with the same off-diagonal formalism but with a fully self-consistent treatment of the quasiparticle wavefunctions and with non-local band g-factor corrections included, and check whether the non-monotonic dips in the Rydberg series survive with the same positions and depths.

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Extended reading notes

Core claim

The central claim is that the non-monotonic Rydberg series of A-exciton g-factors in monolayer WSe2 arises from magnetic-field-induced hybridization captured by off-diagonal matrix elements of the spin and orbital angular momentum operators, both between single-particle bands and between many-body exciton states. Earlier formulations, which retained only diagonal expectation values of these operators, produce a monotonic decreasing trend for the 1s-4s g-factors and cannot reproduce the experimental measurements. In the generalized formalism, the term $\hat{g}\cdot B$ with a non-diagonal exciton g-factor matrix naturally mixes exciton subspaces, and the resulting orbital and spin mixing yields non-monotonic g-factors for the A exciton series, with similar non-monotonic structure predicted for the dark/grey Rydberg excitons. The same mechanism explains the brightening of optically inactive dark and grey excitons under out-of-plane, tilted, and in-plane magnetic fields, as well as the field-induced s-p mixing of high-energy excitons.

Load-bearing premise

The calculation assumes that the orbital contribution to each electron's magnetic response is fully captured by wavefunction-based matrix elements evaluated with a rigid energy-gap correction and a finite band sum, leaving out non-local contributions to the single-particle g-factors that the paper explicitly does not include.

Editorial extensions

If this is right

If the formalism is correct, the non-monotonic 1s-4s g-factor sequence in WSe2 is a genuine magnetic-hybridization effect, so simplified single-band or Wannier models will continue to fail for high-lying Rydberg excitons.
Magnetic brightening of dark and grey excitons under out-of-plane, tilted, and in-plane fields follows from the same off-diagonal angular momentum elements, giving one unified mechanism for several previously separate experimental observations.
High-energy s- and p-like excitons should hybridize under magnetic fields, with nominally dark states brightening as the field direction is tilted; this is shown in the paper for absorption near 1.95 eV in WSe2 and can be tested directly.
Because the framework is built for arbitrary magnetic field directions and arbitrary van der Waals materials, the same machinery can predict g-factors, brightening, and hybridization in heterostructures, chiral excitons, and other two-dimensional systems.
The predicted non-monotonic g-factors for the dark and grey Rydberg exciton series provide a concrete target for future magneto-optical measurements beyond the bright A series.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: the quantitative positions and depths of the non-monotonic dips likely depend on the neglected non-local contributions to single-particle band g-factors, which the paper explicitly sets aside; including those corrections could shift the pattern even if the off-diagonal hybridization mechanism is right.
Inference: the same calculation should yield a similarly non-monotonic Rydberg g-factor sequence in closely related monolayers such as MoSe2 and WS2, with material-specific details set by the band structure; this is a testable transfer prediction.
Inference: field-angle-resolved photoluminescence of the 2s and 3s A-exciton states, which is not reported in the experimental papers cited here, would distinguish the hybridization mechanism from environment-induced renormalization of the exciton wavefunctions.
Inference: the magnetic s-p mixing described here and the electric-field s-p mixing observed in monolayer WSe2 likely share the same orbital angular momentum matrix elements, so combined magneto-electro-optical measurements could isolate the orbital part of the response.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The useful thing to know about this paper is that it adds something the GW-BSE g-factor literature was missing: full off-diagonal spin and orbital angular momentum matrices, in both the single-particle and exciton bases. That is a real generalization of Deilmann et al. and the earlier diagonal-only work. The implementation is careful, and the low-energy validation is solid—Zeeman splitting and dark/grey brightening match known results, and the symmetry analysis is used honestly, including flagging the numerical mixing of degenerate A states. The WIEN2k cross-check of the orbital elements is a good reproducibility practice. I do not see circularity: the g-factors are not fitted to experiment, and the earlier calculations they cite are independent.

The soft spot is the central claim about the non-monotonic Rydberg series. The paper says the off-diagonal terms 'naturally emerge' and cannot be captured by simplified models. That is true against the two-band parabolic model, but that model is monotonic by construction (positive g2). The stress-test point is fair: setting off-diagonal blocks to zero in Eq. (1) leaves a diagonal sum over the full GW-BSE envelopes with the full k-dependent band g-factors. The 2s–4s envelopes spread far from K, and non-parabolic diagonal g(k) alone could produce the wiggle. The paper does not show that omitting off-diagonal terms flattens the series. So 'necessary' is not established; 'sufficient to reproduce' is. That is a meaningful difference, and the abstract's language overstates it.

Two lesser issues. The footnote about neglecting non-local band g-factor corrections is honest but unquantified; if those terms are sizable at the k-points that dominate 2s–4s, the pattern could shift. And no code or data is released, which makes it harder to run the missing control. The experimental comparison is qualitative, but the four data sets disagree anyway, so I would not hold that against the mechanism.

Bottom line: this is a solid paper with a genuine new formalism and a plausible resolution of a known puzzle. It deserves a serious referee. The referee should ask for the diagonal-only control and a discussion of what happens if non-local g-factor terms are included or estimated. I would send it out, and I would expect revision rather than rejection.

For my own work, I would cite it—the generalized formalism is useful regardless of the necessity claim.

Referee Report

3 major / 3 minor

Summary. This manuscript develops a generalized first-principles framework for many-body exciton g-factors in arbitrarily oriented magnetic fields, extending earlier GW-BSE treatments by including off-diagonal matrix elements of the spin and orbital angular momentum operators in both the single-particle and exciton bases. The method is applied to monolayer WSe2, where it reproduces the known low-energy exciton Zeeman splittings and the magnetic-field-induced brightening of dark and grey excitons, and it is used to address the long-standing experimental puzzle of the non-monotonic 1s-4s A-exciton g-factor Rydberg series. The authors also report magnetic-field-induced hybridization of higher-energy s- and p-like excitons and support their calculations with a symmetry analysis based on the D3h point group.

Significance. If the central claim is fully established, this work would provide a valuable and general ab initio tool for predicting magneto-optical responses of excitons in van der Waals materials, going beyond existing diagonal-only treatments. The low-energy validation is convincing: the GW-BSE results satisfy the symmetry constraints, reproduce dark/grey exciton brightening mechanisms, and are cross-checked against WIEN2k all-electron calculations. The explicit demonstration that off-diagonal elements appear in the computed g-factor matrices for higher excitons is also an important step. However, the key Rydberg-series conclusion currently rests on a comparison with a simplified two-band model rather than on a controlled diagonal-only limit of the same GW-BSE calculation, so the necessity part of the claim is not yet established. With the additional control calculation requested below, the paper would be a strong contribution to the field.

major comments (3)

[Fig. 4(b); SM Eq. (S12)] The central claim that off-diagonal spin/orbital matrix elements are responsible for the non-monotonic 1s-4s A-exciton g-factors is not established, because the only comparison made is between the full generalized GW-BSE result and a parabolic two-band model. When all off-diagonal single-particle and excitonic elements are set to zero, Eq. (1) reduces exactly to SM Eq. (S12), ⟨g_z⟩_S = Σ_{vck} |A^S_{vck}|^2 (g^z_{ck} − g^z_{vk}), which still contains the full k-dependent diagonal band g-factors. The two-band model uses g_X(k) = g_X(0) + g_X(2) k^2 with g_X(2) > 0, so its Rydberg series is monotonic by construction rather than as a consequence of neglecting off-diagonal terms. Since the GW-BSE 1s-4s envelopes in Fig. S6 sample progressively larger k, the non-monotonic pattern in Fig. 4(b) could arise from the non-parabolic k-dependence of the diagonal band g-factors alone. Please compute the diagonal-only limit of Eq. (S12) using the same GW-BSE envelopes and full k-dependent band g-factors; if it is still non-monotonic, the necessity of the off-diagonal terms for the Rydberg claim is not supported, and if it is monotonic, showing this explicitly would strengthen the paper.
[Footnote 61; SM Sec. I] The quantitative Rydberg-series prediction relies on orbital angular momentum matrix elements evaluated from DFT wavefunctions with a rigid GW scissor shift applied to the conduction bands, and the authors explicitly neglect non-local contributions to the band g-factors (footnote 61, Refs. [107,108]). The higher 2s-4s exciton envelopes weight larger |k| (Fig. S6), where the scissor approximation and the omitted non-local terms could modify the k-dependence of g^z_{ck} and g^z_{vk} in a state-dependent way. Please quantify this sensitivity, for example by comparing the k-dependent L^z matrix elements against the WIEN2k all-electron results over the k range sampled by the Rydberg envelopes, or by estimating the size of the non-local corrections from Refs. [107,108]; if those corrections are sizable, the predicted non-monotonic pattern could change.
[Fig. 4(a,b)] The experimental data in Fig. 4(a) come from hBN-encapsulated samples, while the GW-BSE calculation in Fig. 4(b) is performed for a vacuum environment; the two-band model is evaluated for both vacuum and hBN, but no GW-BSE calculation including the hBN dielectric environment is presented. Because the spatial extent and envelope k-distribution of Rydberg excitons depend on the dielectric environment, the comparison in support of the central claim is only qualitative. Please either provide a quantitative agreement metric or show explicitly that the dielectric environment does not change the predicted non-monotonic ordering.

minor comments (3)

[Fig. 4 caption] The caption labels both panels as (a); the second panel should be labeled (b).
[Introduction and abstract] There are several typographical errors, including 'perturbativelly' in the Introduction, 'non-m onotonic' in the title, and the panel labels in the Fig. 3 caption, where '(c) s+, (e) s−, and (e) z' should presumably read '(c) s+, (d) s−, and (e) z'.
[SM Sec. III.B] The sentence 'This can only be achieved with in-plane (B ⊥ z) fields' could be misread as applying to all G-D couplings; the D-G brightening under B ∥ z is correctly described elsewhere, so a small clarifying phrase about A-G/A-D mixing would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GW-BSE exciton g-factors are computed ab initio from momentum-matrix elements and BSE amplitudes, and the non-monotonic Rydberg series is compared with, not fitted to, experiment.

full rationale

The central derivation is self-contained. Equation (1) is derived in the Supplemental Material from the BSE Hamiltonian and the angular-momentum operators, and the single-particle g-factor matrix elements are evaluated from momentum matrix elements and band energies rather than fitted to any experimental exciton g-factor. The non-monotonic 1s-4s Rydberg series is not obtained by matching the experimental points in Fig. 4(a); the GW-BSE values are computed and then compared with the measurements of Refs. [36-39]. Earlier works cited for the orbital-angular-momentum formula (Wozniak et al. 2020, Amit et al. 2022) are prior independent numerical implementations, and the present paper re-derives the needed expressions in the Supplemental Material (Eqs. S9-S11), so no load-bearing self-citation is present. The effective two-band model's coefficients g(0) = -4.12 and g(2) = 96.84 are indeed taken from the same ab initio framework, but that model is used only as a simplified contrast and is not the basis of the central claim. The legitimate scientific gap is that the paper does not show a diagonal-only GW-BSE control obtained by setting the off-diagonal blocks to zero in Eq. (1), so the claim that off-diagonal terms are necessary for the non-monotonic series is not strictly proven by the presented comparison against a parabolic model; this is a completeness issue, not circularity. Similarly, footnote 61's statement that non-local band g-factor effects are not considered is a stated approximation, not a circular input. The derivation therefore does not reduce to its inputs by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central ab initio claim introduces no new entities or fitted constants targeting the experimental g-factors. The free parameters that appear are either baseline-model fits or standard approximation choices (scissor shift, symmetry-model splittings). The main assumptions are the validity of GW-BSE, the neglect of non-local and phonon effects, and the accuracy of the orbital angular momentum matrix elements.

free parameters (3)

two-band g(k) coefficients g_X(0), g_X(2) = -4.12 and 96.84
Fitted to the calculated single-particle g-factor dispersion in the effective two-band model; used as a baseline comparison, not in the ab initio central claim.
OAM scissor shift = 1.028 eV (GW gap at K, applied as rigid shift)
Applied to conduction bands when computing orbital angular momentum matrix elements; this choice affects the single-particle L values and hence the exciton g-factors, and it is not derived from the target result.
symmetry-model splittings Delta and delta = 54.8 meV and 2.4 meV
Fitted to GW-BSE exciton energies; used only in the validation symmetry model, not in the ab initio g-factor calculation.

assumptions (5)

domain assumption GW-BSE within the Tamm-Dancoff approximation yields accurate exciton wavefunctions and energies for WSe2.
The central g-factor formula uses BSE exciton amplitudes; the Tamm-Dancoff approximation neglects anti-resonant electron-hole terms.
domain assumption The magnetic coupling is fully described by H = H_BSE + (L + S) . B with g0 approximately 2.
This defines the generalized g-factor operator; it neglects non-local corrections and possible field-induced modifications of the wavefunctions.
domain assumption Orbital angular momentum matrix elements from k.p perturbation theory are converged with a finite band summation and a rigid GW scissor shift.
Underlies Eq. S9/S10; convergence is checked with 600 bands, but no formal bound is given and the scissor shift is an approximation.
domain assumption Direct excitons only, with negligible coupling to phonons or electrons.
The many-body basis excludes indirect excitons and phonon-assisted processes, as stated in Sec. II A of the Supplemental Material.
standard math Group-theory character tables and compatibility relations for D3h and C3h point groups are correct.
Used to classify exciton irreps and determine symmetry-allowed magnetic couplings.

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Cite this review

Pith. "Pith review of Generalized many-body exciton g-factors: magnetic hybridization and non-monotonic Rydberg series in monolayer WSe$_2$." pith.science (2026). https://pith.science/paper/KFA6TWXM

@misc{pith2026250518468,
  author       = {Pith},
  title        = {Pith review of: Generalized many-body exciton g-factors: magnetic hybridization and non-monotonic Rydberg series in monolayer WSe$_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KFA6TWXM}},
  note         = {Machine review of arXiv:2505.18468}
}

abstract

Magneto-optics of low dimensional semiconductors, such as monolayer transition metal dichalcogenides, offers a vast playground for exploring complex quantum phenomena. However, current ab initio approaches fail to capture important experimental observations related to brightening of excitonic levels and their g-factor dependence. Here, we develop a robust and general first principles framework for many-body exciton g-factors by incorporating off-diagonal terms for the spin and orbital angular momenta of single-particle bands and many-body states for magnetic fields pointing in arbitrary spatial directions. We implement our framework using many-body perturbation theory via the GW-Bethe-Salpeter equation (BSE) and supplement our analysis with robust symmetry-based models, establishing a fruitful synergy between many-body GW-BSE and group theory. Focusing on the archetypal monolayer WSe$_2$, we accurately reproduce the known results of the low-energy excitons including the Zeeman splitting and the dark/grey exciton brightening. Furthermore, our theory naturally reveals fundamental physical mechanisms of magnetic-field hybridization of higher-energy excitons (s- and p-like) and resolves the long-standing puzzle of the experimentally measured non-monotonic Rydberg series (1s-4s) of exciton g-factors. Our framework offers a comprehensive approach to investigate, rationalize, and predict the non-trivial interplay between magnetic fields, angular momenta, and many-body exciton physics in van der Waals systems.

Figures

Figures reproduced from arXiv: 2505.18468 by the authors.

**Figure 1.** (a) Top view of the monolayer WSe2. The colored parallelogram indicates the primitive unit cell. (b) First Brillouin zone. The colored triangle indicates the irreducible wedge. The dashed lines in panels (a,b) indicate the mirror planes σv. (c) Low-energy bands at K/−K valleys, including the irreps and spin orientation. (d) Low-energy exciton states at the Γ-point, identified by their labeling and irreps (see main … view at source ↗

**Figure 2.** Zeeman splitting of the low-energy excitons under [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Absolute value of the (a) out-of-plane g-factor ma [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: (a) Experimental g-factors of the Rydberg series [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Forward citations

Cited by 2 Pith papers

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A new GW-BSE-based theory of exciton orbital magnetic moment, with envelope-winding and center-of-mass Berry geometry terms, reproduces measured valley g-factors in biased bilayer graphene.
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Reference graph

Works this paper leans on

151 extracted references · 74 canonical work pages · cited by 2 Pith papers

[1]

For the latter, we need to consider oﬀ-diagonal matrix elements of single-particle operators in the Bloch basis

extends on previous derivations [56, 57] by considering oﬀ-diagonal terms in the exciton g-factor, not only in the excitonic but also in the single- particle electron/hole manifold (band g-factors). For the latter, we need to consider oﬀ-diagonal matrix elements of single-particle operators in the Bloch basis. For in- stance, the orbital angular momentum ...
[2]

We consider the nonzero oscillator strengths only for s-like ex- citon states

Several exciton peaks emerge around 1.95 eV, visible for θ = 45◦, 90◦. We consider the nonzero oscillator strengths only for s-like ex- citon states. pabilities of our approach, we address the long-standing puzzle and conﬂicting experimental reports of the Ry- dberg series of the A exciton g-factors for the so-called 1s–4s states. Fig. 4(a) compiles the a...

2023
[3]

Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Cole- man, and M. S. Strano, Electronics and optoelectron- ics of two-dimensional transition metal dichalcogenides, Nat. Nanotechnol. 7, 699 (2012)

2012
[4]

A. K. Geim and I. V. Grigorieva, Van der Waals het- erostructures, Nature 499, 419 (2013)

2013
[5]

K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. C. Neto, 2D materials and van der Waals het- erostructures, Science 353, aac9439 (2016)

2016
[6]

Y. Liu, N. O. Weiss, X. Duan, H.-C. Cheng, Y. Huang, and X. Duan, Van der Waals heterostructures and de- vices, Nat. Rev. Mater. 1, 16042 (2016)

2016
[7]

Manzeli, D

S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev, and A. Kis, 2D transition metal dichalcogenides, Nat. Rev. Mater. 2, 17033 (2017)

2017
[8]

Radisavljevic, A

B. Radisavljevic, A. Radenovic, J. Brivio, V. Gia- cometti, and A. Kis, Single-layer MoS2 transistors, Nat. Nanotechnol. 6, 147 (2011)

2011

Show all 151 references

[9]

L´ opez-S´ anchez, D

O. L´ opez-S´ anchez, D. Lembke, M. Kayci, A. Raden- ovic, and A. Kis, Ultrasensitive photodetectors based on monolayer MoS 2, Nat. Nanotechnol. 8, 497 (2013)

2013
[10]

Jariwala, V

D. Jariwala, V. K. Sangwan, L. J. Lauhon, T. J. Marks, and M. C. Hersam, Emerging device appli- cations for semiconducting two-dimensional transition metal dichalcogenides, ACS Nano 8, 1102 (2014)

2014
[11]

Pospischil, M

A. Pospischil, M. M. Furchi, and T. Mueller, Solar- energy conversion and light emission in an atomic mono- layer p–n diode, Nat. Nanotechnol. 9, 257 (2014)

2014
[12]

Withers, O

F. Withers, O. Del Pozo-Zamudio, A. Mishchenko, A. P. 6 Rooney, A. Gholinia, K. Watanabe, T. Taniguchi, S. J. Haigh, A. K. Geim, A. I. Tartakovskii, and K. S. Novoselov, Light-emitting diodes by band-structure en- gineering in van der Waals heterostructures, Nat. Mater. 14, 301 (2015)

2015
[13]

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, Valleytronics in 2D materials, Nat. Rev. Mater. 1, 16055 (2016)

2016
[14]

Z. Ye, D. Sun, and T. F. Heinz, Optical manipulation of valley pseudospin, Nat. Phys. 13, 26 (2017)

2017
[15]

K. F. Mak and J. Shan, Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides, Nat. Photonics 10, 216 (2016)

2016
[16]

K. F. Mak, D. Xiao, and J. Shan, Light–valley inter- actions in 2D semiconductors, Nat. Photonics 12, 451 (2018)

2018
[17]

K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Atomically thin MoS 2: A new direct-gap semiconduc- tor, Phys. Rev. Lett. 105, 136805 (2010)

2010
[18]

Splendiani, L

A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli, and F. Wang, Emerging photolumines- cence in monolayer MoS 2, Nano Lett. 10, 1271 (2010)

2010
[19]

D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Optical spectrum of MoS 2: Many-body eﬀects and diversity of exciton states, Phys. Rev. Lett. 111, 216805 (2013)

2013
[20]

Chernikov, T

A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi, Y. Li, B. Aslan, D. R. Reichman, M. S. Hybertsen, and T. F. Heinz, Exciton binding energy and nonhydrogenic rydberg series in monolayer WS2, Phys. Rev. Lett. 113, 076802 (2014)

2014
[21]

G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, Colloquium: Excitons in atomically thin transition metal dichalco- genides, Rev. Mod. Phys. 90, 021001 (2018)

2018
[22]

T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu, and J. Feng, Valley- selective circular dichroism of monolayer molybdenum disulphide, Nat. Commun. 3, 887 (2012)

2012
[23]

Xiao, G.-B

D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Cou- pled spin and valley physics in monolayers of MoS 2 and other group-VI dichalcogenides, Phys. Rev. Lett. 108, 196802 (2012)

2012
[24]

Sallen, L

G. Sallen, L. Bouet, X. Marie, G. Wang, C. R. Zhu, W. P. Han, Y. Lu, P. H. Tan, T. Amand, B. L. Liu, and B. Urbaszek, Robust optical emission polarization in MoS2 monolayers through selective valley excitation, Phys. Rev. B 86, 081301 (2012)

2012
[25]

K. F. Mak, K. He, J. Shan, and T. F. Heinz, Control of valley polarization in monolayer MoS 2 by optical helic- ity, Nat. Nanotechnol. 7, 494 (2012)

2012
[26]

Y. Li, J. Ludwig, T. Low, A. Chernikov, X. Cui, G. Arefe, Y. D. Kim, A. M. van der Zande, A. Rigosi, H. M. Hill, S. H. Kim, J. Hone, Z. Li, D. Smirnov, and T. F. Heinz, Valley splitting and polarization by the Zee- man eﬀect in monolayer MoSe 2, Phys. Rev. Lett. 113, 266804 (2014)

2014
[27]

MacNeill, C

D. MacNeill, C. Heikes, K. F. Mak, Z. Anderson, A. Korm´ anyos, V. Z´ olyomi, J. Park, and D. C. Ralph, Breaking of valley degeneracy by magnetic ﬁeld in monolayer MoSe2, Phys. Rev. Lett. 114, 037401 (2015)

2015
[28]

G. Wang, L. Bouet, M. Glazov, T. Amand, E. Ivchenko, E. Palleau, X. Marie, and B. Urbaszek, Magneto-optics in transition metal diselenide monolayers, 2D Mater. 2, 034002 (2015)

2015
[29]

Srivastava, M

A. Srivastava, M. Sidler, A. V. Allain, D. S. Lembke, A. Kis, and A. Imamo˘ glu, Valley Zeeman eﬀect in el- ementary optical excitations of monolayer WSe 2, Nat. Phys. 11, 141 (2015)

2015
[30]

Aivazian, Z

G. Aivazian, Z. Gong, A. M. Jones, R.-L. Chu, J. Yan, D. G. Mandrus, C. Zhang, D. Cobden, W. Yao, and X. Xu, Magnetic control of valley pseudospin in mono- layer WSe2, Nat. Phys. 11, 148 (2015)

2015
[31]

Mitioglu, P

A. Mitioglu, P. Plochocka, ´A. Granados del Aguila, P. Christianen, G. Deligeorgis, S. Anghel, L. Kulyuk, and D. Maude, Optical investigation of monolayer and bulk tungsten diselenide (WSe2) in high magnetic ﬁelds, Nano Lett. 15, 4387 (2015)

2015
[32]

A. V. Stier, K. M. McCreary, B. T. Jonker, J. Kono, and S. A. Crooker, Exciton diamagnetic shifts and valley Zeeman eﬀects in monolayer WS 2 and MoS2 to 65 tesla, Nat. Commun. 7, 10643 (2016)

2016
[33]

A. V. Stier, N. P. Wilson, G. Clark, X. Xu, and S. A. Crooker, Probing the inﬂuence of dielectric environment on excitons in monolayer WSe2: insight from high mag- netic ﬁelds, Nano Lett. 16, 7054 (2016)

2016
[34]

Schmidt, A

R. Schmidt, A. Arora, G. Plechinger, P. Nagler, A. Granados del ´Aguila, M. V. Ballottin, P. C. M. Christianen, S. Michaelis de Vasconcellos, C. Sch¨ uller, T. Korn, and R. Bratschitsch, Magnetic-ﬁeld-induced rotation of polarized light emission from monolayer WS2, Phys. Rev. ...

2016
[35]

Robert, T

C. Robert, T. Amand, F. Cadiz, D. Lagarde, E. Cour- tade, M. Manca, T. Taniguchi, K. Watanabe, B. Ur- baszek, and X. Marie, Fine structure and lifetime of dark excitons in transition metal dichalcogenide mono- layers, Phys. Rev. B 96, 155423 (2017)

2017
[36]

Zhang, T

X.-X. Zhang, T. Cao, Z. Lu, Y.-C. Lin, F. Zhang, Y. Wang, Z. Li, J. C. Hone, J. A. Robinson, D. Smirnov, S. G. Louie, and T. F. Heinz, Magnetic brightening and control of dark excitons in monolayer WSe 2, Nat. Nan- otech. 12, 883 (2017)

2017
[37]

Arora, M

A. Arora, M. Koperski, A. Slobodeniuk, K. Nogajewski, R. Schmidt, R. Schneider, M. R. Molas, S. M. de Vas- concellos, R. Bratschitsch, and M. Potemski, Zeeman spectroscopy of excitons and hybridization of electronic states in few-layer WSe2, MoSe2 and MoTe2, 2D Mater. 6, 015010 (2018)

2018
[38]

A. V. Stier, N. P. Wilson, K. A. Velizhanin, J. Kono, X. Xu, and S. A. Crooker, Magnetooptics of exciton ry- dberg states in a monolayer semiconductor, Phys. Rev. Lett. 120, 057405 (2018)

2018
[39]

E. Liu, J. van Baren, T. Taniguchi, K. Watanabe, Y.- C. Chang, and C. H. Lui, Magnetophotoluminescence of exciton Rydberg states in monolayer WSe 2, Phys. Rev. B 99, 205420 (2019)

2019
[40]

S.-Y. Chen, Z. Lu, T. Goldstein, J. Tong, A. Chaves, J. Kunstmann, L. S. R. Cavalcante, T. Wo´ zniak, G. Seifert, D. R. Reichman, T. Taniguchi, K. Watan- abe, D. Smirnov, and J. Yan, Luminescent emission of excited rydberg excitons from monolayer WSe 2, Nano Lett. 19, 2464 (2019)

2019
[41]

T. Wang, Z. Li, Y. Li, Z. Lu, S. Miao, Z. Lian, Y. Meng, M. Blei, T. Taniguchi, K. Watanabe, S. Ton- gay, D. Smirnov, C. Zhang, and S.-F. Shi, Giant valley- polarized Rydberg excitons in monolayer WSe2 revealed by magneto-photocurrent spectroscopy, Nano Lett. 20, 7635 (2020). 7

2020
[42]

Robert, H

C. Robert, H. Dery, L. Ren, D. Van Tuan, E. Cour- tade, M. Yang, B. Urbaszek, D. Lagarde, K. Watanabe, T. Taniguchi, T. Amand, and X. Marie, Measurement of conduction and valence bands g-factors in a transi- tion metal dichalcogenide monolayer, Phys. Rev. Lett. 126, 067403 (2021)

2021
[43]

Zinkiewicz, T

M. Zinkiewicz, T. Wo´ zniak, T. Kazimierczuk, P. Kapus- cinski, K. Oreszczuk, M. Grzeszczyk, M. Bartoˇ s, K. No- gajewski, K. Watanabe, T. Taniguchi, C. Faugeras, P. Kossacki, M. Potemski, A. Babi´ nski, and M. R. Mo- las, Excitonic complexes in n-doped WS 2 monolayer, Nano Le...

2021
[44]

Arora, Magneto-optics of layered two-dimensional semiconductors and heterostructures: Progress and prospects, J

A. Arora, Magneto-optics of layered two-dimensional semiconductors and heterostructures: Progress and prospects, J. Appl. Phys. 129, 120902 (2021)

2021
[45]

P. E. Faria Junior, K. Zollner, T. Wo´ zniak, M. Kur- pas, M. Gmitra, and J. Fabian, First-principles insights into the spin-valley physics of strained transition metal dichalcogenides monolayers, New J. Phys. 24, 083004 (2022)

2022
[46]

F. S. Covre, P. E. Faria Junior, V. O. Gordo, C. S. de Brito, Y. V. Zhumagulov, M. D. Teodoro, O. D. D. Couto, L. Misoguti, S. Pratavieira, M. B. Andrade, P. C. M. Christianen, J. Fabian, F. Withers, and Y. Galv˜ ao Gobato, Revealing the impact of strain in the optical propert...

2022
[47]

Blundo, P

E. Blundo, P. E. Faria Junior, A. Surrente, G. Pet- tinari, M. A. Prosnikov, K. Olkowska-Pucko, K. Zoll- ner, T. Wo´ zniak, A. Chaves, T. Kazimierczuk, M. Fe- lici, A. Babi´ nski, M. R. Molas, P. C. M. Christianen, J. Fabian, and A. Polimeni, Strain-induced exciton hy- bridiza...

2022
[48]

P. E. Faria Junior, T. Naimer, K. M. McCreary, B. T. Jonker, J. J. Finley, S. A. Crooker, J. Fabian, and A. V. Stier, Proximity-enhanced valley Zeeman splitting at the WS 2/graphene interface, 2D Mater. 10, 034002 (2023)

2023
[49]

Olkowska-Pucko, T

K. Olkowska-Pucko, T. Wo´ zniak, E. Blundo, N. Za- wadzka, /suppress Lucja Kipczak, P. E. Faria Junior, J. Sz- pakowski, G. Krasucki, S. Cianci, D. Vaclavkova, D. Jana, P. Kapu´ sci´ nski, M. Grzeszczyk, D. Cecchetti, G. Pettinari, I. Antoniazzi, Z. Sofer, I. Plutnarov´ a, K. ...

2025
[50]

Cho, Uniﬁed theory of symmetry-breaking eﬀects on excitons in cubic and wurtzite structures, Phys

K. Cho, Uniﬁed theory of symmetry-breaking eﬀects on excitons in cubic and wurtzite structures, Phys. Rev. B 14, 4463 (1976)

1976
[51]

Venghaus, S

H. Venghaus, S. Suga, and K. Cho, Magnetolumines- cence and magnetoreﬂectance of the A exciton of CdS and CdSe, Phys. Rev. B 16, 4419 (1977)

1977
[52]

M. R. Molas, C. Faugeras, A. O. Slobodeniuk, K. No- gajewski, M. Bartos, D. M. Basko, and M. Potemski, Brightening of dark excitons in monolayers of semicon- ducting transition metal dichalcogenides, 2D Mater. 4, 021003 (2017)

2017
[53]

M. R. Molas, A. O. Slobodeniuk, T. Kazimier- czuk, K. Nogajewski, M. Bartos, P. Kapu´ sci´ nski, K. Oreszczuk, K. Watanabe, T. Taniguchi, C. Faugeras, P. Kossacki, D. M. Basko, and M. Potemski, Probing and manipulating valley coherence of dark excitons in monolayer WSe2, Phys....

2019
[54]

M. S. Hybertsen and S. G. Louie, Electron correlation in semiconductors and insulators: Band gaps and quasi- particle energies, Phys. Rev. B 34, 5390 (1986)

1986
[56]

Albrecht, L

S. Albrecht, L. Reining, R. Del Sole, and G. Onida, Ab initio calculation of excitonic eﬀects in the optical spectra of semiconductors, Phys. Rev. Lett. 80, 4510 (1998)

1998
[57]

Blase, I

X. Blase, I. Duchemin, D. Jacquemin, and P.-F. Loos, The Bethe–Salpeter equation formalism: From physics to chemistry, J. Phys. Chem. Lett. 11, 7371 (2020)

2020
[58]

Deilmann, P

T. Deilmann, P. Kr¨ uger, and M. Rohlﬁng, Ab initio studies of exciton g factors: Monolayer transition metal dichalcogenides in magnetic ﬁelds, Phys. Rev. Lett. 124, 226402 (2020)

2020
[59]

T. Amit, D. Hernang´ omez-P´ erez, G. Cohen, D. Y. Qiu, and S. Refaely-Abramson, Tunable magneto-optical properties in MoS 2 via defect-induced exciton transi- tions, Phys. Rev. B 106, L161407 (2022)

2022
[60]

Kipczak, A

L. Kipczak, A. O. Slobodeniuk, T. Wo´ zniak, M. Bhat- nagar, N. Zawadzka, K. O. Pucko, M. J. Grzeszczyk, K. Watanabe, T. Taniguchi, A. Babinski, and M. Mo- las, Analogy and dissimilarity of excitons in monolayer and bilayer of MoSe 2, 2D Mater. , 025014 (2023)

2023
[61]

It also contains Refs

See Supplemental Material at URL for the computa- tional details, the representation of spin and orbital operators in the excitonic basis, the symmetry analy- sis, additional computational data, and additional ﬁg- ures. It also contains Refs. [ 33, 43, 46, 50–53, 56– 58, 66, 6...
[62]

In this form, the g-factor is simply the total angular mo- mentum, J = L + S with spin-orbit eﬀects incorporated via the basis functions
[63]

We focus on the combined eﬀect of the ex- citonic properties within nearly-degenerate subspaces

Non-local eﬀects in the band g-factors [ 107, 108] are not considered. We focus on the combined eﬀect of the ex- citonic properties within nearly-degenerate subspaces
[64]

Koperski, K

M. Koperski, K. Nogajewski, A. Arora, V. Cherkez, P. Mallet, J.-Y. Veuillen, J. Marcus, P. Kossacki, and M. Potemski, Single photon emitters in exfoliated WSe2 structures, Nat. Nanotech. 10, 503 (2015)

2015
[65]

K.-Q. Lin, C. S. Ong, S. Bange, P. E. Faria Junior, B. Peng, J. D. Ziegler, J. Zipfel, C. B¨ auml, N. Paradiso, K. Watanabe, T. Taniguchi, C. Strunk, B. Monserrat, J. Fabian, A. Chernikov, D. Y. Qiu, S. G. Louie, and J. M. Lupton, Narrow-band high-lying excitons with negative-...

2021
[66]

Dery and Y

H. Dery and Y. Song, Polarization analysis of excitons in monolayer and bilayer transition-metal dichalcogenides, Phys. Rev. B 92, 125431 (2015)

2015
[67]

G. Wang, C. Robert, M. M. Glazov, F. Cadiz, E. Cour- tade, T. Amand, D. Lagarde, T. Taniguchi, K. Watan- abe, B. Urbaszek, and X. Marie, In-plane propagation of light in transition metal dichalcogenide monolayers: Optical selection rules, Phys. Rev. Lett. 119, 047401 (2017)

2017
[68]

G. F. Koster, J. O. Dimmock, R. G. Wheeler, and H. Statz, Properties of the thirty-two point groups , Vol. 24 (MIT press, 1963)

1963
[69]

M. M. Glazov, T. Amand, X. Marie, D. Lagarde, 8 L. Bouet, and B. Urbaszek, Exciton ﬁne structure and spin decoherence in monolayers of transition metal dichalcogenides, Phys. Rev. B 89, 201302 (2014)

2014
[70]

J. V. V. Cassiano, A. de Lelis Ara´ ujo, P. E. Faria Junior, and G. J. Ferreira, Dft2kp: Eﬀective kp models from ab- initio data, SciPost Physics Codebases25, 10.21468/sci- postphyscodeb.25 (2024)

2024 doi
[71]

Zhang, H

S. Zhang, H. Sheng, Z.-D. Song, C. Liang, Y. Jiang, S. Sun, Q. Wu, H. Weng, Z. Fang, X. Dai, et al. , Vasp2kp: k·p models and Land´ e g-factors from ab initio calculations, Chin. Phys. Lett. 40, 127101 (2023)

2023
[72]

In photoluminescence experiments, the optical spectra incorporate the optical selection rules and the exciton occupation (Boltzmann distribution function [ 109] with rapid exponential decay). Therefore, the emission of A, G, and D excitons display similar intensities [ 33, 50,...
[73]

B. Zhu, K. Xiao, S. Yang, K. Watanabe, T. Taniguchi, and X. Cui, In-plane electric-ﬁeld-induced orbital hy- bridization of excitonic states in monolayer WSe2, Phys. Rev. Lett. 131, 036901 (2023)

2023
[74]

J. D. Cao, K. S. Denisov, and I. Zutic, Tunable resonant s-p mixing of excitons in van der Waals heterostructures, arXiv:2503.11927 (2025)

2025
[75]

Mennel, M

L. Mennel, M. M. Furchi, S. Wachter, M. Paur, D. K. Polyushkin, and T. Mueller, Optical imaging of strain in two-dimensional crystals, Nat. Commun. 9, 516 (2018)

2018
[76]

A. Raja, L. Waldecker, J. Zipfel, Y. Cho, S. Brem, J. D. Ziegler, M. Kulig, T. Taniguchi, K. Watanabe, E. Malic, T. F. Heinz, T. C. Berkelbach, and A. Chernikov, Dielectric disorder in two-dimensional materials, Nat. Nanotech. 14, 832 (2019)

2019
[77]

P. V. Kolesnichenko, Q. Zhang, T. Yun, C. Zheng, M. S. Fuhrer, and J. A. Davis, Disentangling the eﬀects of doping, strain and disorder in monolayer WS2 by optical spectroscopy, 2D Mater. 7, 025008 (2020)

2020
[78]

Raiber, P

S. Raiber, P. E. Faria Junior, D. Falter, S. Feldl, P. Marzena, K. Watanabe, T. Taniguchi, J. Fabian, and C. Sch¨ uller, Ultrafast pseudospin quantum beats in multilayer WSe2 and MoSe 2, Nat. Commun. 13, 4997 (2022)

2022
[79]

B. Hou, D. Wang, B. A. Barker, and D. Y. Qiu, Exchange-driven intermixing of bulk and topological surface states by chiral excitons in bi 2se3, Phys. Rev. Lett. 130, 216402 (2023)

2023
[80]

H.-Y. Xie, P. Ghaemi, M. Mitrano, and B. Uchoa, The- ory of topological exciton insulators and condensates in ﬂat chern bands, PNAS 121, e2401644121 (2024)

2024
[81]

T. P. Cysne, L. M. Canonico, M. Costa, R. B. Muniz, and T. G. Rappoport, Orbitronics in two-dimensional materials, arXiv:2502.12339 (2025)

2025
[82]

Schutte, J

W. Schutte, J. De Boer, and F. Jellinek, Crystal struc- tures of tungsten disulﬁde and diselenide, J. Solid State Chem. 70, 207 (1987)

1987
[83]

Camarasa-G´ omez, A

M. Camarasa-G´ omez, A. Ramasubramaniam, J. B. Neaton, and L. Kronik, Transferable screened range- separated hybrid functionals for electronic and optical properties of van der Waals materials, Phys. Rev. Mater. 7, 104001 (2023)

2023
[84]

Giannozzi, S

P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Giron- coli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. ...

2009
[85]

Giannozzi, O

P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. Dal Corso, S. de Gironcoli, P. Delu- gas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Geba...

2017
[86]

Giannozzi, O

P. Giannozzi, O. Baseggio, P. Bonf` a, D. Brunato, R. Car, I. Carnimeo, C. Cavazzoni, S. de Giron- coli, P. Delugas, F. Ferrari Ruﬃno, A. Ferretti, N. Marzari, I. Timrov, A. Urru, and S. Baroni, Quan- tum ESPRESSO toward the exascale, J. Chem. Phys. 152, 154105 (2020)

2020
[87]

J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996)

1996
[88]

M. J. van Setten, M. Giantomassi, E. Bousquet, M. J. Verstraete, D. R. Hamann, X. Gonze, and G.-M. Rig- nanese, The pseudodojo: Training and grading a 85 ele- ment optimized norm-conserving pseudopotential table, Comput. Phys. Commun. 226, 39 (2018)

2018
[89]

Blaha, K

P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. Madsen, and L. D. Marks, Wien2k: An APW+lo pro- gram for calculating the properties of solids, J. Chem. Phys. 152, 074101 (2020)

2020
[90]

Lejaeghere, G

K. Lejaeghere, G. Bihlmayer, T. Bj¨ orkman, P. Blaha, S. Bl¨ ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Du/suppress lak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Gen- ove...

2016
[91]

D. J. Singh and L. Nordstrom, Planewaves, Pseudopo- tentials, and the LAPW method (Springer Science & Business Media, 2006). 9

2006
[92]

Deslippe, G

J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, BerkeleyGW: A mas- sively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures, Comput. Phys. Commun. 183, 1269 (2012)

2012
[94]

Wu, Spin-Orbit Coupling, Broken Time-Reversal Symmetry, and Polarizability Self-Consistency in GW and GW-BSE Theory with Applications to Two- Dimensional Materials , Ph.D

M. Wu, Spin-Orbit Coupling, Broken Time-Reversal Symmetry, and Polarizability Self-Consistency in GW and GW-BSE Theory with Applications to Two- Dimensional Materials , Ph.D. thesis, University of Cal- ifornia, Berkeley (2020)

2020
[95]

B. A. Barker, J. Deslippe, J. Lischner, M. Jain, O. V. Yazyev, D. A. Strubbe, and S. G. Louie, Spinor GW /Bethe-Salpeter calculations in BerkeleyGW: Im- plementation, symmetries, benchmarking, and perfor- mance, Phys. Rev. B 106, 115127 (2022)

2022
[96]

F. H. da Jornada, D. Y. Qiu, and S. G. Louie, Nonuni- form sampling schemes of the Brillouin zone for many- electron perturbation-theory calculations in reduced di- mensionality, Phys. Rev. B 95, 035109 (2017)

2017
[97]

Ismail-Beigi, Truncation of periodic image interac- tions for conﬁned systems, Phys

S. Ismail-Beigi, Truncation of periodic image interac- tions for conﬁned systems, Phys. Rev. B 73, 233103 (2006)

2006
[98]

Wo´ zniak, P

T. Wo´ zniak, P. E. Faria Junior, G. Seifert, A. Chaves, and J. Kunstmann, Exciton g-factors of van der Waals heterostructures from ﬁrst-principles calculations, Phys. Rev. B 101, 235408 (2020)

2020
[100]

L. V. Keldysh, Coulomb interaction in thin semiconduc- tor and semimetal ﬁlms, JETP Lett. 29, 658 (1979)

1979
[101]

Laturia, M

A. Laturia, M. L. Van de Put, and W. G. Vanden- berghe, Dielectric properties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bulk, npj 2D Mater. Appl. 2, 6 (2018)

2018
[102]

Y. G. Gobato, C. S. de Brito, A. Chaves, M. A. Pros- nikov, T. Wo´ zniak, S. Guo, I. D. Barcelos, M. V. Miloˇ sevi´ c, F. Withers, and P. C. M. Christianen, Dis- tinctive g-factor of moir´ e-conﬁned excitons in van der Waals heterostructures, Nano Lett. 22, 8641 (2022)

2022
[103]

Kurpas, M

M. Kurpas, M. Gmitra, and J. Fabian, Spin-orbit cou- pling and spin relaxation in phosphorene: Intrinsic ver- sus extrinsic eﬀects, Phys. Rev. B 94, 155423 (2016)

2016
[104]

Kurpas, P

M. Kurpas, P. E. Faria Junior, M. Gmitra, and J. Fabian, Spin-orbit coupling in elemental two- dimensional materials, Phys. Rev. B 100, 125422 (2019)

2019
[105]

Yafet, g factors and spin-lattice relaxation of conduc- tion electrons, in Solid state physics , Vol

Y. Yafet, g factors and spin-lattice relaxation of conduc- tion electrons, in Solid state physics , Vol. 14 (Elsevier, 1963)

1963
[106]

Bhowal and G

S. Bhowal and G. Vignale, Orbital hall eﬀect as an al- ternative to valley hall eﬀect in gapped graphene, Phys. Rev. B 103, 195309 (2021)

2021
[108]

J. J. Esteve-Paredes, M. A. Garc´ ıa-Bl´ azquez, A. J. Ur´ıa- ´Alvarez, M. Camarasa-G´ omez, and J. J. Palacios, Ex- citons in nonlinear optical responses: shift current in MoS2 and GeS monolayers, npj Comput. Mater. 11, 13 (2025)

2025
[109]

Xuan and S

F. Xuan and S. Y. Quek, Valley Zeeman eﬀect and Lan- dau levels in two-dimensional transition metal dichalco- genides, Phys. Rev. Res. 2, 033256 (2020)

2020
[110]

Xuan and S

F. Xuan and S. Y. Quek, Valley-ﬁlling instability and critical magnetic ﬁeld for interaction-enhanced Zee- man response in doped WSe 2 monolayers, npj Comput. Mater. 7, 198 (2021)

2021
[111]

Y. V. Zhumagulov, A. Vagov, D. R. Gulevich, P. E. Faria Junior, and V. Perebeinos, Trion induced pho- toluminescence of a doped MoS 2 monolayer, J. Chem. Phys. 153, 044132 (2020)

2020
[112]

Dirnberger, J

F. Dirnberger, J. D. Ziegler, P. E. Faria Junior, R. Bushati, T. Taniguchi, K. Watanabe, J. Fabian, D. Bougeard, A. Chernikov, and V. M. Menon, Quasi- 1D exciton channels in strain-engineered 2D materials, Sci. Adv. 7, eabj3066 (2021) . Generalized many-body exciton g-factors:...

2021
[113]

Note how the coeﬃcients depend on the choice of basis. In the numerical calculations within the GW-BSE formalism, we do no t expect the Hamiltonian to take such a intuitive form since the exciton basis will ver y likely a superposition of this well-deﬁned basis. Such “mixed” b...
[114]

This subspace is well-separated of the other excitons, as the next exciton state is ∼ 164 meV above the A exciton

The (bright-grey) A-G splitting is 52.4 meV while the (grey-dark) G-D sp litting is 2.4 meV. This subspace is well-separated of the other excitons, as the next exciton state is ∼ 164 meV above the A exciton. Exciton Energy |P+|2 |P−|2 |Pz|2 Irrep A 1771.58 783.025 4771.892 – Γ...

1900
[115]

Schutte, J

W. Schutte, J. De Boer, and F. Jellinek, Crystal structures of tungsten disulﬁde and diselenide, J. Solid State Chem. 70, 207 (1987)

1987
[116]

Camarasa-G´ omez, A

M. Camarasa-G´ omez, A. Ramasubramaniam, J. B. Neaton, and L. Kronik, Transfer able screened range-separated hybrid functionals for electronic and optic al properties of van der Waals materials, Phys. Rev. Mater. 7, 104001 (2023)

2023
[117]

Giannozzi, S

P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris , G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-S amos, N. Marzari, F. ...

2009
[118]

Giannozzi, O

P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Narde lli, M. Calandra, 23 R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnime o, A. Dal Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fr atesi, G. Fugallo, R. ...

2017
[119]

Giannozzi, O

P. Giannozzi, O. Baseggio, P. Bonf` a, D. Brunato, R. Car, I. Carnimeo, C. C avazzoni, S. de Gironcoli, P. Delugas, F. Ferrari Ruﬃno, A. Ferretti, N. Marzari, I . Timrov, A. Urru, and S. Baroni, Quantum ESPRESSO toward the exascale,J. Chem. Phys. 152, 154105 (2020)

2020
[120]

J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient ap proximation made simple, Phys. Rev. Lett. 77, 3865 (1996)

1996
[121]

M. J. van Setten, M. Giantomassi, E. Bousquet, M. J. Verstraete, D . R. Hamann, X. Gonze, and G.-M. Rignanese, The pseudodojo: Training and grading a 85 element optimized norm- conserving pseudopotential table, Comput. Phys. Commun. 226, 39 (2018)

2018
[122]

Blaha, K

P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. Madsen, and L. D. Mark s, Wien2k: An APW+lo program for calculating the properties of solids, J. Chem. Phys. 152, 074101 (2020)

2020
[123]

Lejaeghere, G

K. Lejaeghere, G. Bihlmayer, T. Bj¨ orkman, P. Blaha, S. Bl¨ ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. D ewhurst, I. D. Marco, C. Draxl, M. Du/suppress lak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genoves...

2016
[124]

D. J. Singh and L. Nordstrom, Planewaves, Pseudopotentials, and the LAPW method 24 (Springer Science & Business Media, 2006)

2006
[125]

Deslippe, G

J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, Berke- leyGW: A massively parallel computer package for the calculation of the qu asiparticle and optical properties of materials and nanostructures,Comput. Phys. Commun. 183, 1269 (2012)

2012
[126]

Rohlﬁng and S

M. Rohlﬁng and S. G. Louie, Electron-hole excitations in semiconductors and insulators, Phys. Rev. Lett. 81, 2312 (1998)

1998
[127]

Rohlﬁng and S

M. Rohlﬁng and S. G. Louie, Electron-hole excitations and optical spectra from ﬁrst principles, Phys. Rev. B 62, 4927 (2000)

2000
[128]

Wu, Spin-Orbit Coupling, Broken Time-Reversal Symmetry, and Polarizabil ity Self- Consistency in GW and GW-BSE Theory with Applications to Two-Di mensional Materials , Ph.D

M. Wu, Spin-Orbit Coupling, Broken Time-Reversal Symmetry, and Polarizabil ity Self- Consistency in GW and GW-BSE Theory with Applications to Two-Di mensional Materials , Ph.D. thesis, University of California, Berkeley (2020)

2020
[129]

M. S. Hybertsen and S. G. Louie, Electron correlation in semiconduct ors and insulators: Band gaps and quasiparticle energies, Phys. Rev. B 34, 5390 (1986)

1986
[130]

B. A. Barker, J. Deslippe, J. Lischner, M. Jain, O. V. Yazyev, D. A. S trubbe, and S. G. Louie, Spinor GW /Bethe-Salpeter calculations in BerkeleyGW: Implementation, sym metries, benchmarking, and performance, Phys. Rev. B 106, 115127 (2022)

2022
[131]

F. H. da Jornada, D. Y. Qiu, and S. G. Louie, Nonuniform sampling schemes of the Brillouin zone for many-electron perturbation-theory calculations in reduced dimensionality, Phys. Rev. B 95, 035109 (2017)

2017
[132]

Ismail-Beigi, Truncation of periodic image interactions for conﬁn ed systems, Phys

S. Ismail-Beigi, Truncation of periodic image interactions for conﬁn ed systems, Phys. Rev. B 73, 233103 (2006)

2006
[133]

Wo´ zniak, P

T. Wo´ zniak, P. E. Faria Junior, G. Seifert, A. Chaves, and J. Kun stmann, Exciton g-factors of van der Waals heterostructures from ﬁrst-principles calculations, Phys. Rev. B 101, 235408 (2020)

2020
[134]

T. Amit, D. Hernang´ omez-P´ erez, G. Cohen, D. Y. Qiu, and S. Refaely-Abramson, Tunable magneto-optical properties in MoS 2 via defect-induced exciton transitions, Phys. Rev. B 106, L161407 (2022)

2022
[135]

P. E. Faria Junior, K. Zollner, T. Wo´ zniak, M. Kurpas, M. Gmitra, and J. Fabian, First- principles insights into the spin-valley physics of strained tr ansition metal dichalcogenides monolayers, New J. Phys. 24, 083004 (2022)

2022
[136]

P. E. Faria Junior, T. Naimer, K. M. McCreary, B. T. Jonker, J. J. Fi nley, S. A. Crooker, 25 J. Fabian, and A. V. Stier, Proximity-enhanced valley Zeeman splitting at the WS 2/graphene interface, 2D Mater. 10, 034002 (2023)

2023
[137]

Kipczak, A

L. Kipczak, A. O. Slobodeniuk, T. Wo´ zniak, M. Bhatnagar, N. Zawadzka, K. O. Pucko, M. J. Grzeszczyk, K. Watanabe, T. Taniguchi, A. Babinski, and M. Molas, Analogy anddissimilarity of excitons in monolayer and bilayer of MoSe 2, 2D Mater. , 025014 (2023)

2023
[138]

N. S. Rytova, Screened potential of a point charge in a thin ﬁlm, Moscow University Physics Bulletin 3, 30 (1967)

1967
[139]

L. V. Keldysh, Coulomb interaction in thin semiconductor and semime tal ﬁlms, JETP Lett. 29, 658 (1979)

1979
[140]

Laturia, M

A. Laturia, M. L. Van de Put, and W. G. Vandenberghe, Dielectric proper ties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bu lk, npj 2D Mater. Appl. 2, 6 (2018)

2018
[141]

Deilmann, P

T. Deilmann, P. Kr¨ uger, and M. Rohlﬁng, Ab initio studies of excit on g factors: Monolayer transition metal dichalcogenides in magnetic ﬁelds, Phys. Rev. Lett. 124, 226402 (2020)

2020
[142]

Y. G. Gobato, C. S. de Brito, A. Chaves, M. A. Prosnikov, T. Wo´ zniak, S. Guo, I. D. Barcelos, M. V. Miloˇ sevi´ c, F. Withers, and P. C. M. Christianen, Distinctive g-factor of moir´ e-conﬁned excitons in van der Waals heterostructures, Nano Lett. 22, 8641 (2022)

2022
[143]

J. Ruan, Z. Li, C. S. Ong, and S. G. Louie, Optically controlled single- valley exciton doublet states with tunable internal spin structures and spin magnetization generation, PNAS 120, e2307611120 (2023)

2023
[144]

J. J. Esteve-Paredes, M. A. Garc´ ıa-Bl´ azquez, A. J. Ur´ ıa-´Alvarez, M. Camarasa-G´ omez, and J. J. Palacios, Excitons in nonlinear optical responses: shift curren t in MoS 2 and GeS mono- layers, npj Comput. Mater. 11, 13 (2025)

2025
[145]

Yafet, g factors and spin-lattice relaxation of conduction electrons , in Solid state physics , Vol

Y. Yafet, g factors and spin-lattice relaxation of conduction electrons , in Solid state physics , Vol. 14 (Elsevier, 1963)

1963
[146]

Bhowal and G

S. Bhowal and G. Vignale, Orbital hall eﬀect as an alternative to valley h all eﬀect in gapped graphene, Phys. Rev. B 103, 195309 (2021)

2021
[147]

A. Urru, O. P. O. Ivo Souza, S. S. Tsirkin, and D. Vanderbilt, Optic al spatial dispersion via wannier interpolation, arXiv:2504.09742 (2025)

2025 arXiv
[148]

G. F. Koster, J. O. Dimmock, R. G. Wheeler, and H. Statz, Properties of the thirty-two point groups, Vol. 24 (MIT press, 1963). 26

1963
[149]

Kurpas, M

M. Kurpas, M. Gmitra, and J. Fabian, Spin-orbit coupling and spin relaxation in phosphorene: Intrinsic versus extrinsic eﬀects, Phys. Rev. B 94, 155423 (2016)

2016
[150]

Kurpas, P

M. Kurpas, P. E. Faria Junior, M. Gmitra, and J. Fabian, Spin-orbit c oupling in elemental two-dimensional materials, Phys. Rev. B 100, 125422 (2019)

2019
[151]

J. V. V. Cassiano, A. de Lelis Ara´ ujo, P. E. Faria Junior, and G. J. Ferreira, Dft2kp: Eﬀective kp models from ab-initio data, SciPost Physics Codebases 25, 10.21468/scipostphyscodeb.25 (2024)

2024 doi
[152]

Zhang, H

S. Zhang, H. Sheng, Z.-D. Song, C. Liang, Y. Jiang, S. Sun, Q. Wu, H. Weng, Z. Fang, X. Dai, et al., Vasp2kp: k·p models and Land´ e g-factors from ab initio calculations,Chin. Phys. Lett. 40, 127101 (2023)

2023
[153]

Robert, T

C. Robert, T. Amand, F. Cadiz, D. Lagarde, E. Courtade, M. Manca, T. Taniguchi, K. Watan- abe, B. Urbaszek, and X. Marie, Fine structure and lifetime of dark excitons in transition metal dichalcogenide monolayers, Phys. Rev. B 96, 155423 (2017)

2017
[154]

M. R. Molas, C. Faugeras, A. O. Slobodeniuk, K. Nogajewski, M. Bartos, D. M. Basko, and M. Potemski, Brightening of dark excitons in monolayers of semiconduc ting transition metal dichalcogenides, 2D Mater. 4, 021003 (2017)

2017
[155]

M. R. Molas, A. O. Slobodeniuk, T. Kazimierczuk, K. Nogajewski, M. B artos, P. Kapu´ sci´ nski, K. Oreszczuk, K. Watanabe, T. Taniguchi, C. Faugeras, P. Kossacki, D. M. Basko, and M. Potemski, Probing and manipulating valley coherence of dark excitons in monolayer WSe2, Phys. ...

2019

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

For the latter, we need to consider oﬀ-diagonal matrix elements of single-particle operators in the Bloch basis

extends on previous derivations [56, 57] by considering oﬀ-diagonal terms in the exciton g-factor, not only in the excitonic but also in the single- particle electron/hole manifold (band g-factors). For the latter, we need to consider oﬀ-diagonal matrix elements of single-particle operators in the Bloch basis. For in- stance, the orbital angular momentum ...

[2] [2]

We consider the nonzero oscillator strengths only for s-like ex- citon states

Several exciton peaks emerge around 1.95 eV, visible for θ = 45◦, 90◦. We consider the nonzero oscillator strengths only for s-like ex- citon states. pabilities of our approach, we address the long-standing puzzle and conﬂicting experimental reports of the Ry- dberg series of the A exciton g-factors for the so-called 1s–4s states. Fig. 4(a) compiles the a...

2023

[3] [3]

Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Cole- man, and M. S. Strano, Electronics and optoelectron- ics of two-dimensional transition metal dichalcogenides, Nat. Nanotechnol. 7, 699 (2012)

2012

[4] [4]

A. K. Geim and I. V. Grigorieva, Van der Waals het- erostructures, Nature 499, 419 (2013)

2013

[5] [5]

K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. C. Neto, 2D materials and van der Waals het- erostructures, Science 353, aac9439 (2016)

2016

[6] [6]

Y. Liu, N. O. Weiss, X. Duan, H.-C. Cheng, Y. Huang, and X. Duan, Van der Waals heterostructures and de- vices, Nat. Rev. Mater. 1, 16042 (2016)

2016

[7] [7]

Manzeli, D

S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev, and A. Kis, 2D transition metal dichalcogenides, Nat. Rev. Mater. 2, 17033 (2017)

2017

[8] [8]

Radisavljevic, A

B. Radisavljevic, A. Radenovic, J. Brivio, V. Gia- cometti, and A. Kis, Single-layer MoS2 transistors, Nat. Nanotechnol. 6, 147 (2011)

2011