Artificial electric field in Fermi Liquids

Leon Balents; Ryuichi Shindou

REVIEW

Artificial electric field in Fermi Liquids

Ryuichi Shindou, Leon Balents

Strongly Correlated Electrons · Submitted 3 Mar 2006 · 15 references

Pith's verdict: A genuinely new many-body correction to quasiparticle dynamics, but the load-bearing derivation sits in an unpublished companion; worth refereeing if referees insist on seeing it.

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arXiv IDcond-mat/0603089 (version 2)

Pith IDU5GFJMZN, Pith's permanent record of this paper and its review

CitationLeon Balents, Ryuichi Shindou. "Artificial electric field in Fermi Liquids." arXiv:cond-mat/0603089 (2006). https://pith.science/paper/U5GFJMZN

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Claim paper

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Editor's note

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Two things to know. This paper introduces an artificial electric field in interacting Fermi liquids, coming from the frequency dependence of the renormalized Bloch functions. That is genuinely new—earlier work on Berry phase dynamics only had the artificial magnetic field. The paper traces physical consequences: modified quasiparticle equation of motion, a correction to the anomalous Hall conductivity, and a spectral-weight renormalization Z' that could in principle be seen in ARPES. The physical idea is clear, and the Keldysh/Wigner-gradient machinery is appropriate.

The paper does some things well. The definitions of the gauge fields from the diagonalization of the Wigner-transformed Lagrangian are natural, and the effective Boltzmann equation (14) is a concrete starting point. The ARPES suggestion is speculative but honest—they say it is no doubt difficult in practice. The treatment of the phonon model in footnote 11 shows the effect is expected to be finite, though it is only a sketch.

The soft spot is real and load-bearing. The central reduced Keldysh equation, Eq. (8), is not derived in this paper. It is introduced with "Following this prescription, we obtain ..." and the derivation is cited to reference [12], "in preparation." That is exactly the step where interband coherence is eliminated and the E field emerges. If that elimination is done carelessly, the E terms could be an artifact of the projection or could be reabsorbed into the energy correction Mα. The paper itself concedes incompleteness: the Ward-identity check of Eq. (2) is also deferred to [12], and the authors explicitly speculate that the results extend to arbitrary U(1) Fermi liquids. The projection assumptions—weak interband coherence, negligible incoherent weight from other bands, neglect of the quasiparticle decay rate—are stated but not quantitatively justified. For a Letter these are acceptable if the full derivation exists somewhere; without it, the referee cannot audit the claim.

None of this makes me think the result is wrong. The structure of the argument is plausible and the authors know the machinery. But the published text alone is not self-contained, and the most interesting claim lives in an unpublished companion. This deserves peer review—a serious referee who requires the derivation to be made available, or the paper revised to include the essential steps. I'd bring it to a reading group, but with a warning that the interesting part is partly a promise.

Referee report

Significance

If the central projection step can be made explicit and auditable, the result is significant: it extends Berry-curvature semiclassical dynamics to interacting Fermi liquids in a systematic Keldysh framework, and it produces concrete, parameter-free predictions: Eq. (2) for the anomalous Hall conductivity and Eq. (3) for the ARPES spectral-weight correction. The idea of an artificial electric field tied to the frequency dependence of the renormalized Bloch functions is original and physically suggestive. The manuscript also gives explicit credit to earlier work by Sundaram and Niu and by Haldane. The main weakness is that the load-bearing derivation is not self-contained: the reduced Keldysh equation, the elimination of interband coherence, and the Ward-identity consistency check for Eq. (2) are all deferred to the unpublished companion paper [12]. As written, the claims cannot be verified from the manuscript alone.

Major objections (4)

[Paragraph preceding Eq. (8)] The central reduced Keldysh equation, Eq. (8), is not derived in the manuscript. The text states 'Following this prescription, we obtain ...' and then cites reference [12], which is marked 'in preparation.' Since Eqs. (1), (2), (3), and (14) all follow from Eq. (8), the entire claim that an artificial electric field E_α modifies quasiparticle dynamics rests on an unavailable derivation. The paper needs to include the full derivation of Eq. (8), or at least a complete, self-contained appendix performing the projection to second order in gradients; a citation to an unpublished companion paper is not sufficient for a load-bearing step.
[Passage on elimination of off-diagonal spectral functions (p. 3)] The assertion that the off-diagonal spectral functions A_αβ are at most first order in the gradient expansion and can be eliminated iteratively in favor of the diagonal components is described only in words. No explicit equations are given for the first-order solution A_αβ or for the second-order substitution into the diagonal equations. Without this calculation one cannot exclude that second-order terms alter the coefficient of ∂_k n in Eq. (14) beyond B_α − E_α × v, or that E_α-dependent contributions are reabsorbed into M_α. This is a load-bearing gap because the effective one-band equation (8) is the foundation of all subsequent results.
[After Eq. (2)] The statement that Eq. (2) 'can be also proven to be consistent with the many-body formulation of the transverse conductivity' by Ward identities is deferred to reference [12], again an unpublished work. Even if Eq. (8) were accepted, the conductivity formula involves current vertex corrections that are not shown in this manuscript. The consistency proof should either be included or the claim should be labeled as unproven.
[Footnote 11] The illustrative phonon model in footnote 11 is only a sketch: it does not show the explicit self-energy, the frequency derivative ∂_ω Σ, the resulting E_α, or the resulting form of Eq. (14). Because the existence of a finite artificial electric field is a central claim, this example should be worked out in enough detail to be checked, or it should be clearly labeled as a heuristic estimate.

Minor objections (5)

[Eq. (2)] The distribution function appears as 'ln_α' in Eq. (2); this is almost certainly a typographical artifact and should be 'n_α'.
[Paragraph preceding Eq. (6)] There is a typo 'diaognal' in 'The diaognal elements of A'; it should read 'diagonal'.
[Reference [12]] Reference [12] is cited as 'in preparation' in several places where it carries the main derivation. The manuscript should either give a complete derivation or provide a publicly available preprint identifier; as it stands the citation is not auditable.
[Header] The header date '(Dated: October 16, 2018)' does not match the arXiv submission date (2006) and should be corrected or removed.
[Passage 'we speculate that our final results naturally extend...' (p. 2)] The text explicitly says the authors 'speculate' that the final results extend to arbitrary U(1) Fermi liquids, which is an admitted limitation. This should be reflected in the abstract or conclusions as a stated restriction, not presented as a settled generalization.

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The paper

The paper claims

The paper argues that interactions generate an artificial electric field in Fermi liquids, alongside the known artificial magnetic field, and that this new field changes quasiparticle transport and spectral weight.

The reading

Using Keldysh Green-function theory, the paper derives the effective Boltzmann equation for a quasiparticle on a single Fermi surface of an interacting Fermi liquid. It finds that interactions introduce a new artificial electric field in frequency–momentum space, in addition to the known artificial magnetic field from momentum-space Berry curvature, and both fields enter the quasiparticle equations of motion. The artificial electric field modifies the transverse current, hence the anomalous Hall conductivity, and it produces a field-dependent correction to the quasiparticle renormalization factor. The paper shows how these artificial fields could be mapped out with angle-resolved photoemission spectroscopy.

What carries the argument

The central object is the renormalized Bloch function $u_\alpha(\omega,k)$, the $\alpha$-th eigenvector of the Wigner-transformed Keldysh Lagrangian $\hat{L}(\omega,k;T,R)$. Its Berry connection $A_{\alpha,i} = \langle u_\alpha|\partial_{k_i}u_\alpha\rangle$ and its frequency component $A_{\alpha,0} = \langle u_\alpha|\partial_\omega u_\alpha\rangle$ define the artificial fields $B_\alpha$ and $E_\alpha$ via $B_{\alpha,i} = i\epsilon_{ijm}\partial_{k_j}A_{\alpha,m}$ and $E_{\alpha,i} = i(\partial_\omega A_{\alpha,i} - \partial_{k_i}A_{\alpha,0})$. The argument proceeds by a second-order gradient expansion of the Keldysh equation: interband coherence is eliminated iteratively, leaving a closed kinetic equation for the active-band spectral function in which the band curvatures $\Omega^\alpha_{Q_jX_k}$ are the only remnant of the other bands.

Extended reading notes

Core claim

On the paper's own terms, the central claim is that a U(1) Fermi liquid with non-degenerate bands is characterized by two on-shell gauge fields, an artificial magnetic field $B_\alpha$ and a new artificial electric field $E_\alpha$, and that both govern quasiparticle dynamics through the equations of motion $$\frac{dR}{dT} = v + (B_\$\alpha$ - E_\$\alpha$ \times v)\times \frac{dk}{dT}, \qquad \frac{dk}{dT} = -e + b\times \frac{dR}{dT}.$$ Because $E_\alpha$ is built from the frequency dependence of the renormalized Bloch functions, it vanishes in a non-interacting system and is inherently a many-body effect. The same fields enter the transverse current, giving the anomalous Hall conductivity $$\$sigma^{{jk}}$ = \$epsilon^{{jkl}}$\frac{$e^{2}$}{\hbar}\sum_\$\alpha$ \int \frac{dk}{(2\pi)^d} (B_\$\alpha$ - E_\$\alpha$ \times v_\$\alpha$)_l\, n_\$\alpha$,$$ and modify the quasiparticle spectral weight as $Z' = 1 - \frac{1}{2}(B_\alpha\cdot b + E_\alpha\cdot e)$, which the paper proposes to detect momentum-resolved through ARPES.

Load-bearing premise

The load-bearing premise is that quasiparticles are sharp and well separated from other bands, so interband spectral weight can be eliminated in a low-order gradient expansion and the decay rate can be neglected; if either assumption fails, the effective Boltzmann equation gains corrections.

Editorial extensions

If this is right

Quasiparticle dynamics in any U(1) Fermi liquid without inversion or time-reversal symmetry must include the artificial electric field; it is not an artifact of a particular model.
The anomalous Hall conductivity of a ferromagnetic metal acquires an $E_\alpha$-dependent term, so measured values can differ from non-interacting Berry-curvature predictions even at low temperature.
Applied electric and magnetic fields change the angle-resolved spectral weight through $Z' = 1-\frac{1}{2}(B_\alpha\cdot b+E_\alpha\cdot e)$, giving an ARPES-based route to map the artificial fields in momentum space.
The effective Boltzmann equation gives a concrete starting point for computing magnetotransport and Hall response of interacting spin-orbit coupled metals.

Reading between the lines

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

The paper does not compute the artificial electric field for any specific material end-to-end; the order-of-magnitude estimate in the text suggests it can be a substantial fraction of the lattice scale, so a systematic scan of strong electron-phonon or electron-magnon coupled metals is a natural next step.
Because only the combination $B_\alpha - E_\alpha\times v$ enters the current, Hall measurements alone cannot separate the two fields; combining them with the proposed ARPES spectral-weight change could, in principle, disentangle $E_\alpha$ from $B_\alpha$.
The derivation assumes U(1) bands and no quasiparticle decay; extending it to degenerate bands would require non-Abelian artificial fields, and extending it to stronger damping would test whether the $E_\alpha$ term survives beyond the sharp-quasiparticle limit.

Original abstract

Based on the Keldysh formalism, we derive an effective Boltzmann equation for a quasi-particle associated with a particular Fermi surface in an interacting Fermi liquid. This provides a many-body derivation of Berry curvatures in electron dynamics with spin-orbit coupling, which has received much attention in recent years in non-interacting models. As is well-known, the Berry curvature in momentum space modifies naive band dynamics via an artificial magnetic field in momentum space. Our Fermi liquid formulation completes the reinvention of modified band dynamics by introducing in addition an "artificial electric field", related to Berry curvature in frequency and momentum space. We show explicitly how the artificial electric field affects the renormalization factor and transverse conductivity of interacting U(1) Fermi liquids with non-degenerate bands. Accordingly, we also propose a method of momentum resolved Berry's curvature detection in terms of angle resolved photoemission spectroscopy (ARPES).

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,994 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.74 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Based on the Keldysh formalism, we derive an effective Boltzmann equation for a quasi-particle associated with a particular Fermi surface in an interacting Fermi liquid. This provides a many-body derivation of Berry curvatures in electron dynamics with spin-orbit coupling, which has received much attention in recent years in non-interacting models. As is well-known, the Berry curvature in momentum space modifies naive band dynamics via an artificial magnetic field in momentum space. Our Fermi liquid formulation completes the reinvention of modified band dynamics by introducing in addition an "artificial electric field", related to Berry curvature in frequency and momentum space. We show explicitly how the artificial electric field affects the renormalization factor and transverse conductivity of interacting U(1) Fermi liquids with non-degenerate bands. Accordingly, we also propose a method of momentum resolved Berry's curvature detection in terms of angle resolved photoemission spectroscopy (ARPES).

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Based on the Keldysh formalism, we derive an effective Boltzmann equation for a quasi-particle associated with a particular Fermi surface in an interacting Fermi liquid.	method 0.70	0.96	0.06	asserted 1.00
s1	This provides a many-body derivation of Berry curvatures in electron dynamics with spin-orbit coupling, which has received much attention in recent years in non-interacting models.	contribution 0.51	0.79	0.11	asserted 0.99
s2	As is well-known, the Berry curvature in momentum space modifies naive band dynamics via an artificial magnetic field in momentum space.	background 1.00	0.05	0.05	asserted 1.00
s3	Our Fermi liquid formulation completes the reinvention of modified band dynamics by introducing in addition an "artificial electric field", related to Berry curvature in frequency and momentum space.	contribution 0.93	0.93	0.54	asserted 1.00
s4	We show explicitly how the artificial electric field affects the renormalization factor and transverse conductivity of interacting U(1) Fermi liquids with non-degenerate bands.	result 0.90	0.95	0.32	asserted 1.00
s5	Accordingly, we also propose a method of momentum resolved Berry's curvature detection in terms of angle resolved photoemission spectroscopy (ARPES).	contribution 0.55	0.73	0.13	asserted 0.58

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (15)

[12]

R. Shindou and L. Balents, in preparation

View on Pith
[1]

Z. Fang, et.al., Science 302 92, (2003)

View on Pith 2003
[2]

Y. Kato, R. C. Myers, A. C. Gossard and D. D. Awschalom, Nature 429 50, (2004)

View on Pith 2004
[3]

J. Wunderlich, B. Kaestner, J. Sinova and T. Jungwirth Phys. Rev. Lett. 94 047204, (2005)

View on Pith 2005
[4]

R. Karplus and J. M. Luttinger Phys. Rev. 95 1154, (1954)

View on Pith 1954
[5]

G. Sundaram and Q. Niu, Phys. Rev. B 59 14915, (1999)

View on Pith 1999
[6]

E. Fradkin, Field Theories of Condensed Matter Systems (Addison-Wesley, Redwood City, 1991)

View on Pith 1991
[7]

J.Rammer and H.Smith, Rev. Mod. Phys. 58 323, (1986)

View on Pith 1986
[8]

F.D.M. Haldane, Phys. Rev. Lett. 93 206602, (2004)

View on Pith 2004
[9]

K. Ishikawa and T. Matsuyama, Z. Phys. C 33 41, (1986)

View on Pith 1986
[10]

L. P. Kadanoﬀ and G. Baym, Quantum Statistical Me- chanics (Perseus Press, Cambridge, 1989)

View on Pith 1989
[11]

As a simplestic example having ﬁnite a.e.f., one may con - sider a lattice-regularized Rashba model under applied magnetic ﬁeld, in which electrons are coupled with the Einstein phonon having a ﬁnite mass ω0. Then, the 2nd order perturbation with respect to this coupling gives rise to a collisional self-energy, with an exponentially small linf-time term ˆ...

View on Pith
[13]

L. D. Landau and E. M. Lifshitz, Statistical Physics, part.2 (Butterworth-Heinemann, Oxford, 2002). 5

View on Pith 2002
[14]

D. Xiao, J. Shi and Q. Niu, Phys. Rev. Lett. 95 137204, (2005)

View on Pith 2005
[15]

Using ˆU ′ ≡ ˆU|ω=ǫ, one could also express B′ α into a “magnetic-ﬁeld” like form; B′ α,j ≡ iǫjlm dkl A′ α,m where A′ α,i ≡ [ ˆU ′† dki ˆU ′]αα. As in Eq. (3), however, the renor- malization factor also acquires the topological correctio n characterized by the inner product between ( e, b) and (Eα, Bα). Thus, we believe that Eα and Bα are more nat- ural s...

View on Pith

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [12]

R. Shindou and L. Balents, in preparation

View on Pith

[2] [1]

Z. Fang, et.al., Science 302 92, (2003)

View on Pith 2003

[3] [2]

Y. Kato, R. C. Myers, A. C. Gossard and D. D. Awschalom, Nature 429 50, (2004)

View on Pith 2004

[4] [3]

J. Wunderlich, B. Kaestner, J. Sinova and T. Jungwirth Phys. Rev. Lett. 94 047204, (2005)

View on Pith 2005

[5] [4]

R. Karplus and J. M. Luttinger Phys. Rev. 95 1154, (1954)

View on Pith 1954

[6] [5]

G. Sundaram and Q. Niu, Phys. Rev. B 59 14915, (1999)

View on Pith 1999

[7] [6]

E. Fradkin, Field Theories of Condensed Matter Systems (Addison-Wesley, Redwood City, 1991)

View on Pith 1991

[8] [7]

J.Rammer and H.Smith, Rev. Mod. Phys. 58 323, (1986)

View on Pith 1986

[9] [8]

F.D.M. Haldane, Phys. Rev. Lett. 93 206602, (2004)

View on Pith 2004

[10] [9]

K. Ishikawa and T. Matsuyama, Z. Phys. C 33 41, (1986)

View on Pith 1986

[11] [10]

L. P. Kadanoﬀ and G. Baym, Quantum Statistical Me- chanics (Perseus Press, Cambridge, 1989)

View on Pith 1989

[12] [11]

As a simplestic example having ﬁnite a.e.f., one may con - sider a lattice-regularized Rashba model under applied magnetic ﬁeld, in which electrons are coupled with the Einstein phonon having a ﬁnite mass ω0. Then, the 2nd order perturbation with respect to this coupling gives rise to a collisional self-energy, with an exponentially small linf-time term ˆ...

View on Pith

[13] [13]

L. D. Landau and E. M. Lifshitz, Statistical Physics, part.2 (Butterworth-Heinemann, Oxford, 2002). 5

View on Pith 2002

[14] [14]

D. Xiao, J. Shi and Q. Niu, Phys. Rev. Lett. 95 137204, (2005)

View on Pith 2005

[15] [15]

Using ˆU ′ ≡ ˆU|ω=ǫ, one could also express B′ α into a “magnetic-ﬁeld” like form; B′ α,j ≡ iǫjlm dkl A′ α,m where A′ α,i ≡ [ ˆU ′† dki ˆU ′]αα. As in Eq. (3), however, the renor- malization factor also acquires the topological correctio n characterized by the inner product between ( e, b) and (Eα, Bα). Thus, we believe that Eα and Bα are more nat- ural s...

View on Pith