Pith. sign in

REVIEW 4 major objections 4 minor 46 references

This paper argues that the Pomeranchuk transition in an isotropic 2D Fermi liquid is controlled by a Gaussian fixed point with dynamical exponent z=2, not by the Landau-damped z=3 theory found in earlier effective-field-theory treatments, a

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-04 00:17 UTC pith:5E6RPR6O

load-bearing objection Plausible and important claim that 2D Pomeranchuk transitions are z=2 Gaussian, but the load-bearing projection to the critical subspace is deferred to an unpublished companion, so the case is not yet closed. the 4 major comments →

arxiv 2511.01725 v2 pith:5E6RPR6O submitted 2025-11-03 cond-mat.str-el hep-th

Critical theory of Pomeranchuk transitions via high-dimensional bosonization

Zhengfei Hu , Jaychandran Padayasi , O\u{g}uz T\"urker , Kun Yang This is my paper
classification cond-mat.str-el hep-th
keywords Pomeranchuk transitionnematic Fermi liquidhigh-dimensional bosonizationdynamical critical exponentGaussian fixed pointupper critical dimensionmarginally irrelevant perturbationFermi surface instability
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to establish the exact critical theory of Pomeranchuk transitions in isotropic two-dimensional Fermi liquids: when a Landau parameter approaches −1 and the Fermi surface spontaneously distorts, the transition is controlled by a Gaussian fixed point with dynamical exponent z=2, not by the Landau-damped z=3 theory that earlier effective-field-theory treatments found. The argument works through high-dimensional bosonization, where the Fermi-liquid interactions that drive the transition enter as quadratic terms and can be treated exactly. The central object is an exact eigenmode Φ whose density profile has the angular-momentum-two shape; near criticality this mode softens with ω∼|q|^2, decoupling from the particle-hole continuum. Since the upper critical dimension is 4−z=2, a 2D system sits exactly at that dimension, so the quartic self-interaction is marginally irrelevant and correlations are mean-field-like up to logarithms. A sympathetic reader would care because it turns this 2D quantum critical point into one of the rare exactly solvable metallic quantum critical points, with sharp, testable predictions.

Core claim

On the paper's own terms, the central discovery is that the Pomeranchuk instability is triggered by the softening of an exact eigenmode Φ of the bosonized Gaussian action, and that this mode—not the Landau-damped particle-hole continuum—carries the criticality. After projecting out non-critical modes, the effective action is S0 = ∫ d²x dt [ (∂_tΦ)^2 − (1+f₂)|∇Φ|^2 − m₀(∇²Φ)^2 ], whose spectrum at f₂=−1 is ω² = m₀|q|⁴, i.e. z=2. Including the leading symmetry-allowed self-interaction |∇Φ|⁴, a one-loop RG calculation gives β(λ₄) = −(9/m₀^{3/2})λ₄², so the quartic term is marginally irrelevant. Hence the critical point is Gaussian at its upper critical dimension d_c=4−z=2, and two-point correla

What carries the argument

High-dimensional bosonization in terms of the phase-space field φ and local Fermi-surface displacement ρ(θ)=nθ·∇φ; splitting even/odd harmonics of the angular variable and Fourier transforming reduces the quadratic action to a family of coupled bosons for odd angular-momentum channels. The coupling matrix M′ has two exact zero eigenvectors when g₂=1+f₂ vanishes; one gives the normalizable critical field Φ=(i(φ′₁−φ′₋₁))/√2, the other is the non-normalizable partner Ψ. Dressing fields by spin factors makes M′ constant and allows projection onto the critical subspace, yielding the z=2 quantum Lifshitz form and the one-loop beta function.

Load-bearing premise

The argument depends on the claim that the non-normalizable zero mode Ψ and all non-critical z=1 modes, once integrated out, never feed back into the critical field Φ through relevant couplings; the explicit projection is deferred to a companion paper. If that coupling is relevant, the Gaussian fixed point would be modified.

What would settle it

Compute (or simulate) the full density-density correlator in a clean 2D fermion model tuned to the ℓ=2 instability, explicitly including couplings to the Ψ sector, and check whether the effective action retains the z=2 form. Concretely: if a numerical Monte Carlo susceptibility at the nematic critical point shows dominant ω∼|q|³ scaling, or if a one-loop calculation of couplings between Φ and the discarded modes produces a relevant operator, the central claim collapses. A simpler experimental check is that the critical mode should sharpen (quality factor growing as g₂^{−3/2}) rather than broad

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • In any even channel ℓ=2n, the 2D Pomeranchuk transition is at its upper critical dimension, so critical exponents are mean-field with logarithmic corrections rather than Landau-damped z=3 values.
  • The previously identified z=3 pole in the density-density propagator is reinterpreted as incoherent particle-hole continuum; it does not govern the critical point.
  • The critical collective mode becomes progressively sharper (Q∼g₂^{−3/2}) as the transition is approached, instead of overdamped, so it may be visible in spectroscopy or sound attenuation experiments.
  • There is a companion sharp z=1 propagating mode at velocity v_F/√2 that survives at criticality and could serve as a distinct experimental signature.
  • Because only one marginal perturbation remains, universal critical properties can be computed perturbatively in λ₄, providing a rare solvable quantum critical point in a 2D metal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the central claim is right, sign-problem-free numerical studies of the Ising nematic transition should see dynamic susceptibility peaked at ω∼|q|² with logarithmic corrections; a clean ω∼|q|³ dominance would falsify it.
  • The exact treatment suggests the standard perturbative expansion around free fermions misses a key cancellation: Landau damping is suppressed by a factor g₂². A direct fermionic RG or high-order self-energy calculation near g₂→0 could expose where that cancellation happens.
  • The companion z=1 mode may mediate temperature-dependent transport or acoustic anomalies near the transition; measuring sound attenuation in candidate materials such as Sr₃Ru₂O₇ could test the predicted underdamped dynamics.
  • Extending the projection to the ordered side could reveal whether the same Gaussian theory controls the Goldstone-mode spectrum and vortex unbinding of the nematic director, connecting to the paper's stated outlook.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript uses high-dimensional bosonization to construct an effective field theory for Pomeranchuk transitions in isotropic 2D Fermi liquids. The authors claim that, in any even angular-momentum channel ℓ ≥ 2, the critical mode is a single real scalar Φ with the quantum Lifshitz action Eq. (11), dynamical exponent z = 2, and upper critical dimension d_c = 2. They further claim that the only marginal perturbation is λ₄|∇Φ|⁴, that its beta function is β(λ₄) = −(9/m₀^{3/2})λ₄², and that the transition is therefore governed by a Gaussian fixed point with mean-field correlators and logarithmic corrections. The paper contrasts this with the Hertz–Millis z = 3 result, attributing the latter to non-critical particle-hole continuum contributions.

Significance. If the central claim is correct, the paper resolves a long-standing discrepancy between Hertz–Millis predictions and numerical/experimental indications of z = 2 scaling in nematic and ferromagnetic quantum critical points, and it provides an exactly solvable quantum critical point in a 2D metal. The manuscript has clear strengths: the quadratic bosonized action Eqs. (1)–(5) is derived carefully, the zero modes of the coupling matrix are computed exactly in Appendix A, and the special case g_{ℓ≠2} = 1 is solved exactly for the Gaussian effective action in Eq. (8), albeit with details deferred. The RG calculation of Eq. (14) is explicit and, modulo the issues below, gives a falsifiable prediction of logarithmic corrections to mean-field behavior. The paper is concise and the main physical picture is compelling.

major comments (4)
  1. [Critical theory and Higher-order terms and RG analysis, after Eq. (6) and before Eq. (10)] The central claim that the critical sector is closed under RG rests on the projection to the Φ subspace, but the projection details are deferred to the unpublished companion Ref. [36]. The dismissal of the non-normalizable zero mode Ψ and of the infinite set of non-critical z = 1 modes is based on (i) Ψ having vanishing overlap with ρ₂ and (ii) power counting under z = 2 scaling. Neither demonstrates that nonlinear vertices do not generate relevant or marginal couplings between Φ and the discarded modes. For example, a term like (D_θ Φ)²(D_θ Ψ)² would be marginal under z = 2 and, if generated, would make Eq. (14) incomplete. The text says the full Gaussian action is in Ref. [36], but the closedness of the projection—not just the Gaussian action—is what is needed. This is load-bearing because the whole fixed-point structure depends on the absence of such couplings.
  2. [Eqs. (10)–(11) and Eq. (14)] The sign of the q⁴ term is internally inconsistent. As written, S₀ = ∫ [ω² − (1+f_{2n})|q|² − m₀|q|⁴]|Φ|², and the real-space form is ˙Φ² − (1+f_{2n})|∇Φ|² − m₀(∇²Φ)². If m₀ > 0, the action is unbounded below at f_{2n} = −1, so there is no stable Gaussian fixed point. However, the beta function Eq. (14) contains m₀^{−3/2}, which requires m₀ > 0. The intended stabilizing term must be +m₀(∇²Φ)², or one must adopt the convention m₀ < 0. This sign error changes the sign of the quadratic part and the sign of β(λ₄), and it must be corrected before the fixed-point analysis can be assessed.
  3. [Higher-order terms and RG analysis, Eqs. (12)–(14)] The one-loop beta function Eq. (14) is computed from a theory that contains only Φ and λ₄|∇Φ|⁴. The justification that no other marginal operators appear relies on (i) the Φ → −Φ symmetry, which excludes odd powers of Φ, and (ii) the assumed irrelevance of all couplings to non-critical modes. Symmetry arguments do rule out odd Φ terms, but they do not rule out marginal four-derivative terms involving the non-critical modes. The statement that non-critical z = 1 modes become infinitely stiff under z = 2 scaling is a free-field scaling argument; it does not prove that interaction-generated vertices between Φ and those modes are irrelevant. Without an explicit demonstration (or at least a detailed summary from Ref. [36]) that the projected nonlinear action yields no additional marginal operators, Eq. (14) is not established as the complete β-function.
  4. [Eq. (8) and Appendix B] The exact result for the effective Φ action, Eq. (8), is central to the claim that the critical mode is underdamped with ω ∼ √g₂ v_F |q|, but its derivation is entirely relegated to Ref. [36]. Appendix B provides a Gaussian analysis in the special case g_{ℓ≠2} = 1, which supports the z = 2 pole and the interpretation of the z = 3 contribution as incoherent particle-hole pairs. However, this is still a Gaussian analysis of a truncated chain; it does not include the nonlinear terms that could couple Φ to the non-critical modes. The conclusion 'This completes our proof' in the final paragraph overstates what has been shown within the Letter itself.
minor comments (4)
  1. [Abstract] Typo: 'Pomeranchuck' should be 'Pomeranchuk'. The same typo appears in the abstract and possibly elsewhere.
  2. [Eq. (10)] Φ is described as a real scalar, but Eq. (10) writes Φ†Φ. Please use Φ² or clarify the complex notation.
  3. [Eq. (13)] The loop integral is written with a factor 36 and the sign of m₀ inside the denominator; after fixing the sign convention of Eq. (10), please verify that the numerical coefficient is consistent with the normalization of the |∇Φ|⁴ vertex.
  4. [References] Ref. [36] is given as 'To be published'. Since the manuscript relies on it for several load-bearing equations, it would be helpful to update the reference or state explicitly which results are taken from it.

Circularity Check

1 steps flagged

Central z=2 critical-sector reduction is deferred to same-author unpublished companion [36]; otherwise the RG calculation is self-contained.

specific steps
  1. self citation load bearing [Critical theory section (around Eq. (6) and Eq. (8)); 'Higher-order terms and RG analysis' before Eq. (12)]
    "Upon projection to the critical subspace (details in Ref. [36]), we have: ρ(θ,q)=|q| sin(2(θ−θ q))Φ(q)/√2 ... Integrating out the non-critical modes exactly in the Gaussian action, we get the effective Lagrangian for Φ in Euclidean signature as [36] ... We first note that any coupling between the non-critical z=1 Fermi liquid modes and the critical field Φ beyond the quadratic |q|² level is irrelevant, as under z=2 RG scaling these modes become infinitely stiff (or equivalently, the Fermi velocity flows to infinity) [29, 36]."

    The load-bearing chain that selects Φ as the sole critical field, gives the z=2 effective Lagrangian, and closes the action under RG is not derived in this Letter. Eq. (6) is explicitly assigned to a projection 'details in Ref. [36]'; Eq. (8), whose expansion produces the z=2 dispersion and the absence of Landau damping, is assigned to '[36]'; and the irrelevance of all non-critical couplings is justified by '[29, 36]'. Ref. [29] is earlier work by one of the present authors, and Ref. [36] is an unpublished companion by the same group. Thus the central claim reduces to an unverified same-author citation chain rather than to the bosonized action Eq. (1). If that projection is assumed, the Gaussian fixed point follows, but the derivation is not self-contained.

full rationale

The paper is not constructionally circular: no parameter is fitted to the predicted quantity, no result is renamed as a prediction, and the one-loop beta function Eq. (14) and the zero-mode diagonalization in Appendix A are explicit, checkable calculations. What raises the score is the self-citation load-bearing step: the reduction of the full bosonized action to a single critical field Φ, the exact effective Lagrangian Eq. (8), and the statement that non-critical z=1 modes never feed back into Φ at relevant or marginal order are all deferred to Refs. [29] and [36], both of which share authors with this paper, with [36] 'to be published'. This is a missing-proof / self-reference gap rather than an equation-level equivalence, so I do not assign 6 or higher. The q^4 stabilizer is also introduced by citing the authors' prior work [27,29]; that is an added modeling assumption, not a circular step. Apart from this deferred projection chain, the RG analysis of the assumed critical action is self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The central claim rests on the standard Landau-Fermi-liquid assumption plus the exactness of HDB as represented by Eq (1); the q^4 term m0 is introduced by hand; the projection that discards Psi is deferred to a companion. No genuinely new microscopic entities are introduced.

free parameters (1)
  • m0 (coefficient of (laplacian Phi)^2 / q^4 term) = not determined; assumed positive
    Introduced ad hoc in Eqs (10)-(11) to stabilize the critical action when the q^2 coefficient vanishes at f_2=-1. The paper says the term arises from |gradient rho|^2 interactions but does not compute m0; the beta function and upper-critical-dimension analysis depend on it.
axioms (4)
  • domain assumption The HDB action Eq (1) is the exact low-energy action of a 2D isotropic Fermi liquid, with Landau interactions entering only quadratically.
    Used as the starting point in the section 'Gaussian action'. The paper relies on the HDB program (Refs [15-25]) to treat Fermi-liquid interactions exactly at quadratic level; if non-quadratic terms are relevant at the PI, the z=2 derivation fails.
  • ad hoc to paper The non-normalizable zero mode Psi and the rest of the non-critical z=1 modes do not couple relevantly to Phi; the projection to the Phi subspace is valid.
    After Eq (6): 'Psi is not normalizable and has vanishing overlap with rho_2... Psi can be safely ignored.' The projection details are relegated to Ref [36]. This is the key reduction from an infinite family of bosons to a single critical mode.
  • ad hoc to paper The next-order gradient term |gradient rho|^2 (coefficient m0) has the same coupling structure as the q^2 term and is positive, producing the z=2 action Eq (10).
    The q^4 term is 'introduced' to stabilize the action near the PI; m0 is not computed from microscopic parameters. The beta function (Eq (14)) depends on m0^{-3/2}.
  • domain assumption The only allowed marginal self-interaction of Phi is |gradient Phi|^4; terms odd in Phi are forbidden by the Phi -> -Phi symmetry, and terms with bare Phi powers are excluded by shift symmetry.
    Used to restrict the higher-order action to Eq (12). Footnotes [42,43] address two candidate odd terms, but no complete operator analysis is given; if another marginal or relevant term is generated, the fixed point changes.

pith-pipeline@v1.3.0-alltime-deepseek · 11407 in / 20132 out tokens · 215597 ms · 2026-08-04T00:17:08.252585+00:00 · methodology

0 comments
read the original abstract

We use high-dimensional bosonization to derive an effective field theory that describes the Pomeranchuck transition in isotropic two-dimensional Fermi liquids. We find that the transition is triggered by the softening of an eigenmode that leads to spontaneous Fermi surface distortion. The resultant theory in terms of this critical mode has dynamical critical exponent $z = 2$ and the upper critical dimension is $d_c = 4-z= 2$. As a result the system is at the upper critical dimension in 2D, resulting in a Gaussian fixed point with a marginally irrelevant quartic perturbation.

Figures

Figures reproduced from arXiv: 2511.01725 by Jaychandran Padayasi, Kun Yang, O\u{g}uz T\"urker, Zhengfei Hu.

Figure 1
Figure 1. Figure 1: FIG. 1: Fluctuations and instabilities of the Fermi [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Schematic representations of Fermi liquid modes close to the Pomeranchuk instability. (a) After scaling out [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

46 extracted references · 4 canonical work pages

  1. [1]

    Sachdev,Quantum Phase Transitions, 2nd ed

    S. Sachdev,Quantum Phase Transitions, 2nd ed. (Cambridge University Press, 2011)

    2011

  2. [2]

    Nozieres and D

    P. Nozieres and D. Pines,Theory Of Quantum Liquids, Advanced Books Classics (Avalon Publishing, 1999)

    1999

  3. [3]

    Giuliani and G

    G. Giuliani and G. Vignale,Quantum Theory of the Electron Liquid, Masters Series in Physics and Astronomy (Cambridge University Press, 2005)

    2005

  4. [4]

    S. M. Girvin and K. Yang,Modern Condensed Matter Physics(Cambridge University Press, 2019)

    2019

  5. [5]

    Fradkin, S

    E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein, and A. P. MacKenzie, Nematic Fermi Fluids in Condensed Matter Physics, Annual Review of Condensed Matter Physics1, 153 (2010), arXiv:0910.4166 [cond- mat.str-el]

    Pith/arXiv arXiv 2010

  6. [6]

    R. M. Fernandes and A. V. Chubukov, Low-energy microscopic models for iron-based superconductors: a review, Reports on Progress in Physics80, 014503 (2016)

    2016

  7. [7]

    Oganesyan, S

    V. Oganesyan, S. A. Kivelson, and E. Fradkin, Quantum theory of a nematic fermi fluid, Phys. Rev. B64, 195109 (2001)

    2001

  8. [8]

    J. A. Hertz, Quantum critical phenomena, Phys. Rev. B 14, 1165 (1976)

    1976

  9. [9]

    A. J. Millis, Effect of a nonzero temperature on quantum critical points in itinerant fermion systems, Phys. Rev. B 48, 7183 (1993)

    1993

  10. [10]

    D. L. Maslov and A. V. Chubukov, Fermi liquid near Pomeranchuk quantum criticality, Phys. Rev. B81, 045110 (2010), arXiv:0911.1251 [cond-mat.str-el]

    Pith/arXiv arXiv 2010

  11. [11]

    M. A. Metlitski and S. Sachdev, Quantum phase transitions of metals in two spatial dimensions. i. ising- nematic order, Phys. Rev. B82, 075127 (2010)

    2010

  12. [12]

    Schattner, S

    Y. Schattner, S. Lederer, S. A. Kivelson, and E. Berg, Ising nematic quantum critical point in a metal: A monte carlo study, Phys. Rev. X6, 031028 (2016)

    2016

  13. [13]

    E. Berg, S. Lederer, Y. Schattner, and S. Trebst, Monte carlo studies of quantum critical metals, Annual Review of Condensed Matter Physics10, 63–84 (2019)

    2019

  14. [14]

    In this Letter, we approach the PI using high- dimensional bosonization (HDB) [15–25] which avoids the issues encountered in the Hertz-Millis treatment

    using sign-problem free quantum Monte Carlo see regimes where the susceptibility followsz= 2scaling which is inconsistent with the corresponding Hertz-Millis type theory. In this Letter, we approach the PI using high- dimensional bosonization (HDB) [15–25] which avoids the issues encountered in the Hertz-Millis treatment. In one-dimensional (1D) systems, ...

    Pith/arXiv arXiv 2025

  15. [15]

    Y. Liu, W. Jiang, A. Klein, Y. Wang, K. Sun, A. V. Chubukov, and Z. Y. Meng, Dynamical exponent of a quantum critical itinerant ferromagnet: A monte carlo study, Physical Review B105, 10.1103/physrevb.105.l041111 (2022)

    work page doi:10.1103/physrevb.105.l041111 2022

  16. [16]

    Luther, Tomonaga fermions and the dirac equation in three dimensions, Physical Review B19, 320–330 (1979)

    A. Luther, Tomonaga fermions and the dirac equation in three dimensions, Physical Review B19, 320–330 (1979)

    1979

  17. [17]

    Enrico Fermi,

    F. D. M. Haldane, Luttinger’s theorem and Bosonization of the Fermi surface, inPerspectives in Many-Particle Physics, Proceedings of the International School of Physics “Enrico Fermi,” Course CXXI, edited by 6 R. Broglia and S. J.R. (North-Holland, Amsterdam,

  18. [18]

    Houghton, H.-J

    A. Houghton, H.-J. Kwon, and J. Marston, Stability and single-particle properties of bosonized fermi liquids, Physical Review B50, 1351 (1994)

    1994

  19. [19]

    Houghton and J

    A. Houghton and J. B. Marston, Bosonization and fermion liquids in dimensions greater than one, Phys. Rev. B48, 7790 (1993), arXiv:cond-mat/9210007

    Pith/arXiv arXiv 1993

  20. [20]

    A. H. Castro Neto and E. Fradkin, Bosonization of the low energy excitations of fermi liquids, Phys. Rev. Lett. 72, 1393 (1994)

    1994

  21. [21]

    Houghton, H

    A. Houghton, H. J. Kwon, and J. B. Marston, Multidimensional bosonization, Adv. Phys.49, 141 (2000), arXiv:cond-mat/9810388

    Pith/arXiv arXiv 2000

  22. [22]

    D. V. Khveshchenko, Bosonization of current-current interactions, Physical Review B49, 16893–16898 (1994)

    1994

  23. [23]

    A. H. Castro Neto and E. Fradkin, Bosonization of fermi liquids, Phys. Rev. B49, 10877 (1994)

    1994

  24. [24]

    L. V. Delacretaz, Y.-H. Du, U. Mehta, and D. T. Son, Nonlinear bosonization of Fermi surfaces: The method of coadjoint orbits, Phys. Rev. Res.4, 033131 (2022), arXiv:2203.05004 [cond-mat.str-el]

    Pith/arXiv arXiv 2022

  25. [25]

    D. V. Khveshchenko, Geometrical approach to bosonization of D>1 dimensional (non)-fermi liquids, Physical Review B52, 4833–4841 (1995)

    1995

  26. [26]

    V. G. Kac,Infinite-Dimensional Lie Algebras (Cambridge University Press, 1990)

    1990

  27. [27]

    Chen and L

    S. Chen and L. V. Delacretaz, Quantizing bosonized fermisurfaces(2025),arXiv:2510.07583[cond-mat.str-el]

    arXiv 2025

  28. [28]

    W. Ding, A. Seidel, and K. Yang, Entanglement entropy of fermi liquids via multidimensional bosonization, Physical Review X2, 10.1103/physrevx.2.011012 (2012)

    work page doi:10.1103/physrevx.2.011012 2012

  29. [29]

    Yang, Ferromagnetic transition in one-dimensional itinerant electron systems, Phys

    K. Yang, Ferromagnetic transition in one-dimensional itinerant electron systems, Phys. Rev. Lett.93, 066401 (2004)

    2004

  30. [30]

    Shankar, Renormalization-group approach to interacting fermions, Reviews of Modern Physics66, 129 (1994)

    R. Shankar, Renormalization-group approach to interacting fermions, Reviews of Modern Physics66, 129 (1994). [31]ρ R andρ L are actually the 1D counterpart ofρ(θ)in 2D, as there are only two directions in 1D, while there is a continuum of them in 2D parameterized byθ

    1994

  31. [31]

    K. Yang, Quantum Theory of Pomeranchuck Transition in Two Dimensional Fermi Liquids via High Dimensional Bosonization, arXiv e-prints , cond-mat/0502270 (2005), arXiv:cond-mat/0502270 [cond-mat.str-el]

    Pith/arXiv arXiv 2005

  32. [32]

    Giamarchi,Quantum Physics in One Dimension (Oxford University Press, 2003)

    T. Giamarchi,Quantum Physics in One Dimension (Oxford University Press, 2003). [33]D θ is odd underθ→θ+π

    2003

  33. [34]

    Either parametrization admits a polarization and allows a configuration space form of the action Eq

    Fieldsφand˜φare not independent in the sense that (φ,Πφ ∼ Dθ ˜φ)and( ˜φ,Π˜φ∼ Dθφ)are merely two different conjugate pairs to parametrize the phase space of fields. Either parametrization admits a polarization and allows a configuration space form of the action Eq. (2). Here we choose( ˜φ,Π˜φ∼ Dθφ)since we want to study even channel densities. For odd chan...

  34. [35]

    Symmetry group is the diagonal subgroupU(1) d of SO(2)×U(1)

  35. [36]

    Z. Hu, J. Padayasi, and K. Yang, To be published

  36. [37]

    A. O. Caldeira and A. J. Leggett, Quantum tunnelling in a dissipative system, Annals of physics149, 374 (1983)

    1983

  37. [38]

    There is never a lower order term in perturbative expansion ofg2 sincesappears only as powers ofg2/a(s)

  38. [39]

    P. M. Chaikin and T. C. Lubensky,Principles of Condensed Matter Physics(Cambridge University Press, 1995)

    1995

  39. [40]

    Ardonne, P

    E. Ardonne, P. Fendley, and E. Fradkin, Topological order and conformal quantum critical points, Annals of Physics310, 493–551 (2004)

    2004

  40. [41]

    ThisisschematicbecauseEq.(6)givesanonlocalrelation between ˆnand∇Φ

  41. [42]

    The only other term that could be marginal is (∇2Φ)(∇Φ)2, however it is forbidden because it is not invariant under theΦ(x)→ −Φ(x)global rotation of the nematic order parameter, which is a particle-hole transformation

  42. [43]

    There is a marginal term that might be generated from RG flow,∼ ˙Φ(∇Φ)2 which is again disallowed because the term is odd underΦ(x,t)→−Φ(x,t)

  43. [44]

    M. J. Lawler, D. G. Barci, V. Fernández, E. Fradkin, andL.Oxman,Nonperturbativebehaviorofthequantum phasetransitiontoanematicfermifluid,PhysicalReview B73, 10.1103/physrevb.73.085101 (2006)

    work page doi:10.1103/physrevb.73.085101 2006

  44. [45]

    Nilsson and A

    J. Nilsson and A. H. Castro Neto, Heat bath approach to landau damping and pomeranchuk quantum critical points, Physical Review B72, 10.1103/physrevb.72.195104 (2005)

    work page doi:10.1103/physrevb.72.195104 2005

  45. [46]

    Cai and M

    K.-L. Cai and M. Cheng, Disorder operators in two-dimensional fermi and non-fermi liquids through multidimensional bosonization, Phys. Rev. B112, 155123 (2025). 7 END MA TTER Appendix A: Zero modes of the Hamiltonian at the Pomeranchuk instability The coupling matrix that appeared in Eq. (3), Mℓℓ′(q) = ∫π 0 dθ π ei(ℓ′−ℓ)θ cos2(θ−θ q) + ∫π 0 dθdθ′ π2 ei(ℓ′...

    2025

  46. [1994]

    arXiv:cond-mat/0505529

    Pith/arXiv arXiv