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Holomorphic Structure and Quantum Critical Points in Supersymmetric Lifshitz Field Theories

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Pith's one-line read Supersymmetric Lifshitz theories with holomorphic superpotentials have exact lines of quantum critical points, one for every coupling value.

desk verdict Genuinely new N=2 holomorphic Lifshitz SUSY models with a clean non-renormalization argument, but the strong-coupling fixed-line claim goes beyond what the paper actually controls. read the letter →

arxiv 1908.03220 v1 pith:S6QAJRGL submitted 2019-08-08 hep-th cond-mat.str-el

Igal Arav , Yaron Oz , Avia Raviv-Moshe This is my paper · ORCID
classification hep-thcond-mat.str-el
keywords SupersymmetryLifshitzscalingquantumcriticalpointnon-renormalizationtheoremholomorphicsuperpotentialtimedomaindynamicalexponentWess-Zuminomodel
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 constructs non-relativistic field theories whose supersymmetry closes on the Hamiltonian and therefore gives four real supercharges but no boost symmetry. The theories are built from a holomorphic superpotential $W(\Phi)$, and the paper proves a non-renormalization theorem: quantum corrections do not alter $W$. When the interaction is marginal, in $d=6$ with $n=3$, $d=4$ with $n=4$, and $d=3$ with $n=6$, the dimensionless coupling $\lambda_n=f_n g^{-n/2}$ has identically vanishing $\beta$ function. Hence every value of $\lambda_n$ labels a Lifshitz quantum critical point with exact scale invariance and a coupling-dependent dynamical critical exponent $z=2+2\gamma_\Phi(\lambda_n)$, without relying on perturbative control. This gives explicit families of strongly coupled, non-Lorentzian fixed points that still admit exact statements.

What carries the argument

The load-bearing object is the holomorphic superfield $\Phi(t,x,\theta,\theta^\dagger)$ defined by $D^\dagger_{\dot\alpha}\Phi=0$, with superpotential $W(\Phi)=\int d^dx\, (G(\Phi)\partial_i\Phi\partial_i\Phi+F(\Phi))$. Holomorphicity restricts the effective action to terms that can be expanded in non-negative powers of $\Phi$ and the couplings, and the $U(1)\times U(1)_R$ charge assignments then force every loop correction back to the classical tree-level form. Supergraph identities of the Lifshitz covariant derivatives reduce every closed loop to a single $d^4\theta$ integral, which cannot be converted into the $d^2\theta$ integral of a superpotential term without introducing a time derivative; this is the technical mechanism that protects $W$ from renormalization. What carries the result through to the marginal cases is the ratio invariance $\lambda_n=f_n g^{-n/2}$ under the common field-strength renormalization.

What would settle it

Look for a nonzero two-loop $\beta$ function for $\lambda_n$ in one of the three marginal models, or for a non-vanishing instanton tunneling amplitude between the trivial vacuum and a soliton vacuum; either would remove the exact fixed line.

Watch

Extended reading notes

Core claim

The paper's central claim is that in each of the three marginal cases the dimensionless coupling $\lambda_n=f_n g^{-n/2}$ is exactly marginal: $\beta_{\lambda_n}=0$ for every value, not only near zero coupling. The proof combines the holomorphicity of the superpotential with two $U(1)\times U(1)_R$ symmetries, following the relativistic non-renormalization argument: any possible correction to the superpotential would need a homogeneous factor that forces it back to a tree-level single-vertex contribution, so the quantum effective superpotential equals the classical one. Since after canonical normalization both $g$ and $f_n$ pick up the same power of the field-strength factor $Z_\Phi$, their ratio is invariant and the $\beta$ function vanishes. The anomalous dimension $\gamma_\Phi(\lambda_n)$ coming from the Kähler potential is generically nonzero, so the dynamical exponent $z=2+2\gamma_\Phi(\lambda_n)$ varies continuously along the fixed line. The paper stresses that the non-renormalization argument is non-perturbative and applies at strong coupling, provided supersymmetry and the global symmetries remain unbroken and the infrared physics is described by the same degrees of freedom.

Load-bearing premise

The conclusion rests on the symmetry pairing bosons with fermions remaining unbroken and on no new low-energy degrees of freedom appearing; if either fails, the superpotential's protected form and the fixed line are lost.

Editorial extensions

If this is right

  • In $d=6, n=3$; $d=4, n=4$; and $d=3, n=6$, every real value of $\lambda_n$ is a scale-invariant fixed point, so these are one-parameter families of interacting Lifshitz conformal field theories rather than isolated fixed points.
  • The dynamical critical exponent $z=2+2\gamma_\Phi(\lambda_n)$ changes along the line; the one-loop calculation gives $\gamma_\Phi>0$ and hence $z>2$ in all three marginal cases at weak coupling.
  • The non-renormalization theorem fixes the moduli space of vacua exactly: it is the solution set of the differential equation $\delta W/\delta\varphi=0$, including non-constant soliton vacua that can break spatial translation symmetry.
  • For the gapless case with real positive $f_2$, the superpotential remains unrenormalized while the effective coupling $\tilde\lambda_n=f_n v^{-n/2}\mu_s^{-1}$ is relevant, so the theory is strongly coupled in the infrared and may flow to a different Lifshitz fixed point.

Reading between the lines

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

  • If the exact fixed lines survive strong-coupling checks, they provide a rare example of a non-Lorentzian fixed manifold with continuously varying $z$; an analogous holomorphic non-renormalization could be sought in models with more supercharges or with vector and matrix degrees of freedom.
  • A concrete test is to evaluate the Witten index or the instanton action of the soliton vacua discussed in the paper; a nonzero tunneling amplitude into a non-supersymmetric ground state would invalidate the fixed line at strong coupling.
  • Since the $d=3$, $n=6$ case lives in $3+1$ dimensions, it could serve as a toy arena for condensed-matter quantum critical phenomena with $z\neq1$, a connection the paper does not itself develop.
  • The scheme-independence of $z$ and $\gamma_\Phi$ in the dual-scale formalism suggests that extracting these quantities from numerical or lattice probes may require care; computing them beyond one loop in a scheme-independent way would be a useful cross-check.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper constructs a class of non-relativistic (Lifshitz) supersymmetric field theories in d+1 dimensions with four real supercharges and a holomorphic superpotential, following earlier work on N=1 time-domain supersymmetry. The authors prove a non-renormalization theorem for the superpotential using holomorphy, global symmetries and a weak-coupling smoothness assumption, then identify three classically marginal cases (d=6, n=3; d=4, n=4; d=3, n=6). For each case they argue that the dimensionless coupling lambda_n has an identically vanishing beta function, so every value of lambda_n labels an exact Lifshitz scale-invariant fixed point with dynamical critical exponent z = 2 + 2 gamma_Phi(lambda_n). They compute the one-loop anomalous dimension in the three marginal cases and find it positive, and they discuss the gapless singular case f2 > 0, including its IR singularities and the role of non-renormalization in protecting the singular sphere. The paper closes with an extended discussion of non-perturbative effects, especially soliton-like vacua and instanton corrections, and explicitly lists these as open issues.

Significance. If the main claim holds, the paper provides rare examples of interacting Lifshitz field theories with exact scale invariance at arbitrary coupling, and it demonstrates that holomorphic non-renormalization can operate outside relativistic supersymmetry. The construction is explicit and the supporting calculations are concrete: the non-renormalization proof carefully adapts Seiberg's holomorphy argument, the one-loop anomalous dimension computations for the three marginal cases are shown in detail, and the dual-scale RG formalism in Section 3.2 is a useful contribution in its own right. The cross-check between time-first regularization with a spatial cutoff and split dimensional regularization in Section 3.5 strengthens the perturbative part of the paper. However, the strong-coupling portion of the central claim is conditional on assumptions that the paper itself shows are not guaranteed, and this gap must be addressed before the result can be accepted as stated.

major comments (3)
  1. [Abstract and Section 3.5 vs. Section 4] The claim that beta_n(lambda_n) = 0 identically and that every value of lambda_n gives an exact Lifshitz quantum critical point at strong coupling is not established by the argument given. The non-renormalization theorem of Section 3.3 relies on assumptions 1-3: unbroken supersymmetry and global symmetries, smoothness in the weak-coupling limit, and faithful description of the IR by microscopic degrees of freedom. Section 4 then shows that these assumptions can fail in the strong-coupling regime: soliton-like vacua exist (Eq. 4.1), and the tunneling amplitude bound in Eq. (4.2) is e^{-S_E} <= e^{-C |lambda_n|^{-2/(n-2)}}, which tends to 1 as |lambda_n| -> infinity. Thus instanton corrections to W_eff are not exponentially suppressed exactly where the strong-coupling claim is made. The abstract and Section 3.5 should either prove that such contributions vanish or explicitly state the line-of-fixed-points result as conditional on assumptions 1-3, with the strong-coupling regime left as an open problem.
  2. [Section 3.3, paragraph after Eq. (3.48)] The proof of W_eff = W_tree excludes non-perturbative contributions by invoking the weak-coupling smoothness assumption together with the expansion in non-negative powers of the couplings. This is legitimate for small couplings but cannot be used to justify the same conclusion at large |lambda_n|. The authors acknowledge this in the same paragraph and in Section 4, but the statement of the non-renormalization theorem and the subsequent use of Eq. (3.75) in Section 3.5 present the result as unconditional. The paper should separate the theorem, which holds under the stated assumptions, from the strong-coupling corollary, which does not follow from the given proof.
  3. [Section 3.6] The gapless singular case analysis relies on the same non-renormalization theorem to conclude that Im(f2) remains zero and that the singular sphere radius k0 is not renormalized. If dynamical supersymmetry breaking occurs through the soliton or instanton effects described in Section 4, this protection mechanism fails and the IR analysis in Section 3.6 is not valid. The conjectural nature of the IR fixed point is already acknowledged, but the dependence of the earlier conclusions on the unproven non-perturbative assumptions should be stated more prominently in this subsection as well.
minor comments (5)
  1. [Various] There are several typographical issues: 'T able' in the caption of Table 1, 'Kähler' appears with inconsistent spelling in a few places, and reference [7] contains 'La 1.825Sr0.175CuO4' with the subscript notation not typeset correctly.
  2. [Eq. (3.85)] The symbol k is used both as a spatial momentum vector and as its modulus; please clarify the notation, for example by writing |k| explicitly in denominators such as (k-q)^4.
  3. [Section 3.5, after Eq. (3.84)] The statement that 'these equations then have an infinite set of solutions' is confusing because it is followed by a discussion of scheme dependence. It would be clearer to say that Eqs. (3.83)-(3.84) are consistent only when Eq. (3.80) holds, and that the remaining freedom in gamma_s and gamma_t reflects renormalization-scheme dependence.
  4. [Section 3.6, Eq. (3.96)] The notation delta(k1+...+k_{n-1}) is used with an absolute value inside a delta function; please define whether this is | |k1+...+k_{n-1}| - k0 | and explain how the domain restriction is implemented in the measure.
  5. [Introduction] The analogy with the quenched-disorder result in reference [52] is mentioned only in passing in the Discussion; a sentence explaining the precise connection to the relation z = 2 + gamma_g would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fixed-line conclusion follows from an in-paper non-renormalization proof, not from fitted inputs or self-citation.

full rationale

The paper's central claim (Section 3.5) that beta_n(lambda_n)=0 in the marginal cases is derived from the non-renormalization theorem proved in Section 3.3, which in turn rests on holomorphy, the stated symmetries, and an explicit weak-coupling smoothness assumption. Equation (3.74) is an algebraic identity following from W_eff=W_tree and canonical rescaling: lambda_cn_n = f_cn_n (g_cn)^(-n/2) = f_n Z_Phi^(n/2) g^(-n/2) Z_Phi^(-n/2) = lambda_n. This is a theorem-driven relation, not a parameter fitted to data and then renamed a prediction. The relation z=2+2gamma_Phi is obtained from the Callan-Symanzik equation and independent scaling homogeneity, not from imposing the advertised conclusion. No self-citation is load-bearing: references [22] and [35] supply background and an alternative regularization method, while the same one-loop result is obtained with a spatial cutoff, so the split-dimensional computation is a cross-check rather than the foundation. The acknowledged strong-coupling caveat (Section 4, soliton/instanton contributions to W_eff) is a limitation on the validity of the non-renormalization theorem's assumptions, not a circular step, because the paper explicitly flags it rather than silently assuming the conclusion. No step in the derivation chain reduces to its own input, so the circularity score is 0.

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

No numbers are fitted to data; the dimensionless coupling lambda_n is a genuine physical parameter labeling the family of fixed points. The derivation relies on standard superspace techniques plus the stated physical assumptions about unbroken supersymmetry and infrared completeness, which the paper itself flags as uncontrolled non-perturbatively.

assumptions (5)
  • domain assumption Supersymmetry and the relevant global symmetries are non-anomalous and remain unbroken by quantum corrections.
    Section 3.3, assumption 1. The non-renormalization theorem and the fixed-point argument both require unbroken supersymmetry in the full quantum theory.
  • domain assumption The system is smooth in the weak coupling limit, so the effective action can be expanded in non-negative powers of the fields and couplings.
    Section 3.3, assumption 2. Used to rule out non-perturbative contributions to the effective superpotential in the Seiberg-style argument.
  • domain assumption The infrared physics of the system can be faithfully described by the microscopic degrees of freedom.
    Section 3.3, additional assumption. Needed for the Wilsonian effective action to have the same field content as the ultraviolet theory.
  • domain assumption A supersymmetric vacuum state exists in the quantum theory.
    Assumed in Sections 2.1, 2.2, and 3.5. The paper notes in Section 4 that soliton-like vacua and instantons could cause dynamical supersymmetry breaking, undermining this assumption.
  • standard math The Wilsonian effective action is local and can be written in terms of a single chiral superfield with a Kahler potential and a holomorphic superpotential.
    Standard superspace background, used throughout Section 3.3 to analyze the form of the effective action.

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Cite this review

Pith. "Pith review of Holomorphic Structure and Quantum Critical Points in Supersymmetric Lifshitz Field Theories." pith.science (2026). https://pith.science/paper/S6QAJRGL

@misc{pith2026190803220,
  author       = {Pith},
  title        = {Pith review of: Holomorphic Structure and Quantum Critical Points in Supersymmetric Lifshitz Field Theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6QAJRGL}},
  note         = {Machine review of arXiv:1908.03220}
}
read the original abstract

We construct supersymmetric Lifshitz field theories with four real supercharges in a general number of space dimensions. The theories consist of complex bosons and fermions and exhibit a holomorphic structure and non-renormalization properties of the superpotential. We study the theories in a diverse number of space dimensions and for various choices of marginal interactions. We show that there are lines of quantum critical points with an exact Lifshitz scale invariance and a dynamical critical exponent that depends on the coupling constants.

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Reference graph

Works this paper leans on

55 extracted references · 38 canonical work pages · cited by 2 Pith papers

  1. [1]

    Quantum criticality

    P. Coleman and A. J. Schofield,“Quantum Criticality,“ Nature 433, 226 (2005), arXiv:cond-mat/0503002 [cond-mat.str-el]

    work page Pith review arXiv 2005

  2. [2]

    Quantum Criticality,

    S. Sachdev and B. Keimer, “Quantum Criticality,” Phys. Today 64N2, 29 (2011), arXiv:1102.4628 [cond-mat.str-el]

    arXiv 2011

  3. [3]

    Quantum criticality in heavy-fermion metals,

    P. Gegenwart, Q. Si, and F. Steglich, “Quantum criticality in heavy-fermion metals,” Nature Physics 4 (Mar., 2008) 186-197, arXiv:0712.2045 [cond-mat.str-el]

    arXiv 2008

  4. [4]

    Quantum Phase Transitions,

    S. Sachdev, “Quantum Phase Transitions,” Cambridge University Press (2011)

    work page 2011

  5. [5]

    Topological order and conformal quantum critical points,

    E. Ardonne, P. Fendley and E. Fradkin, “Topological order and conformal quantum critical points,” Annals Phys. 310, 493 (2004), arXiv:cond-mat/0311466 [cond-mat.str-el]

    arXiv 2004

  6. [6]

    Anisotropic sine-gordon model and infinite-order phase transitions in three dimensions,

    G. Grinstein, “Anisotropic sine-gordon model and infinite-order phase transitions in three dimensions,” Phys. Rev. B 23 (May, 1981) 4615-4630

    work page 1981

  7. [7]

    Resistivity of La 1.825Sr0.175CuO4 and YBa2Cu3O7 to 1100 K: Absence of saturation and its implications,

    M. Gurvitch and A. T. Fiory, “Resistivity of La 1.825Sr0.175CuO4 and YBa2Cu3O7 to 1100 K: Absence of saturation and its implications,” Phys. Rev. Lett.5 (Sep, 1987) 1337-1340

    work page 1987

  8. [8]

    YbRh 2Si2: Pronounced Non-Fermi-Liquid Effects above a Low-Lying Magnetic Phase Transition,

    O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer, F. M. Grosche, P. Gegenwart, M. Lang, G. Sparn, and F. Steglich, “YbRh 2Si2: Pronounced Non-Fermi-Liquid Effects above a Low-Lying Magnetic Phase Transition,” Phys. Rev. Lett. 85 (Jul, 2000) 626-629

    work page 2000

Show all 55 references
  1. [9]

    Similarity of Scattering Rates in Metals Showing T-Linear Resistivity,

    J. A. N. Bruin, H. Sakai, R. S. Perry, and A. P. Mackenzie, “Similarity of Scattering Rates in Metals Showing T-Linear Resistivity,” Science 339 (Feb., 2013) 804-807

    2013

  2. [10]

    Critical Behavior at the Onset of → k-Space Instability on the λ Line,

    R. M. Hornreich, M. Luban and S. Shtrikman, “Critical Behavior at the Onset of → k-Space Instability on the λ Line,” Phys. Rev. Lett. 35, 1678 (1975)

    1975

  3. [11]

    Naturalness versus supersymmetric nonrenormalization theorems,

    N. Seiberg, “Naturalness versus supersymmetric nonrenormalization theorems,” Phys. Lett. B 318, 469 (1993), arXiv:hep-ph/9309335 [hep-th]

    1993 arXiv

  4. [12]

    Improved Methods for Supergraphs,

    M. T. Grisaru, W. Siegel and M. Rocek, “Improved Methods for Supergraphs,” Nucl. Phys. B 159, 429 (1979)

    1979

  5. [13]

    Non-relativistic Supersymmetry,

    W. Xue, “Non-relativistic Supersymmetry,” arXiv:1008.5102 [hep-th]

    arXiv

  6. [14]

    Horava-Lifshitz-like extensions of supersymmetric theories,

    M. Gomes, J. R. Nascimento, A. Y. Petrov and A. J. da Silva, “Horava-Lifshitz-like extensions of supersymmetric theories,” Phys. Rev. D 90, no. 12, 125022 (2014), arXiv:1408.6499 [hep-th]

    2014 arXiv

  7. [15]

    On Non-Relativistic Supersymmetry and its Spontaneous Breaking,

    A. Meyer, Y. Oz and A. Raviv-Moshe, “On Non-Relativistic Supersymmetry and its Spontaneous Breaking,” JHEP 1706, 128 (2017), arXiv:1703.04740 [hep-th]

    2017 arXiv

  8. [16]

    On Lorentz-Violating Supersymmetric Quantum Field Theories,

    D. Redigolo, “On Lorentz-Violating Supersymmetric Quantum Field Theories,” Phys. Rev. D 85 (2012) 085009, arXiv:1106.2035 [hep-th]

    2012 arXiv

  9. [17]

    Lorentz breaking supersymmetry and Horava-Lifshitz-like models,

    M. Gomes, J. Queiruga and A. J. da Silva, “Lorentz breaking supersymmetry and Horava-Lifshitz-like models,” Phys. Rev. D 92 (2015) no.2, 025050, arXiv:1506.01331 [hep-th]

    2015 arXiv

  10. [18]

    N = 1D = 3 Lifshitz-Wess-Zumino model: A paradigm of reconciliation between Lifshitz-like operators and supersymmetry,

    E. A. Gallegos, “N = 1D = 3 Lifshitz-Wess-Zumino model: A paradigm of reconciliation between Lifshitz-like operators and supersymmetry,” Phys. Lett. B 793 (2019) 372, arXiv:1806.01481 [hep-th]

    2019 arXiv

  11. [19]

    Renormalization properties of a Galilean Wess-Zumino model,

    R. Auzzi, S. Baiguera, G. Nardelli and S. Penati, “Renormalization properties of a Galilean Wess-Zumino model,” JHEP 1906 (2019) 048, arXiv:1904.08404 [hep-th]. – 52 –

    2019 arXiv

  12. [20]

    On the Perturbative Expansion around a Lifshitz Point,

    D. Orlando and S. Reffert, “On the Perturbative Expansion around a Lifshitz Point,” Phys. Lett. B 683, 62 (2010), arXiv:0908.4429 [hep-th]

    2010 arXiv

  13. [21]

    Relating Field Theories via Stochastic Quantization,

    R. Dijkgraaf, D. Orlando and S. Reffert, “Relating Field Theories via Stochastic Quantization,” Nucl. Phys. B 824, 365 (2010), arXiv:0903.0732 [hep-th]

    2010 arXiv

  14. [22]

    On Supersymmetric Lifshitz Field Theories,

    S. Chapman, Y. Oz and A. Raviv-Moshe, “On Supersymmetric Lifshitz Field Theories,” JHEP 1510, 162 (2015), arXiv:1508.03338 [hep-th]

    2015 arXiv

  15. [23]

    Supersymmetric Field Theories and Stochastic Differential Equations,

    G. Parisi and N. Sourlas, “Supersymmetric Field Theories and Stochastic Differential Equations,” Nucl. Phys. B 206, 321 (1982)

    1982

  16. [24]

    Constraints on Supersymmetry Breaking,

    E. Witten, “Constraints on Supersymmetry Breaking,” Nucl. Phys. B 202, 253 (1982)

    1982

  17. [25]

    Dynamical Breaking of Supersymmetry,

    E. Witten, “Dynamical Breaking of Supersymmetry,” Nucl. Phys. B 188, 513 (1981)

    1981

  18. [26]

    Introduction to supersymmetry in condensed matter physics,

    N. Sourlas, “Introduction to supersymmetry in condensed matter physics,” Physica D: Nonlinear Phenomena 15 (1985) 115-122

    1985

  19. [27]

    Supersymmetry and Morse theory,

    E. Witten, “Supersymmetry and Morse theory,” J. Diff. Geom. 17 (1982) no.4, 661

    1982

  20. [28]

    Stochastic Quantization,

    P. H. Damgaard and H. Huffel, “Stochastic Quantization,” Phys. Rept. 152, 227 (1987)

    1987

  21. [29]

    Renormalization of Lorentz violating theories,

    D. Anselmi and M. Halat, “Renormalization of Lorentz violating theories,” Phys. Rev. D 76, 125011 (2007), arXiv:0707.2480 [hep-th]

    2007 arXiv

  22. [30]

    Weighted scale invariant quantum field theories,

    D. Anselmi, “Weighted scale invariant quantum field theories,” JHEP 0802 (2008) 051, arXiv:0801.1216 [hep-th]

    2008 arXiv

  23. [31]

    Nonrelativistic field theoretic scale anomaly,

    O. Bergman, “Nonrelativistic field theoretic scale anomaly,” Phys. Rev. D 46, 5474 (1992)

    1992

  24. [32]

    A Supersymmetry primer,

    S. P. Martin, “A Supersymmetry primer,” Adv. Ser. Direct. High Energy Phys. 21 (2010) 1 [Adv. Ser. Direct. High Energy Phys. 18 (1998) 1], arXiv:hep-ph/9709356 [hep-th]

    2010 arXiv

  25. [33]

    Holomorphic Supersymmetric Quantum Mechanics, Generalized Supersymmetry, and Parasupersymmetry,

    A. D. Dolgallo, K. N. Ilinski, “Holomorphic Supersymmetric Quantum Mechanics, Generalized Supersymmetry, and Parasupersymmetry,” Annals Phys. 236 2 (1994) 219-240

    1994

  26. [34]

    Ground State Structure in Supersymmetric Quantum Mechanics,

    A. M. Jaffe, A. Lesniewski and M. Lewenstein, “Ground State Structure in Supersymmetric Quantum Mechanics,” Annals Phys. 178 (1987) 313

    1987

  27. [35]

    Lifshitz Anomalies, Ward Identities and Split Dimensional Regularization,

    I. Arav, Y. Oz and A. Raviv-Moshe, “Lifshitz Anomalies, Ward Identities and Split Dimensional Regularization,” JHEP 1703, 088 (2017), arXiv:1612.03500 [hep-th]

    2017 arXiv

  28. [36]

    Lorentz symmetry breaking as a quantum field theory regulator,

    M. Visser, “Lorentz symmetry breaking as a quantum field theory regulator,” Phys. Rev. D 80 (2009) 025011, arXiv:0902.0590 [hep-th]

    2009 arXiv

  29. [37]

    Tree-Level Unitarity and Renormalizability in Lifshitz Scalar Theory,

    T. Fujimori, T. Inami, K. Izumi and T. Kitamura, “Tree-Level Unitarity and Renormalizability in Lifshitz Scalar Theory,” PTEP 2016 (2016) no.1, 013B08, arXiv:1510.07237 [hep-th]

    2016 arXiv

  30. [38]

    A New Theory of Anyons,

    A. L. Fitzpatrick, S. Kachru, J. Kaplan, E. Katz and J. G. Wacker, “A New Theory of Anyons,” arXiv:1205.6816 [hep-th]

    arXiv

  31. [39]

    Lifshitz-type Quantum Field Theories in Particle Physics,

    J. Alexandre, “Lifshitz-type Quantum Field Theories in Particle Physics,” Int. J. Mod. Phys. A 26, 4523 (2011), arXiv:1109.5629 [hep-ph]

    2011 arXiv

  32. [40]

    Fermion effective dispersion relation for z = 2 Lifshitz QED,

    J. Alexandre and J. Brister, “Fermion effective dispersion relation for z = 2 Lifshitz QED,” Phys. Rev. D 88, no. 6, 065020 (2013), arXiv:1307.7613 [hep-th]

    2013 arXiv

  33. [41]

    An Elementary Proof of Dyson’s Power Counting Theorem,

    Y. Hahn and W. Zimmermann, “An Elementary Proof of Dyson’s Power Counting Theorem,” Commun. Math. Phys. 10 (1968) 330. – 53 –

    1968

  34. [42]

    High-energy behavior in quantum field theory,

    S. Weinberg, “High-energy behavior in quantum field theory,” Phys. Rev. 118, 838 (1960)

    1960

  35. [43]

    Split dimensional regularization for the Coulomb gauge,

    G. Leibbrandt and J. Williams, “Split dimensional regularization for the Coulomb gauge,” Nucl. Phys. B 475, 469 (1996), arXiv:hep-th/9601046 [hep-th]

    1996 arXiv

  36. [44]

    The three point function in split dimensional regularization in the Coulomb gauge,

    G. Leibbrandt, “The three point function in split dimensional regularization in the Coulomb gauge,” Nucl. Phys. B 521, 383 (1998), arXiv:hep-th/9804109 [hep-th]

    1998 arXiv

  37. [45]

    Supersymmetry and supergravity,

    J. Wess and J. Bagger, “Supersymmetry and supergravity,” Princeton, USA: Univ. Pr. (1992)

    1992

  38. [46]

    Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry,

    H. K. Dreiner, H. E. Haber and S. P. Martin, “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry,” Phys. Rept. 494, 1 (2010), arXiv:0812.1594 [hep-ph]

    2010 arXiv

  39. [47]

    Renormalization group approach to interacting fermions,

    R. Shankar, “Renormalization group approach to interacting fermions,” Rev. Mod. Phys. 66 (1994) 129, arXiv:cond-mat/9307009 [cond-mat]

    1994 arXiv

  40. [48]

    Effective field theory and the Fermi surface,

    J. Polchinski, “Effective field theory and the Fermi surface,” in “Proceedings, Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles : Boulder, USA, June 1-26, 1992,” arXiv:hep-th/9210046 [hep-th]

    1992 arXiv

  41. [49]

    Compactness results and applications to some

    A. Azzollini and A. Pomponio, “Compactness results and applications to some ”zero mass” elliptic problems,” Nonlinear Anal. 69 (2008) 3559-3576, arXiv:math/0601410 [math.AP]

    2008 arXiv

  42. [50]

    Nonlinear scalar field equations, I existence of a ground state,

    H. Berestycki and P. -L. Lions, “Nonlinear scalar field equations, I existence of a ground state,” Arch. Rational Mech. Anal. 82 (1983) 313-345

    1983

  43. [51]

    Nonlinear scalar field equations, II existence of infinitely many solutions,

    H. Berestycki, P. -L. Lions, “Nonlinear scalar field equations, II existence of infinitely many solutions,” Arch. Rational Mech. Anal. 82 (1983) 347-375

    1983

  44. [52]

    Renormalization group flow in field theories with quenched disorder,

    O. Aharony and V. Narovlansky, “Renormalization group flow in field theories with quenched disorder,” Phys. Rev. D 98, no. 4, 045012 (2018), arXiv:1803.08534 [hep-th]

    2018 arXiv

  45. [53]

    The dynamics of charge-density waves,

    G. Gruner, “The dynamics of charge-density waves,” Rev. Mod. Phys. 60 (1988) 1129

    1988

  46. [54]

    The dynamics of spin-density waves,

    G. Gruner, “The dynamics of spin-density waves,” Rev. Mod. Phys. 66 (1994) 1

    1994

  47. [55]

    Lattice symmetry breaking in cuprate superconductors: Stripes, nematics, and superconductivity,

    M. Vojta, “Lattice symmetry breaking in cuprate superconductors: Stripes, nematics, and superconductivity,” Adv. Phys. 58 (2009) 699-820. – 54 –

    2009

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.