Depinning transition of charge-density waves: mapping onto $O(n)$ symmetric $\phi^4$ theory with $n\to -2$ and loop-erased random walks

Andrei A. Fedorenko; Kay Joerg Wiese

REVIEW 3 major objections 5 minor 1 cited by

Depinning transition of charge-density waves: mapping onto $O(n)$ symmetric $\phi^4$ theory with $n\to -2$ and loop-erased random walks

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The depinning transition of charge-density waves is exactly captured by $O(n)$ symmetric $\phi^4$ theory with $n\to -2$, the same field theory that describes loop-erased random walks.

desk verdict A genuinely new and likely important mapping from CDW depinning to O(n=-2) phi^4 theory, with the main caveat being the unproven all-orders quadratic fixed-point form for the disorder correlator. read the letter →

arxiv 1908.11721 v2 pith:44W4FW4X submitted 2019-08-30 cond-mat.dis-nn cond-mat.stat-mech

Kay Joerg Wiese , Andrei A. Fedorenko This is my paper · ORCID

classification cond-mat.dis-nncond-mat.stat-mech

keywords charge-densitywavesdepinningtransitionfunctionalrenormalizationgroupO(n)-symmetricphi^4theorynegative-componentlimitloop-erasedrandomwalksfractaldimensionepsilonexpansion

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 the critical behavior of driven periodic elastic systems, exemplified by charge-density waves at the depinning transition, is exactly captured by the much simpler $O(n)$-symmetric $\phi^4$ field theory in the formal limit $n\to -2$. If correct, this unifies a glassy nonequilibrium transition, a standard equilibrium field theory, and the fractal geometry of loop-erased random walks. The authors provide a perturbative diagrammatic proof, a nonperturbative supersymmetry-based proof, and an explicit four-loop calculation showing that the $\beta$ function and the critical exponents $z$, $\nu = 1/2$, and $\eta = 0$ coincide between the two theories. This yields the value $z = 1.6243 \pm 0.001$ for the dynamic exponent in three dimensions, consistent with the most precise numerical simulations, and the exact value $z = 5/4$ in two dimensions.

What carries the argument

The engine of the argument is the two-replica supersymmetric representation of the disorder average. Writing two copies of the elastic system and passing to center-of-mass coordinates transforms the charge-density-wave action into a $\phi^4$-type theory with one complex boson and two complex fermions, which is equivalent to complex $\phi^4$ theory with $N = -1$, or real $O(n)$ theory with $n\to -2$. The fractal dimension of the loop-erased walk is extracted from the crossover operator $O(y) = \Phi_1^*\Phi_1 - \Phi_2^*\Phi_2$, which measures the length of the blue backbone after loops have been erased; the same operator renormalizes the friction term in the dynamics, tying the depinning exponent $z$ to the crossover exponent of the $\phi^4$ theory.

What would settle it

Compute the five-loop functional renormalization-group flow for the periodic fixed point of the charge-density-wave action: if a term of order $u^3$ or higher appears in the running disorder correlator at the fixed point, the quadratic closure fails and the $\beta$ function will deviate from the $O(n)$ $\phi^4$ result with $n\to -2$. Alternatively, a measurement of the loop-erased random walk fractal dimension in $d=3$ that disagrees with $z = 1.6243 \pm 0.001$ by more than the combined error bars would falsify the common-sector prediction.

Watch

Extended reading notes

Core claim

The central discovery is an equivalence of three sectors: the depinning transition of charge-density waves as described by the functional renormalization group, the $O(n)$ symmetric $\phi^4$ theory at $n\to -2$, and loop-erased random walks in arbitrary dimension. The authors show that the disorder correlator of the charge-density-wave problem flows to a fixed point with a purely quadratic cusped shape, reducing the functional RG to a single coupling $g$, and that the resulting flow is identical to the $\beta$ function of the $\phi^4$ theory. They then identify the dynamic critical exponent $z$ of depinning with the crossover exponent that measures the length of the loop-erased backbone in the $\phi^4$ theory. The identity is verified explicitly at four loops; the $\varepsilon$-expansion through fifth order, resummed from six-loop data, matches the best simulations in $d=3$ and the exact SLE$_2$ value in $d=2$.

Load-bearing premise

The mapping depends on the fixed-point disorder correlator having exactly the quadratic cusped form $\Delta(u) = \Delta(0) - \frac{g}{2}u(1-u)$ for $u\in[0,1]$, which has only been verified to three-loop order; if higher-order terms appear at four or more loops, the flow no longer closes on the single coupling $g$ and the equivalence to $\phi^4$ theory fails.

Editorial extensions

If this is right

The dynamic critical exponent of charge-density waves at depinning in $d = 3$ is predicted as $z = 1.6243 \pm 0.001$, matching the most precise numerical value; in $d = 2$ it is exactly $z = 5/4$.
Within the sector where the two theories can be compared, all renormalization-group functions and critical exponents coincide, giving a nonperturbative statement and not merely a few matching loop orders.
The result supports the earlier conjecture that pinned charge-density waves belong to the same universality class as Abelian sandpiles and loop-erased random walks.
The equivalence yields the correction-to-scaling exponent $\omega = 0.83 \pm 0.01$ for loop-erased random walks, with a proposed measurement protocol: erase loops with probability $p<1$ in simulations.
The same effective-theory route could be applied to other disordered elastic systems such as random-field magnets, replacing the technically demanding functional RG with standard $\phi^4$ methods.

Reading between the lines

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

If the equivalence is exact, the depinning transition becomes a physical realization of a negative-component field theory, and experiments on driven vortex lattices or Wigner crystals could in principle probe the common critical sector through dynamic response measurements.
The identification of the dynamic exponent with a crossover exponent suggests that high-precision simulations of the crossover exponent in $O(n)$ models at $n = -2$ across dimensions would serve as an independent check of the mapping away from $d = 3$ and $d = 2$.
The paper explicitly leaves the two theories non-isomorphic: observables such as the full two-point dynamic correlation function are not shared, so any experiment probing avalanche statistics could confront the claim that the equivalence holds only on the restricted sector.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Kay, here's my read on arXiv:1908.11721.

The paper claims something genuinely new: CDW depinning is exactly equivalent to O(n=-2) phi^4 theory, and both describe LERWs in any dimension. That is a big deal if it holds, because it replaces the cumbersome FRG with textbook phi^4 machinery and produces a sharp z=1.6243±0.001 in d=3. The 5-loop expansion and the agreement with Wilson's numerics (1.62400±0.00005) are striking. The diagrammatic rewriting of the crossing cancellation is elegant, and the authors are clear that the two theories agree only in a specific sector, not as full isomorphisms. They earn credit for stating exactly what they proved and what they deferred.

The soft spot is exactly where the stress-test points. The reduction to a single coupling g depends on the fixed-point disorder correlator being exactly quadratic, Delta(u)=Delta(0)-(g/2)u(1-u), and the paper says this is "confirmed explicitly to three-loop order and presumably holds to all orders." That is the load-bearing step. If beyond three loops a u^3 or u^4 term appears, the effective action acquires interactions beyond quartic, and the phi^4 equivalence fails. The paper defers the all-orders proof to the companion [28]. In a Letter that's acceptable, but the word "presumably" is doing real work. The "nonperturbative proof" in the text also uses the same quadratic replacement, so I would not call it fully nonperturbative until the fixed-point shape is settled.

Two smaller things. The Borel-resummation error bar on z is quoted without derivation; I'd like to see it or a reference to where it is derived. And the claim that the result was "overlooked for decades" is a bit strong, given that the authors' own earlier work conjectured the LERW connection. Not a flaw, just a tone.

On circularity: no. They are not fitting z to the simulations; the exponent comes from standard phi^4 epsilon expansion. The agreement with numerics is a test, not an input. The self-citations are appropriate here, given that the FRG machinery is theirs.

Who is this for: anyone working on depinning, LERWs, or the O(n) model. I'd bring it to our reading group. It deserves a serious referee; the main request should be a clear statement of the status of the fixed-point proof in [28] and, if possible, a check of whether the quadratic form is exact in the periodic case.

Referee Report

3 major / 5 minor

Summary. The Letter claims to establish an exact equivalence between the depinning transition of charge-density waves (CDWs) in random media and O(n)-symmetric phi^4 theory in the formal limit n -> -2, a theory that also describes loop-erased random walks (LERWs). The authors start from the functional-RG description of CDWs, use the periodic fixed-point form of the disorder correlator, and then transform the disorder-averaged dynamical action via a two-copy supersymmetry construction into a phi^4-type theory with one complex boson and two complex fermions, which they identify with real phi^4 theory at n = -2. The dynamic exponent z is identified with the crossover exponent associated with the operator O(y) = phi_1^* phi_1 - phi_2^* phi_2, and the Letter reports the epsilon expansion of z to five loops, Eq. (14), with Borel-resummed values z(d=2) = 5/4 and z(d=3) = 1.6243 ± 0.001. The authors state that both theories yield identical beta functions and exponents nu = 1/2 and eta = 0, and that the equivalence is checked to four-loop order and proven both perturbatively and nonperturbatively, with details deferred to Ref. [28].

Significance. If the claimed exact equivalence holds, this is a significant result: it connects a glassy, disordered, dynamical critical phenomenon to an ordinary phi^4 field theory, provides high-precision exponents for LERWs from six-loop phi^4 results, and gives strong support to the Narayan-Middleton conjecture. The paper makes explicit falsifiable predictions, including the exact value z(d=2) = 5/4 and the high-precision value z(d=3) = 1.6243 ± 0.001, which agrees strikingly with numerical simulations. The identification of a crossover operator that measures the LERW backbone and its relation to the friction renormalization is an elegant and potentially powerful construction. However, the manuscript's own caveats, in particular the statement that the exact quadratic form of the disorder correlator is only "presumably" valid to all orders, leave the central exact-equivalence claim conditional rather than fully established in the present Letter.

major comments (3)

[after Eq. (7) and Supplemental Eq. (21)] The exact equivalence between CDWs and O(n=-2) phi^4 theory rests on the claim that the depinning fixed-point disorder correlator has the exact cusped quadratic form Delta(u) = Delta(0) - (g/2) u(1-u) on [0,1]. The Letter states that this form has been confirmed only to three-loop order and "presumably holds to all orders," citing Refs. [26,27], which are three-loop papers. Since the Letter itself notes that all Taylor coefficients of Delta are relevant couplings for d < 4, the appearance of any higher-order coefficient at the fixed point would generate interactions beyond quartic in the effective action and invalidate the mapping to phi^4 theory. The present Letter does not provide a proof of the all-orders quadratic form; it defers to Ref. [28]. The authors must either supply or precisely cite such a proof, or explicitly qualify the equivalence as holding only to the order to which it has been checked and adjust the exactness statements accordingly.
[Supplemental Eqs. (20)-(22) and Eq. (10)] The reduction to the phi^4 action requires replacing Delta(u) by Delta(0) + (g/2) u^2, which discards the linear term -g/2 u present in the cusped quadratic fixed point. The authors note in the Supplemental Material that including this linear term would produce a term ilde u(x) sum_a ar psi_a psi_a, renormalize Delta(0), and lead to breaking of supersymmetry. However, the Letter does not demonstrate that this term cannot feed back into the renormalization of the effective coupling g or into the crossover operator O(y) at higher orders. Because the cusp and the associated Delta'(0+) are central to the depinning fixed point, the claim that this sector is inert for the quantities identified with the phi^4 theory needs an explicit argument or a precise reference; without it, the proof that the beta functions and exponents coincide is incomplete.
[Eq. (14) and surrounding text] The central quantitative claim, z(d=3) = 1.6243 ± 0.001, is obtained by Borel resummation of the six-loop phi^4 crossover exponent from Ref. [62], but the Letter's own verification of the CDW/phi^4 mapping is stated to be at four-loop order. The text says the diagrams for O(y) were generated at five-loop order and the coupling renormalization at four-loop order, using results from Refs. [50,51], and it says the all-orders proof is in Ref. [28]. The reader cannot verify from the Letter whether the five-loop expression in Eq. (14) follows from a complete five-loop calculation within the mapped theory or from a combination of known phi^4 results with the mapping checked only to four loops. The authors should state explicitly which parts of Eq. (14) are proven in the present Letter and which are inherited from the assumed exact equivalence.

minor comments (5)

[Supplemental Material, Section B title] The word "equivalance" in the title "Proof for the equivalance of phi^4-theory at N=-1 and CDWs" is a typo and should read "equivalence."
[Main text near Eq. (10)] The sentence "It is easy to check, that while u(x) and \tilde u(x) have nontrivial expectations" contains a comma splice; it should read "It is easy to check that, while u(x) and \tilde u(x) have nontrivial expectations, ...".
[Eq. (12) and surrounding diagrams] The ASCII diagrams are difficult to read, especially the two-line arrow diagram in Eq. (12); the authors should consider replacing them with properly typeset Feynman diagrams or moving the detailed diagrammatic proof entirely to the Supplemental Material.
[Eq. (14)] The epsilon^4 bracket in Eq. (14) is visually ambiguous because of the leading minus sign outside a large bracket containing both positive and negative terms; rewriting the expression with explicit parentheses would improve readability.
[Main text, third paragraph] The phrase "the choice g = 1 cancels the first two terms, while the last one is absent at N = 0" is slightly imprecise: at N = 0 the second term is present but the last term is absent, so the cancellation is between the first and second terms; the wording should be adjusted to avoid confusion.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the CDW-to-O(n=-2) mapping is derived from the FRG fixed-point structure, a supersymmetry transformation, and an external LERW theorem, while the reported exponent is benchmarked against independent simulations.

full rationale

The paper's derivation chain is not circular. The central mapping of CDW depinning onto O(n=-2) phi^4 theory is obtained from: (i) the FRG fixed-point form Delta(u)=Delta(0)-(g/2)u(1-u), which is cited to three-loop FRG calculations and explicitly described as 'presumably' holding to all orders rather than as an established theorem; (ii) the supersymmetric disorder-averaging action (Eq. 8/17) followed by the replacement in Supplemental Eq. (21) that converts the quadratic cusp into a phi^4 interaction; and (iii) an external mathematical result, the Sapozhnikov-Shiraishi theorem (Ref. [54]), RW = LERW plus loop soup, which independently anchors the LERW connection. No parameter is fitted to the target exponents: the dynamic exponent z is obtained from the standard phi^4 epsilon expansion, Eq. (14), using diagrams from the independent references [50,51], and the resulting z(d=3)=1.6243 is compared with, not derived from, Wilson's numerical value 1.62400. Some self-citations are present, notably [28] for the all-orders proof details and [62] for the six-loop extension, but they are not the sole justification: the Letter contains a proof sketch, a four-loop check, and an external numerical benchmark. The least secure link is the unproven exact quadratic shape of the FRG fixed point beyond three loops; the paper itself flags this as 'presumably' true, so it is a correctness/assumption risk rather than a circular reduction. Overall, the central claim retains independent content and is externally falsifiable.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The derivation rests on the assumed exact quadratic cusp form of the disorder correlator at the fixed point, the validity of the two-replica supersymmetry method for the second cumulant, and the external loop-soup theorem. No new entities are introduced and no parameters are fitted to numerical data.

assumptions (4)

domain assumption The disorder correlator at the periodic fixed point has the exact cusped quadratic form Delta(u) = Delta(0) - (g/2) u(1-u) for u in [0,1], with the FRG flow closing in the space of polynomials of degree two.
Used to reduce the full FRG to a single coupling g and to derive the equivalence to phi^4 theory. The paper states this form has been confirmed to three-loop order and 'presumably holds to all orders' (main text after Eq. (7)).
domain assumption The supersymmetry method with two replicas (r=2) correctly captures the second cumulant of disorder and resolves dimensional reduction, reproducing the FRG flow.
Invoked in the main text and Supplemental Material B, relying on Refs. [59-61], including self-cited Ref. [61]. If this method fails, the mapping to phi^4 theory fails.
standard math The O(n) phi^4 theory with n -> -2 is equivalent to a theory with one complex boson and two complex fermions, with fermion loop signs giving n = -2.
Standard Feynman diagram bookkeeping for Grassmann fields, used to rewrite the theory. Could be verified, but not a new mathematical theorem.
standard math The theorem that a random walk is the union of a LERW and an independent loop soup of intensity 2 (RW = LERW (+) LS(2)), from Ref. [54].
Used in the sketch of the nonperturbative proof in Supplemental Material A. It is an external mathematical theorem.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Depinning transition of charge-density waves: mapping onto $O(n)$ symmetric $\phi^4$ theory with $n\to -2$ and loop-erased random walks." pith.science (2026). https://pith.science/paper/44W4FW4X

@misc{pith2026190811721,
  author       = {Pith},
  title        = {Pith review of: Depinning transition of charge-density waves: mapping onto $O(n)$ symmetric $\phi^4$ theory with $n\to -2$ and loop-erased random walks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/44W4FW4X}},
  note         = {Machine review of arXiv:1908.11721}
}

abstract

Driven periodic elastic systems such as charge-density waves (CDWs) pinned by impurities show a non-trivial, glassy dynamical critical behavior. Their proper theoretical description requires the functional renormalization group. We show that their critical behavior close to the depinning transition is related to a much simpler model, $O(n)$-symmetric $\phi^4$ theory in the unusual limit of $n\to -2$. We demonstrate that both theories yield identical results to 4-loop order and give both a perturbative and a non-perturbative proof of their equivalence. As we show, both theories can be used to describe loop-erased random walks (LERWs), the trace of a random walk where loops are erased as soon as they are formed. Remarkably, two famous models of non-self-intersecting random walks, self-avoiding walks (SAWs) and LERWs, can both be mapped onto $\phi^4$ theory taken, with formally $n=0$ and $n\to -2$ components. This mapping allows us to compute the dynamic critical exponent of CDWs at the depinning transition and the fractal dimension of LERWs in $d=3$ with unprecedented accuracy, $z(d=3)= 1.6243 \pm 0.001$, in excellent agreement with the estimate $z = 1.624 00 \pm 0.00005$ of numerical simulations.

Figures

Figures reproduced from arXiv: 1908.11721 by the authors.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Fractal dimension of critical curves in the $O(n)$-symmetric $\phi^4$-model and crossover exponent at 6-loop order: Loop-erased random walks, self-avoiding walks, Ising, XY and Heisenberg models
cond-mat.stat-mech 2019-08 accept novelty 7.0 of 10

A six-loop field-theoretic calculation gives the fractal dimension of critical curves and the crossover exponent in O(n) models, with estimates for LERW, SAW, Ising, XY and Heisenberg systems.

Reference graph

Works this paper leans on

68 extracted references · 66 canonical work pages · cited by 1 Pith paper

[28]

Wiese and A.A

K.J. Wiese and A.A. Fedorenko, Field theories for loop-erased random walks, Nucl. Phys. B 946,114696 (2019)

work page 2019
[62]

Wiese, Supersymmetry breaking in disordered systems and relation to functional renormalization and replica-symmetry breaking, J

K.J. Wiese, Supersymmetry breaking in disordered systems and relation to functional renormalization and replica-symmetry breaking, J. Phys.: Condens. Matter. 17, S1889 (2005)

work page 2005
[1]

Gr ¨uner, The dynamics of charge-density waves, Rev

G. Gr ¨uner, The dynamics of charge-density waves, Rev. Mod. Phys. 60, 1129 (1988)

work page 1988
[2]

Fukuyama and P.A

H. Fukuyama and P.A. Lee, Dynamics of the charge-density wave. I. Impurity pinning in a single chain , Phys. Rev. B 17, 535 (1978)

work page 1978
[3]

P. A. Lee and T. M. Rice, Electric-ﬁeld depinning of charge- density waves, Phys. Rev. B 19, 3970 (1979)

work page 1979
[4]

Blatter, M.V

G. Blatter, M.V . Feigel’man, V .B. Geshkenbein, A.I. Larkin, and V .M. Vinokur,Vortices in high-temperature superconduc- tors, Rev. Mod. Phys. 66, 1125 (1994)

work page 1994
[5]

Nattermann and S

T. Nattermann and S. Scheidl, Vortex-glass phases in type-II superconductors, Adv. Phys. 49, 607 (2000)

work page 2000
[6]

Le Doussal and T

P. Le Doussal and T. Giamarchi, Moving glass theory of driven lattices with disorder, Phys. Rev. B 57, 11356 (1998)

1998

Show all 68 references

[7]

Klein, I

T. Klein, I. Joumard, S. Blanchard, J. Marcus, R. Cubitt, T. Gi- amarchi, and P. Le Doussal, A Bragg glass phase in the vortex lattice of a type II superconductor, Nature 413, 404 (2001)

2001
[8]

Monceau, Electronic crystals: an experimental overview , Adv

P. Monceau, Electronic crystals: an experimental overview , Adv. Phys. 61, 325 (2012)

2012
[9]

Reichhardt, C.J

C. Reichhardt, C.J. Olson, N. Grønbech-Jensen, and F. Nori, Moving Wigner glasses and smectics: dynamics of disordered Wigner crystals, Phys. Rev. Lett. 86, 4354 (2001)

2001
[10]

Chitra, T

R. Chitra, T. Giamarchi, and P. Le Doussal, Dynamical Prop- erties of the Pinned Wigner Crystal, Phys. Rev. Lett. 80, 3827 5 (1998)

1998
[11]

Larkin, Effect of inhomogeneities on the structure of the mixed state of superconductors , Sov

A.I. Larkin, Effect of inhomogeneities on the structure of the mixed state of superconductors , Sov. Phys. JETP 31, 784 (1970)

1970
[12]

Middleton, Thermal rounding of the charge-density-wave depinning transition, Phys

A.A. Middleton, Thermal rounding of the charge-density-wave depinning transition, Phys. Rev. B 45, 9465 (1992)

1992
[13]

Middleton and D.S

A.A. Middleton and D.S. Fisher, Critical behavior of charge- density waves below threshold: Numerical and scaling analy- sis, Phys. Rev. B 47, 3530 (1993)

1993
[14]

Duemmer and W

O. Duemmer and W. Krauth, Critical exponents of the driven elastic string in a disordered medium, Phys. Rev. E 71, 061601 (2005)

2005
[15]

Di Scala, E

N. Di Scala, E. Olive, Y . Lansac, Y . Fily, and J.C. Soret, The elastic depinning transition of vortex lattices in two dimensions, New J. Phys. 14 14, 123027 (2012)

2012
[16]

Bustingorry, A.B

S. Bustingorry, A.B. Kolton, and T. Giamarchi, Random- manifold to random-periodic depinning of an elastic interface , Phys. Rev. B 82, 094202 (2010)

2010
[17]

Fisher, Sliding charge-density waves as a dynamical criti- cal phenomena, Phys

D.S. Fisher, Sliding charge-density waves as a dynamical criti- cal phenomena, Phys. Rev. B 31, 1396 (1985)

1985
[18]

Narayan and D.S

O. Narayan and D.S. Fisher, Critical behavior of sliding charge-density waves in 4-epsilon dimensions , Phys. Rev. B 46, 11520 (1992)

1992
[19]

Narayan and D.S

O. Narayan and D.S. Fisher, Dynamics of Sliding Charge- Density Waves in 4-epsilon Dimensions , Phys. Rev. Lett. 68, 3615 (1992)

1992
[20]

Leschhorn, T

H. Leschhorn, T. Nattermann, S. Stepanow, and L.-H. Tang, Driven interface depinning in a disordered medium, Ann. Phys. 509, 1 (1997)

1997
[21]

Nattermann, S

T. Nattermann, S. Stepanow, L.-H. Tang, and H. Leschhorn, Dynamics of interface depinning in a disordered medium , J. Phys. II (France) 2, 1483 (1992)

1992
[22]

Larkin and Y .N

A.I. Larkin and Y .N. Ovchinnikov,Pinning in type II supercon- ductors, J. Low Temp. Phys. 34, 409 (1979)

1979
[23]

Le Doussal, K.J

P. Le Doussal, K.J. Wiese, and P. Chauve,Functional renormal- ization group and the ﬁeld theory of disordered elastic systems, Phys. Rev. E 69, 026112 (2004)

2004
[24]

Le Doussal, K.J

P. Le Doussal, K.J. Wiese, and P. Chauve, 2-loop functional renormalization group analysis of the depinning transition , Phys. Rev. B 66, 174201 (2002)

2002
[25]

Chauve, P

P. Chauve, P. Le Doussal, and K.J. Wiese, Renormalization of Pinned Elastic Systems: How Does it Work Beyond One Loop?, Phys. Rev. Lett. 86, 1785 (2001)

2001
[26]

Wiese, C

K.J. Wiese, C. Husemann, and P. Le Doussal, Field theory of disordered elastic interfaces at 3-loop order: The β-function, Nucl. Phys. B 932, 540 (2018)

2018
[27]

Husemann and K.J

C. Husemann and K.J. Wiese, Field theory of disordered elas- tic interfaces to 3-loop order: Critical exponents and scaling functions, Nucl. Phys. B 932, 589 (2018)

2018
[29]

De Gennes, Exponents for the excluded volume problem as derived by the Wilson method, Phys

P.-G. De Gennes, Exponents for the excluded volume problem as derived by the Wilson method, Phys. Lett. A 38, 339 (1972)

1972
[30]

Majumdar, Exact Fractal Dimension of the Loop-Erased Self-Avoiding Walk in Two Dimensions , Phys

S.N. Majumdar, Exact Fractal Dimension of the Loop-Erased Self-Avoiding Walk in Two Dimensions , Phys. Rev. Lett. 68, 2329 (1992)

1992
[31]

Dhar, Theoretical studies of self-organized criticality, Phys- ica A 369, 29 (2006)

D. Dhar, Theoretical studies of self-organized criticality, Phys- ica A 369, 29 (2006)

2006
[32]

J. W. Lyklema, C. Evertsz, and L. Pietronero, The Laplacian random walk, EPL 2, 77 (1986)

1986
[33]

Lawler, The Laplacian-b random walk and the Schramm- Loewner evolution, Illinois J

G.F. Lawler, The Laplacian-b random walk and the Schramm- Loewner evolution, Illinois J. Math. 50, 701 (2006)

2006
[34]

Lawler, A self-avoiding random walk, Duke Math

G.F. Lawler, A self-avoiding random walk, Duke Math. J. 47, 655 (1980)

1980
[35]

Kozma, The scaling limit of loop-erased random walk in three dimensions, Act

G. Kozma, The scaling limit of loop-erased random walk in three dimensions, Act. Math. 199, 29 (2007)

2007
[36]

Guttmann and R.J

A.J. Guttmann and R.J. Bursill, Critical exponent for the loop erased self-avoiding walk by monte carlo methods, J. Stat. Phys 59, 1 (1990)

1990
[37]

Agrawal and D

H. Agrawal and D. Dhar, Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions , Phys. Rev. E 63, 056115 (2001)

2001
[38]

Grassberger, Scaling of loop-erased walks in 2 to 4 dimen- sions, J

P. Grassberger, Scaling of loop-erased walks in 2 to 4 dimen- sions, J. Stat. Phys. 136, 399 (2009)

2009
[39]

Wilson, Dimension of the loop-erased random walk in three dimensions, Phys

D.B. Wilson, Dimension of the loop-erased random walk in three dimensions, Phys. Rev. E 82, 062102 (2010)

2010
[40]

Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J

O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118, 221 (2000)

2000
[41]

Lawler, O

G.F. Lawler, O. Schramm, and W. Werner, Conformal invari- ance of planar loop-erased random walks and uniform span- ning trees, Ann. Probab. 32, 939 (2004)

2004
[42]

Nienhuis, Exact Critical Point and Critical Exponents of O(n) Models in Two Dimensions , Phys

B. Nienhuis, Exact Critical Point and Critical Exponents of O(n) Models in Two Dimensions , Phys. Rev. Lett. 49, 1062 (1982)

1982
[43]

Duplantier, Loop-erased self-avoiding walks in two dimen- sions: exact critical exponents and winding numbers , Physica A 191, 516 (1992)

B. Duplantier, Loop-erased self-avoiding walks in two dimen- sions: exact critical exponents and winding numbers , Physica A 191, 516 (1992)

1992
[44]

If this conjecture holds, then the φ4 theory atn→− 2 has to reproduce the FRG picture for CDWs, at least for observ- ables related to LERWs

agree with rigorous mathematical bounds, and have been tested against numerical simulations at the upper critical di- mensionduc = 4 [38], where it was found that they correctly reproduce the leading and subleading logarithmic corrections. If this conjecture holds, then the φ4...

1944
[45]

Fedorenko, P

A.A. Fedorenko, P. Le Doussal, and K.J. Wiese, Field theory conjecture for loop-erased random walks , J. Stat. Phys. 133, 805 (2008)

2008
[46]

Narayan and A.A

O. Narayan and A.A. Middleton, Avalanches and the renormalization-group for pinned charge-density waves, Phys. Rev. B 49, 244 (1994)

1994
[47]

Rosso, P

A. Rosso, P. Le Doussal, and K.J. Wiese, Avalanche-size dis- tribution at the depinning transition: A numerical test of the theory, Phys. Rev. B 80, 144204 (2009)

2009
[48]

Kaspar and M

D.C. Kaspar and M. Mungan, Subthreshold behavior and avalanches in an exactly solvable charge density wave system , EPL 103, 46002 (2013)

2013
[49]

Zinn-Justin, Phase transitions and renormalization group , Oxford University Press, Oxford, (2007)

J. Zinn-Justin, Phase transitions and renormalization group , Oxford University Press, Oxford, (2007)

2007
[50]

Kleinert and V

H. Kleinert and V . Schulte-Frohlinde,Critical properties ofφ4- theories, World Scientiﬁc Publishing, (2001)

2001
[51]

Kleinert, J

H. Kleinert, J. Neu, N. Schulte-Frohlinde, and S.A. Larin, Five- loop renormalization group functions of O(N )-symmetricφ4- theory and ε-expansion of critical exponents up to ε5, Phys. Lett. B 272, 39 (1991)

1991
[52]

Kompaniets and E

M.V . Kompaniets and E. Panzer,Minimally subtracted six-loop renormalization ofO(n)-symmetricφ4 theory and critical ex- ponents, Phys. Rev. D 96, 036016 (2017)

2017
[53]

See Supplemental material for details on relations between LERWs,O(n =−2) symmetricφ4 theory and CDWs, which includes Ref. [53]
[54]

Wu, Number of spanning trees on a lattice , J

F.Y . Wu, Number of spanning trees on a lattice , J. Phys. A: Math. Gen. 10, L113 (1977)

1977
[55]

Sapozhnikov and D

A. Sapozhnikov and D. Shiraishi, On Brownian motion, simple paths, and loops, Probab. Th. Rel. Fields 172, 615 (2018)

2018
[56]

Amit and V

D.J. Amit and V . Martin-Mayor, Field theory, the renormaliza- tion group, and critical phenomena , World Scientiﬁc, Singa- pore, 3rd edition, (1984)

1984
[57]

Kirkham, Calculation of crossover exponent from Heisen- berg to Ising behaviour using the fourth-order ε expansion, J

J.E. Kirkham, Calculation of crossover exponent from Heisen- berg to Ising behaviour using the fourth-order ε expansion, J. Phys. A 14, L437(1981)

1981
[58]

Shimada and S

H. Shimada and S. Hikami, Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3d conformal boot- strap, J. Stat. Phys 165, 1006 (2016)

2016
[59]

Le Doussal and K.J

P. Le Doussal and K.J. Wiese, Avalanche dynamics of elastic interfaces, Phys. Rev. E 88, 022106 (2013). 6

2013
[60]

Parisi and N

G. Parisi and N. Sourlas, Random Magnetic Fields, Super- symmetry, and Negative Dimensions, Phys. Rev. Lett. 43, 744 (1979)

1979
[61]

Parisi and N

G. Parisi and N. Sourlas, Supersymmetric ﬁeld theories and stochastic differential equations , Nucl. Phys. B 206, 321 (1982)

1982
[63]

Kompaniets and K.J

M. Kompaniets and K.J. Wiese, Fractal dimension of critical curves in the O(n)-symmetric φ4-model and crossover expo- nent at 6-loop order: Loop-erased random walks, self-avoiding walks, Ising, XY and Heisenberg models, arXiv:1908.07502

1908 arXiv
[64]

Feldman, Critical Exponents of the Random-FieldO(N ) Model, Phys

D.E. Feldman, Critical Exponents of the Random-FieldO(N ) Model, Phys. Rev. Lett. 88, 177202 (2002)

2002
[65]

Tarjus and M

G. Tarjus and M. Tissier, Nonperturbative Functional Renor- malization Group for Random-Field Models: the Way Out of Dimensional Reduction, Phys. Rev. Lett. 93, 267008 (2004)

2004
[66]

Le Doussal and K.J

P. Le Doussal and K.J. Wiese, Random-Field Spin Models Be- yond 1 Loop: A Mechanism for Decreasing the Lower Critical Dimension, Phys. Rev. Lett. 96, 197202 (2006)

2006
[67]

Le Doussal and K.J

P. Le Doussal and K.J. Wiese, An Exact Mapping of the Stochastic Field Theory for Manna Sandpiles to Interfaces in Random Media, Phys. Rev. Lett. 114, 110601 (2015)

2015
[68]

Wiese, Coherent-state path integral versus coarse-grained effective stochastic equation of motion: From reaction diffusion to stochastic sandpiles, Phys

K.J. Wiese, Coherent-state path integral versus coarse-grained effective stochastic equation of motion: From reaction diffusion to stochastic sandpiles, Phys. Rev. E 93, 042117 (2016). 7 SUPPLEMENTAL MA TERIAL A. Details on the mapping of LERW on O(n =−2) model We use that in ...

2016

Pith tools

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

[1] [28]

Wiese and A.A

K.J. Wiese and A.A. Fedorenko, Field theories for loop-erased random walks, Nucl. Phys. B 946,114696 (2019)

work page 2019

[2] [62]

Wiese, Supersymmetry breaking in disordered systems and relation to functional renormalization and replica-symmetry breaking, J

K.J. Wiese, Supersymmetry breaking in disordered systems and relation to functional renormalization and replica-symmetry breaking, J. Phys.: Condens. Matter. 17, S1889 (2005)

work page 2005

[3] [1]

Gr ¨uner, The dynamics of charge-density waves, Rev

G. Gr ¨uner, The dynamics of charge-density waves, Rev. Mod. Phys. 60, 1129 (1988)

work page 1988

[4] [2]

Fukuyama and P.A

H. Fukuyama and P.A. Lee, Dynamics of the charge-density wave. I. Impurity pinning in a single chain , Phys. Rev. B 17, 535 (1978)

work page 1978

[5] [3]

P. A. Lee and T. M. Rice, Electric-ﬁeld depinning of charge- density waves, Phys. Rev. B 19, 3970 (1979)

work page 1979

[6] [4]

Blatter, M.V

G. Blatter, M.V . Feigel’man, V .B. Geshkenbein, A.I. Larkin, and V .M. Vinokur,Vortices in high-temperature superconduc- tors, Rev. Mod. Phys. 66, 1125 (1994)

work page 1994

[7] [5]

Nattermann and S

T. Nattermann and S. Scheidl, Vortex-glass phases in type-II superconductors, Adv. Phys. 49, 607 (2000)

work page 2000

[8] [6]

Le Doussal and T

P. Le Doussal and T. Giamarchi, Moving glass theory of driven lattices with disorder, Phys. Rev. B 57, 11356 (1998)

1998