Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0011124 · Physical review. B, Condensed matter

Breakdown of conformal invariance at strongly random critical points

M. B. Hastings, Shivaji L. Sondhi

Published Aug 14, 2001• 9 citations
arXivHTMLPDF

Abstract

We consider the breakdown of conformal and scale invariance in random systems with strongly random critical points. Extending previous results on one-dimensional systems, we provide an example of a three-dimensional system that has a strongly random critical point. The average correlation functions of this system demonstrate a breakdown of conformal invariance, while the typical correlation functions demonstrate a breakdown of scale invariance. The breakdown of conformal invariance is due to the vanishing of the correlation functions at the infinite disorder fixed point, causing the critical correlation functions to be controlled by a dangerously irrelevant operator describing the approach to the fixed point. We relate the computation of average correlation functions to a problem of persistence in the renormalization group flow.

DOI
10.1103/physrevb.64.094204

Related papers

Renormalization group in quantum critical theories with Harris-marginal disorder

Koushik Ganesan, Andrew Lucas, Leo Radzihovsky

We develop a renormalization group for weak Harris-marginal disorder in otherwise strongly interacting quantum critical theories, focusing on systems which have emergent conformal invariance. Using conformal perturbation theory, we argue that previously proposed random lines of fixed points with Lifshitz scaling in fact flow toward other universal fixed points, and this flow is captured by a ``one-loop'' analysis. Our approach appears best controlled in theories with only a few operators with low scaling dimension. In this regime, we compare our predictions for the flow of disorder to holographic models, and find complete agreement.

arXiv:2110.11978•Physical review. D/Physical review. D.•Mar 28, 2022•8 citations

Effect of randomness on surface critical phenomena

Kaoru Ohno, Yutaka Okabe

We study surface critical phenomena of the ${\mathit{S}}^{4}$ model having Gaussian randomness. Using the renormalization-group 4-d expansion, we obtain surface critical exponents indicating the decay of the correlation function, ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\parallel}}}^{\mathrm{random}}$ and ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\perp}}}^{\mathrm{random}}$, at the random fixed point. These exponents are evaluated for both the ordinary and special transitions up to the second order, and the surface scaling relation ${\mathrm{\ensuremath{\eta}}}^{\mathrm{random}}$=2${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\perp}}}^{\mathrm{random}}$-${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\parallel}}}^{\mathrm{random}}$ is confirmed in this random system. We also confirm the conformal invariance of the real-space correlation function at criticality.

Physical review. B, Condensed matter•Sep 1, 1992•7 citations

Infinite-randomness quantum Ising critical fixed points

Olexei I. Motrunich, Siun-Chuon Mau, David A. Huse +1 more

We examine the ground state of the random quantum Ising model in a transverse field using a generalization of the Ma-Dasgupta-Hu renormalization group (RG) scheme. For spatial dimensionality $d=2,$ we find that at strong randomness the RG flow for the quantum critical point is towards an infinite-randomness fixed point, as in one dimension. This is consistent with the results of a recent quantum Monte Carlo study by Pich et al. [Phys. Rev. Lett. 81, 5916 (1998)], including estimates of the critical exponents from our RG that agree well with those from the quantum Monte Carlo. The same qualitative behavior appears to occur for three dimensions; we have not yet been able to determine whether or not it persists to arbitrarily high d. Some consequences of the infinite-randomness fixed point for the quantum critical scaling behavior are discussed. Because frustration is irrelevant in the infinite-randomness limit, the same fixed point should govern both ferromagnetic and spin-glass quantum critical points. This RG maps the random quantum Ising model with strong disorder onto a novel type of percolation/aggregation process.

arXiv:cond-mat/9906322•Physical review. B, Condensed matter•Jan 1, 2000•269 citations

Nonperturbative functional renormalization group for random field models and related disordered systems. IV. Supersymmetry and its spontaneous breaking

Matthieu Tissier, Gilles Tarjus

We apply the nonperturbative functional renormalization group (NP-FRG) in the superfield formalism that we have developed in the preceding paper to study long-standing issues concerning the critical behavior of the random field Ising model. Through the introduction of an appropriate regulator and a supersymmetry-compatible nonperturbative approximation, we are able to follow the supersymmetry, more specifically, the superrotational invariance first unveiled by Parisi and Sourlas [Phys. Rev. Lett. 43, 744 (1979)] and its spontaneous breaking along the RG flow. Breaking occurs below a critical dimension ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, and the supersymmetry-broken fixed point that controls the critical behavior then leads to a breakdown of the ``dimensional-reduction'' property. We solve the NP-FRG flow equations numerically and determine the critical exponents as a function of dimension down to $d\ensuremath{\lesssim}3$, with a good agreement in $d=3$ and 4 with the existing numerical estimates.

arXiv:1110.5500•Physical Review B•Mar 2, 2012•64 citations

Critical limit and anisotropy in the two-point correlation function of three-dimensional O(N) models

Massimo Campostrini B, A Pelissetto, Paolo Rossi +1 more

In three-dimensional O( N ) models, we investigate the low-momentum behavior of the two-point Green's function G ( x ) in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotational-invariant fixed point. In non-rotational invariant physical systems with O( N )-invariant interactions, the vanishing of space-anisotropy approaching the rotational-invariant fixed point is described by a critical exponent ρ, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At N = ∞ one finds ρ = 2. We show that, for all values of N ⩾ 0, ρ ≃ 2. Non-Gaussian corrections to the universal low-momentum behavior of G ( x ) are evaluated, and found to be very small.

arXiv:cond-mat/9612164•Europhysics Letters (EPL)•Jun 10, 1997•19 citations

Breakdown of the Perturbative Renormalization Group at Certain Quantum Critical Points

D. Belitz, T. R. Kirkpatrick, Jörg Rollbühler

It is shown that the presence of multiple time scales at a quantum critical point can lead to a breakdown of the loop expansion for critical exponents, since coefficients in the expansion diverge. Consequently, results obtained from finite-order perturbative renormalization-group treatments may not be an approximation in any sense to the true asymptotic critical behavior. This problem manifests itself as a nonrenormalizable field theory, or, equivalently, as the presence of a dangerous irrelevant variable. The quantum ferromagnetic transition in disordered metals provides an example.

arXiv:cond-mat/0406350•Physical Review Letters•Oct 4, 2004•17 citations
We consider the breakdown of conformal and scale invariance in random systems with strongly random critical points. Extending previous results on one-dimensional systems, we provide an example of a three-dimensional system that has a strongly random critical point. The average correlation functions of this system demonstrate a breakdown of conformal invariance, while the typical correlation functions demonstrate a breakdown of scale invariance. The breakdown of conformal invariance is due to the vanishing of the correlation functions at the infinite disorder fixed point, causing the critical correlation functions to be controlled by a dangerously irrelevant operator describing the approach to the fixed point. We relate the computation of average correlation functions to a problem of persistence in the renormalization group flow.
We develop a renormalization group for weak Harris-marginal disorder in otherwise strongly interacting quantum critical theories, focusing on systems which have emergent conformal invariance. Using conformal perturbation theory, we argue that previously proposed random lines of fixed points with Lifshitz scaling in fact flow toward other universal fixed points, and this flow is captured by a ``one-loop'' analysis. Our approach appears best controlled in theories with only a few operators with low scaling dimension. In this regime, we compare our predictions for the flow of disorder to holographic models, and find complete agreement.
We study surface critical phenomena of the S4{\mathit{S}}^{4}S4 model having Gaussian randomness. Using the renormalization-group 4-d expansion, we obtain surface critical exponents indicating the decay of the correlation function, ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\parallel}}}^{\mathrm{random}}$ and ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\perp}}}^{\mathrm{random}}$, at the random fixed point. These exponents are evaluated for both the ordinary and special transitions up to the second order, and the surface scaling relation ${\mathrm{\ensuremath{\eta}}}^{\mathrm{random}}$=2${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\perp}}}^{\mathrm{random}}$-${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\parallel}}}^{\mathrm{random}}$ is confirmed in this random system. We also confirm the conformal invariance of the real-space correlation function at criticality.
We examine the ground state of the random quantum Ising model in a transverse field using a generalization of the Ma-Dasgupta-Hu renormalization group (RG) scheme. For spatial dimensionality d=2,d=2,d=2, we find that at strong randomness the RG flow for the quantum critical point is towards an infinite-randomness fixed point, as in one dimension. This is consistent with the results of a recent quantum Monte Carlo study by Pich et al. [Phys. Rev. Lett. 81, 5916 (1998)], including estimates of the critical exponents from our RG that agree well with those from the quantum Monte Carlo. The same qualitative behavior appears to occur for three dimensions; we have not yet been able to determine whether or not it persists to arbitrarily high d. Some consequences of the infinite-randomness fixed point for the quantum critical scaling behavior are discussed. Because frustration is irrelevant in the infinite-randomness limit, the same fixed point should govern both ferromagnetic and spin-glass quantum critical points. This RG maps the random quantum Ising model with strong disorder onto a novel type of percolation/aggregation process.
We apply the nonperturbative functional renormalization group (NP-FRG) in the superfield formalism that we have developed in the preceding paper to study long-standing issues concerning the critical behavior of the random field Ising model. Through the introduction of an appropriate regulator and a supersymmetry-compatible nonperturbative approximation, we are able to follow the supersymmetry, more specifically, the superrotational invariance first unveiled by Parisi and Sourlas [Phys. Rev. Lett. 43, 744 (1979)] and its spontaneous breaking along the RG flow. Breaking occurs below a critical dimension ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, and the supersymmetry-broken fixed point that controls the critical behavior then leads to a breakdown of the ``dimensional-reduction'' property. We solve the NP-FRG flow equations numerically and determine the critical exponents as a function of dimension down to $d\ensuremath{\lesssim}3$, with a good agreement in d=3d=3d=3 and 4 with the existing numerical estimates.
In three-dimensional O( N ) models, we investigate the low-momentum behavior of the two-point Green's function G ( x ) in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotational-invariant fixed point. In non-rotational invariant physical systems with O( N )-invariant interactions, the vanishing of space-anisotropy approaching the rotational-invariant fixed point is described by a critical exponent ρ, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At N = ∞ one finds ρ = 2. We show that, for all values of N ⩾ 0, ρ ≃ 2. Non-Gaussian corrections to the universal low-momentum behavior of G ( x ) are evaluated, and found to be very small.
It is shown that the presence of multiple time scales at a quantum critical point can lead to a breakdown of the loop expansion for critical exponents, since coefficients in the expansion diverge. Consequently, results obtained from finite-order perturbative renormalization-group treatments may not be an approximation in any sense to the true asymptotic critical behavior. This problem manifests itself as a nonrenormalizable field theory, or, equivalently, as the presence of a dangerous irrelevant variable. The quantum ferromagnetic transition in disordered metals provides an example.