Morphology of renormalization-group flow for the de Almeida--Thouless--Gardner universality class

Morphology of renormalization-group flow for the de Almeida–Thouless–Gardner universality class

Patrick Charbonneau, Yi Hu, Archishman Raju, James P. Sethna, and Sho Yaida

Phys. Rev. E 99, 022132 – Published 21 February 2019

Abstract

A replica-symmetry-breaking phase transition is predicted in a host of disordered media. The criticality of the transition has, however, long been questioned below its upper critical dimension, six, due to the absence of a critical fixed point in the renormalization-group flows at one-loop order. A recent two-loop analysis revealed a possible strong-coupling fixed point, but given the uncontrolled nature of perturbative analysis in the strong-coupling regime, debate persists. Here we examine the nature of the transition as a function of spatial dimension and show that the strong-coupling fixed point can go through a Hopf bifurcation, resulting in a critical limit cycle and a concomitant discrete scale invariance. We further investigate a different renormalization scheme and argue that the basin of attraction of the strong-coupling fixed point (or limit cycle) may stay finite for all dimensions.

Received 4 August 2018
Revised 25 January 2019

DOI:https://doi.org/10.1103/PhysRevE.99.022132

Physics Subject Headings (PhySH)

Phase transitions Spontaneous symmetry breaking

Glassy systems

Renormalization group Replica methods

Condensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsPolymers & Soft MatterParticles & Fields

Authors & Affiliations

Patrick Charbonneau^1,2, Yi Hu¹, Archishman Raju³, James P. Sethna³, and Sho Yaida¹

¹Department of Chemistry, Duke University, Durham, North Carolina 27708, USA
²Department of Physics, Duke University, Durham, North Carolina 27708, USA
³Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 99, Iss. 2 — February 2019

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Zeros of $β^{I} = 0$ (green solid) and $β^{II} = 0$ (black dashed) for $d = 3$ . (a) At one-loop order, curves do not intersect except at the unstable Gaussian fixed point (red dot) at the origin. (b) At two-loop order, curves further intersect at the stable strong-coupling fixed point (blue dot).
Reuse & Permissions
Figure 2
Flows in the space of couplings for minimal two-loop RG equations. (a) At $d = 5.43 \in (d_{H}, d_{cas})$ , a limit cycle (thick orange line) is observed around the strong coupling fixed point (blue dot). Its basin of attraction is delineated by thick black lines. (b) At $d = 5.50 \in [d_{cas}, d_{u}]$ , an infinite-size remnant of the limit cycle is observed, but no stable fixed point. Within the remnant, the flow emanates from the strong-coupling fixed point and then circles around and approaches this infinite remnant. (c) At $d = 6.005 \in (d_{u}, d_{col})$ , the Gaussian fixed point (red dot) is stable with a finite basin of attraction. On the boundary of the basin, two unstable fixed points (green dots) are observed along with their mirror images. (d) At $d = 6.05 \geq d_{col}$ , the finite Gaussian basin and the infinite-size remnant of the limit cycle have merged. The Gaussian basin is now semi-infinite.
Reuse & Permissions
Figure 3
Flows in the space of couplings for the coordinate-transformed two-loop RG scheme, with $λ_{1} = - 0.55$ and $λ_{2} = 0$ . (a) At $d = 3$ , a limit cycle (thick orange line) is observed around the strong coupling fixed point. Its basin of attraction is delineated by thick black lines. (b) At $d = 6.005$ , two basins of attraction are observed: one for the Gaussian fixed point and the other for the strong-coupling fixed point.
Reuse & Permissions
Figure 4
Critical parameters at the nontrivial fixed point derived within the minimal two-loop (solid lines) and nonperturbative (dashed lines) RG schemes as functions of the spatial dimensions $d$ . (a) Real parts of the stability exponents around the fixed point within the critical surface, $λ_{1}$ and $λ_{2}$ . (b) Critical exponents, $ν$ (cyan) and $η$ (navy-blue). (c) Fixed-point values of running couplings, $g^{I}$ (red) and $g^{II}$ (orange) are of order unity in all $d$ . Note that within the two-loop (nonperturbative) RG scheme, the nature of the fixed point changes at $d_{0} \approx 4.8$ (5.2) and $d_{H} \approx 5.4$ (5.8). Namely, two stability exponents merge for $d > d_{0}$ , at which point they acquire imaginary parts. The flow thus spirals into the fixed point, but remains stable. Right above $d = d_{H}$ , the real part of these eigenvalues becomes negative and the flow spirals out of the fixed point to the limit cycle.
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics