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Efimov-Type Discrete Scale Invariance at the Multicritical Point of a Two-Dimensional Disordered Class D Topological Superconductor

Vaitheeswaran Gnanaraj, Balakrishanan Vellaikannan, P Balamanikandan

Published Apr 21, 2026• 0 citations
Publisher

Abstract

Abstract The interplay between disorder, topology, and many-body interactions in two-dimensional superconducting systems continues to reveal unexpected critical phenomena beyond the reach of conventional universality theory. We propose and characterize a novel universality class — the Discrete Scale Invariance Multicritical Point (DSI-MCP) — emerging from Efimov-type discrete scale invariance at the tricritical point of a two-dimensional disordered Class D topological superconductor. Using a two-loop renormalization group analysis within the replica framework, we demonstrate that when disorder and three-body couplings simultaneously exceed their critical thresholds, the renormalization group flow bypasses the conventional tricritical point and spirals into a new stable fixed point with complex eigenvalues. The imaginary component of these eigenvalues — absent at any conventional critical point with real scaling exponents — encodes log-periodic scaling and generates an infinite geometric tower of bound states, establishing the condensed matter analogue of the atomic Efimov effect. The DSI-MCP is distinguished from the tricritical point of Pan et al. by three independent criteria: a localization exponent that is fixed by symmetry and independent of the dimensional regularization parameter, a two-parameter multicritical threshold requiring simultaneous supercriticality in both disorder and three-body coupling, and the presence of log-periodic oscillations in the correlation length that are strictly forbidden at any conventional critical point. We further identify a competition between the DSI mechanism and Quantum Griffiths Singularity physics, governed by the statistical character of the disorder distribution, and construct a unified phase diagram delineating three experimentally distinguishable regimes: activated scaling, log-periodic power-law modulation, and a coexistence corridor exhibiting a doubled oscillation frequency with no precedent in the existing literature on disordered topological systems. All predictions are parameter-free once the operating point in coupling space is specified, and are directly accessible to STM spectroscopy, mesoscopic transport, and cold-atom platforms simulating Class D symmetry. These findings establish the DSI-MCP as a new entry in the classification of critical phenomena in disordered topological matter, bridging Efimov physics, quantum Griffiths singularities, and multicritical disorder-driven phase transitions into a unified theoretical framework.

DOI
10.21203/rs.3.rs-9455979/v1

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Abstract The interplay between disorder, topology, and many-body interactions in two-dimensional superconducting systems continues to reveal unexpected critical phenomena beyond the reach of conventional universality theory. We propose and characterize a novel universality class — the Discrete Scale Invariance Multicritical Point (DSI-MCP) — emerging from Efimov-type discrete scale invariance at the tricritical point of a two-dimensional disordered Class D topological superconductor. Using a two-loop renormalization group analysis within the replica framework, we demonstrate that when disorder and three-body couplings simultaneously exceed their critical thresholds, the renormalization group flow bypasses the conventional tricritical point and spirals into a new stable fixed point with complex eigenvalues. The imaginary component of these eigenvalues — absent at any conventional critical point with real scaling exponents — encodes log-periodic scaling and generates an infinite geometric tower of bound states, establishing the condensed matter analogue of the atomic Efimov effect. The DSI-MCP is distinguished from the tricritical point of Pan et al. by three independent criteria: a localization exponent that is fixed by symmetry and independent of the dimensional regularization parameter, a two-parameter multicritical threshold requiring simultaneous supercriticality in both disorder and three-body coupling, and the presence of log-periodic oscillations in the correlation length that are strictly forbidden at any conventional critical point. We further identify a competition between the DSI mechanism and Quantum Griffiths Singularity physics, governed by the statistical character of the disorder distribution, and construct a unified phase diagram delineating three experimentally distinguishable regimes: activated scaling, log-periodic power-law modulation, and a coexistence corridor exhibiting a doubled oscillation frequency with no precedent in the existing literature on disordered topological systems. All predictions are parameter-free once the operating point in coupling space is specified, and are directly accessible to STM spectroscopy, mesoscopic transport, and cold-atom platforms simulating Class D symmetry. These findings establish the DSI-MCP as a new entry in the classification of critical phenomena in disordered topological matter, bridging Efimov physics, quantum Griffiths singularities, and multicritical disorder-driven phase transitions into a unified theoretical framework.
Two-dimensional (2D) disordered superconductor (SC) in class D exhibits a disorder-induced quantum multicritical phenomenon among diffusive thermal metal (DTM), topological superconductor (TS), and conventional localized (AI) phases. To characterize the quantum tricritical point where these three phases meet, we carry out a two-loop renormalization group (RG) analysis for 2D Dirac fermion with random mass in terms of the $\ensuremath{\epsilon}$-expansion in the spatial dimension $d=2\ensuremath{-}\ensuremath{\epsilon}$. In two dimensions $(\ensuremath{\epsilon}=0)$, the random mass is marginally irrelevant around a clean-limit fixed point of the gapless Dirac fermion, while there exists an IR unstable fixed point at finite disorder strength that corresponds to the tricritical point. The critical exponent, dynamical exponent, and scaling dimension of the (uniform) mass term are evaluated around the tricritical point by the two-loop RG analysis. Using a mapping between an effective theory for the 2D random-mass Dirac fermion and the (1+1)(1+1)(1+1)-dimensional Gross-Neveu model, we further deduce the four-loop evaluation of the critical exponent, and the scaling dimension of the uniform mass around the tricritical point. Both the two-loop and four-loop results suggest that criticalities of a AI-DTM transition line as well as TS-DTM transition line are controlled by other saddle-point fixed point(s) at finite uniform mass.
A generic two-dimensional disordered topological superconductor in symmetry class D exhibits rich phenomenology and multiple phases: diffusive thermal metal (DTM), Anderson insulator (AI), and thermal quantum Hall (TQH) phase (a topological superconductor). We numerically investigate the phase diagram of a lattice model of such class-D superconductor, specifically focusing on transitions between the phases and the associated universal critical behaviors. We confirm the existence of a tricritical point and its repulsive nature at the point on the phase diagram where the three phases meet. We characterize the critical behaviors at various critical points and the tricritical point using numerical evaluation of the localization length, the conductance (or conductivity), and the density of states. We conclude that the two metal-insulator transitions (DTM-TQH and DTM-AI) belong to the same universality class, whereas the tricritical point represents a distinct universality class.
Recently topological states of matter have witnessed a new physical phenomenon where both edge modes and gapless bulk coexist at topological quantum criticality. The presence and absence of edge modes on a critical line can lead to an unusual class of topological phase transition between the topological and nontopological critical phases. We explore the existence of this new class of topological phase transitions in a generic model representing the topological insulators and superconductors and we show that such transition occurs at a multicritical point, i.e., at the intersection of two critical lines. To characterize these transitions we reconstruct the theoretical frameworks, which include bound state solution of the Dirac equation, winding number, correlation factors, and scaling theory of the curvature function to work for the criticality. Critical exponents and scaling laws are discussed to distinguish between the multicritical points, which separate the critical phases. Entanglement entropy and its scaling in the real space provide further insights into the unique transition at criticality revealing the interplay between fixed point and critical point at the multicriticalities.
Quantum multicritical points (QMCPs) emerge at the junction of two or more quantum phase transitions due to the interplay of disparate fluctuations, leading to novel universality classes. While quantum critical points have been well characterized, our understanding of QMCPs is much more limited, even though they might be less elusive to study experimentally than quantum critical points. Here, we characterize the QMCP of an interacting heterogeneous quantum system in two and three dimensions, the ferromagnetic random transverse-field Ising model (RTIM). The QMCP of the RTIM emerges due to both geometric and quantum fluctuations, studied here numerically by the strong disorder renormalization group method. The QMCP of the RTIM is found to exhibit ultraslow, activated dynamic scaling, governed by an infinite disorder fixed point. This ensures that the obtained multicritical exponents tend to the exact values at large scales, while also being universal---i.e., independent of the form of disorder---providing a solid theoretical basis for future experiments.
An important yet largely unsolved problem in the statistical mechanics of disordered quantum systems is to understand how quenched disorder affects quantum phase transitions in systems of itinerant fermions. In the clean limit, continuous quantum phase transitions of the symmetry-breaking type in Dirac materials such as graphene and the surfaces of topological insulators are described by relativistic (2+1)-dimensional quantum field theories of the Gross-Neveu-Yukawa (GNY) type. We study the universal critical properties of the chiral Ising, XY, and Heisenberg GNY models perturbed by quenched random-mass disorder, both uncorrelated or with long-range power-law correlations. Using the replica method combined with a controlled triple epsilon expansion below four dimensions, we find a variety of new finite-randomness critical and multicritical points with nonzero Yukawa coupling between low-energy Dirac fields and bosonic order parameter fluctuations, and compute their universal critical exponents. Analyzing bifurcations of the renormalization-group flow, we find instances of the fixed-point annihilation scenario—continuously tuned by the power-law exponent of long-range disorder correlations and associated with an exponentially large crossover length—as well as the transcritical bifurcation and the supercritical Hopf bifurcation. The latter is accompanied by the birth of a stable limit cycle on the critical hypersurface, which represents the first instance of fermionic quantum criticality with emergent discrete scale invariance.
The quantum phase transition between two clean, noninteracting topologically distinct gapped states in three dimensions is governed by a massless Dirac fermion fixed point, irrespective of the underlying symmetry class, and this constitutes a remarkably simple example of superuniversality. For a sufficiently weak disorder strength, we show that the massless Dirac fixed point is at the heart of the robustness of superuniversality. We establish this by considering both perturbative and nonperturbative effects of disorder. The superuniversality breaks down at a critical strength of disorder, beyond which the topologically distinct localized phases become separated by a delocalized diffusive phase. In the global phase diagram, the disorder controlled fixed point where superuniversality is lost, serves as a multicritical point, where the delocalized diffusive and two topologically distinct localized phases meet and the nature of the localization-delocalization transition depends on the underlying symmetry class. Based on these features, we construct the global phase diagrams of noninteracting, dirty topological systems in three dimensions. We also establish a similar structure of the phase diagram and the superuniversality for weak disorder in higher spatial dimensions. By noting that 1/r21/{r}^{2}1/r2 power-law correlated disorder acts as a marginal perturbation for massless Dirac fermions in any spatial dimension ddd, we have established a general renormalization group framework for addressing disorder driven critical phenomena for fixed spatial dimension d>2d>2d>2.