--- title: Topological Quantum Critical Points url: https://www.emergentmind.com/topics/topological-quantum-critical-points-tqcp type: topic --- # Topological Quantum Critical Points A topological quantum critical point (TQCP) is a locus in parameter space at which the topology of a ground state—rather than a local symmetry or order parameter—changes in a quantum many-body system. Unlike conventional quantum critical points, TQCPs are not described by symmetry breaking but instead by singular changes in global invariants (e.g., Fermi surface topology, Chern number, winding number) under the protection of a symmetry group. The critical theory at a TQCP is frequently characterized by unique scaling exponents, emergent excitations, or nontrivial interplay of topology, disorder, and interactions. ## 1. General Definition and Mechanistic Framework A TQCP separates two quantum phases which exhibit identical local symmetry-breaking patterns, but distinct values of a quantized topological invariant. At a TQCP, the system is generically gapless, with critical exponents and field content determined by the microscopic symmetry and topological class. Unlike Landau-Ginzburg–Wilson paradigms, which are diagnosed by local order parameters and spontaneous symmetry breaking, a TQCP instead features a discontinuous jump in a nonlocal or quantized index—such as the Chern number, the $Z_2$ index, mirror Chern number, or the winding number associated to the bulk Hamiltonian—which cannot be identified by any local correlator or response [1511.02213, 2007.07273, 2310.12252, 2510.25959]. The critical low-energy theory often reflects this topological structure: - In Fermi liquids, the TQCP can be a Lifshitz-type singularity at which the Fermi surface changes topology and the Landau quasiparticle mass diverges, with single-particle residue vanishing as $Z \to 0$ and density of states $N(0) \to \infty$ [1511.02213]. - In noncentrosymmetric or inversion broken 3D insulators, the TQCP hosts an anisotropic band touching with quadratic dispersion in one direction and linear in the other two, associated with vanishing chirality and a $Z_2$ index jump [1207.6151]. - In topological superconductors (class DIII), the TQCP is a gap-closing where the winding number changes by an even integer and the system can host emergent gapless Majorana or Weyl modes [2510.25959, 2110.08332]. ## 2. Effective Field Theories and Universal Scaling Laws The nature of the critical theory at a TQCP is dictated by symmetry, dimension, and the structure of the protecting group $G_p$. Effective field theories describing TQCPs exhibit several key features: - Gap closing at isolated Fermi points with nontrivial topology (e.g., Dirac or Weyl nodes, quadratic band touchings, or flat bands) [1406.2766, 1511.02213, 1604.01805, 1207.6151]. - Dynamical exponent $z$ and critical exponents ($\nu$, $\eta$) are in general not specified solely by symmetry but also by the topological class and may be emergent properties—e.g., $z=2$ quadratic touchings can transition to $z=1$ Dirac points if strong interactions are present [2503.01512]. - At multicritical TQCPs (e.g., Lifshitz points), non-conformal scaling, $z=2$ or higher, and unconventional energy-momentum scaling are possible [1207.6151, 2403.03716, 2503.01512]. - Renormalization group flows may reveal multiple fixed points with distinct symmetry content, e.g., Lorentz invariant, rotational, or only discrete crystalline symmetry (as in $j=3/2$ Dirac electrons in antiperovskite crystalline topological insulators) [1604.01805]. - The global phase diagram exhibits phase boundaries distinguished not by local order, but by topological quantum numbers and possible transitions between distinct critical points (e.g., Ising vs. Ising* criticality in 1d chains [2403.03716, 2008.02285]). ## 3. Disorder, Interactions, and Emergent Critical Manifolds TQCPs can be robust or fragile with respect to disorder and interactions: - Disorder that preserves parity-density or Berry-phase structure can lead to complete cancellation of certain backscattering processes (symplectic class), providing partial topological protection to the TQCP against localization [1001.2442]. - For generic non-magnetic disorder, absolute stability at the TQCP is enforced by bulk-edge correspondence, even when cancellation does not occur for all processes. Once a threshold disorder is exceeded, an intervening metallic phase may emerge between topological and trivial insulators [1001.2442]. - Strong-coupling fixed points may terminate continuous TQCP lines and produce either supersymmetric interacting CFTs (e.g., 2D class D TQCP with emergent N=1 SUSY) or simple direct sum of free boson and fermion field theories (in 3d), with possible first-order transitions replacing TQCPs when the protecting symmetry is broken [2110.08332]. - In lower dimensions, disorder and strong symmetry (e.g., time-reversal, $\mathbb{Z}_2$) can yield symmetry-enriched infinite-randomness critical points, supporting robust topological edge modes and nonlocal order at criticality, while the bulk remains in conventional universality classes [2008.02285]. ## 4. Probing and Characterizing Topological Transitions Conventional probes such as local order parameters fail at a TQCP. Instead, universal diagnostics include: - The entanglement-based $c$-function, constructed as (directional) derivatives of slab entanglement entropy, which exhibits a peak or discontinuity at the TQCP and tracks the universal count of low-energy modes independent of microscopic details [2007.07273]. - Nonlocal string order parameters, such as those diagnosing SPT order or topological invariants in spin chains, remain nonzero or undergo quantized jumps at a TQCP [1807.08490, 2403.03716, 2106.15457, 2008.02285]. - Quantum transport reveals unique violation or modification of the Wiedemann–Franz law, e.g., the appearance of an extra “zeron” channel for heat conduction on the Fermi-liquid side, or enhanced thermal resistivity on the flat-band side, each with distinct behavior of Lorenz ratio $L(T)$ relative to its universal Sommerfeld value [1511.02213]. - Measurement of low-energy AC Josephson impedance, especially in mesoscopic hybrid systems, provides scaling signatures of the gap closing and of dissipative finite-frequency response unique to TQCPs. For instance, a “dip-peak-dip” in $\chi_2/T^2$ as a function of Zeeman field across the TQCP directly reveals the universal scaling function $\mathcal{F}(\omega/T, \delta/T)$ [1106.5506]. ## 5. Case Studies: Model Classes and Phenomenology ### Fermi Surface TQCP—Lifshitz and Fermion Condensation Points In strongly correlated Fermi liquids, the TQCP is marked by divergence of the effective mass $M^* \sim |g-g_c|^{-\alpha}$ and $N(0) \to \infty$, leading to $v_F \to 0$ and complete breakdown of Fermi liquid theory. The kinetic equation admits a new gapless transverse zero-sound (“zeron”) mode on the Fermi-liquid side, while on the non-Fermi-liquid side a flat band emerges supporting additional thermal resistivity but not contributing to charge resistivity—manifested experimentally by distinct behavior in thermal and electrical transport coefficients [1511.02213]. ### 3D Topological Insulator TQCP In 3D $Z_2$ topological insulators, the TQCP between topological and trivial insulator is described by a four-component Dirac Hamiltonian with mass $m$. Disorder can lead to a metallic phase intervening between the two topological sectors if the disorder strength exceeds a critical value; the bulk-edge correspondence ensures delocalization at $m=0$ [1001.2442]. Similar criticality arises in noncentrosymmetric systems, but the bands touch at zero chirality points with quadratic-linear anisotropy, leading to $E^{3/2}$ density of states scaling and distinctive non-Fermi-liquid power laws in thermodynamic quantities [1207.6151, 1406.2766]. ### Strong Interaction Limit and Emergent Symmetry Strong-coupling TQCPs can support new universality classes, e.g., supersymmetric fixed points, Gross–Neveu CFTs, or manifolds of interacting CFTs connected by duality rather than continuous emergent symmetries [2110.08332, 2310.12252]. The availability of such a manifold depends on whether the low-energy fixed point is isolated or not; only the former supports continuous emergent symmetry as a consequence of deformability between topologically distinct gapped phases [2310.12252]. ### Criticality with Disorder and Subsystem Symmetry Random quantum spin chains and Floquet systems can realize TQCPs at infinite-randomness fixed points, enriched by nontrivial action of symmetries (e.g., time-reversal) on disorder operators or string order parameters. The resulting critical points can support protected zero modes and unique entanglement scaling even in the presence of strong disorder [2008.02285]. Similarly, transitions between higher-order topological and trivial Mott insulators can be governed by fracton criticalities with subsystem U(1) symmetry and $L \ln L$ entanglement rather than area law [2008.01746]. ## 6. Extensions, Anomalous Phenomena, and Open Problems Topological quantum criticality extends beyond standard insulator-superconductor or Lifshitz transitions: - Quantum critical points themselves can be classified by topological invariants, with direct transitions between critical lines of distinct topological character (e.g., Ising vs. Ising* sectors distinguished by the transformation properties of disorder operators) [2403.03716]. - Interacting systems can exhibit transitions between two gapless phases, both topologically distinct and without local order parameters, mediated by multicritical Lifshitz points [2403.03716]. - Tensor network constructions allow for direct interpolation between distinct $\mathbb{Z}_2$ topological phases, where critical points are described by compact boson CFTs with exactly computable compactification radii and operator content, supporting both anyon condensation transitions and symmetry-enriched criticality [1807.08490]. - In 3+1d, tQCPs with minimal change of winding number $\delta N_w = 2$ correspond under duality to emergent single Weyl points on the lattice, revealing deep connections between superconducting and chiral semimetal physics [2510.25959]. - Open questions include the classification of TQCP universality classes in presence of strong interactions, disorder, and crystalline or subsystem symmetries, and the general applicability of entanglement-based probes such as generalized $c$-functions as universal diagnostics [2007.07273]. The extension to higher-order topological and fracton phases is an active domain. ## 7. Summary Table: Selected Prototypical TQCPs | Main Class | Key Topological Change | Gapless Field Content | Universal Scaling/Signature | Reference | |--------------------|-----------------------|------------------------------------------|---------------------------------------|-----------------| | Fermi-liquid TQCP | Fermi surface change | FL breakdown, divergence of $M^*$ | Zeron mode, WF law violation | [1511.02213] | | 3D TI TQCP | $Z_2$ index jump | Dirac fermion at $m=0$ | Density of states $\sim E^{3/2}$ | [1001.2442] | | TSC TQCP (DIII) | Winding $\delta N_w$ | Majorana or Weyl node | Emergent SU(2)/U(1) charge algebra | [2510.25959] | | HOTI–trivial QCP | Higher-order invariant| Fracton/dipole liquids | Entanglement $L\ln L$ scaling | [2008.01746] | | 1D random spin TQCP| SPT/distinction in $T\mu T$| Edge zero-modes, infinite randomness | Boundary exponents, string order | [2008.02285] | | 2D Chern transition| Chern number change | Interfaces, emergent surface modes | c-function peak, Hall response | [2007.07273] | Precise scaling exponents and collective modes must be determined in context of the particular fixed point and symmetry class. --- The topological quantum critical point is thus a singularity at which topology "condenses" into a new phase via nontrivial gapless modes, modified scaling, and robust, sometimes discontinuous, changes in response and entanglement. Its theoretical and experimental signatures fundamentally transcend symmetry-based paradigms and challenge the conventional frameworks for phase transitions in quantum matter. Source: https://www.emergentmind.com/topics/topological-quantum-critical-points-tqcp