The interplay between topology and quantum criticality has established symmetry enriched criticality as a new paradigm beyond conventional frameworks in condensed matter and statistical physics. However, the conditions under which such criticality emerges remain poorly understood. In this work, we show that a broad class of symmetry enriched critical points separating symmetry protected topological (SPT) and spontaneous symmetry breaking (SSB) phases is governed by Kennedy Tasaki (KT) self duality. At these criticalities, the KT transformation that interchanges the SPT and SSB phases acts as a self-dual noninvertible symmetry, endowing the critical points with nontrivial topological characteristics. We numerically demonstrate this mechanism in $\mathbb{Z}_{n} \times \mathbb{Z}_{n}$ cluster qudit chains and further argue that it naturally extends to higher dimensions, providing a duality based criterion for identifying symmetry-enriched criticality across dimensions.The interplay between topology and quantum criticality has established symmetry enriched criticality as a new paradigm beyond conventional frameworks in condensed matter and statistical physics. However, the conditions under which such criticality emerges remain poorly understood. In this work, we show that a broad class of symmetry enriched critical points separating symmetry protected topological (SPT) and spontaneous symmetry breaking (SSB) phases is governed by Kennedy Tasaki (KT) self duality. At these criticalities, the KT transformation that interchanges the SPT and SSB phases acts as a self-dual noninvertible symmetry, endowing the critical points with nontrivial topological characteristics. We numerically demonstrate this mechanism in $\mathbb{Z}_{n} \times \mathbb{Z}_{n}$ cluster qudit chains and further argue that it naturally extends to higher dimensions, providing a duality based criterion for identifying symmetry-enriched criticality across dimensions.

arXiv:2610.27666Oct 20, 20260 citations

Quantum criticality of topological phase transitions in three-dimensional interacting electronic systems

Bohm‐Jung Yang, Eun-Gook Moon, Hiroki Isobe +1 more

arXiv:1406.2766Nature PhysicsAug 24, 201486 citations

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

N. G. Jones, Ruben Verresen

Topological phases protected by symmetry can occur in gapped and—surprisingly—in critical systems. We consider non-interacting fermions in one dimension with spinless time-reversal symmetry. It is known that the phases are classified by a topological invariant $\omega$ ω and a central charge c. We investigate the correlations of string operators, giving insight into the interplay between topology and criticality. In the gapped phases, these non-local string order parameters allow us to extract $\omega$ ω . Remarkably, ratios of correlation lengths are universal. In the critical phases, the scaling dimensions of these operators serve as an order parameter, encoding $\omega$ ω and c. We derive exact asymptotics of these correlation functions using Toeplitz determinant theory. We include physical discussion, e.g., relating lattice operators to the conformal field theory. Moreover, we discuss the dual spin chains. Using the aforementioned universality, the topological invariant of the spin chain can be obtained from correlations of local observables.Topological phases protected by symmetry can occur in gapped and—surprisingly—in critical systems. We consider non-interacting fermions in one dimension with spinless time-reversal symmetry. It is known that the phases are classified by a topological invariant $$\omega $$ ω and a central charge c. We investigate the correlations of string operators, giving insight into the interplay between topology and criticality. In the gapped phases, these non-local string order parameters allow us to extract $$\omega $$ ω . Remarkably, ratios of correlation lengths are universal. In the critical phases, the scaling dimensions of these operators serve as an order parameter, encoding $$\omega $$ ω and c. We derive exact asymptotics of these correlation functions using Toeplitz determinant theory. We include physical discussion, e.g., relating lattice operators to the conformal field theory. Moreover, we discuss the dual spin chains. Using the aforementioned universality, the topological invariant of the spin chain can be obtained from correlations of local observables.

arXiv:1805.06904Journal of Statistical PhysicsApr 8, 201943 citations

Interplay of Disorder and Interaction in Majorana Quantum Wires

Alejandro M. Lobos, Roman M. Lutchyn, S. Das Sarma

We study the interplay between disorder and interaction in one-dimensional topological superconductors which carry localized Majorana zero-energy states. Using Abelian bosonization and the perturbative renormalization group approach, we obtain the renormalization group flow and the associated scaling dimensions of the parameters and identify the critical points of the low-energy theory. We predict a quantum phase transition from a topological superconducting phase to a nontopological localized phase, and obtain the phase boundary between these two phases as a function of the electron-electron interaction and the disorder strength in the nanowire. Based on an instanton analysis which incorporates the effect of disorder, we also identify a large regime of stability of the Majorana-carrying topological phase in the parameter space of the model.We study the interplay between disorder and interaction in one-dimensional topological superconductors which carry localized Majorana zero-energy states. Using Abelian bosonization and the perturbative renormalization group approach, we obtain the renormalization group flow and the associated scaling dimensions of the parameters and identify the critical points of the low-energy theory. We predict a quantum phase transition from a topological superconducting phase to a nontopological localized phase, and obtain the phase boundary between these two phases as a function of the electron-electron interaction and the disorder strength in the nanowire. Based on an instanton analysis which incorporates the effect of disorder, we also identify a large regime of stability of the Majorana-carrying topological phase in the parameter space of the model.

arXiv:1202.2837Physical Review LettersOct 5, 2012149 citations

Three studies in quantum criticality

R. R. Ramazashvili

Jan 1, 19980 citations

\mathbb{Z}_{n} \times \mathbb{Z}_{n}

cluster qudit chains and further argue that it naturally extends to higher dimensions, providing a duality based criterion for identifying symmetry-enriched criticality across dimensions.

\omega

ω and a central charge c. We investigate the correlations of string operators, giving insight into the interplay between topology and criticality. In the gapped phases, these non-local string order parameters allow us to extract

\omega

ω . Remarkably, ratios of correlation lengths are universal. In the critical phases, the scaling dimensions of these operators serve as an order parameter, encoding

\omega

ω and c. We derive exact asymptotics of these correlation functions using Toeplitz determinant theory. We include physical discussion, e.g., relating lattice operators to the conformal field theory. Moreover, we discuss the dual spin chains. Using the aforementioned universality, the topological invariant of the spin chain can be obtained from correlations of local observables.