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

Berry Curvature Renormalization

Wei Chen

Published May 20, 2015• 0 citations
OpenAlex

Abstract

A scaling scheme for the Berry curvature in topological insulators and superconductors is proposed, from which the topological phase transition can be identified from the flow of the energy scales (hopping, chemical potential, etc.) that control the topology under scaling. At an infinitesimal operation, one obtains the renormalization group (RG) equations for the energy scales in the form of differential equations. A length scale defined from the Berry curvature is suggested to characterize the scale invariance at the critical point.

Related papers

Scaling theory of topological phase transitions

Wei Chen

Topologically ordered systems are characterized by topological invariants that are often calculated from the momentum space integration of a certain function that represents the curvature of the many-body state. The curvature function may be Berry curvature, Berry connection, or other quantities depending on the system. Akin to stretching a messy string to reveal the number of knots it contains, a scaling procedure is proposed for the curvature function in inversion symmetric systems, from which the topological phase transition can be identified from the flow of the driving energy parameters that control the topology (hopping, chemical potential, etc) under scaling. At an infinitesimal operation, one obtains the renormalization group (RG) equations for the driving energy parameters. A length scale defined from the curvature function near the gap-closing momentum is suggested to characterize the scale invariance at critical points and fixed points, and displays a universal critical behavior in a variety of systems examined.

arXiv:1505.05345•Journal of Physics Condensed Matter•Jan 20, 2016•46 citations

Scaling theory of Z 2 topological invariants

Wei Chen, Manfred Sigrist, Andreas P Schnyder

For inversion-symmetric topological insulators and superconductors characterized by [Formula: see text] topological invariants, two scaling schemes are proposed to judge topological phase transitions driven by an energy parameter. The scaling schemes renormalize either the phase gradient or the second derivative of the Pfaffian of the time-reversal operator, through which the renormalization group flow of the driving energy parameter can be obtained. The Pfaffian near the time-reversal invariant momentum is revealed to display a universal critical behavior for a great variety of models examined.

arXiv:1604.07662•Journal of Physics Condensed Matter•Jul 12, 2016•37 citations

Curvature function renormalization, topological phase transitions, and multicriticality

Faruk Abdulla, Priyanka Mohan, Sumathi Rao

A recently proposed curvature renormalization group scheme for topological phase transitions defines a generic ``curvature function'' as a function of the parameters of the theory and shows that topological phase transitions are signaled by the divergence of this function at certain parameter values, called critical points, in analogy with usual phase transitions. A renormalization group procedure was also introduced as a way of flowing away from the critical point toward a fixed point, where an appropriately defined correlation function goes to zero and topological quantum numbers characterizing the phase are easy to compute. In this paper, using two independent models, a model in the AIII symmetry class and a model in the BDI symmetry class, in one dimension as examples, we show that there are cases where the fixed-point curve and the critical-point curve appear to intersect, which turn out to be multicritical points, and focus on understanding its implications.

arXiv:2003.10190•Physical review. B./Physical review. B•Dec 14, 2020•11 citations

Weakly interacting topological insulators: Quantum criticality and the renormalization group approach

Wei Chen

For $D$-dimensional weakly interacting topological insulators in certain symmetry classes, the topological invariant can be calculated from a $D$- or $(D+1)$-dimensional integration over a certain curvature function that is expressed in terms of single-particle Green's functions. Based on the divergence of curvature function at the topological phase transition, we demonstrate how a renormalization group approach circumvents these integrations and reduces the necessary calculation to that for the Green's function alone, rendering a numerically efficient tool to identify topological phase transitions in a large parameter space. The method further unveils a number of statistical aspects related to the quantum criticality in weakly interacting topological insulators, including correlation function, critical exponents, and scaling laws, that can be used to characterize the topological phase transitions driven by either interacting or noninteracting parameters. We use 1D class BDI and 2D class A Dirac models with electron-electron and electron-phonon interactions to demonstrate these principles and find that interactions may change the critical exponents of the topological insulators.

arXiv:1801.00697•Physical review. B./Physical review. B•Mar 14, 2018•38 citations

Universality classes of topological phase transitions with higher-order band crossing

Wei Chen, Andreas P. Schnyder

Abstract In topological insulators and topological superconductors, the discrete jump of the topological invariant upon tuning a certain system parameter defines a topological phase transition. A unified framework is employed to address the quantum criticality of the topological phase transitions in one to three spatial dimensions, which simultaneously incorporates the symmetry classification, order of band crossing, m -fold rotational symmetry, correlation functions, critical exponents, scaling laws, and renormalization group approach. We first classify higher-order Dirac models according to the time-reversal, particle-hole, and chiral symmetries, and determine the even–oddness of the order of band crossing in each symmetry class. The even–oddness further constrains the rotational symmetry m permitted in a symmetry class. Expressing the topological invariant in terms of a momentum space integration over a curvature function, the order of band crossing determines the critical exponent of the curvature function, as well as that of the Wannier state correlation function introduced through the Fourier transform of the curvature function. The conservation of topological invariant further yields a scaling law between critical exponents. In addition, a renormalization group approach based on deforming the curvature function is demonstrated for all dimensions and symmetry classes. Through clarification of how the critical quantities, including the jump of the topological invariant and critical exponents, depend on the nonspatial and the rotational symmetry, our work introduces the notion of universality class into the description of topological phase transitions.

arXiv:1901.11468•New Journal of Physics•Jun 17, 2019•62 citations

Topological phase transition between non-high symmetry critical phases and curvature function renormalization group

Ranjith Kumar R, Y R Kartik, Sujit Sarkar

Abstract The interplay between topology and criticality has been a recent interest of study in condensed matter physics. A unique topological transition between certain critical phases has been observed as a consequence of the edge modes living at criticalities. In this work, we generalize this phenomenon by investigating possible transitions between critical phases which are non-high symmetry (non-HS) in nature. We find the triviality and non-triviality of these critical phases in terms of the decay length of the edge modes and also characterize them using the winding numbers. The distinct non-HS critical phases are separated by multicritical points with linear dispersion at which the winding number exhibits the quantized jump, indicating a change in the topology (number of edge modes) at the critical phases. Moreover, we reframe the scaling theory based on the curvature function, i.e. curvature function renormalization group method to efficiently address the non-HS criticalities and multicriticalities. Using this we identify the conventional topological transition between gapped phases through non-HS critical points, and also the unique topological transition between critical phases through multicritical points. The renormalization group flow, critical exponents and correlation function of Wannier states enable the characterization of non-HS criticalities along with multicriticalities.

arXiv:2211.03320•New Journal of Physics•Aug 1, 2023•12 citations
A scaling scheme for the Berry curvature in topological insulators and superconductors is proposed, from which the topological phase transition can be identified from the flow of the energy scales (hopping, chemical potential, etc.) that control the topology under scaling. At an infinitesimal operation, one obtains the renormalization group (RG) equations for the energy scales in the form of differential equations. A length scale defined from the Berry curvature is suggested to characterize the scale invariance at the critical point.
Topologically ordered systems are characterized by topological invariants that are often calculated from the momentum space integration of a certain function that represents the curvature of the many-body state. The curvature function may be Berry curvature, Berry connection, or other quantities depending on the system. Akin to stretching a messy string to reveal the number of knots it contains, a scaling procedure is proposed for the curvature function in inversion symmetric systems, from which the topological phase transition can be identified from the flow of the driving energy parameters that control the topology (hopping, chemical potential, etc) under scaling. At an infinitesimal operation, one obtains the renormalization group (RG) equations for the driving energy parameters. A length scale defined from the curvature function near the gap-closing momentum is suggested to characterize the scale invariance at critical points and fixed points, and displays a universal critical behavior in a variety of systems examined.
For inversion-symmetric topological insulators and superconductors characterized by [Formula: see text] topological invariants, two scaling schemes are proposed to judge topological phase transitions driven by an energy parameter. The scaling schemes renormalize either the phase gradient or the second derivative of the Pfaffian of the time-reversal operator, through which the renormalization group flow of the driving energy parameter can be obtained. The Pfaffian near the time-reversal invariant momentum is revealed to display a universal critical behavior for a great variety of models examined.
A recently proposed curvature renormalization group scheme for topological phase transitions defines a generic ``curvature function'' as a function of the parameters of the theory and shows that topological phase transitions are signaled by the divergence of this function at certain parameter values, called critical points, in analogy with usual phase transitions. A renormalization group procedure was also introduced as a way of flowing away from the critical point toward a fixed point, where an appropriately defined correlation function goes to zero and topological quantum numbers characterizing the phase are easy to compute. In this paper, using two independent models, a model in the AIII symmetry class and a model in the BDI symmetry class, in one dimension as examples, we show that there are cases where the fixed-point curve and the critical-point curve appear to intersect, which turn out to be multicritical points, and focus on understanding its implications.
For DDD-dimensional weakly interacting topological insulators in certain symmetry classes, the topological invariant can be calculated from a DDD- or (D+1)(D+1)(D+1)-dimensional integration over a certain curvature function that is expressed in terms of single-particle Green's functions. Based on the divergence of curvature function at the topological phase transition, we demonstrate how a renormalization group approach circumvents these integrations and reduces the necessary calculation to that for the Green's function alone, rendering a numerically efficient tool to identify topological phase transitions in a large parameter space. The method further unveils a number of statistical aspects related to the quantum criticality in weakly interacting topological insulators, including correlation function, critical exponents, and scaling laws, that can be used to characterize the topological phase transitions driven by either interacting or noninteracting parameters. We use 1D class BDI and 2D class A Dirac models with electron-electron and electron-phonon interactions to demonstrate these principles and find that interactions may change the critical exponents of the topological insulators.
Abstract In topological insulators and topological superconductors, the discrete jump of the topological invariant upon tuning a certain system parameter defines a topological phase transition. A unified framework is employed to address the quantum criticality of the topological phase transitions in one to three spatial dimensions, which simultaneously incorporates the symmetry classification, order of band crossing, m -fold rotational symmetry, correlation functions, critical exponents, scaling laws, and renormalization group approach. We first classify higher-order Dirac models according to the time-reversal, particle-hole, and chiral symmetries, and determine the even–oddness of the order of band crossing in each symmetry class. The even–oddness further constrains the rotational symmetry m permitted in a symmetry class. Expressing the topological invariant in terms of a momentum space integration over a curvature function, the order of band crossing determines the critical exponent of the curvature function, as well as that of the Wannier state correlation function introduced through the Fourier transform of the curvature function. The conservation of topological invariant further yields a scaling law between critical exponents. In addition, a renormalization group approach based on deforming the curvature function is demonstrated for all dimensions and symmetry classes. Through clarification of how the critical quantities, including the jump of the topological invariant and critical exponents, depend on the nonspatial and the rotational symmetry, our work introduces the notion of universality class into the description of topological phase transitions.
Abstract The interplay between topology and criticality has been a recent interest of study in condensed matter physics. A unique topological transition between certain critical phases has been observed as a consequence of the edge modes living at criticalities. In this work, we generalize this phenomenon by investigating possible transitions between critical phases which are non-high symmetry (non-HS) in nature. We find the triviality and non-triviality of these critical phases in terms of the decay length of the edge modes and also characterize them using the winding numbers. The distinct non-HS critical phases are separated by multicritical points with linear dispersion at which the winding number exhibits the quantized jump, indicating a change in the topology (number of edge modes) at the critical phases. Moreover, we reframe the scaling theory based on the curvature function, i.e. curvature function renormalization group method to efficiently address the non-HS criticalities and multicriticalities. Using this we identify the conventional topological transition between gapped phases through non-HS critical points, and also the unique topological transition between critical phases through multicritical points. The renormalization group flow, critical exponents and correlation function of Wannier states enable the characterization of non-HS criticalities along with multicriticalities.