[Paper Review] Topological transitions in Ising models

Somenath Jalal, Rishabh Khare+1 more|arXiv (Cornell University)|Oct 31, 2016

Quantum many-body systems3 citations

TL;DR

This paper establishes a holographic connection between critical Ising models across dimensions via the universal effective Hamiltonian of the 1D transverse-field Ising model (TFIM), revealing that its critical point hosts a Lifshitz transition marked by a Fermi-surface topology change and a topological quantum number shift (e.g., Chern number). The transition is probed via Thouless charge pumping and identified through emergent SU(2) symmetry and topological invariants, extending beyond the Ginzburg-Landau-Wilson paradigm to include classical, quantum, and lattice gauge theories.

ABSTRACT

The thermal dynamics of the two-dimensional Ising model and quantum dynamics of the one-dimensional transverse-field Ising model (TFIM) are mapped to one another through the transfer-matrix formalism. We show that the fermionised TFIM undergoes a Fermi-surface topology-changing Lifshitz transition at its critical point. We identify the degree of freedom which tracks the Lifshitz transition via changes in topological quantum numbers (e.g., Chern number, Berry phase etc.). An emergent $SU(2)$ symmetry at criticality is observed to lead to a topological quantum number different from that which characterises the ordered phase. The topological transition is also understood via a spectral flow thought-experiment in a Thouless charge pump, revealing the bulk-boundary correspondence across the transition. The duality property of the phases and their entanglement content are studied, revealing a holographic relation with the entanglement at criticality. The effects of a non-zero longitudinal field and interactions that scatter across the singular Fermi surface are treated within the renormalisation group (RG) formalism. The analysis reveals that the critical point of the 1D TFIM and the 1D spin-1/2 Heisenberg chain are connected via a line of $SU(2)$-symmetric theories. We extend our analysis to show that the classical to quantum correspondence links the critical theories of Ising models in various dimensions holographically through the universal effective Hamiltonian that describes the Lifshitz transition of the 1D TFIM. We obtain in this way a unified perspective of transitions in Ising models that lie beyond the traditional Ginzburg-Landau-Wilson paradigm. We discuss the consequences of our results for similar topological transitions observed in classical spin models, topological insulators, superconductors and lattice gauge-field theories which are related to the Ising universality class.

Motivation & Objective

• To establish a unified framework for phase transitions in Ising models beyond the Ginzburg-Landau-Wilson paradigm.
• To identify topological transitions in the 1D transverse-field Ising model (TFIM) via spectral flow and topological invariants such as the Chern number.
• To demonstrate the holographic equivalence of critical theories in various dimensions through the universal effective Hamiltonian of the 1D TFIM.
• To explore the role of emergent SU(2) symmetry and topological order at criticality, including entanglement and duality properties.
• To extend results to classical Ising models, topological insulators, and lattice gauge theories, revealing shared topological features.

Proposed method

• Mapping the 2D classical Ising model to the 1D quantum TFIM using the transfer-matrix formalism.
• Fermionising the 1D TFIM to reveal a massive Dirac fermion description and identify a Fermi-surface topology change at criticality.
• Applying Thouless adiabatic charge pumping on a two-torus to probe the bulk-boundary correspondence and anomalous charge transport.
• Using the renormalisation group (RG) formalism to analyze the effects of longitudinal fields and scattering at the singular Fermi surface.
• Identifying topological invariants such as the Chern number and Berry phase as indicators of the Lifshitz transition in the fermionic spectrum.
• Establishing a holographic relation between critical theories in 1D, 2D, and 3D Ising models via the universal effective Hamiltonian of the 1D TFIM.

Experimental results

Research questions

• RQ1 How does the critical point of the 1D transverse-field Ising model exhibit a Lifshitz transition with a topology-changing Fermi surface?
• RQ2 What topological quantum numbers (e.g., Chern number, Berry phase) signal the transition, and how do they differ between ordered and critical phases?
• RQ3 How does emergent SU(2) symmetry at criticality lead to a distinct topological invariant compared to the ordered phase?
• RQ4 What is the role of the Thouless charge pump in revealing bulk-boundary correspondence across the Lifshitz transition?
• RQ5 How are the critical theories of Ising models in different dimensions holographically related through the 1D TFIM’s universal effective Hamiltonian?

Key findings

• The 1D TFIM undergoes a Lifshitz transition at criticality, marked by a topology change in the Fermi surface of its fermionised description.
• The transition is tracked by a change in topological quantum numbers, including a non-zero Chern number at criticality, distinct from the ordered phase.
• An emergent SU(2) symmetry at the critical point leads to a topological invariant different from that of the ordered phase, indicating a new universality class.
• Thouless charge pumping reveals bulk-boundary correspondence across the transition, with quantised charge transport linked to the Chern number.
• The critical theory of the 1D TFIM is holographically related to those of the 2D TFIM, 3D Ising model, and 2D quantum Ising lattice gauge theory via a universal effective Hamiltonian.
• The 2D TFIM hosts a Weyl-semimetal Lifshitz transition between topological and non-topological superconducting phases, and another between two topological SC phases involving a gapless Weyl stripe metal.