[Paper Review] Fracton Critical Point in Higher-Order Topological Phase Transition

Yizhi You, Julian Bibo+2 more|arXiv (Cornell University)|Aug 4, 2020

Quantum many-body systems4 citations

TL;DR

This paper proposes a novel 2D quantum critical point (QCP) between a higher-order topological insulator (HOTI) and a trivial Mott insulator, driven by percolating domain walls that host spinon zero modes. Fluctuations of these modes induce fracton dynamics via a critical dipole liquid theory with subsystem U(1) symmetry, leading to a logarithmic entanglement entropy scaling of $L\ln(L)$, which violates the area law and marks a new paradigm in topological quantum criticality beyond the Ginzburg-Landau-Wilson framework.

ABSTRACT

The theory of quantum phase transitions separating different phases with distinct symmetry patterns at zero temperature is one of the foundations of modern quantum many-body physics. In this paper we demonstrate that the existence of a 2D topological phase transition between a higher-order topological insulator (HOTI) and a trivial Mott insulator with the same symmetry eludes this paradigm. We present a theory of this quantum critical point (QCP) driven by the fluctuations and percolation of the domain walls between a HOTI and a trivial Mott insulator region. Due to the spinon zero modes that decorate the rough corners of the domain walls, the fluctuations of the phase boundaries trigger a spinon-dipole hopping term with fracton dynamics. Hence we find the QCP is characterized by a critical dipole liquid theory with subsystem $U(1)$ symmetry and the breakdown of the area law entanglement entropy which exhibits a logarithmic enhancement: $L \ln(L)$. Using the density matrix renormalization group (DMRG) method, we analyze the dipole stiffness together with structure factor at the QCP which provide strong evidence of a critical dipole liquid with a Bose surface. These numerical signatures further support the fracton dynamics of the QCP, and suggest a new paradigm for 2D quantum criticality proximate to a topological phase.

Motivation & Objective

• To investigate the nature of a quantum phase transition between a higher-order topological insulator (HOTI) and a trivial Mott insulator with identical symmetries.
• To understand how topological features of the HOTI phase influence the critical behavior at the transition point.
• To explore whether the critical point inherits fracton dynamics and violates the area law for entanglement entropy.
• To establish a new paradigm for 2D quantum criticality that transcends the Ginzburg-Landau-Wilson framework.

Proposed method

• Modeling the transition via a 2D square lattice Hamiltonian with four spin-1/2 degrees of freedom per unit cell, including XY interactions and ring-exchange terms.
• Mapping spin operators to hardcore bosons to describe the system in terms of dipole degrees of freedom and subsystem U(1) symmetry.
• Using the density matrix renormalization group (DMRG) method to compute the dipole stiffness and static structure factor on infinite cylinders.
• Analyzing the flux dependence of the ground state energy to extract the dipole stiffness $D_d$ via second symmetric derivatives.
• Evaluating entanglement entropy scaling to detect logarithmic enhancement $L\ln(L)$ as a signature of critical dipole liquid behavior.
• Identifying a Bose surface in the structure factor with nodal lines along $k_x$ and $k_y$ axes, indicating criticality.

Experimental results

Research questions

• RQ1 Can a quantum phase transition occur between a HOTI and a trivial Mott insulator when both break the same symmetries?
• RQ2 How do the topological zero modes at domain wall corners influence the critical behavior of the phase transition?
• RQ3 Does the critical point exhibit fracton dynamics due to constrained spinon-dipole motion?
• RQ4 Is the entanglement entropy at the critical point enhanced beyond the area law, and if so, with what scaling?
• RQ5 Can the critical theory be described as a critical dipole liquid with a Bose surface and emergent subsystem symmetry?

Key findings

• The quantum critical point is characterized by a critical dipole liquid theory with subsystem U(1) symmetry and fracton dynamics.
• The entanglement entropy scales as $L\ln(L)$, violating the conventional area law, which is a hallmark of long-range mutual information.
• Numerical DMRG results show a non-vanishing dipole stiffness $D_d$ only near the critical point, confirming the existence of a critical phase.
• The static structure factor exhibits a Bose surface with nodal lines along the $k_x$ and $k_y$ axes, supporting the critical dipole liquid description.
• The critical point is identified at $\lambda_c \approx 2.5$ using both correlation length and dipole stiffness, with minor numerical deviations attributed to entanglement and finite-size effects.
• The system shows algebraically decaying correlations and diverging correlation length only within stripes, indicating quasi-1D dynamics that enable access to the 2D critical point via finite-cylinder simulations.