[Paper Review] Fractonic critical point proximate to a higher-order topological insulator: How does UV blend with IR?

Yizhi You, Julian Bibo+2 more|arXiv (Cornell University)|Jan 5, 2021

Quantum many-body systems17 citations

TL;DR

This paper proposes a novel quantum critical point (QCP) between a higher-order topological insulator (HOTI) and a trivial Mott insulator, characterized by a critical dipole liquid theory with subsystem U(1) symmetry. Using a coupled-stripe construction and bosonization, the authors show the QCP hosts a Bose surface along the $k_x, k_y$ axes, leading to fracton dynamics and a breakdown of the area law for entanglement entropy, with strong UV-IR mixing dominating low-energy physics due to high-momentum zero modes.

ABSTRACT

We propose an unconventional topological quantum phase transition connecting a higher-order topological insulator (HOTI) and a featureless Mott insulator sharing the same symmetry patterns. We construct an effective theory description of the quantum critical point (QCP) by combining a bosonization approach and the coupled-stripe construction of 1D critical spin ladders. The phase transition theory is characterized by a critical dipole liquid theory with subsystem $U(1)$ symmetry whose low energy modes contain a Bose surface along the $k_x,k_y$ axis. Such a quantum critical point manifests fracton dynamics and the breakdown of the area law entanglement entropy due to the existence of a Bose surface. We numerically confirm our findings by measuring the entanglement entropy, topological rank-2 Berry phase, and the static structure factor throughout the topological transition and compare it with our previous approach obtained from the percolation picture. A significant new element of our phase transition theory is that the infrared~(IR) effective theory is controlled by short wave-length fluctuations with peculiar UV-IR mixing.

Motivation & Objective

• To understand the nature of a topological quantum phase transition between a higher-order topological insulator (HOTI) and a trivial Mott insulator, both sharing the same symmetry but distinct topological order.
• To address the limitations of the Landau-Ginzburg-Wilson (LGW) paradigm in describing such transitions, especially when topological order is involved.
• To develop a controlled field-theoretic description of the critical point that captures emergent fracton dynamics and non-trivial entanglement structure.
• To investigate how the infrared (IR) behavior of the critical point is influenced by ultraviolet (UV) degrees of freedom, particularly through UV-IR mixing.

Proposed method

• Construct a 1D critical theory using a one-dimensional spin ladder (stripe) with subsystem U(1) and time-reversal symmetry, modeling the HOTI-Mott insulator transition.
• Apply bosonization to map the 1D critical spin ladder to a $c=1$ conformal field theory (CFT) with fluctuating critical dipoles along the stripe.
• Implement a coupled-stripe construction to lift the 1D critical theory to a 2D critical dipole liquid theory, recovering the original microscopic Hamiltonian from Ref. [3].
• Use the coupled-stripe framework to demonstrate that the 2D critical theory supports a Bose surface with zero-energy states along the $k_x$ and $k_y$ axes.
• Confirm the critical theory via numerical methods: DMRG and exact diagonalization to compute entanglement entropy, rank-2 Berry phase, and static structure factor.
• Analyze the four-point correlation function to reveal double logarithmic scaling in real space, indicating strong UV-IR mixing due to subsystem symmetry-induced zero modes at high momentum.

Experimental results

Research questions

• RQ1 Can a quantum critical point between a HOTI and a trivial Mott insulator be described by a field theory that captures fracton dynamics and subsystem symmetry?
• RQ2 How does the presence of a Bose surface affect the entanglement entropy and its scaling at criticality?
• RQ3 To what extent does UV-IR mixing alter the conventional renormalization group picture in this critical theory?
• RQ4 Why does the dipole correlation function exhibit a non-power-law decay with double logarithmic scaling, and what does this imply for the universality class?
• RQ5 How does the inter-stripe coupling in the coupled-stripe construction preserve criticality without opening a gap?

Key findings

• The critical point is described by a 2D critical dipole liquid theory with a Bose surface along the $k_x$ and $k_y$ axes, supporting zero-energy modes at high momentum.
• The entanglement entropy at the critical point scales as $L \ln L$, violating the area law due to the Bose surface and non-local correlations.
• The four-point correlation function $\langle S^+(0,0)S^-(r_x,0)S^+(r_x,r_y)S^-(0,r_y)\rangle$ decays as $e^{-a(\ln r_x \ln r_y)}$, indicating double logarithmic scaling from UV-IR mixing.
• The dipole correlation function along a stripe decays as $r_y^{-2}$, with a scaling dimension that depends on the UV cut-off, confirming UV-IR mixing.
• The critical theory is dominated by short-wavelength fluctuations due to a subextensive number of zero modes on the $k_x, k_y$ axes, invalidating standard coarse-graining procedures.
• Numerical results from DMRG and exact diagonalization confirm the presence of a Bose surface, fracton dynamics, and the $L \ln L$ entanglement scaling, validating the analytical framework.