Boundary criticality for the Gross-Neveu-Yukawa models

TL;DR

The paper addresses boundary criticality in Gross-Neveu-Yukawa models by combining lattice DQMC on a honeycomb system with armchair boundaries and a $4-\epsilon$ RG framework. It demonstrates ordinary, special, and extraordinary boundary transitions at the bulk CDW quantum critical point, assigning Dirichlet boundary conditions to Dirac fermions and Dirichlet/Neumann conditions to the bosonic order parameter depending on the transition. The work provides explicit one-loop boundary RG factors and anomalous dimensions, maps out the boundary exponents for the chiral Ising class, and discusses extensions to the chiral XY class, thereby offering a versatile framework for boundary criticality in GNY universality classes with potential experimental relevance. The findings advance understanding of edge-critical behavior in Dirac systems and offer predictions suitable for edge-sensitive probes such as scanning tunneling spectroscopy. All mathematical expressions are formulated in a way that can be directly used for replication and further theoretical exploration, including the $4-\epsilon$ expansion and explicit boundary conditions at the edge.

Abstract

We study the boundary criticality for the Gross-Neveu-Yukawa (GNY) models. Employing interacting Dirac fermions on a honeycomb lattice with armchair boundaries, we use determinant quantum Monte Carlo simulation to uncover rich boundary criticalities at the quantum phase transition to a charge density wave (CDW) insulator, including the ordinary, special, and extraordinary transitions. The Dirac fermions satisfy a Dirichlet boundary condition, while the boson field, representing the CDW order, obeys Dirichlet and Neumann conditions at the ordinary and special transitions, respectively, thereby enriching the critical GNY model. We develop a perturbative $4-ε$ renormalization group approach to compute the boundary critical exponents. Our framework generalizes to other GNY universality class variants and provides theoretical predictions for experiments.

Figures (5)

• Figure 1: (a) Illustration of the honeycomb lattice ($L_x=7$) with armchair boundaries. The black (pink) link denotes the bulk (boundary) bond. The red and green sites denote the two sublattices, respectively. (b) Phase diagram of the lattice model \ref{['eq:tV-model']}. The inset illustrates the crossing of the RG-invariant quantity $R_\text{bulk}$, where OT (ET) denotes the ordinary transition (extraordinary transition). (c) The crossing of the boundary RG-invariant quantity $R_\text{bdy}$ at the edge transition, with $V_\text{bulk}=1.3$ fixed.
• Figure 2: Feynman diagrams relevant to the boundary field. The dashed (solid) line represents the boson (fermion) propagator. The vertex $\otimes$ represents the boundary boson or fermion field and $\diamond$ represents the boundary mass term. (a) Correction to the boundary boson. (b) Correction to the boundary fermion. (c) Correction to the boundary mass term. (d) Tadpole diagram for the correction to the boundary fermion.
• Figure S1: (a) Illustration of the unit cell in a honeycomb lattice with armchair boundaries, along with the corresponding order parameters. The interaction strength in black (red) bonds correspond to $V_\text{bulk}$ ($V_\text{bdy}$). The gray dashed line delineates a unit cell. $\phi$ denotes the CDW order parameter, where $\phi_\text{bdy}$ ($\phi_\text{bulk}$) represents the order on the boundary (in the bulk). The lattice is periodic along the $y$ axis, while the lattice length along the $x$ axis is chosen to be $L=4$ for an illustration. In the mean field calculation, $L$ ranges from 25 to 49. (b) Phase diagram of lattice model \ref{['eq:lattice_sm']}. (c) Boundary order parameter as a function of the boundary interaction $V_\text{bdy}$ strength at the bulk critical point $V_\text{bulk}^*$. OT, ST, and ET represent ordinary, special, and extraordinary transition, respectively.
• Figure S2: Illustration of the bulk and edge unit cells used in the $R_\text{bulk}$ and $R_\text{bdy}$ calculations in a lattice with $L_x=7$ and $L_y=14$, which is periodic in the $y$ direction. Only correlators between shaded sites are included in the corresponding calculation.
• Figure S3: The two-loop "rainbow" diagram. The dashed (solid) line represents the boson (fermion) propagator. The vertex $\bigotimes$ denotes the boundary fermion.