[Paper Review] On bipartite Rokhsar-Kivelson points and Cantor deconfinement

Eduardo Fradkin, David A. Huse+3 more|arXiv (Cornell University)|Nov 14, 2003

Quantum many-body systems28 references83 citations

TL;DR

This paper investigates quantum dimer models near Rokhsar-Kivelson (RK) points on bipartite lattices, showing that weak perturbations in 2+1D lead to a devil's staircase of incommensurate valence bond crystals exhibiting gapless photons and deconfinement—termed 'Cantor deconfinement'—due to a finite-measure set of critical states. In 3+1D, a continuous transition occurs between a U(1) RVB phase and a staggered valence bond crystal.

ABSTRACT

Quantum dimer models on bipartite lattices exhibit Rokhsar-Kivelson (RK) points with exactly known critical ground states and deconfined spinons. We examine generic, weak, perturbations around these points. In d=2+1 we find a first order transition between a ``plaquette'' valence bond crystal and a region with a devil's staircase of commensurate and incommensurate valence bond crystals. In the part of the phase diagram where the staircase is incomplete, the incommensurate states exhibit a gapless photon and deconfined spinons on a set of finite measure, almost but not quite a deconfined phase in a compact U(1) gauge theory in d=2+1! In d=3+1 we find a continuous transition between the U(1) resonating valence bond (RVB) phase and a deconfined staggered valence bond crystal. In an appendix we comment on analogous phenomena in quantum vertex models, most notably the existence of a continuous transition on the triangular lattice in d=2+1.

Motivation & Objective

• To understand the criticality of Rokhsar-Kivelson (RK) points in quantum dimer models on bipartite lattices.
• To analyze the effects of generic weak perturbations on RK fixed points in d = 2+1 and d = 3+1 dimensions.
• To investigate the emergence of incommensurate valence bond crystals and their deconfining properties near RK points.
• To establish the existence of a 'Cantor deconfinement' phase in 2+1D, where deconfinement occurs on a set of finite but non-zero measure.
• To clarify the relationship between RK fixed points and deconfined quantum critical points in U(1) gauge theories.

Proposed method

• Analyzes the height action representation of quantum dimer models on square and honeycomb lattices using a Gaussian field theory with Lagrangian L = 1/2(∂τh)² + 1/2ρ²(∇h)² + 1/2ρ⁴(∇²h)² + λ cos(2πh).
• Identifies RK fixed points as multicritical points in d = 2+1 with two relevant symmetric operators, and as a critical point with one relevant operator in d = 3+1.
• Uses the height field formalism to map dimer configurations to electric fluxes via a Gauss law constraint, enabling a field-theoretic description.
• Applies renormalization group analysis to study the stability of RK fixed points under perturbations.
• Examines the phase diagram via the emergence of a devil's staircase of incommensurate and commensurate valence bond crystals under weak perturbations.
• Compares results to quantum vertex models, particularly on the triangular lattice, to generalize findings to related models with stable RK fixed points.

Experimental results

Research questions

• RQ1 What is the nature of criticality at Rokhsar-Kivelson points in quantum dimer models on bipartite lattices in d = 2+1 and d = 3+1?
• RQ2 How do generic weak perturbations modify the phase diagram near RK points in 2+1D?
• RQ3 Can incommensurate valence bond crystals near RK points exhibit deconfinement and gapless modes, and what is the measure of such states?
• RQ4 What is the origin and structure of the devil's staircase of valence bond crystals in 2+1D quantum dimer models?
• RQ5 How does the 'Cantor deconfinement' phenomenon relate to deconfined quantum criticality in U(1) gauge theories?

Key findings

• In d = 2+1, the RK fixed point on bipartite lattices (e.g., square and honeycomb) is a multicritical point with two relevant symmetric operators.
• Weak perturbations in 2+1D lead to a devil's staircase of commensurate and incommensurate valence bond crystals, with incommensurate states exhibiting a gapless photon and deconfinement.
• The incommensurate phases form a generalized Cantor set of finite measure near the RK point, leading to the phenomenon of 'Cantor deconfinement'.
• In d = 3+1, a continuous transition occurs between the U(1) resonating valence bond (RVB) phase and a staggered valence bond crystal, with the RK fixed point acting as a critical point.
• The gap in the confining commensurate phases is extremely small, and the deconfining incommensurate phases dominate the phase diagram in the weak-perturbation limit.
• Quantum vertex models on the triangular lattice also exhibit a stable RK fixed point in d = 2+1, supporting a similar Cantor deconfinement scenario without requiring tuning of an extra parameter.