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arXiv:1808.07981 · Physical review. B./Physical review. B

Ising ferromagnet to valence bond solid transition in a one-dimensional spin chain: Analogies to deconfined quantum critical points

Shenghan Jiang, Olexei I. Motrunich

Published Feb 1, 2019• 82 citations
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Abstract

We study a one-dimensional (1D) system that shows many analogies to proposed two-dimensional (2D) deconfined quantum critical points (DQCP). Our system is a translationally invariant spin-$\frac{1}{2}$ chain with onsite ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry and time-reversal symmetry. It undergoes a direct continuous transition from a ferromagnet (FM), where one of the ${Z}_{2}$ symmetries and the time reversal are broken, to a valence bond solid (VBS), where all onsite symmetries are restored while the translation symmetry is broken. The other ${Z}_{2}$ symmetry remains unbroken throughout, but its presence is crucial for both the direct transition (via specific Berry phase effect on topological defects, also related to a Lieb-Schultz-Mattis--type theorem) and the precise characterization of the VBS phase (which has crystalline-symmetry-protected-topological--like property). The transition has a description in terms of either two domain-wall species that ``fractionalize'' the VBS order parameter or in terms of two partons that ``fractionalize'' the FM order parameter, with each picture having its own ${Z}_{2}$ gauge theory structure. The two descriptions are dual to each other and, at long wavelengths, take the form of a self-dual gauged Ashkin-Teller model, reminiscent of the self-dual easy-plane noncompact ${\mathrm{CP}}^{1}$ model that arises in the description of the 2D easy-plane DQCP. We also find an exact reformulation of the transition that leads to a simple field-theory description that explicitly unifies the FM and VBS order parameters; this reformulation can be interpreted as a new parton approach that does not attempt to fractionalize either of the FM and VBS order parameters but instead encodes them in instanton operators. Aside from providing explicit realizations of many ideas proposed in the context of the 2D DQCP, here in the simpler and fully tractable 1D setting with continuous transition, our study also suggests a possible line of approach to the 2D DQCP.

DOI
10.1103/physrevb.99.075103

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We study spin-1/21/21/2 chains with long-range power-law decaying unfrustrated (bipartite) Heisenberg exchange Jr∝r−αJ_r \propto r^{-α}Jr​∝r−α and multi-spin interactions QQQ favoring a valence-bond solid (VBS) ground state. Employing quantum Monte Carlo techniques and Lanczos diagonalization, we analyze order parameters and excited-state level crossings to characterize quantum states and phase transitions in the (α,Q)(α,Q)(α,Q) plane. For weak QQQ and sufficiently slowly decaying Heisenberg interactions (small ααα), the system has a long-range-ordered antiferromagnetic (AFM) ground state, and upon increasing ααα there is a continuous transition into a quasi long-range ordered (QLRO) critical state of the type in the standard Heisenberg chain. For rapidly decaying long-range interactions, there is transition between QLRO and VBS ground states of the same kind as in the frustrated J1J_1J1​-J2J_2J2​ Heisenberg chain. Our most important finding is a direct continuous quantum phase transition between the AFM and VBS states - a close analogy to the 2D deconfined quantum-critical point. In previous 1D analogies the ordered phases both have gapped fractional excitations, and the critical point is a conventional Luttinger Liquid. In our model the excitations fractionalize upon transitioning from the AFM state, changing from spin waves to deconfined spinons. We extract critical exponents at the AFM-VBS transition and use order-parameter distributions to study emergent symmetries. We find emergent O(444) symmetry of the O(333) AFM and scalar VBS order parameters. Thus, the order parameter fluctuations exhibit the covariance of a uniaxially deformed O(444) sphere (an "elliptical" symmetry). This unusual quantum phase transition does not yet have any known field theory description, and our detailed results can serve to guide its construction. We discuss possible experimental realizations.
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