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arXiv:2307.06366

Two-dimension to three-dimension transition of chiral spin liquid and fractional quantum Hall phases

Xiaofan Wu, Ya-Hui Zhang

Published Jul 12, 2023• 1 citation
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Abstract

There have been lots of interest in two-dimensional (2D) fractional phases with an emergent U (1) gauge field. However, many experimental realizations are actually in three-dimensional (3D) systems with infinitely stacked 2D layers. Then a natural question arises: starting from the decoupling limit with 2+1d U (1) gauge field in each layer, how does the gauge field become 3+1d when increasing inter-layer coupling? Here we propose a 2D to 3D transition through condensing inter-layer exciton. The Goldstone mode of the condensation becomes the missing az component in the 3D phase. As a simple example, we construct a 3D chiral spin liquid (CSL) from infinitely stacked 2D CSL. The 3D CSL has a gapless photon mode with dispersion $ω\sim q_z^2$ in the z-direction. The same theory also applies to the fractional quantum Hall phase. At the 2D to the 3D transition point, there are gapless modes at each $q_z$ along a line $q = (0,0,q_z)$ in momentum space, in contrast to a conventional critical point with gapless mode only at one momentum. Meanwhile, the scaling dimension $Δ(q_z)$ has $q_z$ dependence, indicating a more non-trivial structure than a simple decoupled fixed point. Our theory can also be generalized to a critical point between a generic infinite component Chern-Simons-Maxwell theory (iCSM) with both intra-layer and inter-layer Chern Simons term and a 3D gapless phase. Certain iCSM theories have recently been shown to describe gapped non-foliated fracton orders. Therefore we have a continuous transition between a gapped fracton order and a 3D gapless phase.

DOI
10.48550/arxiv.2307.06366

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There have been lots of interest in two-dimensional (2D) fractional phases with an emergent U (1) gauge field. However, many experimental realizations are actually in three-dimensional (3D) systems with infinitely stacked 2D layers. Then a natural question arises: starting from the decoupling limit with 2+1d U (1) gauge field in each layer, how does the gauge field become 3+1d when increasing inter-layer coupling? Here we propose a 2D to 3D transition through condensing inter-layer exciton. The Goldstone mode of the condensation becomes the missing az component in the 3D phase. As a simple example, we construct a 3D chiral spin liquid (CSL) from infinitely stacked 2D CSL. The 3D CSL has a gapless photon mode with dispersion ω∼qz2ω\sim q_z^2ω∼qz2​ in the z-direction. The same theory also applies to the fractional quantum Hall phase. At the 2D to the 3D transition point, there are gapless modes at each qzq_zqz​ along a line q=(0,0,qz)q = (0,0,q_z)q=(0,0,qz​) in momentum space, in contrast to a conventional critical point with gapless mode only at one momentum. Meanwhile, the scaling dimension Δ(qz)Δ(q_z)Δ(qz​) has qzq_zqz​ dependence, indicating a more non-trivial structure than a simple decoupled fixed point. Our theory can also be generalized to a critical point between a generic infinite component Chern-Simons-Maxwell theory (iCSM) with both intra-layer and inter-layer Chern Simons term and a 3D gapless phase. Certain iCSM theories have recently been shown to describe gapped non-foliated fracton orders. Therefore we have a continuous transition between a gapped fracton order and a 3D gapless phase.
Gauge theories form a large class of interacting conformal field theories in 3D, among which critical Chern-Simons-matter theories play a prominent role. Here, we study a simple instance: a gapless complex scalar coupled to a U(1)_{2} Chern-Simons gauge field. This theory is conjectured to be dual to three other Chern-Simons-matter theories and to describe the transition between a Kalmeyer-Laughlin chiral spin liquid (a ν=1/2 bosonic Laughlin state) and a trivially gapped phase. Using the fuzzy-sphere regularization, we realize the corresponding quantum phase transition between a ν_{f}=2 fermionic integer quantum Hall state and a ν_{b}=1/2 bosonic fractional quantum Hall state. We show that the transition is continuous and governed by a conformal field theory with SO(3) global symmetry. The operator spectrum contains only one relevant Lorentz scalar, the SO(3) singlet with scaling dimension Δ_{S}=1.52(18), which tunes the transition. Our nonperturbative results identify the universality class naturally associated with the U(1)_{2} Chern-Simons-Higgs theory and show that the conjectured dualities remain self-consistent.
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The fractional quantum Hall effect (FQHE) realized in two-dimensional electron systems under a magnetic field is one of the most remarkable discoveries in condensed matter physics. Interestingly, it has been proposed that FQHE can also emerge in time-reversal invariant spin systems, known as the chiral spin liquid (CSL) characterized by the topological order and the emerging of the fractionalized quasiparticles. A CSL can naturally lead to the exotic superconductivity originating from the condense of anyonic quasiparticles. Although CSL was highly sought after for more than twenty years, it had never been found in a spin isotropic Heisenberg model or related materials. By developing a density-matrix renormalization group based method for adiabatically inserting flux, we discover a FQHE in a spin-½ isotropic kagome Heisenberg model. We identify this FQHE state as the long-sought CSL with a uniform chiral order spontaneously breaking time reversal symmetry, which is uniquely characterized by the half-integer quantized topological Chern number protected by a robust excitation gap. The CSL is found to be at the neighbor of the previously identified Z2 spin liquid, which may lead to an exotic quantum phase transition between two gapped topological spin liquids.
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