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Physical review. B./Physical review. B

Series expansion studies of the J 1 − J 2 Heisenberg bilayer

Erik Wagner, Wolfram Brenig

Published Sep 14, 2023• 0 citations
Publisher

Abstract

We study a bilayer of the frustrated ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice. Starting from the dimer limit at strong interlayer coupling, we perform series expansions using the perturbative continuous unitary transformation, based on the flow equation method, in order to determine the spectrum up to the two-triplon sector. From the one-triplon dispersion, we obtain quantum critical lines for transitions from the dimer phase into either N\'eel or collinear magnetic order. For low to intermediate frustration these transitions are consistent with existing findings, based on the magnetic phases. In the region of strongest frustration, i.e., ${J}_{2}/{J}_{1}\ensuremath{\sim}0.5$, we provide an estimate for the stability of the anticipated single-layer quantum spin-liquids against finite interlayer coupling. In the two-triplon sector, we find a set of well defined (anti)bound states, which can be classified according to total spin and in-plane rotational symmetry. For vanishing frustration, these states agree with the previous series expansion analysis. For ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.5$, we provide evidence for a close-by condensation of one-triplon and two-triplon singlet bound states, suggesting that between the dimer and the collinear state additional phases may intervene.

DOI
10.1103/physrevb.108.115131

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We study a bilayer of the frustrated J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ Heisenberg model on the square lattice. Starting from the dimer limit at strong interlayer coupling, we perform series expansions using the perturbative continuous unitary transformation, based on the flow equation method, in order to determine the spectrum up to the two-triplon sector. From the one-triplon dispersion, we obtain quantum critical lines for transitions from the dimer phase into either N'eel or collinear magnetic order. For low to intermediate frustration these transitions are consistent with existing findings, based on the magnetic phases. In the region of strongest frustration, i.e., J2/J1\ensuremath∼0.5{J}_{2}/{J}_{1}\ensuremath{\sim}0.5J2​/J1​\ensuremath∼0.5, we provide an estimate for the stability of the anticipated single-layer quantum spin-liquids against finite interlayer coupling. In the two-triplon sector, we find a set of well defined (anti)bound states, which can be classified according to total spin and in-plane rotational symmetry. For vanishing frustration, these states agree with the previous series expansion analysis. For J2/J1\ensuremath≳0.5{J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.5J2​/J1​\ensuremath≳0.5, we provide evidence for a close-by condensation of one-triplon and two-triplon singlet bound states, suggesting that between the dimer and the collinear state additional phases may intervene.
Series expansion studies of the J1J_1J1​-J2J_2J2​-Heisenberg bilayer
We study a bilayer of the frustrated J1J_1J1​-J2J_2J2​ Heisenberg-model on the square lattice. Starting from the dimer limit at strong interlayer coupling, we perform series expansions using the perturbative Continuous Unitary Transformation, based on the flow equation method, in order to determine the spectrum up to the two-triplon sector. From the one-triplon dispersion we obtain quantum critical lines for transitions from the dimer phase into either Néel or collinear magnetic order. For low to intermediate frustration these transitions are consistent with existing findings, based on the magnetic phases. In the region of strongest frustration, i.e. J2/J1∼0.5J_2/J_1\sim 0.5J2​/J1​∼0.5, we provide an estimate for the stability of the anticipated single-layer quantum spin-liquids against finite interlayer coupling. In the two-triplon sector we find a set of well defined (anti)bound states, which can be classified according to total spin and in-plane rotational symmetry. For vanishing frustration these states agree with previous series expansion analysis. For J2/J1≳0.5J_2/J_1\gtrsim 0.5J2​/J1​≳0.5 we provide evidence for a close-by condensation of one-triplon and two-triplon singlet bound states, suggesting that between the dimer and the collinear state, additional phases may intervene.
We study the spectrum of a bilayer of Kitaev magnets on the honeycomb lattice, coupled by Heisenberg exchange in the quantum-dimer phase at strong interlayer coupling. Using the perturbative continuous unitary transformation we perform series expansion starting from the fully dimerized limit, to evaluate the elementary excitations, reaching up to and focusing on the two-triplon sector. In stark contrast to conventional bilayer quantum magnets, and because of the broken SU(2)SU(2)SU(2) invariance, encoded in the intralayer directional compass exchange, the bilayer Kitaev magnet is shown to exhibit a rich structure of two-triplon scattering-state continua, as well as several collective two-triplon (anti)bound states. Direct physical pictures for the occurrence of the latter are provided and the (anti)bound states are studied versus the stacking type, the spin components, and the exchange parameters. In addition to the two-triplon spectra, we investigate a corresponding experimental probe by evaluating the magnetic Raman-scattering intensity. We find a very strong sensitivity of this intensity on the two-triplon interactions and the scattering geometry, however, a signal from the (anti)bound states appears only in very close proximity to the continuum.
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We analyze the phase diagram of the frustrated Heisenberg antiferromagnet, the ${J}_{1}{\ensuremath{-}J}_{2}$ model, in two dimensions. Two quantum phase transitions in the model are already known: the second-order transition from the N\'eel state to the spin liquid state at ${(J}_{2}{/J}_{1}{)}_{c2}\ensuremath{\approx}0.38,$ and the first-order transition from the spin liquid state to the collinear state at ${(J}_{2}{/J}_{1}{)}_{c4}\ensuremath{\approx}0.60.$ We have found evidence for two second-order phase transitions: the transition from the spin columnar dimerized state to the state with plaquette-type modulation at ${(J}_{2}{/J}_{1}{)}_{c3}=0.50\ifmmode\pm\else\textpm\fi{}0.02,$ and the transition from the simple N\'eel state to the N\'eel state with spin columnar dimerization at ${(J}_{2}{/J}_{1}{)}_{c1}=0.34\ifmmode\pm\else\textpm\fi{}0.04.$ We also present an independent calculation of ${(J}_{2}{/J}_{1}{)}_{c2}\ensuremath{\approx}0.38$ using another approach.
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