Gutzwiller projected states for the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the Kagome lattice: Achievements and pitfalls

Gutzwiller projected states for the $J_{1} - J_{2}$ Heisenberg model on the Kagome lattice: Achievements and pitfalls

Yasir Iqbal, Francesco Ferrari, Aishwarya Chauhan, Alberto Parola, Didier Poilblanc, and Federico Becca

Phys. Rev. B 104, 144406 – Published 6 October 2021

Abstract

We assess the ground-state phase diagram of the $J_{1} − J_{2}$ Heisenberg model on the Kagome lattice by employing Gutzwiller-projected fermionic wave functions. Within this framework, different states can be represented, defined by distinct unprojected fermionic Hamiltonians that include hopping and pairing terms, as well as a coupling to local Zeeman fields to generate magnetic order. For $J_{2} = 0$ , the so-called U(1) Dirac state, in which only hopping is present (such as to generate a $π$ -flux in the hexagons), has been shown to accurately describe the exact ground state [Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Phys. Rev. B 87, 060405(R) (2013); Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Phys. Rev. X 7, 031020 (2017).]. Here we show that its accuracy improves in the presence of a small antiferromagnetic superexchange $J_{2}$ , leading to a finite region where the gapless spin liquid is stable; then, for $J_{2} / J_{1} = 0.11 (1)$ , a first-order transition to a magnetic phase with pitch vector $q = (0, 0)$ is detected by allowing magnetic order within the fermionic Hamiltonian. Instead, for small ferromagnetic values of $| J_{2} | / J_{1}$ , the situation is more contradictory. While the U(1) Dirac state remains stable against several perturbations in the fermionic part (i.e., dimerization patterns or chiral terms), its accuracy clearly deteriorates on small systems, most notably on 36 sites where exact diagonalization is possible. Then, on increasing the ratio $| J_{2} | / J_{1}$ , a magnetically ordered state with $\sqrt{3} \times \sqrt{3}$ periodicity eventually overcomes the U(1) Dirac spin liquid. Within the ferromagnetic $J_{2}$ regime, evidence is shown in favor of a first-order transition at $J_{2} / J_{1} = - 0.065 (5)$ .

Received 5 August 2021
Revised 19 September 2021
Accepted 24 September 2021

DOI:https://doi.org/10.1103/PhysRevB.104.144406

Physics Subject Headings (PhySH)

Frustrated magnetism Quantum spin liquid

Antiferromagnets Kagome lattice Quantum spin models

Exact diagonalization Heisenberg model Monte Carlo methods Variational approach

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yasir Iqbal ^1,*,†, Francesco Ferrari ^2,†, Aishwarya Chauhan ^1,†, Alberto Parola³, Didier Poilblanc⁴, and Federico Becca⁵

¹Department of Physics and Quantum Centers in Diamond and Emerging Materials (QuCenDiEM) group, Indian Institute of Technology Madras, Chennai 600036, India
²Institute for Theoretical Physics, Goethe University Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany
³Dipartimento di Scienza e Alta Tecnologia, Università dell'Insubria, Via Valleggio 11, I-22100 Como, Italy
⁴Laboratoire de Physique Théorique UMR-5152, CNRS and Université de Toulouse, F-31062 Toulouse, France
⁵Dipartimento di Fisica, Università di Trieste, Strada Costiera 11, I-34151 Trieste, Italy

^*yiqbal@physics.iitm.ac.in
^†These authors contributed equally to this work.

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Vol. 104, Iss. 14 — 1 October 2021

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Figure 1
The antiferromagnetic order parameter for the $J_{1}$ – $J_{2}$ Heisenberg model on the Kagome lattice. Different colors indicate the various ground-state phases: Magnetic phases with $q = (0, 0)$ (red) or $\sqrt{3} \times \sqrt{3}$ periodicity (blue) and gapless spin liquid (yellow). The values are estimated in the thermodynamic limit (the error bars are smaller than the symbols), as shown below. The spin patterns for the two magnetically ordered phases are also shown. The U(1) Dirac state is expected to represent the paramagnetic region for $J_{2} > 0$ ; instead, for $J_{2} < 0$ the situation is more controversial. The exact nature of the $J_{2} / J_{1} < 0$ non-magnetic phase is still controversial (see text) as labelled by the question mark.
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Figure 2
Finite-size scalings of the fictitious Zeeman field $h$ , see Eq. (4), and the square of the sublattice magnetization $m^{2}$ [panels (a) and (b) refer to the $q = (0, 0)$ order; panels (c) and (d) refer to the $\sqrt{3} \times \sqrt{3}$ order with $q = (4 π / 3 a, 0)$ ]. For $J_{2} > 0$ , we employed $3 \times L \times L$ clusters with $L = 4 n$ ( $n = 1, \dots, 6$ ); for $J_{2} < 0$ , we used $L = 6 n$ ( $n = 1, \dots, 4$ ). The hopping structure of the fermionic Hamiltonian (4) reproduces the U(1) Dirac state for $J_{2} / J_{1} ⩾ - 0.06$ , while it gives the $[π, π]$ state for $J_{2} / J_{1} ⩽ - 0.07$ . The insets in (b) and (d) show the finite-size scaling of $m^{2}$ within the spin-liquid regime, as a function of $1 / L^{2}$ , which is consistent with a power-law decay of the spin-spin correlations of the U(1) Dirac state [38].
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Figure 3
Variational energy as a function of the fictitious Zeeman field $h$ . The energy landscape has been computed for $J_{2} / J_{1} = 0.10$ (a) and $J_{2} / J_{1} = 0.12$ (b). Clusters with $3 \times L \times L$ sites have been used, with $L = 12$ , 16, 20, and 24.
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Figure 4
Variational energies as a function of $J_{2} / J_{1}$ on the $9 \times 8 \times 8$ cluster. Different wave functions are considered, including a VBC state with 36 sites in the unit cell. When two local minima in the energy landscape are present (with small and large Zeeman fields, as in Fig. 3), both variational energies are shown. We note that the energy landscape of the $[π, π]$ state with a $(4 π / 3 a, 0)$ field has a single minimum as a function of $h$ .
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Figure 5
Overlap between the symmetrized U(1) Dirac state $| Ψ_{sym} 〉$ and few low-energy exact eigenstates on the 36-site cluster, obtained by Lanczos diagonalization. Both the area and the colors of the circles represent the value of the overlap. On the horizontal axis we report the value of $J_{2} / J_{1}$ , while on the vertical axis we show the energy gap $Δ$ of the exact eigenstates with respect to the ground state, in units of $J_{1}$ . Notice that the variational Ansatz $| Ψ_{sym} 〉$ has a finite overlap only with the exact eigenstates belonging to the same symmetry sector. In the inset, analogous results on the 12-site cluster are reported.
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Figure 6
Energy $E_{p}$ [see Eq. (9)] versus variance $σ_{p}^{2}$ [see Eq. (10)] starting from the initial state given by the symmetrized U(1) Dirac Ansatz for three values of the ratio $J_{2} / J_{1}$ . The energy (variance) is given in units of $J_{1}$ ( $J_{1}^{2}$ ). Calculations are done for a cluster with 36 sites.
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Physical Review B

covering condensed matter and materials physics

Gutzwiller projected states for the $J_{1} - J_{2}$ Heisenberg model on the Kagome lattice: Achievements and pitfalls

Yasir Iqbal, Francesco Ferrari, Aishwarya Chauhan, Alberto Parola, Didier Poilblanc, and Federico Becca

Phys. Rev. B 104, 144406 – Published 6 October 2021

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Physical Review B

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Gutzwiller projected states for the J1−J2 Heisenberg model on the Kagome lattice: Achievements and pitfalls

Yasir Iqbal, Francesco Ferrari, Aishwarya Chauhan, Alberto Parola, Didier Poilblanc, and Federico Becca

Phys. Rev. B 104, 144406 – Published 6 October 2021

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Gutzwiller projected states for the $J_{1} - J_{2}$ Heisenberg model on the Kagome lattice: Achievements and pitfalls