Classical and quantum phases of the pyrochlore <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:math> magnet with Heisenberg and Dzyaloshinskii-Moriya interactions
Vincent Noculak, Daniel Lozano‐Gómez, J. Oitmaa, Rajiv R. P. Singh, Yasir Iqbal, Michel J. P. Gingras, Johannes Reuther
Abstract
We investigate the ground state and critical temperature (${T}_{c}$) phase diagrams of the classical and quantum $S=\frac{1}{2}$ pyrochlore lattice with nearest-neighbor Heisenberg and Dzyaloshinskii-Moriya interactions (DMI). We consider ferromagnetic and antiferromagnetic Heisenberg exchange interaction as well as direct and indirect DMI. At the classical level, three ground states are found: all-in/all-out, ferromagnetic, and a locally ordered $XY$ phase, known as ${\mathrm{\ensuremath{\Gamma}}}_{5}$, which displays an accidental classical U(1) degeneracy at the mean-field level. Quantum zero-point energy fluctuations computed to order $1/S$ are found to lift the classical ground-state degeneracy and select the so-called ${\ensuremath{\psi}}_{3}$ state out of the degenerate manifold in most parts of the ${\mathrm{\ensuremath{\Gamma}}}_{5}$ regime. Likewise, thermal fluctuations treated classically at the Gaussian level entropically select the ${\ensuremath{\psi}}_{3}$ state at $T={0}^{+}$. In contrast to this low-temperature state-selection behavior, classical Monte Carlo simulations find that the system orders at ${T}_{c}$ in the noncoplanar ${\ensuremath{\psi}}_{2}$ state of ${\mathrm{\ensuremath{\Gamma}}}_{5}$ for antiferromagnetic Heisenberg exchange and indirect DMI with a transition from ${\ensuremath{\psi}}_{2}$ to ${\ensuremath{\psi}}_{3}$ at a temperature ${T}_{{\mathrm{\ensuremath{\Gamma}}}_{5}}<{T}_{c}$. The same method finds that the system orders via a single transition at ${T}_{c}$ directly into the ${\ensuremath{\psi}}_{3}$ state for most of the region with ferromagnetic Heisenberg exchange and indirect DMI. Such ordering behavior at ${T}_{c}$ for the $S=\frac{1}{2}$ quantum model is corroborated by high-temperature series expansion. To investigate the $T=0$ quantum ground state of the model, we apply the pseudo-fermion functional renormalization group (PFFRG). The quantum paramagnetic phase of the pure antiferromagnetic $S=\frac{1}{2}$ Heisenberg model is found to persist over a finite region in the phase diagram for both direct or indirect DMI. Interestingly, we find that a combined ferromagnetic Heisenberg and indirect DMI, near the boundary of ferromagnetism and ${\mathrm{\ensuremath{\Gamma}}}_{5}$ antiferromagnetism, may potentially realize a $T=0$ quantum ground state lacking conventional magnetic order. Otherwise, for the largest portion of the phase diagram, PFFRG finds the same long-range ordered phases (all-in/all-out, ferromagnetic, and ${\mathrm{\ensuremath{\Gamma}}}_{5}$) as in the classical model.