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arXiv:2112.08104 · SciPost Physics

Quantitative functional renormalization for three-dimensional quantum Heisenberg models

Nils Niggemann, Johannes Reuther, Björn Sbierski

Published May 11, 2022• 1 citation
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Abstract

We employ a recently developed variant of the functional renormalization group method for spin systems, the so-called pseudo Majorana functional renormalization group, to investigate three-dimensional spin-1/2 Heisenberg models at finite temperatures. We study unfrustrated and frustrated Heisenberg systems on the simple cubic and pyrochlore lattices. Comparing our results with other quantum many-body techniques, we demonstrate a high quantitative accuracy of our method. Particularly, for the unfrustrated simple cubic lattice antiferromagnet ordering temperatures obtained from finite-size scaling of one-loop data deviate from error controlled quantum Monte Carlo results by \sim5\% ∼ 5 % and we confirm consistency of our data with established critical exponents \nu ν and \eta η of the three-dimensional Heisenberg universality class. As the PMFRG yields results in good agreement with QMC, but remains applicable when the system is frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a paradigmatic magnetically disordered system and find nearly perfect agreement of our two-loop static homogeneous susceptibility with other methods. We further investigate the broadening of pinch points in the spin structure factor as a result of quantum and thermal fluctuations and confirm a finite width in the extrapolated limit T\rightarrow0 T → 0 . While extensions towards higher loop orders \ell ℓ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing \ell ℓ in the presence of magnetic order. Overall, the pseudo Majorana functional renormalization group is established as a powerful many-body technique in quantum magnetism with a wealth of possible future applications.

DOI
10.21468/scipostphys.12.5.156

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We employ a recently developed variant of the functional renormalization group method for spin systems, the so-called pseudo Majorana functional renormalization group, to investigate three-dimensional spin-1/2 Heisenberg models at finite temperatures. We study unfrustrated and frustrated Heisenberg systems on the simple cubic and pyrochlore lattices. Comparing our results with other quantum many-body techniques, we demonstrate a high quantitative accuracy of our method. Particularly, for the unfrustrated simple cubic lattice antiferromagnet ordering temperatures obtained from finite-size scaling of one-loop data deviate from error controlled quantum Monte Carlo results by \sim5% ∼ 5 % and we confirm consistency of our data with established critical exponents \nu ν and \eta η of the three-dimensional Heisenberg universality class. As the PMFRG yields results in good agreement with QMC, but remains applicable when the system is frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a paradigmatic magnetically disordered system and find nearly perfect agreement of our two-loop static homogeneous susceptibility with other methods. We further investigate the broadening of pinch points in the spin structure factor as a result of quantum and thermal fluctuations and confirm a finite width in the extrapolated limit T\rightarrow0 T → 0 . While extensions towards higher loop orders \ell ℓ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing \ell ℓ in the presence of magnetic order. Overall, the pseudo Majorana functional renormalization group is established as a powerful many-body technique in quantum magnetism with a wealth of possible future applications.
We employ a recently developed variant of the functional renormalization group method for spin systems, the so-called pseudo Majorana functional renormalization group, to investigate three-dimensional spin-1/2 Heisenberg models at finite temperatures. We study unfrustrated and frustrated Heisenberg systems on the simple cubic and pyrochlore lattices. Comparing our results with other quantum many-body techniques, we demonstrate a high quantitative accuracy of our method. Particularly, for the unfrustrated simple cubic lattice antiferromagnet ordering temperatures obtained from finite-size scaling of one-loop data deviate from error controlled quantum Monte Carlo results by ∼5%\sim5\%∼5% and we further confirm the established values for the critical exponent ν\nuν and the anomalous dimension η\etaη. As the PMFRG yields results in good agreement with QMC, but remains applicable when the system is frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a paradigmatic magnetically disordered system and find nearly perfect agreement of our two-loop static homogeneous susceptibility with other methods. We further investigate the broadening of pinch points in the spin structure factor as a result of quantum and thermal fluctuations and confirm a finite width in the extrapolated limit T→0T\rightarrow0T→0. While extensions towards higher loop orders ℓ\ellℓ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing ℓ\ellℓ in the presence of magnetic order. Overall, the pseudo Majorana functional renormalization group is established as a powerful many-body technique in quantum magnetism with a wealth of possible future applications.
The pseudofermion functional renormalization group (PFFRG) method has proven to be a powerful numerical approach to treat frustrated quantum spin systems. In its usual implementation, however, the complex fermionic representation of spin operators introduces unphysical Hilbert-space sectors which render an application at finite temperatures inaccurate. In this work we formulate a general functional renormalization group approach based on Majorana fermions to overcome these difficulties. We, particularly, implement spin operators via an SO(3)\text{SO}(3)SO(3) symmetric Majorana representation which does not introduce any unphysical states and, hence, remains applicable to quantum spin models at finite temperatures. We apply this scheme, dubbed pseudo-Majorana functional renormalization group (PMFRG) method, to frustrated Heisenberg models on small spin clusters as well as square and triangular lattices. Computing the finite-temperature behavior of spin correlations and thermodynamic quantities such as free energy and heat capacity, we find good agreement with exact diagonalization and the high-temperature series expansion down to moderate temperatures. We observe a significantly enhanced accuracy of the PMFRG compared to the PFFRG at finite temperatures. More generally, we conclude that the development of functional renormalization group approaches with Majorana fermions considerably extends the scope of applicability of such methods.
We calculate the magnetic phase diagram of the spin-1/21/21/2 nearest neighbor XXZ pyrochlore model using the pseudo-Majorana functional renormalization group in the temperature flow formalism. Our phase diagram as a function of temperature and coupling ratio, allowing both longitudinal and transverse couplings to be ferromagnetic and antiferromagnetic, reveals a large non-magnetic regime at low temperatures, which includes the quantum spin ice phase near the antiferromagnetic Ising model, as well as the antiferromagnetic Heisenberg and XY models. We are able to detect magnetic phase transitions via critical finite size scaling down to temperatures two orders of magnitude smaller than the spin interactions, demonstrating the remarkably good performance of our method upon approaching the ground state. Specifically, the low temperature transition from the zero-flux quantum spin ice phase into the transverse ferromagnetic phase shows very good agreement with previous quantum Monte Carlo results. Comparing our findings with classical results, we identify a quantum order-by-disorder effect near the antiferromagnetic XY model. In magnetically disordered regimes, we find characteristic patterns of broadened pinch points in the spin structure factor and investigate their evolution when approaching magnetically ordered phases. We also compute linear responses to lattice symmetry breaking perturbations and identify a possible lattice nematic ground state of the antiferromagnetic XY model.
We calculate the magnetic phase diagram of the spin- 1 / 2 nearest neighbor XXZ pyrochlore model using the pseudo-Majorana functional renormalization group in the temperature flow formalism. Our phase diagram as a function of temperature and coupling ratio, allowing both longitudinal and transverse couplings to be ferromagnetic and antiferromagnetic, reveals a large nonmagnetic regime at low temperatures, which includes the quantum spin ice phase near the antiferromagnetic Ising model, as well as the antiferromagnetic Heisenberg and XY models. We are able to detect magnetic phase transitions via critical finite size scaling down to temperatures two orders of magnitude smaller than the spin interactions, demonstrating the remarkably good performance of our method upon approaching the ground state. Specifically, the low-temperature transition from the zero-flux quantum spin ice phase into the transverse ferromagnetic phase shows very good agreement with previous quantum Monte Carlo results. Comparing our findings with classical results, we identify a quantum order-by-disorder effect near the antiferromagnetic XY model. In magnetically disordered regimes, we find characteristic patterns of broadened pinch points in the spin structure factor and investigate their evolution when approaching magnetically ordered phases. We also compute linear responses to lattice symmetry breaking perturbations and identify a possible lattice nematic ground state of the antiferromagnetic XY model.
The numerical study of high-dimensional frustrated quantum magnets remains a challenging problem. Here we present an extension of the pseudo-Majorana functional renormalization group to spin-1/2 XXZ-type Hamiltonians with field or magnetization along spin- Z direction at finite temperature. We consider a U(1) symmetry-adapted fermionic spin representation and derive the diagrammatic framework and its renormalization group flow equations. We discuss benchmark results and application to two antiferromagnetic triangular lattice materials recently studied in experiments with applied magnetic fields: First, we numerically reproduce the magnetization data measured for CeMgAl 11 O 19 confirming model parameters previously estimated from inelastic neutron spectrum in high fields. Second, we showcase the accuracy of our method by studying the thermal phase transition into the spin solid up-up-down phase of Na 2 BaCo ( PO 4 ) 2 in good agreement with experiment.
We implement the temperature flow scheme first proposed by Honerkamp and Salmhofer in Phys.~Rev.~B 64, 184516 (2001) into the pseudo-Majorana functional renormalization group method for quantum spin systems. Since the renormalization group parameter in this approach is a physical quantity -- the temperature TTT -- the numerical efficiency increases significantly compared to more conventional renormalization group parameters, especially when computing finite temperature phase diagrams. We first apply this method to determine the finite temperature phase diagram of the J1J_1J1​-J2J_2J2​ Heisenberg model on the simple cubic lattice where our findings support claims of a vanishingly small nonmagnetic phase around the high frustration point J2=0.25J1J_2=0.25J_1J2​=0.25J1​. Perhaps most importantly, we find the temperature flow scheme to be advantageous in detecting finite temperature phase transitions as, by construction, a phase transition is never encountered at an artificial, unphysical cutoff parameter. Finally, we apply the temperature flow scheme to the dipolar XXZ model on the square lattice where we find a rich phase diagram with a large non-magnetic regime down to the lowest accessible temperatures. Wherever a comparison with error-controlled (quantum) Monte Carlo methods is applicable, we find excellent quantitative agreement with less than 5%5\%5% deviation from the numerically exact results.