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arXiv:0908.1225 · Physical Review B

Microscopic model versus systematic low-energy effective field theory for a doped quantum ferromagnet

U. Gerber, Christoph P. Hofmann, F. Kämpfer, U.‐J. Wiese

Published Feb 17, 2010• 9 citations
arXivHTMLPDF

Abstract

We consider a microscopic model for a doped quantum ferromagnet as a test case for the systematic low-energy effective field theory for magnons and holes, which is constructed in complete analogy to the case of quantum antiferromagnets. In contrast to antiferromagnets, for which the effective field theory approach can be tested only numerically, in the ferromagnetic case, both the microscopic and the effective theory can be solved analytically. In this way, the low-energy parameters of the effective theory are determined exactly by matching to the underlying microscopic model. The low-energy behavior at half-filling as well as in the single- and two-hole sectors is described exactly by the systematic low-energy effective field theory. In particular, for weakly bound two-hole states the effective field theory even works beyond perturbation theory. This lends strong support to the quantitative success of the systematic low-energy effective field theory method not only in the ferromagnetic but also in the physically most interesting antiferromagnetic case.

DOI
10.1103/physrevb.81.064414

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We consider a microscopic model for a doped quantum ferromagnet as a test case for the systematic low-energy effective field theory for magnons and holes, which is constructed in complete analogy to the case of quantum antiferromagnets. In contrast to antiferromagnets, for which the effective field theory approach can be tested only numerically, in the ferromagnetic case, both the microscopic and the effective theory can be solved analytically. In this way, the low-energy parameters of the effective theory are determined exactly by matching to the underlying microscopic model. The low-energy behavior at half-filling as well as in the single- and two-hole sectors is described exactly by the systematic low-energy effective field theory. In particular, for weakly bound two-hole states the effective field theory even works beyond perturbation theory. This lends strong support to the quantitative success of the systematic low-energy effective field theory method not only in the ferromagnetic but also in the physically most interesting antiferromagnetic case.
Identifying the correct low-energy effective theory for magnons and holes in an antiferromagnet has remained an open problem for a long time. In analogy to the effective theory for pions and nucleons in QCD, based on a symmetry analysis of Hubbard and $t\text{\ensuremath{-}}J$-type models, we construct a systematic low-energy effective field theory for magnons and holes located inside pockets centered at lattice momenta $(\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a},\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a})$. The effective theory is based on a nonlinear realization of the spontaneously broken spin symmetry and makes model-independent universal predictions for the entire class of lightly doped antiferromagnetic precursors of high-temperature superconductors. The predictions of the effective theory are exact, order by order in a systematic low-energy expansion. We derive the one-magnon exchange potentials between two holes in an otherwise undoped system. Remarkably, in some cases the corresponding two-hole Schr\"odinger equations can even be solved analytically. The resulting bound states have ddd-wave characteristics. The ground state wave function of two holes residing in different hole pockets has a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-like symmetry, while for two holes in the same pocket the symmetry resembles dxy{d}_{xy}dxy​.
Based on a symmetry analysis of the microscopic Hubbard and ttt-JJJ models, a systematic low-energy effective field theory is constructed for hole-doped antiferromagnets on the honeycomb lattice. In the antiferromagnetic phase, doped holes are massive due to the spontaneous breakdown of the SU(2)s\text{SU}{(2)}_{s}SU(2)s​ symmetry, just as nucleons in Quantum Chromodynamics (QCD) pick up their mass from spontaneous chiral symmetry breaking. In the broken phase, the effective action contains a single-derivative term, similar to the Shraiman-Siggia term in the square lattice case. Interestingly, an accidental continuous spatial rotation symmetry arises at leading order. As an application of the effective field theory, we consider one-magnon exchange between two holes and the formation of two-hole bound states. As an unambiguous prediction of the effective theory, the wave function for the ground state of two holes bound by magnon exchange exhibits fff-wave symmetry.
A review of the Contractor Renormalization (CORE) method, as a systematic derivation of the low energy effective hamiltonian, is given, with emphasis on its differences and advantages over traditional perturbative (weak/strong links) real space RG. For the low energy physics of the 2D Hubbard model, we derive the plaquette bosons (projected SO(5)) model which connects the microscopic model to phases and phenomenology of high‐Tc cuprates. For the S = 1/2 Pyrochlore and Kagomé antiferromagnets, the effective hamiltonians predict spin‐disordered, lattice symmetry breaking, ground states with a large density of low energy singlets as found by exact diagonalization of small clusters.
Field-theoretic methods are used to investigate the large-UUU Hubbard model on the honeycomb lattice at half-filling and in the hole-doped regime. Within the framework of a functional-integral approach, we obtain the Lagrangian density associated with the charge and spin degrees of freedom. The Hamiltonian related to the charge degrees of freedom is exactly diagonalized. In the strong-coupling regime, we derive a perturbative low-energy theory suitable to describe the quantum antiferromagnetic phase (AF) as a function of hole doping. At half-filling, we deal with the underlying spin degrees of freedom of the quantum AF Heisenberg model by employing a second-order spin-wave analysis, in which case we have calculated the ground-state energy and the staggered magnetization; the results are in very good agreement with previous studies. Further, in the continuum, we derive a nonlinear $\ensuremath{\sigma}$ model with a topological Hopf term that describes the AF-VBS (valence bond solid) competition. Lastly, in the challenging doped regime, our approach allows the derivation of a $t\text{\ensuremath{-}}J$ Hamiltonian, and the analysis of the role played by charge and spin quantum fluctuations on the ground-state energy and, particularly, on the breakdown of the AF order at a critical hole doping; the results are benchmarked against recent Grassmann tensor product state simulations.
The systematic effective Lagrangian method was first formulated in the context of the strong interaction; chiral perturbation theory (CHPT) is the effective theory of quantum chromodynamics (QCD). It was then pointed out that the method can be transferred to the nonrelativistic domain—in particular, to describe the low-energy properties of ferromagnets. Interestingly, whereas for Lorentz-invariant systems the effective Lagrangian method fails in one spatial dimension (ds=1) , it perfectly works for nonrelativistic systems in ds=1 . In the present brief review, we give an outline of the method and then focus on the partition function for ferromagnetic spin chains, ferromagnetic films, and ferromagnetic crystals up to three loops in the perturbative expansion—an accuracy never achieved by conventional condensed matter methods. We then compare ferromagnets in ds=1 , 2, 3 with the behavior of QCD at low temperatures by considering the pressure and the order parameter. The two apparently very different systems (ferromagnets and QCD) are related from a universal point of view based on the spontaneously broken symmetry. In either case, the low-energy dynamics is described by an effective theory containing Goldstone bosons as basic degrees of freedom.