Low-temperature properties of ferromagnetic spin chains in a magnetic field

Abstract

The thermodynamic properties of ferromagnetic spin chains have been analyzed with a variety of microscopic methods over the years: Bethe ansatz, spin-wave theory, Schwinger-boson mean-field theory, Green functions and renormalization group methods. Surprisingly, in all these different studies, to the best of our knowledge, the manifestation of the spin-wave interaction in the low-temperature series for the thermodynamic quantities has been ignored. In the present work, we address this problem by following a different path, based on the systematic effective Lagrangian method. We evaluate the partition function up to two-loop order and derive the lowtemperature expansion of the energy density, entropy density, heat capacity, magnetization and susceptibility in the presence of a weak external magnetic field. Remarkably, the spin-wave interaction only manifests itself beyond two-loop order. In particular, there is no term of order T 2 in the low-temperature series of the free energy density. This is the analog of Dyson's statement that, in the case of three-dimensional ideal ferromagnets, there is no term of order T 4 in the low-temperature series of the free energy density. The range of validity of our series is critically examined in view of the Mermin-Wagner theorem. We also compare our results with the condensed matter literature and point out that there are some misleading statements.

Magnon–magnon interactions in O(3) ferromagnets and equations of motion for spin operators

“…Strictly speaking, effective field theory is not directly applicable to one-dimensional O(3) ferromagnet in the absence of external magnetic field, since magnons acquire nonperturbatively generated gap. How-ever, a careful analysis from [64,65,109,110] reveals that first few terms in low-temperature series for free energy are well defined even in case of zero external field, and KYA yields results in agreement with thermal Betheansatz in this extreme low-T sector [111].…”

“…Conventional effective field theory (EFT) combines derivative and loop expansions to set up perturbative calculations to given order in some energy scale [53][54][55]. The relevant scale for ferromagnets is determined by temperature and EFT enables one to systematically calculate free energy in powers of T [51,[59][60][61][62][63][64][65]. Yet, as emphasized in the Introduction, some issues concerning ordered magnets are adequately treated by the perturbation theory for lattice magnon fields which preserves the full symmetry of Heisenberg Hamiltonian.…”

“…The perturbative calculations may be conducted in a systematic manner, free of the upper mentioned ambiguities characterizing spin-operator oriented approaches. The general program of [45] is successfully applied on the calculation of the free energy up to two [59,60] and three loops [61] and is subsequently extended to low dimensional ferromagnets [62][63][64][65]. If one is concerned only with low-temperature series of the free energy and related quantities, which are strongly controlled by internal symmetry of the Heisenberg model, the continuum field-theoretic methods of [45,56,[59][60][61][62][63][64][65] will suffice.…”

“…The general program of [45] is successfully applied on the calculation of the free energy up to two [59,60] and three loops [61] and is subsequently extended to low dimensional ferromagnets [62][63][64][65]. If one is concerned only with low-temperature series of the free energy and related quantities, which are strongly controlled by internal symmetry of the Heisenberg model, the continuum field-theoretic methods of [45,56,[59][60][61][62][63][64][65] will suffice. However, the comparison with experimental data and other spin-operator-based methods often requires for the discrete symmetry of the lattice to be included also.…”

Effective Lagrangians for quantum many-body systems

“…where d v is now the total order of the vertex v, taking into account the difference between spatial and temporal indices. The same argument asserting the existence of a well-defined power counting as above applies, except that now one has a valid derivative expansion even at d = 2 [41]. The Lagrangian L (s,t) eff is correspondingly assigned the total order s + 2t.…”