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Resummation of fluctuations near ferromagnetic quantum critical points

C. J. Pedder, F. Krüger, and A. G. Green
Phys. Rev. B 88, 165109 – Published 7 October 2013
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  • INTRODUCTION
  • QUANTUM ORDER-BY-DISORDER FRAMEWORK
  • RESUMMATION OF FLUCTUATIONS
  • PHASE DIAGRAMS
  • CONCLUSIONS AND DISCUSSION
  • ACKNOWLEDGEMENTS
  • APPENDICES
    • References

    Abstract

    We present a detailed analysis of the nonanalytic structure of the free energy for the itinerant ferromagnet near the quantum critical point in two and three dimensions. We analyze a model of electrons with an isotropic dispersion interacting through a contact repulsion. A fermionic version of the quantum order-by-disorder mechanism allows us to calculate the free energy as a functional of the dispersion in the presence of homogeneous and spiraling magnetic order. We resum the leading divergent contributions to derive an algebraic expression for the nonanalytic contribution to free energy from quantum fluctuations. Using a recursion which relates subleading divergences to the leading term, we calculate the full T=0 contribution in d=3. We propose an interpolating functional form, which allows us to track phase transition lines at temperatures far below the tricritical point and down to T=0. In d=2, quantum fluctuations are stronger, and nonanalyticities are more severe. Using a similar resummation approach, we find that despite the different nonanalytic structures, the phase diagrams in two and three dimensions are remarkably similar, exhibiting an incommensurate spiral phase near the avoided quantum critical point.

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    • Received 26 July 2013

    DOI:https://doi.org/10.1103/PhysRevB.88.165109

    ©2013 American Physical Society

    Authors & Affiliations

    C. J. Pedder1,*, F. Krüger2,†, and A. G. Green1,‡

    • 1London Centre for Nanotechnology, University College London, 17-19 Gordon Street, London WC1H 0AH, United Kingdom
    • 2SUPA, Department of Physics, University of St. Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom

    • *c.pedder@ucl.ac.uk
    • †frank.kruger@st-andrews.ac.uk
    • ‡andrew.green@ucl.ac.uk

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    Vol. 88, Iss. 16 — 15 October 2013

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    • Figure 1
      Figure 1
      Phase diagram in d=3 as a function of inverse electron repulsion 1/g and temperature T/μ. Above the tricritical point (shown in red) we find a continuous transition between the ferromagnet at large g and a paramagnet at small g. The spiral forms below the tricritical point between the paramagnet and the ferromagnet. The effect of the resummation of leading divergences is illustrated. (i) The dashed line shows the case without resummation, which shows unphysical behavior as T→0. (ii) Using the resummation of the leading divergences ΔF1, the spiral region becomes invisible on the scale of this phase diagram, collapsing into a region very close to the homogeneous magnetic state. (iii) The solid line shows the phase boundary of the spiral found using the form for ΔF in Eq. (23) and gives the exact T=0 intercept and the correct behavior in the vicinity of the tricritical point.Reuse & Permissions
    • Figure 2
      Figure 2
      Intermediate phase diagrams in d=2 for Q=0. (a) Continuous phase boundaries between the ferromagnet and the paramagnet obtained from the condition α=0, first using just the mean field α=αmf (brown line) and then including the effects of fluctuations α=αmf+αfl (green line). In the regime M≪T≪μ, αfl=c+T (c+<0), leading to a stabilization of ferromagnet order. Around T≃μ fluctuations saturate, causing reentrant behavior. We interpolate between the two regimes, indicated by a dashed green line. (b) The leading fluctuation correction ΔF1(M)=c−M3lnM in the regime T≪M causes a first-order transition at low temperatures (solid red line). The first-order line at higher temperatures is obtained by interpolation (dashed red line) to the tip of the reentrant α=0 line, where one expects the location of the tricritical point.Reuse & Permissions
    • Figure 3
      Figure 3
      Phase diagram in d=2 as a function of inverse electron repulsion 1/g and temperature T/μ. Dashed lines are interpolations between different asymptotic regimes. The phase diagram has the same topology as the one in d=3 (see Fig. 1) and exhibits a spiral phase below the tricritical point, which is shown in red. As in d=3, the transition between the ferromagnet and the spiral is of the Lifshitz type, while the spiral/paramagnet transition is first order.Reuse & Permissions
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