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Low-temperature properties of the Hubbard model on highly frustrated one-dimensional lattices

O. Derzhko, J. Richter, A. Honecker, M. Maksymenko, and R. Moessner
Phys. Rev. B 81, 014421 – Published 26 January 2010
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  • INTRODUCTION AND MOTIVATION
  • SUMMARY OF RESULTS
  • NONINTERACTING ELECTRONS. TRAPPED…
  • TRAPPED ELECTRON GROUND STATES FOR U>0
  • LOW-TEMPERATURE THERMODYNAMICS
  • MAGNETIC PROPERTIES
  • RELATION TO THE XXZ MODEL
  • CONCLUSIONS
  • ACKNOWLEDGEMENTS
  • APPENDICES
    • References

    Abstract

    We consider the repulsive Hubbard model on three highly frustrated one-dimensional lattices—sawtooth chain and two kagomé chains—with completely dispersionless (flat) lowest single-electron bands. We construct the complete manifold of exact many-electron ground states at low electron fillings and calculate the degeneracy of these states. As a result, we obtain closed-form expressions for low-temperature thermodynamic quantities around a particular value of the chemical potential μ0. We discuss specific features of thermodynamic quantities of these ground-state ensembles such as residual entropy, an extra low-temperature peak in the specific heat, and the existence of ferromagnetism and paramagnetism. We confirm our analytical results by comparison with exact-diagonalization data for finite systems.

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    • Received 18 November 2009

    DOI:https://doi.org/10.1103/PhysRevB.81.014421

    ©2010 American Physical Society

    Authors & Affiliations

    O. Derzhko1,2,3, J. Richter3, A. Honecker4, M. Maksymenko1, and R. Moessner2

    • 1Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii Street, L’viv 79011, Ukraine
    • 2Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
    • 3Institut für Theoretische Physik, Universität Magdeburg, P.O. Box 4120, 39016 Magdeburg, Germany
    • 4Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

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    Vol. 81, Iss. 1 — 1 January 2010

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    • Figure 1
      Figure 1
      (Color online) Three one-dimensional lattices considered in this paper: (a) the sawtooth chain, (b) the kagomé chain I, and (c) the kagomé chain II. For the sawtooth Hubbard chain the hopping parameter along the zigzag path t′ is 2 times larger than the hopping parameter t>0 along the base line. For the kagomé Hubbard chains all the hopping parameters t>0 are identical. Bold (red) lines denote the minimal trapping cells for localized electrons.Reuse & Permissions
    • Figure 2
      Figure 2
      (Color online) Illustration of a multicluster state. For each cluster the total spin of the cluster can be flipped independently by applying the cluster spin-flip operator Sni−=∑j∊cluster nicj,↓†cj,↑.Reuse & Permissions
    • Figure 3
      Figure 3
      (Color online) Ground-state degeneracies for n=1,…,nmax for the sawtooth chain (N=7, empty diamonds; N=8, empty squares), the kagomé chain I (N=7, empty diamonds; N=8, filled squares), and the kagomé chain II (N=7, filled diamonds; N=8, filled squares).Reuse & Permissions
    • Figure 4
      Figure 4
      (Color online) Canonical residual entropy S(n,N)/N=ln[DN(n)]/N versus n/N for system sizes N=32,64,128,256,512,1024.Reuse & Permissions
    • Figure 5
      Figure 5
      (Color online) Average number of electrons in the ground state n¯(0,μ,N)/N versus chemical potential μ/μ0 for the sawtooth chain [N=12 (solid) and N=20 (long dashed)], the kagomé chain I [N=18 (short dashed)], the kagomé chain II [N=20 (dotted)] with U→∞. The result for n¯(0,μ,N)/N which follows from Eqs. (5.3, 5.5) is given by θ(1−μ/μ0) and [(N+1)/N]θ(1−μ/μ0), respectively.Reuse & Permissions
    • Figure 6
      Figure 6
      (Color online) Entropy S(T,μ,N)/N and grand-canonical specific heat C(T,μ,N)/N for the sawtooth chain (N=12, triangles) and the kagomé chains I (N=9, diamonds) with t=1, U→∞. (a) S(T,μ,N)/N versus temperature T/μ0 at μ=0.98μ0,μ0,1.02μ0 (from top to bottom). (b) C(T,μ,N)/N versus temperature T/μ0 at μ=0.98μ0,μ0,1.02μ0 (from top to bottom). We also show the hard-dimer result for N→∞ as it follows from Eq. (5.6) (thin solid lines) as well as the results which follow from Eq. (5.3) for N=3 (dotted lines) and N=6 (dashed lines). Note that for these systems no additional leg states exist. Note further that for μ=μ0 and μ=1.02μ0 the hard-dimer results for N=3, 6, and ∞ are indistinguishable.Reuse & Permissions
    • Figure 7
      Figure 7
      (Color online) Entropy S(T,μ,N)/N and grand-canonical specific heat C(T,μ,N)/N for the kagomé chain I (N=12, diamonds) and the kagomé chain II (N=15, the sectors with up to seven electrons were taken into account, pentagons) with t=1, U→∞. (a) S(T,μ,N)/N versus temperature T/μ0 at μ=0.98μ0,μ0,1.02μ0 (from top to bottom). (b) C(T,μ,N)/N versus temperature T/μ0 at μ=0.98μ0,μ0,1.02μ0 (from top to bottom). We also show the hard-dimer result for N→∞ as it follows from Eq. (5.6) (thin solid lines) as well as the finite-size results for N=3 (dash-dotted lines) and N=4 (dotted lines) which follow from Eq. (5.5) and include the contribution of the leg states.Reuse & Permissions
    • Figure 8
      Figure 8
      (Color online) C(T,μ,N)/N versus T/μ0 for the kagomé chain I with N=12 sites and t=1 for different values of U [U=1 (empty circles), U=10 (filled diamonds), and U→∞ (empty diamonds)]. The results for finite U were obtained taking into account the sectors with up to eight electrons. We also show the result which follows from Eq. (5.5) for N=4 (lines).Reuse & Permissions
    • Figure 9
      Figure 9
      (Color online) Canonical specific heat C(T,n,N)/n versus T for ideal (td=tL=1) and distorted (td=1 and tL=1.01) kagomé chains I with n=N/6 electrons; U→∞.Reuse & Permissions
    • Figure 10
      Figure 10
      (Color online) Average ground-state magnetic moment of the sawtooth-Hubbard chain: ⟨S2⟩n/N2 versus n/N for N=8,16,32,64,128,256. For saturated ferromagnetic ground states ⟨S2⟩n/N2 would equal (n/N)2/4 (thin line) in the thermodynamic limit N→∞.Reuse & Permissions
    • Figure 11
      Figure 11
      (Color online) Average ground-state magnetic moment of the kagomé-Hubbard chains: ⟨S2⟩n/N2 versus n/N for N=8,16,32,64,128,256. For saturated ferromagnetic ground states ⟨S2⟩n/N2 would equal (n/N)2/4 (thin line) in the thermodynamic limit N→∞.Reuse & Permissions
    • Figure 12
      Figure 12
      (Color online) Uniform magnetic susceptibility 3Tχ(T,n,N)/Cn for the sawtooth chain (t=1) with n=4 electrons and N=16 sites with U=4 (filled triangles and circles) as well as N=24 sites with U→∞ (empty triangles and circles). Triangles correspond to t′=2 and circles correspond to t′=1. Here we use for the normalization of the vertical axis Cn=4 for N=8 and Cn=60/17 for N=12.Reuse & Permissions
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