Kondo Lattice Model with Finite Temperature Lanczos Method

Burkhard Schmidt; Ivica Zerec; Peter Thalmeier

REVIEW

Kondo Lattice Model with Finite Temperature Lanczos Method

Ivica Zerec, Burkhard Schmidt, Peter Thalmeier

Strongly Correlated Electrons · Submitted 1 Mar 2006 · 17 references

Pith's verdict: Solid finite-cluster FTLM data with a clean PBC/OBC comparison; the two-scale thermodynamic story is a plausible conjecture, not proven.

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arXiv IDcond-mat/0603021 (version 2)

Pith IDNP6CE44V, Pith's permanent record of this paper and its review

CitationBurkhard Schmidt, Ivica Zerec, Peter Thalmeier. "Kondo Lattice Model with Finite Temperature Lanczos Method." arXiv:cond-mat/0603021 (2006). https://pith.science/paper/NP6CE44V

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Claim paper

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Editor's note

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Colleague,

Read Zerec, Schmidt & Thalmeier, 'Kondo Lattice Model with Finite Temperature Lanczos Method' (cond-mat/0603021). Worth your time if you care about finite-cluster numerics for the Kondo lattice.

What's new: a systematic FTLM study of the 8-site KLM at half filling, comparing periodic and open boundary conditions as proxies for different single-particle DOS shapes near the Fermi level. The boundary-condition comparison and the resulting phase diagrams are new, and the data collapse in Figs. 4-5 is clean. The OBC results reproduce the (J_K/t)_c ≈ 1.4 crossover from QMC and mean-field, and the OBC correlations match earlier QMC/FTLM results on larger clusters. No fitted parameters—the J_K and J_K^2 scales are read off the collapsed curves. That is honest, exact small-system work.

The soft spot is the leap from finite cluster to thermodynamic limit. For PBC, the linear-in-J_K onsite scale is explicitly tied to the 8-site cluster's sixfold degenerate Fermi level (Sec. III A). A degenerate level is a delta-function DOS, not a smooth 'strongly peaked' DOS; in a dense metal with continuous spectrum, the single-impurity Kondo scale is exponential in 1/(Jρ), not linear. The paper demonstrates two extreme limits—exact degeneracy (PBC) and an OBC level distribution with no states within J_K of the Fermi level—and then extrapolates to a finite-width peak without studying intermediate cases. The abstract states the two-scale picture more strongly than the body's caveats justify.

Minor issue: Sec. III B says OBC weak coupling has two characteristic temperatures (one finite-size, one additional onsite screening), but Sec. V says OBC has only one characteristic temperature for all J_K/t. Real, but small. No code or raw data; not fatal for an exact-diagonalization paper, but it would help.

Bottom line: the finite-cluster data are solid and the PBC/OBC comparison is a useful demonstration. The thermodynamic conclusion about DOS shape controlling Kondo vs RKKY is a plausible conjecture, not proven. I'd send it to a serious referee requesting larger clusters or an analytic argument for a peaked but non-singular DOS, plus reconciliation of the OBC two-scales statement. I'd cite the OBC benchmark data in my own work; I would not cite the DOS-shape claim as established.

Yours

Referee report

Significance

If the two-scale picture survives contact with the thermodynamic limit, this would be a useful, physically transparent contribution to the Kondo-lattice discussion, complementing QMC and mean-field studies. The paper's strength is that the FTLM data are exact for the finite cluster with no uncontrolled approximations, and the scaling analyses of the correlation functions (Figs. 4 and 5) are clearly presented and reproducible from the stated cluster data. The comparison between PBC and OBC conditions is an interesting way to expose finite-size spectral effects. However, the significance of the work hinges on whether the PBC linear-in-J_K onsite scale is a genuine property of a strongly peaked but smooth DOS, rather than an accident of the 8-site cluster's degenerate Fermi level; on this point the paper currently offers only an assertion, not evidence.

Major objections (3)

[Sec. III A, Fig. 4] The linear-in-J_K scaling of the onsite correlation is explicitly tied, in the text and in Fig. 1, to the sixfold degenerate Fermi level of the PBC cluster: 6 of 8 conduction electrons occupy a single degenerate level. A degenerate level is a delta-function DOS, not the 'strongly peaked' smooth DOS invoked in the abstract and conclusion. The paper provides no calculation for a cluster with a peak of finite width, so the first regime of the abstract—that a finite, strongly peaked DOS produces a linear-in-J_K onsite scale—is not demonstrated. This is load-bearing because the two-scale picture is the paper's central claim.
[Sec. V] The thermodynamic-limit conclusion that 'a strongly peaked DOS near the Fermi level favors onsite screening' rests entirely on the PBC cluster's accidental degeneracy. The paper itself acknowledges in Sec. III A that the jump in <τ1·S1> and the associated scale are consequences of degenerate itinerant degrees of freedom. Since a dense metal has a continuous spectrum and a single-impurity Kondo scale that is exponentially small in 1/(Jρ) for finite ρ, the extrapolation requires either an explicit intermediate calculation with a peak of finite width or a clear statement that the linear scale is a cluster-degeneracy effect rather than a generic DOS effect. As written, the Sec. V conclusions overreach the evidence.
[Sec. IV, Fig. 6] The definition of the crossover coupling (J_K/t)_C via <μ_loc^2> = <μ_loc^2>_0/2 gives (J_K/t)_C ≈ 1.4 for OBC and 0 for PBC. The paper notes that OBC is more representative of the thermodynamic limit, but then uses the PBC result to argue for a DOS-dependent competition. This is consistent with the two-regime picture only if the PBC linear scale is physical; otherwise the comparison of crossover couplings is a comparison of a degeneracy artifact with a finite-size-gap artifact. The argument would be strengthened by showing the crossover behavior for a cluster with a nondegenerate but peaked DOS.

Minor objections (4)

[Sec. II, Fig. 2 caption] The caption states 'The legend is valid for all the figures in this work, if not explicitly specified,' but several figures (e.g., Figs. 4 and 5) show curves with different normalization and fewer J_K values, so the statement is misleading and should be qualified.
[Sec. IV] The phrase 'greaterorsimilar' appears in the text ('JK/t /greaterorsimilar2' twice) and is clearly a LaTeX rendering error; it should read ≳.
[Sec. III B] The 'new low energy scale' for OBC is described as a finite-size effect attributable to level splitting and as a lattice-induced screening scale; the distinction between these two interpretations is not clearly drawn in the text, and an explicit statement of which numbers in Fig. 5 correspond to which effect would help.
[References] Reference 17 is a self-citation to an 'in press' paper with no arXiv or DOI; given that the comparison to the generalized Kondo necklace model is used in the conclusions, a more complete citation or a brief description of the model would improve verifiability.

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The paper

The paper claims

Whether the Kondo lattice shows one or two low-energy scales depends on the Fermi-level density of states: screening scales with $J_K$, RKKY with $J_K^2$.

The reading

This paper studies the Kondo lattice model—itinerant electrons hopping on a two-dimensional eight-site cluster and interacting with localized spins—using a finite-temperature Lanczos method. By comparing periodic and open boundary conditions, it shows that the competition between onsite Kondo screening and intersite RKKY magnetism depends sharply on whether the non-interacting conduction electron spectrum has states within an energy $J_K$ of the Fermi level. When it does, two distinct energy scales appear: a screening scale linear in $J_K$ and an RKKY scale quadratic in $J_K$. When it does not, only the RKKY scale survives. The authors conclude that a strongly peaked density of states at the Fermi level favors Kondo screening, while the RKKY scale is comparatively insensitive to the shape of the density of states.

What carries the argument

The key object is the 8-site square-lattice Kondo lattice model at half filling, solved exactly by the Finite Temperature Lanczos Method under periodic and open boundary conditions. The boundary conditions are the controlling variable: for periodic boundary conditions the non-interacting single-particle spectrum has six of its eight levels exactly at the Fermi level, whereas for open boundary conditions the levels are distributed and shifted below it. The argument is carried by two static spin correlation functions, the onsite $\langle \tau_i \cdot S_i \rangle$ and the nearest-neighbor intersite $\langle S_i \cdot S_j \rangle$, whose temperature dependence—normalized to ground-state values and plotted against $T/J_K$ and $T/(J_K^2/t)$—collapses onto single curves that reveal the linear and quadratic energy scales. The phase diagrams then associate the specific-heat maxima and the correlation inflection points with these scales.

Extended reading notes

Core claim

The central claim is that the competition between onsite Kondo screening and intersite RKKY magnetism in the Kondo lattice model is controlled by the distribution of non-interacting conduction-electron levels around the Fermi level, and that on the eight-site cluster the two boundary conditions realize the two regimes. With periodic boundary conditions, six of the eight conduction electrons sit exactly at the Fermi surface, mimicking a sharply peaked density of states; there the onsite correlation $\langle \tau_1 \cdot S_1 \rangle$ develops on a temperature scale proportional to $J_K$ while the nearest-neighbor intersite correlation $\langle S_1 \cdot S_5 \rangle$ develops on a scale proportional to $J_K^2/t$, and the two processes are independent. With open boundary conditions the levels are spread and shifted below the Fermi level, so no states lie within $J_K$ of it; onsite screening sets in only gradually, the $J_K^2/t$ scale is the only low-energy scale, and the total local moment evolves continuously with $J_K$. The paper concludes that in the thermodynamic limit a strongly peaked DOS favors Kondo screening, while the RKKY interaction is comparatively insensitive to the form of the DOS.

Load-bearing premise

The load-bearing premise is that the periodic-boundary eight-site cluster's highly degenerate Fermi-level states, with six of its eight conduction electrons sitting exactly on the Fermi surface, faithfully represent a metal whose density of states is strongly peaked at the Fermi energy, so that the linear-in-$J_K$ screening scale is a genuine physical energy scale rather than an accidental splitting of a small-cluster degeneracy.

Editorial extensions

If this is right

In a metal whose conduction-electron density of states is strongly peaked at the Fermi level, onsite Kondo screening develops on a temperature scale linear in $J_K$, independent of the intersite RKKY scale.
The RKKY intersite correlation scale is insensitive to the shape of the density of states and scales as $J_K^2/t$, so the magnetic ordering temperature should be comparatively insensitive to band-structure details near the Fermi energy.
When no conduction-electron states lie within $J_K$ of the Fermi level, the RKKY scale is the only low-energy scale, governing both the formation of intersite correlations and the weak-coupling onset of onsite screening.
The crossover from a magnetic to a nonmagnetic ground state, defined by the total local moment dropping to half its bare value, occurs at $(J_K/t)_C\approx 1.4$ for open boundary conditions, in agreement with quantum Monte Carlo and mean-field results.
In the strong-coupling limit the boundary conditions become irrelevant: a single scale linear in $J_K$ describes onsite singlet formation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the linear-in-$J_K$ screening scale is a genuine bulk effect of a peaked DOS rather than a finite-size degeneracy splitting, heavy-fermion materials with a van Hove singularity or flat band near the Fermi energy should show a specific-heat feature whose position tracks $T/J_K$, not the exponential single-impurity Kondo scale.
A direct test would be to repeat the same FTLM analysis on larger clusters while tuning a narrow, non-degenerate DOS peak at the Fermi level; the linear scale should persist and eventually cross over to the conventional exponential Kondo scale as the peak broadens.
The open-boundary results suggest that at half filling the Kondo lattice's spin dynamics may be quantitatively captured by the generalized Kondo necklace, a spin-only model; comparing the two on identical cluster geometries at finite temperature could sharpen this mapping.
The two-scale picture implies that proximity to a heavy-fermion quantum critical point might be tunable by band-structure engineering of the Fermi-level DOS, not only by the ratio $J_K/W$, which is a testable prediction for materials with pressure- or doping-tunable van Hove singularities.

Original abstract

We investigate the Kondo Lattice Model on 2D clusters using the Finite Temperature Lanczos Method. The temperature dependence of thermodynamic and correlations functions are systematically studied for various Kondo couplings JK. The ground state value of the total local moment is presented as well. Finally, the phase diagrams of the finite clusters are constructed for periodic and open boundary conditions. For the two boundary conditions, two different regimes are found for small JK/t, depending on the distribution of non-interacting conduction electron states. If there are states within JK around the Fermi level, two energy scales, linear and quadratic in JK, exist. The former is associated with the onsite screening and the latter with the RKKY interaction. If there are no states within JK around the Fermi level, the only energy scale is that of the RKKY interaction. Our results imply that the form of the electron density of states (DOS) plays an important role in the competition between the Kondo screening and the RKKY interaction. The former is stronger if the DOS is larger around the Fermi level, while the latter is less sensitive to the form of the DOS.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,388 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We investigate the Kondo Lattice Model on 2D clusters using the Finite Temperature Lanczos Method. The temperature dependence of thermodynamic and correlations functions are systematically studied for various Kondo couplings JK. The ground state value of the total local moment is presented as well. Finally, the phase diagrams of the finite clusters are constructed for periodic and open boundary conditions. For the two boundary conditions, two different regimes are found for small JK/t, depending on the distribution of non-interacting conduction electron states. If there are states within JK around the Fermi level, two energy scales, linear and quadratic in JK, exist. The former is associated with the onsite screening and the latter with the RKKY interaction. If there are no states within JK around the Fermi level, the only energy scale is that of the RKKY interaction. Our results imply that the form of the electron density of states (DOS) plays an important role in the competition between the Kondo screening and the RKKY interaction. The former is stronger if the DOS is larger around the Fermi level, while the latter is less sensitive to the form of the DOS.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate the Kondo Lattice Model on 2D clusters using the Finite Temperature Lanczos Method.	contribution 0.77	0.96	0.03	asserted 0.96
s1	The temperature dependence of thermodynamic and correlations functions are systematically studied for various Kondo couplings JK.	contribution 0.94	0.95	0.04	asserted 1.00
s2	The ground state value of the total local moment is presented as well.	result 0.80	0.93	0.03	asserted 0.99
s3	Finally, the phase diagrams of the finite clusters are constructed for periodic and open boundary conditions.	contribution 0.71	0.95	0.05	asserted 1.00
s4	For the two boundary conditions, two different regimes are found for small JK/t, depending on the distribution of non-interacting conduction electron states.	result 1.00	0.95	0.11	asserted 1.00
s5	If there are states within JK around the Fermi level, two energy scales, linear and quadratic in JK, exist.	result 1.00	0.86	0.09	asserted 1.00
s6	The former is associated with the onsite screening and the latter with the RKKY interaction.	interpretation 0.82	0.76	0.07	asserted 1.00
s7	If there are no states within JK around the Fermi level, the only energy scale is that of the RKKY interaction.	result 1.00	0.89	0.08	asserted 1.00
s8	Our results imply that the form of the electron density of states (DOS) plays an important role in the competition between the Kondo screening and the RKKY interaction.	interpretation 1.00	0.87	0.78	hedged possibility 0.82
s9	The former is stronger if the DOS is larger around the Fermi level, while the latter is less sensitive to the form of the DOS.	interpretation 0.83	0.82	0.55	asserted 1.00

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: DOS for the tight-binding model in 2D is shown

**Figure 2.** FIG. 2: (Color online) The specific heat calculated for vario

**Figure 3.** FIG. 3: (Color online) The onsite correlations in the lat

**Figure 5.** FIG. 5: (Color online) Correlation functions for OBC. Above

**Figure 6.** FIG. 6: (Color online) The scaled magnetic susceptibility,

Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:cond-mat/0603021 by the authors.

References

Works cited (17)

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author H. Tsunetsugu , author M. Sigrist , and author K. Ueda , journal Rev. Mod. Phys. volume 69 , pages 803 ( year 1997 )

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author P. Sun and author G. Kotliar , journal Phys. Rev. Lett. volume 95 , pages 016402 ( year 2005 )

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author S. Paschen , author T. L\"uhmann , author S. Wirth , author P. Gegenwart , author O. Trovarelli , author C. Geibel , author F. Steglich , author P. Coleman , and author Q. Si , journal Nature volume 432 , pages 881 ( year 2004 )

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[6]

author J. Custers , author P. Gegenwart , author H. Wilhelm , author K. Neumeier , author Y. Tokiwa , author O. Trovarelli , author C. Geibel , author F. Steglich , author C. Pepin , and author P. Coleman , journal Nature volume 424 , pages 524 ( year 2003 )

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[7]

author Q. Si , author S. Rabello , author K. Ingersent , and author J. Smith , journal Nature volume 413 , pages 804 ( year 2001 )

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[8]

author A.Schr\"oder , author G. Aeppli , author R. Coldea , author M. Adams , author O. Stockert , author H. L\"ohneysen , author E. Bucher , author R. Ramazashvili , and author P. Coleman , journal Nature volume 407 , pages 351 ( year 2000 )

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author N. Mathur , author F. Grosche , author S. Julian , author I. Walker , author D. Freye , author R. Haselwimmer , and author G. Lonzarich , journal Nature volume 394 , pages 39 ( year 1998 )

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[10]

author F. Steglich , author B. Buschinger , author P. Gegenwart , author M. Lohmann , author R. Helfrich , author C. Langhammer , author P. Hellmann , author L. Donnevert , author S. Thomas , author A. Link , et al. , journal J. Phys.: Condens. Matter volume 8 , pages 9909 ( year 1996 )

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[11]

author G.-M. Zhang and author L. Yu , journal Phys. Rev. B volume 62 , pages 76 ( year 2000 )

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[12]

author S. Capponi and author F.F. Assaad , journal Phys. Rev. B volume 63 , pages 155114 ( year 2001 )

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[13]

author J. Jaklic and author P. Prelovsek , journal Adv. Phys. volume 49 , pages 1 ( year 2000 )

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[14]

author K. Haule , author J. Bonca , and author P. Prelovsek , journal Phys. Rev. B volume 61 , pages 2482 ( year 2000 )

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[15]

author Y. Luo , author C. Verdozzi , and author N. Kioussis , journal Phys. Rev. B volume 71 , pages 033304 ( year 2005 )

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[16]

author G.-M. Zhang , author Q. Gu , and author L. Yu , journal Phys. Rev. B volume 62 , pages 69 ( year 2000 )

View on Pith 2000
[17]

author I. Zerec , author B. Schmidt , and author P. Thalmeier ( year 2006 ), note Physica B , in press; Proceedings of SCES'05

View on Pith 2006

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

author H. Tsunetsugu , author M. Sigrist , and author K. Ueda , journal Rev. Mod. Phys. volume 69 , pages 803 ( year 1997 )

View on Pith 1997

[2] [2]

author S. Doniach , journal Physica B volume 91 , pages 231 ( year 1977 )

View on Pith 1977

[3] [3]

author J. Rech , author P. Coleman , author G. Zarand , and author O. Parcollet , note cond-mat/0507001

View on Pith Pith review arXiv

[4] [4]

author P. Sun and author G. Kotliar , journal Phys. Rev. Lett. volume 95 , pages 016402 ( year 2005 )

View on Pith 2005

[5] [5]

author S. Paschen , author T. L\"uhmann , author S. Wirth , author P. Gegenwart , author O. Trovarelli , author C. Geibel , author F. Steglich , author P. Coleman , and author Q. Si , journal Nature volume 432 , pages 881 ( year 2004 )

View on Pith 2004

[6] [6]

author J. Custers , author P. Gegenwart , author H. Wilhelm , author K. Neumeier , author Y. Tokiwa , author O. Trovarelli , author C. Geibel , author F. Steglich , author C. Pepin , and author P. Coleman , journal Nature volume 424 , pages 524 ( year 2003 )

View on Pith 2003

[7] [7]

author Q. Si , author S. Rabello , author K. Ingersent , and author J. Smith , journal Nature volume 413 , pages 804 ( year 2001 )

View on Pith 2001

[8] [8]

author A.Schr\"oder , author G. Aeppli , author R. Coldea , author M. Adams , author O. Stockert , author H. L\"ohneysen , author E. Bucher , author R. Ramazashvili , and author P. Coleman , journal Nature volume 407 , pages 351 ( year 2000 )

View on Pith 2000

[9] [9]

author N. Mathur , author F. Grosche , author S. Julian , author I. Walker , author D. Freye , author R. Haselwimmer , and author G. Lonzarich , journal Nature volume 394 , pages 39 ( year 1998 )

View on Pith 1998

[10] [10]

author F. Steglich , author B. Buschinger , author P. Gegenwart , author M. Lohmann , author R. Helfrich , author C. Langhammer , author P. Hellmann , author L. Donnevert , author S. Thomas , author A. Link , et al. , journal J. Phys.: Condens. Matter volume 8 , pages 9909 ( year 1996 )

View on Pith 1996

[11] [11]

author G.-M. Zhang and author L. Yu , journal Phys. Rev. B volume 62 , pages 76 ( year 2000 )

View on Pith 2000

[12] [12]

author S. Capponi and author F.F. Assaad , journal Phys. Rev. B volume 63 , pages 155114 ( year 2001 )

View on Pith 2001

[13] [13]

author J. Jaklic and author P. Prelovsek , journal Adv. Phys. volume 49 , pages 1 ( year 2000 )

View on Pith 2000

[14] [14]

author K. Haule , author J. Bonca , and author P. Prelovsek , journal Phys. Rev. B volume 61 , pages 2482 ( year 2000 )

View on Pith 2000

[15] [15]

author Y. Luo , author C. Verdozzi , and author N. Kioussis , journal Phys. Rev. B volume 71 , pages 033304 ( year 2005 )

View on Pith 2005

[16] [16]

author G.-M. Zhang , author Q. Gu , and author L. Yu , journal Phys. Rev. B volume 62 , pages 69 ( year 2000 )

View on Pith 2000

[17] [17]

author I. Zerec , author B. Schmidt , and author P. Thalmeier ( year 2006 ), note Physica B , in press; Proceedings of SCES'05

View on Pith 2006