Strongly Inhomogeneous Phases and Non-Fermi Liquid Behavior in Randomly Depleted Kondo Lattices

Matthias Vojta; Ribhu K. Kaul

REVIEW

Strongly Inhomogeneous Phases and Non-Fermi Liquid Behavior in Randomly Depleted Kondo Lattices

Ribhu K. Kaul, Matthias Vojta

Strongly Correlated Electrons · Submitted 28 Feb 2006 · 15 references

Pith's verdict: A genuinely new nf≈nc organizing line in depleted Kondo lattices, but the 'non-Fermi liquid' label outruns the numerical evidence—likely an extremely small coherence scale rather than a true non-FL state.

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arXiv IDcond-mat/0603002 (version 2)

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CitationMatthias Vojta, Ribhu K. Kaul. "Strongly Inhomogeneous Phases and Non-Fermi Liquid Behavior in Randomly Depleted Kondo Lattices." arXiv:cond-mat/0603002 (2006). https://pith.science/paper/T7EQDT5C

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Claim paper

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Editor's note

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Colleague,

Quick take: the paper is worth reading for the nf≈nc dividing line and the mechanism of collective screening producing rare weakly screened moments. That is a fresh result. But the title's 'non-Fermi liquid behavior' is more than the body's evidence actually supports. The non-saturating χ_ff could just as well be a finite-size artifact in a system that is an ordinary Fermi liquid with an exponentially small coherence scale.

What is actually new: the authors give a simple strong-coupling argument that nf=nc is special—at that concentration, all conduction electrons are tied up in local singlets, so the heavy Fermi liquid cannot form. Then they solve SU(N) mean-field saddle-point equations on 20x20 lattices and show a sharp crossover in T* near nf≈nc, a bimodal distribution of local hybridizations, and island formation in the LDOS. This goes beyond the earlier CPA and two-fluid treatments by addressing disorder explicitly and showing that the broad TK distribution comes from collective screening, not bare disorder in the couplings. That is a legitimate microscopic mechanism. The paper is also self-contained; nothing is fitted to experiment, and the authors are candid in the body about the saturation caveat.

Where it softens: the central 'non-FL' claim is exactly the part that is under-supported. Figure 5(a) shows χ_ff not saturating for nf≈nc=0.4, but the lowest T is limited by finite size and the authors explicitly say they cannot rule out saturation. With the static large-N saddle point, every occupied f-site eventually has a nonzero b_r; a site with very small b_r just has a very low local TK. At temperatures above that TK it looks Curie-like, and the bulk would appear non-saturating over many decades while the true ground state is an ordinary FL. Without a system-size dependence of the data, this is not a demonstrable non-FL. The strong-coupling gap argument only holds for JK>>t, and only one coupling (JK/D=1.25) is simulated, so the phase diagram's generality at weaker coupling is open. These are addressable caveats, not a hole below the waterline.

Bottom line: the nf≈nc geometry and the inhomogeneity mechanism are solid enough to be taken seriously and deserve a referee. The paper will interest anyone working on Ce-based heavy fermions or disordered Kondo systems. In revision, I would ask for a tempering of the non-FL language or a direct finite-size scaling test, but I would not reject it.

Referee report

Significance

If the non-Fermi-liquid claim is sustained, the paper provides a microscopic and largely parameter-free mechanism for the destruction of Fermi-liquid coherence in depleted Kondo lattices, distinct from earlier Kondo-disorder models: the inhomogeneity emerges from the collective screening of randomly positioned moments rather than from bare disorder in the Kondo couplings. The main strengths are the self-contained derivation of the crossover scale T* from the self-consistency equations, the absence of experimental fitting, and the falsifiable spatial predictions (island LDOS, bimodal b_r distribution, broad χ_loc distribution) that could be probed by STM and thermodynamic measurements. The significance is conditional, however, on whether the non-saturating susceptibility survives the thermodynamic and T→0 limits, since all numerical evidence is obtained within the static large-N saddle-point approximation at a single value of the Kondo coupling.

Major objections (2)

[Numerical results, Fig. 5(a)] The central non-Fermi-liquid claim rests on the non-saturation of χ_ff(T) at nf≈nc=0.4, but the manuscript explicitly states in the same paragraph that "we cannot rule out saturation of χ_ff at extremely low temperatures" and that finite-size effects prevent lower-temperature calculations. No system-size dependence is shown, so the non-saturation could be a finite-size artifact: in a finite 20×20 system the spectrum is discrete, and the small local Kondo scales in the bimodal b_r distribution can push the saturation temperature below the numerically accessible window even if the thermodynamic limit is an ordinary Fermi liquid. To support the title and abstract claims, the authors should either provide χ_ff(T) for several lattice sizes (e.g., 8×8, 12×12, 16×16, 20×20) demonstrating that the non-saturation persists with increasing L, or soften the claim to "strongly suppressed Fermi-liquid coherence" in the accessible temperature range.
[Strong-coupling analysis and Numerical results] The nf=nc dividing line is derived in the JK≫t limit, and the authors themselves note that the gap at nf=nc is "not expected to survive for JK<~t." All numerical results are for a single value JK/D=1.25, so the phase diagram in Fig. 1, drawn as a generic feature of the depleted Kondo lattice, is not established outside the strong-coupling regime. The authors should show results at at least one additional value of JK (either smaller or larger) or explicitly restrict the phase diagram to the strong-coupling/large-JK regime; as written, the generality of the central claim is not supported.

Minor objections (3)

[Abstract and Conclusions] The title and abstract state "non-Fermi liquid behavior" as an established property, whereas the body repeatedly qualifies the evidence (e.g., "FL behavior is strongly suppressed" and "we cannot rule out saturation"); please align the language with the stated caveat, for example by using "non-Fermi-liquid-like down to the lowest accessible temperatures."
[Fig. 2(a)] The inset axis label "b_r^2/<b_r^2>" is not defined in the caption; please define the disorder average and specify the number of disorder realizations used.
[Inter-moment correlations] The section states that "preliminary results (not shown)" support the phase diagram in Fig. 1b, yet Fig. 1b is used to make contact with Ce_xLa_1-xCoIn_5; without displaying those data, the experimental comparison is difficult to evaluate.

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The paper

The paper claims

Randomly depleting magnetic moments in a Kondo lattice does not smoothly connect the dense heavy-fermion liquid to the dilute single-impurity liquid; the paper argues the crossover is organized by the concentration equality $n_f = n_c$…

The reading

The paper asks what happens to a heavy-fermion metal when magnetic $f$-moments are randomly removed, leaving nonmagnetic sites behind. Its central claim is that the two known Fermi-liquid limits—the dense coherent heavy Fermi liquid and the dilute collection of independent Kondo impurities—are separated by a sharp concentration line $n_f = n_c$, where $n_f$ is the fraction of occupied moment sites and $n_c$ is the conduction-electron filling per site. Near this line the low-temperature state is strongly inhomogeneous and non-Fermi-liquid: the local density of states forms islands, local Kondo scales are broadly distributed, and a small fraction of weakly screened moments keeps the bulk susceptibility from saturating. This matters because real materials such as Ce$_x$La$_{1-x}$CoIn$_5$ show exactly this loss of Fermi-liquid behavior away from both limits, and the paper provides a concrete microscopic scenario for it.

What carries the argument

The load-bearing object is the SU($N$) slave-boson mean-field Hamiltonian $H_{\mathrm{mf}} = \sum_k \epsilon_k c^\dagger_{k\alpha} c_{k\alpha} + \sum_{\{r\}} \mu_r f^\dagger_{r\alpha} f_{r\alpha} - \sum_{\{r\}} (b_r c^\dagger_{r\alpha} f_{r\alpha} + \mathrm{h.c.})$, supplemented by the self-consistency conditions $1 = \langle f^\dagger_{r\alpha} f_{r\alpha}\rangle$ and $2b_r = J_K \langle c^\dagger_{r\alpha} f_{r\alpha}\rangle$. The local fields $b_r$ measure the Kondo screening amplitude at each moment site, and their disorder average defines the scale $T^*$ that plays the role of a coherence temperature. The strong-coupling argument does the conceptual work: at $n_f = n_c$ all conduction electrons are consumed by local singlets, so the ground state is gapped and cannot be a Fermi liquid; the numerics then show how this special concentration organizes the crossover.

Extended reading notes

Core claim

On its own terms, the paper establishes that the evolution from dense Kondo lattice to dilute impurities is not gradual. Strong-coupling counting shows that at $n_f = n_c$ every conduction electron is bound into a local singlet, leaving a gap of order $J_K$ rather than a Fermi surface; the paper takes this as the dividing line between the two Fermi-liquid regimes. Numerical solution of the SU($N$) slave-boson saddle-point equations on up to $20\times20$ lattices at $J_K/D = 1.25$ confirms the picture: the mean-field coherence scale $T^* = \langle b_r^2\rangle/D$ stays roughly constant up to $n_f \approx n_c$ and then drops sharply, the distribution of the local Kondo amplitudes $b_r$ becomes bimodal, and the low-energy local density of states develops island-like structures. In this inhomogeneous regime the $f$-spin susceptibility $\chi_{ff}(T)$ does not saturate down to the lowest temperatures studied, because a subset of moments remains weakly screened; the authors attribute this to the collective screening of randomly positioned moments rather than to disorder in the bare couplings.

Load-bearing premise

The argument depends on the large-N slave-boson saddle point, which neglects temporal fluctuations of the screening fields, faithfully describing the low-temperature physics of the physical spin-1/2 Kondo lattice, and on the $n_f = n_c$ dividing line surviving away from the strong-coupling limit where the gap argument explicitly fails.

Editorial extensions

If this is right

At $n_f \approx n_c$, the uniform $f$-spin susceptibility does not saturate to a constant down to the lowest accessible temperatures, a hallmark of suppressed Fermi-liquid behavior.
Local probes of partially depleted Kondo lattices should see a bimodal distribution of local Kondo scales, with strongly screened islands coexisting with nearly free moments.
In the nearly dense limit the low-energy density of states is suppressed at vacancy sites, while in the dilute limit it is suppressed at moment sites, with the crossover between the two pictures occurring near $n_f \approx n_c$.
Strong inter-moment exchange enhances the coherence temperature, shifts the inhomogeneous region toward smaller $n_f$, and can quench the rare unquenched moments at larger $n_f$ while leaving islands and broad local-susceptibility distributions intact.
The picture gives a microscopic rationale for why Ce$_x$La$_{1-x}$CoIn$_5$ loses Fermi-liquid behavior away from both limits and why Ce$_x$La$_{1-x}$Pb$_3$ looks single-impurity-like up to relatively large $x$.

Reading between the lines

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

My inference: the $n_f = n_c$ separatrix suggests that conduction-electron filling, not just moment concentration, is a control parameter; materials in which $n_c$ can be shifted by pressure or chemical substitution should cross between the two Fermi-liquid regimes at fixed moment depletion.
My inference: island formation in the low-energy local density of states implies a concrete scanning-tunneling signature—patchy, strongly varying tunneling conductance at the Fermi level, with bright regions tracking depleted-moment clusters—that could be sought on cleaved heavy-fermion surfaces.
My inference: the same rare weakly screened moments responsible for the non-saturating susceptibility should also produce a non-saturating NMR spin-lattice relaxation rate divided by temperature and enhanced low-frequency magnetic noise in the inhomogeneous regime; the paper does not compute these transport or dynamic responses.

Original abstract

We investigate the low-temperature behavior of Kondo lattices upon random depletion of the local $f$-moments, by using strong-coupling arguments and solving SU($N$) saddle-point equations on large lattices. For a large range of intermediate doping levels, between the coherent Fermi liquid of the dense lattice and the single-impurity Fermi liquid of the dilute limit, we find strongly inhomogeneous states that exhibit distinct non-Fermi liquid characteristics. In particular, the interplay of dopant disorder and strong interactions leads to rare weakly screened moments which dominate the bulk susceptibility. Our results are relevant to compounds like Ce_{x}La_{1-x}CoIn_5 and Ce_{x}La_{1-x}Pb_3

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, 3,719 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.18 · randomized assignment 0.02 · states quantitative result 0.06 · compares with prior results 0.31 · 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 low-temperature behavior of Kondo lattices upon random depletion of the local $f$-moments, by using strong-coupling arguments and solving SU($N$) saddle-point equations on large lattices. For a large range of intermediate doping levels, between the coherent Fermi liquid of the dense lattice and the single-impurity Fermi liquid of the dilute limit, we find strongly inhomogeneous states that exhibit distinct non-Fermi liquid characteristics. In particular, the interplay of dopant disorder and strong interactions leads to rare weakly screened moments which dominate the bulk susceptibility. Our results are relevant to compounds like Ce_{x}La_{1-x}CoIn_5 and Ce_{x}La_{1-x}Pb_3

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 low-temperature behavior of Kondo lattices upon random depletion of the local $f$-moments, by using strong-coupling arguments and solving SU($N$) saddle-point equations on large lattices.	contribution 0.63	0.95	0.05	asserted 0.79
s1	For a large range of intermediate doping levels, between the coherent Fermi liquid of the dense lattice and the single-impurity Fermi liquid of the dilute limit, we find strongly inhomogeneous states that exhibit distinct non-Fermi liquid characteristics.	result 1.00	0.96	0.80	asserted 1.00
s2	In particular, the interplay of dopant disorder and strong interactions leads to rare weakly screened moments which dominate the bulk susceptibility.	result 0.86	0.82	0.26	asserted 1.00
s3	Our results are relevant to compounds like Ce_{x}La_{1-x}CoIn_5 and Ce_{x}La_{1-x}Pb_3	interpretation 0.95	0.22	0.05	asserted 0.73

What kind of paper the classifier reports this is: theoretical result 0.79 · evidence from simulation or computation 0.67.

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: Schematic phase diagram of the depleted Kondo

**Figure 2.** FIG. 2: (color online) Summary of the mean-field results:

**Figure 3.** FIG. 3: (color online) Spatial distribution of the slave bos

**Figure 4.** FIG. 4: (color online) Spatial distribution of the LDOS,

**Figure 5.** FIG. 5: (color online) Temperature dependence of

Figures reproduced from arXiv:cond-mat/0603002 by the authors.

References

Works cited (15)

[1]

A. C. Hewson, The Kondo Problem to Heavy Fermions , Cambridge University Press, Cambridge (1997)

View on Pith 1997
[2]

C. L. Lin et al. , Phys. Rev. Lett. 58, 1232 (1987)

View on Pith 1987
[3]

S. Nakatsuji et al. , Phys. Rev. Lett. 89, 106402 (2002)

View on Pith 2002
[4]

S. Nakatsuji et al. , Phys. Rev. Lett. 92, 016401 (2004)

View on Pith 2004
[5]

V. Barzykin, cond-mat/0601682 (2006)

View on Pith Pith review arXiv 2006
[6]

Z. Li and Y. Qiu, Phys. Rev. B 43, 12906 (1991); T. Mutou, Phys. Rev. B 64, 245102 (2001)

View on Pith 1991
[7]

Consistent with the experimental data of Refs. 2, 3, we assume upon random f -spin depletion the paramagnetic HFL does not enter a magnetic state

View on Pith
[8]

RKKY interaction possibly leads to spin glass behavior for small nf , with a low ordering temperature ∝ nf

View on Pith
[9]

V. Dobrosavljevic et al. , Phys. Rev. Lett. 69, 1113 (1992); O. O. Bernal et al. , ibid. 75, 2023 (1995); E. Miranda et al. , ibid. 78, 290 (1997); A. H. Castro Neto et al. , Phys. Rev. B, 62, 14975 (2000)

View on Pith 1992
[10]

The ﬁnite- T phase transition is an artifact of mean-ﬁeld theory; however, the low- T physics is known to be cap- tured accurately by this approach

View on Pith
[11]

S. Burdin et al. , Phys. Rev. Lett. 85, 1048 (2000)

View on Pith 2000
[12]

T. Pruschke et al. , Phys. Rev. B 61, 12799 (2000)

View on Pith 2000
[13]

We expect the physics discussed here to be generic to both two- and three-dimensional systems

View on Pith
[14]

R. K. Kaul and M. Vojta, to be published

View on Pith
[15]

T. Senthil et al. , Phys. Rev. B 69, 035111 (2004)

View on Pith 2004

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]

A. C. Hewson, The Kondo Problem to Heavy Fermions , Cambridge University Press, Cambridge (1997)

View on Pith 1997

[2] [2]

C. L. Lin et al. , Phys. Rev. Lett. 58, 1232 (1987)

View on Pith 1987

[3] [3]

S. Nakatsuji et al. , Phys. Rev. Lett. 89, 106402 (2002)

View on Pith 2002

[4] [4]

S. Nakatsuji et al. , Phys. Rev. Lett. 92, 016401 (2004)

View on Pith 2004

[5] [5]

V. Barzykin, cond-mat/0601682 (2006)

View on Pith Pith review arXiv 2006

[6] [6]

Z. Li and Y. Qiu, Phys. Rev. B 43, 12906 (1991); T. Mutou, Phys. Rev. B 64, 245102 (2001)

View on Pith 1991

[7] [7]

Consistent with the experimental data of Refs. 2, 3, we assume upon random f -spin depletion the paramagnetic HFL does not enter a magnetic state

View on Pith

[8] [8]

RKKY interaction possibly leads to spin glass behavior for small nf , with a low ordering temperature ∝ nf

View on Pith

[9] [9]

V. Dobrosavljevic et al. , Phys. Rev. Lett. 69, 1113 (1992); O. O. Bernal et al. , ibid. 75, 2023 (1995); E. Miranda et al. , ibid. 78, 290 (1997); A. H. Castro Neto et al. , Phys. Rev. B, 62, 14975 (2000)

View on Pith 1992

[10] [10]

The ﬁnite- T phase transition is an artifact of mean-ﬁeld theory; however, the low- T physics is known to be cap- tured accurately by this approach

View on Pith

[11] [11]

S. Burdin et al. , Phys. Rev. Lett. 85, 1048 (2000)

View on Pith 2000

[12] [12]

T. Pruschke et al. , Phys. Rev. B 61, 12799 (2000)

View on Pith 2000

[13] [13]

We expect the physics discussed here to be generic to both two- and three-dimensional systems

View on Pith

[14] [14]

R. K. Kaul and M. Vojta, to be published

View on Pith

[15] [15]

T. Senthil et al. , Phys. Rev. B 69, 035111 (2004)

View on Pith 2004