Probing Non-Fermi-Liquid Behaviour of Composite Fermi Liquid via Efficient Thermal Simulations

Bin-Bin Chen; Hongyu Lu; Zi Yang Meng

REVIEW 3 major objections 5 minor 2 cited by

Probing Non-Fermi-Liquid Behaviour of Composite Fermi Liquid via Efficient Thermal Simulations

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper reports thermal tensor network simulations of the half-filled lowest Landau level showing that the composite Fermi liquid's specific heat scales as T^(2/3), providing thermodynamic evidence of non-Fermi-liquid behavior driven by

desk verdict A genuine methodological first for LLL thermodynamics, but the headline T^(2/3) exponent needs more evidence than a single cylinder width and an undocumented fit. read the letter →

arxiv 2509.02218 v3 pith:F7WW5V7E submitted 2025-09-02 cond-mat.str-el

Bin-Bin Chen , Hongyu Lu , Zi Yang Meng This is my paper

classification cond-mat.str-el

keywords compositeFermiliquidnon-FermispecificheatthermaltensornetworklowestLandaulevelemergentgaugefieldquantumHalleffecttanTRG

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks what defines the thermodynamics of the composite Fermi liquid (CFL), the metallic state of electrons at half filling of the lowest Landau level. Using thermal tensor network simulations of the LLL-projected Hamiltonian, it reports that the low-temperature specific heat follows c_V ~ T^α with α close to 2/3, clearly below the linear-T law of a conventional Fermi liquid. The paper interprets this as evidence that composite fermions are strongly coupled to an emergent gauge field, the non-Fermi-liquid mechanism central to Halperin-Lee-Read and QED3 descriptions. It also shows that the guiding-center density correlation develops a 2k_F circle at the same temperature scale, marking the formation of the composite Fermi surface. If the extracted exponent survives the thermodynamic limit, CFL would be a numerically controlled example of a two-dimensional non-Fermi liquid with a clear thermodynamic fingerprint.

What carries the argument

The load-bearing machinery is the tangent-space tensor renormalization group (tanTRG) applied to a matrix product operator representation of the thermal density operator, with the Hamiltonian projected onto lowest Landau level orbitals on a cylinder. The central observable is the specific heat c_V, extracted from the internal energy, together with the guiding-center density correlation D̄(q)=exp(q²/2)⟨n_q n_{-q}⟩_T, whose 2k_F ring and low-q q^3 behavior characterize the composite Fermi surface. To reach low temperatures with feasible bond dimensions, the evolution uses controlled bond expansion (CBE).

What would settle it

Calculate the specific heat with the same method on cylinders with larger circumference (Ly = 16 or 20) and longer screening length (λ = 20 or more); if the fitted exponent in the window above the incompressible region drifts away from 2/3 toward 1 as Ly grows, the claimed non-Fermi-liquid scaling is a finite-size effect.

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Extended reading notes

Core claim

The central claim is that, at half filling of the lowest Landau level, the thermodynamic response of the composite Fermi liquid is not that of a Fermi liquid. Using the tangent-space tensor renormalization group on a cylinder with circumference Ly=12 and a screened Coulomb interaction of screening length λ=10, the paper observes that at temperatures above the finite-size gap (T ≳ 10^-2) the specific heat follows a power law with an exponent that converges to 2/3 for system sizes N=36 and 48. This matches the value expected when composite fermions are coupled to a dynamical emergent gauge field. The same simulations show the guiding-center density-density correlation D̄(q) developing a ring o

Load-bearing premise

The paper assumes the temperature window just above a small-system energy gap is wide enough, and the screened interaction is close enough to the true Coulomb interaction, that the fitted exponent 2/3 is what a large system would show.

Editorial extensions

If this is right

The ν=1/2 state is a metal with gapless excitations, but its thermodynamic response is not that of a Fermi liquid; c_V ~ T^(2/3) is a distinguishing signature.
The result supports the Halperin-Lee-Read picture of composite fermions coupled to an emergent gauge field over a conventional Fermi-liquid description.
The composite Fermi surface forms below T ~ 10^-1, and the same temperature scale governs the onset of the non-Fermi-liquid specific heat, tying Fermi-surface formation to the anomalous thermodynamics.
Small-momentum density correlations behave as q^3 at low temperature, showing the composite fermions form a Fermi sea; the anomalous specific heat is attributed to their coupling to the gauge field.
The thermal tensor network pipeline (tanTRG with CBE in the LLL basis) can be carried over to fractional Chern insulators and moiré systems at half filling.

Reading between the lines

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

A direct check would be to vary the screening length λ toward the pure Coulomb limit and the circumference Ly beyond 12; if α moves away from 2/3, the exponent is a finite-size or screening artifact rather than universal thermodynamics.
If the T^(2/3) scaling is genuine, the same thermal tensor network approach should predict a matching exponent for half-filled Chern bands in moiré materials, where the composite Fermi liquid has recently been observed.
The combination of c_V ~ T^(2/3) and D(q) ~ q^3 suggests a way to separate the gauge-field contribution from the fermionic part, and could be used to extract an effective gauge-field coupling strength from finite-size data.
Since specific heat of a 2DEG is hard to measure directly, the more accessible prediction is the temperature dependence of the compressibility or thermal conductivity, both of which should show related power-law corrections in the same window.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this paper is worth reading and deserves referee time, but the central exponent claim is not as solid as the title implies. The genuinely new piece is the first thermal tensor-network simulation of the LLL-projected half-filled Landau level, benchmarked against ED at N=16 with errors that drop as the bond dimension grows, and pushed to N=48. That is a real methodological advance. The observation that the specific heat looks like T^(2/3) rather than T is exactly the HLR pointer you'd want to see, and the 2kF ring in the guiding-center correlator at low T is a nice consistency check.

The soft spots are real. The cylinder circumference is fixed at Ly=12 for all system sizes; increasing N only makes the cylinder longer. At T~0.01-0.1, the transverse momentum spacing 2π/Ly≈0.52 is larger than the thermal momentum scale, so the low-T window sits in a quasi-1D regime. The observed 2/3 might be a crossover number, not the 2D thermodynamic limit. To believe it, I'd need at least one larger Ly and a demonstration that the fitted exponent doesn't move.

Second, the power-law fit itself is under-documented. The paper doesn't give the fit interval, error bars on α, or the bond dimensions used for the N=48 runs. Given the claim is quantitative, that's a fixable but necessary omission.

Third, the interaction is a screened Yukawa with λ=10, not the pure Coulomb limit. The authors show λ=2 data are worse, but they don't show that λ→∞ leaves α unchanged. The screening dependence is a load-bearing parameter, and right now it's only tested at two values.

The compressibility check is a good idea, and I believe their finite-size gap assignment. The q^3 small-q behavior reported in the SM is also a nice consistency check that the CFs themselves look Fermi-liquid-like.

So: the methodology is a genuine step forward, and the result is plausible and important if true. But the quantitative case needs more evidence. I'd send it to review, with the expectation that the referees require Ly scaling and honest fit errors.

Referee Report

3 major / 5 minor

Summary. The paper applies tangent-space tensor renormalization group (tanTRG) simulations to the lowest-Landau-level projected Coulomb-Yukawa Hamiltonian at filling ν=1/2 on a cylinder with circumference Ly=12 and up to N=48 orbitals. After benchmarking against exact diagonalization for N=16, the authors compute the finite-temperature specific heat and report a low-temperature power law c_V ∼ T^α with α close to 2/3, in contrast to the linear-T behavior of a conventional Fermi liquid. They also compute the guiding-center density correlations, which show a 2k_F ring at low temperatures and a low-q q^3 behavior in the small-q limit. The central claim is that this thermodynamic response provides direct evidence for a composite Fermi liquid governed by coupling to an emergent gauge field, as in the Halperin-Lee-Read theory.

Significance. If the result is robust, this is a significant step: it would be one of the first controlled finite-temperature numerical demonstrations of non-Fermi-liquid thermodynamics in the half-filled Landau level, with a concrete quantitative prediction (α≈2/3) that can be compared with analytic gauge-field theories. The paper's strengths include a careful ED benchmark with errors that decrease with bond dimension, the use of a state-of-the-art thermal tensor-network method, and a complementary analysis of the guiding-center correlations that supports the formation of a composite Fermi surface. The central quantitative claim, however, rests on a single cylinder circumference and a power-law fit whose window and error bars are not reported; this currently limits the conclusiveness of the T^{2/3} identification.

major comments (3)

[§4, Fig. 3(a), Fig. S1] The central T^{2/3} claim is extracted from a single cylinder circumference Ly=12. The transverse momentum spacing is Δk_y=2π/Ly ≈ 0.52 (in ℓ=1 units), while the fitting window T ∈ [0.01,0.1] corresponds to thermal momenta q_y ∼ T/v_F ≲ 0.1, well below Δk_y. Hence only the q_y=0 gauge-field mode contributes at the low-temperature end, and the system is in a quasi-1D regime; the gauge-field contribution to the specific heat in such a quasi-1D setup is expected to have a different temperature exponent from the 2D T^{2/3} HLR result. The compressibility in Fig. S1 indicates the finite-size gap at T≈0.01, so the fitted window sits immediately above this crossover. Without a scan in Ly (e.g., Ly=16 or 20) or an explicit finite-size scaling analysis, the observed 2/3 cannot be distinguished from a quasi-1D crossover artifact. This is load-bearing because it is the only quantitative support for
[§4, Fig. 3(b)] The paper does not report the exact fitting interval, the fitted exponents with statistical errors, or the bond dimension and truncation errors for the N=36 and N=48 production runs. The statement that the power 'converges' to 2/3 is therefore not quantitatively substantiated. In addition, Fig. 3(b) shows that the λ=2 data exhibit severe finite-size effects and only 'tend to approach' 2/3; since λ=10 is the only screening length used for the main result, the robustness of the exponent to the screened-Coulomb model choice is not demonstrated. Please provide a table of αFit with confidence intervals, the fit range, and convergence checks in bond dimension D and circumference Ly.
[SM Section II / Fig. 3(a)] The specific heat is plotted per orbital c_V(T) for N=24, 36, 48 at fixed Ly=12. On a finite cylinder the thermodynamic limit is anisotropic: Lx ∝ N at fixed Ly, and the order of limits (N→∞ at fixed Ly, then Ly→∞, or simultaneous) matters for the exponent of a quasi-1D system. The paper should state the normalization explicitly and justify that c_V per orbital at fixed Ly is the appropriate quantity for extracting a 2D thermodynamic exponent; otherwise the finite-Ly scaling may mix one-dimensional and two-dimensional contributions.

minor comments (5)

[Throughout] Typos and grammar: 'closed to' should be 'close to', 'deviated' should be 'deviates', 'pleateau' should be 'plateau', 'consindered' should be 'considered', 'renomalized' should be 'renormalized'.
[Fig. 2 caption] The inset description says 'the same as in (a)' for panel (c), but the data refer to panel (b); please correct.
[Benchmark and production runs] The bond dimensions, Trotter step, and truncation parameters for the production N=24, 36, 48 runs are not reported. Please include these in the main text or SM.
[Model definition] The screening length is introduced as λ=10 in the main text, while the benchmark uses λ=10×2πℓ²/L_y. Please clarify the units and the relation between these choices.
[SM Section II] The q^3 fit in Fig. S2 reports α=3.01(6), which is good; the same reporting standard (with error bars and fit ranges) should be applied to the specific-heat exponent in Fig. 3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the T^{2/3} specific-heat exponent is a simulated output compared to an external HLR prediction, not an input.

full rationale

The paper's central claim—that the specific heat of the half-filled lowest Landau level scales as T^{2/3}—is obtained from thermal tensor-network simulations of the projected Coulomb-Yukawa Hamiltonian (Eqs. 1-2). The exponent is extracted by power-law fitting of the simulated specific heat (Fig. 3) and then compared with the external Halperin-Lee-Read prediction of 2/3 (Ref. [42]). The value 2/3 is not an input to the Hamiltonian, the algorithm, or the fitting procedure; it is an externally predicted value used only as a benchmark. The numerical method (tanTRG, Ref. [82]) is self-cited by one of the authors, but it is independently benchmarked in Fig. 2 against exact diagonalization for N=16 with controlled bond-dimension convergence, so the method citation is not load-bearing in a circular way. The interpretation that the 2/3 exponent indicates coupling to an emergent gauge field is post-hoc and drawn from external theory, not a definitional consequence of the simulation. The finite-size/temperature-window concern (Ly=12, gap at T~10^{-2}) is a potential systematic error affecting the fitted exponent, but a correctness risk is not circularity. No equation or fitted parameter is identified that reduces to the claimed result by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a standard LLL projection, a screened model interaction with two hand-chosen lengths, and a fitting window chosen after excluding low-temperature data. No new physical entities are introduced; composite fermions and the emergent gauge field are taken from prior literature.

free parameters (3)

Screening length lambda = 10 units of 2*pi*ell^2/Ly, with lambda=2 for comparison
Introduced to make the interaction tractable in the MPO framework; the physical Coulomb limit is lambda to infinity. The paper argues the extracted exponent is insensitive but presents no systematic extrapolation in lambda.
Fitting window for the specific heat power law = not specified numerically
Data below T ~ 10^-2 are excluded based on vanishing compressibility; the upper bound of the window is not given. The extracted exponent depends on this hand-chosen interval.
Specific heat exponent alpha = approximately 2/3 for N=36 and N=48
The central output, obtained from a linear fit of log c_V versus log T. No error bars are reported in the main text.

assumptions (4)

domain assumption Single-particle wavefunctions are restricted to the lowest Landau level (LLL projection).
Standard at high magnetic field; the paper uses the LLL basis throughout, see Eq. (2).
domain assumption The neutralizing background, the -1/2 in Eq. (1), ensures half filling in the grand-canonical ensemble.
This is the standard jellium description of a two-dimensional electron gas.
domain assumption Tangent-space tensor renormalization with fixed bond dimension D converges to the exact thermal density matrix.
Benchmarked against exact diagonalization for N=16, but for N=48 the bond dimensions are not systematically extrapolated; the claimed precision relies on this assumption.
domain assumption The HLR prediction C proportional to T^(2/3) is the correct reference theory for the exponent.
The paper uses this as the reference for its comparison, citing Ref. [42].

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Pith. "Pith review of Probing Non-Fermi-Liquid Behaviour of Composite Fermi Liquid via Efficient Thermal Simulations." pith.science (2026). https://pith.science/paper/F7WW5V7E

@misc{pith2026250902218,
  author       = {Pith},
  title        = {Pith review of: Probing Non-Fermi-Liquid Behaviour of Composite Fermi Liquid via Efficient Thermal Simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F7WW5V7E}},
  note         = {Machine review of arXiv:2509.02218}
}

abstract

The physics of two-dimensional electron gas in a perpendicular magnetic field, i.e., the quantum Hall system, is remarkably rich. At half filling of the lowest Landau level, it has been predicted that "composite fermions"---emergent quasiparticles consisting of an electron attached to two magnetic flux quanta---experience zero net magnetic field and form a Fermi sea, dubbed composite Fermi liquid (CFL). However, despite its seemingly simple appearance, CFL is a strongly correlated quantum many-body state in disguise, and solving it in a controlled manner is extremely difficult, to the extent that the thermodynamic properties of CFL remain largely unknown. In this work, we perform state-of-the-art thermal tensor network simulations of the $\nu=1/2$ Landau level system and observe low-temperature power-law behaviour of the specific heat, signaling the gapless nature of CFL. More importantly, the power is extracted to be close to $2/3$, clearly deviating from the ordinary linear-$T$ behaviour of Fermi liquid, suggesting coupling between the CFs and the dynamical emergent gauge field and thereby revealing the quantum many-body nature of the CFL state.

Figures

Figures reproduced from arXiv: 2509.02218 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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