REVIEW 4 major objections 5 minor 85 references
The two-particle density matrix of a Luttinger liquid
T0 review · 4 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash
Pith's one-line read The two-particle density matrix of a Luttinger liquid has a closed finite-size form governed by a second exponent, lambda=(K^{-1}-K)/2, which controls off-diagonal correlations and emerges only beyond the one-particle level.
desk verdict A genuinely useful closed-form 2-RDM for a Luttinger liquid with a plausible new chiral exponent, but the exponential cutoff ansatz and thin DMRG validation leave the quantitative claim conditional. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central mechanism is constructive bosonization with an explicit ultraviolet cutoff. The fermion field is decomposed into right and left movers, and the two-particle correlator is reduced to exponentials of boson commutators and Gaussian expectations; the load-bearing objects are the chord functions d_epsilon(x) and the two interaction exponents defined through the regularized momentum sums, cosh^2 theta_q + sinh^2 theta_q - 1 ≈ gamma^2 e^{-epsilon|q|} and -2sinh theta_q cosh theta_q ≈ lambda e^{-epsilon|q|}. The resulting expression Eq. (34) combines six chiral contributions into three real terms, with gamma^2 in all three and lambda only in the two anomalous terms, and the diagonal limi
What would settle it
Perform exact diagonalization or DMRG on a finite chain with a next-nearest-neighbor or long-range interaction, fit epsilon from the 1-RDM, and compare Eq. (34) to the numerically obtained 2-RDM elements: a systematic deviation beyond the short-distance scale would show that the single-exponential regulator ansatz is insufficient. Alternatively, measure the 2k_F decay of g2(r) in a cold-atom realization: if the exponent differs from 2K, the lambda structure is not the full story.
Extended reading notes
Core claim
For spinless fermions on a ring with periodic boundary conditions, the paper establishes Eq. (34): a closed finite-size expression for the equal-time two-particle reduced density matrix, built from chord functions d_epsilon(x)=(L/pi)sin(pi(x+i epsilon)/L). The normal (same-chirality) boson correlators contribute the exponent gamma^2=(K+K^{-1}-2)/2, while the anomalous (opposite-chirality) correlators contribute a second exponent lambda=(K^{-1}-K)/2, which is absent from the 1-RDM and can be negative for attractive interactions. Because the expression retains an explicit ultraviolet cutoff epsilon, it can be matched to a microscopic lattice model: with epsilon fixed from the 1-RDM, the analyt
Load-bearing premise
The derivation assumes that a single exponential ultraviolet cutoff epsilon, fitted to the one-particle density matrix, also reproduces the full momentum dependence of the normal and anomalous two-particle correlators; if the true q-dependence of the couplings differs from e^{-epsilon|q|}, the quantitative agreement would fail even though the universal exponents remain correct.
Editorial extensions
If this is right
- Two-particle observables of a Luttinger liquid can now be computed on a ring at finite size without fitting amplitudes: the only nonuniversal parameter is the cutoff epsilon.
- The static structure factor follows from the diagonal of the 2-RDM and reproduces the linear small-momentum behavior s(q) ~ K|q|/2k_F, connecting universal scaling to measurable scattering response.
- Off-diagonal coherences carry information about the orbital symmetry of ordered correlations: an even, node-free particle-hole eigenvector signals an s-wave CDW, while an antisymmetric particle-particle eigenvector signals p-wave pairing.
- Because the expression is finite-size and cutoff-regularized, it can be used directly to compute lattice observables such as the ground-state energy of the J-V chain.
- The new exponent lambda enters only for n>=2 density matrices, so the 2-RDM contains information about interactions that is not present in the 1-RDM, and the method is stated to generalize to higher n.
Reading between the lines
- The subleading 1/N contributions to particle-partition Rényi entropies for n>1, which the paper connects to the 2-RDM structure, could be explicitly traced to lambda; computing the n-RDM cumulant would make this quantitative.
- The same constructive bosonization should yield closed n-particle density matrices, and the coefficients of lambda in those expressions could be checked against exact diagonalization of small chains.
- A time-dependent version of Eq. (34) after an interaction quench is a natural extension; the two-particle coherence growth predicted by it could be compared with the existing quench results for the 1-RDM.
- The assumption of a single exponential regulator epsilon could be tested by applying the formula to models with long-range interactions, where the momentum dependence of the couplings deviates from e^{-epsilon|q|}; any quantitative failure would pinpoint where the regulator ansatz breaks.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper derives a closed, finite-size expression for the equal-time two-particle reduced density matrix of a spinless Tomonaga-Luttinger liquid on a ring, using constructive bosonization with an explicit ultraviolet cutoff. The central result, Eq. (34), combines three chiral terms controlled by the familiar exponent γ²=(K+K^{-1}-2)/2 and a second exponent λ=(K^{-1}-K)/2 that arises from anomalous right-left correlators. The diagonal limit gives the density-density correlation function and static structure factor; off-diagonal elements are analyzed for CDW and p-wave pairing signatures. After fixing one interaction cutoff ϵ from the 1-RDM, the authors compare Eq. (34) with DMRG data for the J-V chain and report quantitative agreement. The appendices contain detailed bosonization algebra, limits, and a data/code availability statement.
Significance. If the quantitative match holds, this is a valuable analytic benchmark for two-particle correlations in interacting one-dimensional systems, going beyond the well-known 1-RDM. The derivation is substantial, reproduces the free-fermion limit and diagonal limits, and identifies λ as a genuinely new structural exponent. The paper also provides reproducible code and data, which strengthens the contribution. However, the central quantitative claim rests on an unproven exponential regulator ansatz and on the transfer of a single fitted parameter from the normal to the anomalous channel; the numerical support is narrow. These issues are addressable but currently leave the precision of Eq. (34) less certain than the universal power laws it contains.
major comments (4)
- [Section III, Eqs. (25) and (28)] The exponential regulator ansatz cosh²θ_q+sinh²θ_q−1 ≈ γ² e^{-ϵ|q|} and −2 sinhθ_q coshθ_q ≈ λ e^{-ϵ|q|} is introduced without derivation from the microscopic J-V interaction. The 1-RDM, Eq. (33), depends only on γ² and ϵ; therefore fitting ϵ to the 1-RDM constrains only the normal channel. Equation (34) then applies the same ϵ to the anomalous channel that controls λ. This transfer is load-bearing for the quantitative claim. Please provide evidence that the single-ϵ ansatz is valid, for example by testing λ-sensitive cuts of the 2-RDM in the attractive regime (K>1, where λ<0 and the sign of the anomalous contribution changes), or by extracting ϵ independently from the 1-RDM and from a λ-sensitive DMRG observable and showing consistency.
- [Section V.B, Figs. 11-12] The full 2-RDM benchmark is presented for a single interaction value, V/J=-0.5, and a single system size, L=50. The text claims agreement 'throughout the Luttinger liquid regime', but no repulsive (K<1) 2-RDM comparison is shown, and no error bars or quantitative discrepancy measure are reported. Please extend the benchmark to at least one repulsive case (and ideally one close to the K→1/2 CDW boundary), and report a concrete measure of the agreement, such as the maximum or mean absolute deviation between DMRG and Eq. (34).
- [Section V.C, Fig. 13] In the structure-factor calculation, the text states: 'We extract the interaction cutoff ϵ from a fit to g2.' Since g2 is the diagonal limit of the 2-RDM, this makes the agreement of s(q) a fit to the same observable, not an independent test of Eq. (34). This also conflicts with the abstract's claim that the cutoff is fixed from the one-particle density matrix. Please present the structure factor using the 1-RDM-fitted ϵ of Fig. 11, and assess the sensitivity of the result to the advertised trace correction of g2.
- [Section IV.D, Eq. (49)-(50)] The orbital-character analysis of the CDW coherence contains a gap. Equation (50) states that the residual δρ(ξ) equals the dominant eigenvector φ(ξ) of the particle-hole block, but the rank-one assumption is not demonstrated; the text asserts it is confirmed by a shift of the reference bond, but no data or figure is shown. Since this section makes a concrete physical claim (site-centered s-wave CDW), please provide the supporting comparison or mark the assertion as an observation from Eq. (34) rather than a proven property.
minor comments (5)
- [Section IV.B, Eq. (42)] The text refers to 'Eq. (45)' when quoting the free-fermion result; the cross-reference should point to the actual free-fermion expression, and the numbering around Eqs. (43)-(45) should be checked.
- [Section IV] The choice ϵ=1 for most analytical plots is presented without explanation. Since ϵ is later tied to microscopic parameters, a brief comment on the expected dependence of the plotted features on ϵ would improve interpretability.
- [Section IV.D, Eq. (50)] The asymptotic arrow notation in Eq. (50) is garbled ('− − − − − − →'). Use a standard symbol such as '→' with the stated limit.
- [Section VI] Typo: 'Tomonage-Luttinger' should be 'Tomonaga-Luttinger'. Also, the author affiliation line contains raw LaTeX ('F¨ ur') that should be rendered properly.
- [Fig. 13] The inset labels N=10 and N=25 with L=50 correspond to fillings 1/5 and 1/2; state this explicitly in the caption or main text to avoid ambiguity.
Circularity Check
Central 2-RDM derivation is self-contained; the main benchmark is a genuine out-of-sample test. Minor circularity appears only in the structure-factor application, where the cutoff is fit to g2 and s(q) is then recomputed from that same g2.
-
fitted input called prediction
[Section V.C.1 (Static Structure Factor), Eqs. (54)-(55), Fig. 13]
"We compute the structure factor for filling fractions 1/2 and 1/5 from the analytical expression, Eq. (34), by discretizing it on the L lattice sites using that g2(r)=ρ2(r,0;0,r)/n20 for r>0. We extract the interaction cutoff ϵ from a fit to g2."
In this application, s(q) is defined as the Fourier transform of the very same g2 used for the fit: s(q)=1+n0 Σ [g2(j)-1] e^{-iqj} (Eq. 55). Fitting the single cutoff ϵ to g2 at the same parameters and then presenting s(q) computed from that g2 as 'excellent agreement' with DMRG is a consistency check of the discretization and transform, not an independent prediction of the two-particle density matrix. This does not affect the central λ claim, and K is independently known from Bethe ansatz, so the circularity is partial and confined to validation.
full rationale
The core derivation is not circular. The exponents γ²=(K+K^{-1}-2)/2 and λ=(K^{-1}-K)/2 are obtained analytically from the Bogoliubov angle via K=lim e^{2θ_q} (Eqs. 25-29), with K itself taken from Bethe ansatz/XXZ results (Eq. 52), independent of the fitted cutoff. The 1-RDM and 2-RDM are not the same object: the 2-RDM contains λ-dependent anomalous-channel terms that are absent from the 1-RDM, so fixing ϵ from the 1-RDM and then matching the full 2-RDM (Figs. 11-12) is an out-of-sample test rather than a parameter renaming. The exponential regulator ansatz (Eqs. 25 and 28) is an assumption, but not a circular one: it is a stated approximation for the q-dependence of the couplings, and the universal power laws and free-fermion limit provide independent anchors. Self-citations to Ref. [37] for the 1-RDM form are not load-bearing because that expression is also attributed to independent earlier work [34-36]. The only genuine circular step is in the structure-factor subsection, where ϵ is fit to g2 and s(q), being a linear transform of the same g2, is then presented as agreement with DMRG; this is a minor, non-central validation issue. Overall score 2 reflects this limited circularity while the central claim retains independent content.
Assumptions & free parameters
free parameters (1)
- ϵ (interaction UV cutoff) =
0.81 for V/J=-0.5, L=50; also fitted to g2 for structure factor at other fillings
assumptions (4)
- domain assumption Low-energy linearization of the dispersion around ±k_F; only right- and left-moving chiral fields Ψ± are retained.
- ad hoc to paper Couplings ω0(q), m(q), g2(q) are taken as linear in |q| with momentum-independent coefficients, and the UV regularization is the exponential ansatz of Eqs. (25) and (28).
- standard math Bogoliubov transformation diagonalizes the quadratic bosonic Hamiltonian, and the ground state is the vacuum of the quasiparticle operators a_q.
- domain assumption For the J-V chain at half filling, K is obtained from the Bethe-ansatz/XXZ mapping K=π/(2 cos^{-1}(-V/2J)); at 1/5 filling, K is computed numerically from Ref. [31].
Cite this review
Pith. "Pith review of The two-particle density matrix of a Luttinger liquid." pith.science (2026). https://pith.science/paper/VD3MQ5HL
@misc{pith2026260714402,
author = {Pith},
title = {Pith review of: The two-particle density matrix of a Luttinger liquid},
year = {2026},
howpublished = {\url{https://pith.science/paper/VD3MQ5HL}},
note = {Machine review of arXiv:2607.14402}
}
abstract
Two-particle coherence is the first level of the reduced-density-matrix hierarchy that contains correlations inaccessible to single-particle observables, yet analytic two-particle density matrices are rare even in one dimension. We derive a closed, finite-size expression for the equal-time two-particle reduced density matrix of spinless fermions in a Tomonaga-Luttinger liquid using constructive bosonization with an explicit ultraviolet cutoff. In addition to the familiar Luttinger parameter $K$-dependent exponent $\gamma^2=(K+K^{-1}-2)/2$ which governs the spatial decay of matrix elements, the result exposes a second exponent, $\lambda=(K^{-1}-K)/2$, which encodes correlations between opposite chiralities and controls the off-diagonal structure. The diagonal limit of the two-particle reduced density matrix yields density correlations and the static structure factor, while its coherences resolve algebraic $2k_F$ charge-density-wave correlations for repulsion and odd-parity p-wave pairing correlations for attraction. After fixing the cutoff from the one-particle density matrix, the analytic result quantitatively reproduces density matrix renormalization group calculations of the interacting J-V chain within the Luttinger liquid regime. The result connects universal Luttinger liquid scaling with observables in finite microscopic systems.
Figures
Figures from the paper (10 more)
Reference graph
Works this paper leans on
-
[1]
From the diagonal limit of the 2-RDM, Eq
Onset of CDW correlations A CDW is a condensate of particle-hole pairs at wavevector 2kF , and it can be diagnosed in the diago- nal (density–density) sector. From the diagonal limit of the 2-RDM, Eq. (42), the 2k F component of the pair correlation function decays as a power law fixed by the Luttinger parameter [cf. Eq. (44)], C(r)≡ g2(r)−1 2kF = (n0ϵ)2K...
-
[2]
and thus the resulting 2-RDM can be obtained from⟨Ψ †(x′ 2)ˆn(x1)Ψ(x2)⟩by anticommuting the field op- erators. The intersection between hyperplanesx ′ 1 =x 1 and x′ 2 =x 2 gives rise to the diagonal elements of the 2-RDM,ρ 2(x2, x1;x 1, x2), while the intersection between hyperplanesx ′ 2 =x 1 andx ′ 1 =x 2 generates a negative copy of the diagonal elemen...
-
[3]
Orbital character The density-density limit establishes that phase with algebraically enhanced CDW correlations forms, but not which one: the condensing particle-hole pair may be on- site (s-wave) or bond-centered (p-wave), and these are distinguished only by the off-diagonal sector [38, 42]. We therefore introduce the off-diagonal coherence ρ(ξ) = D Ψ†(0...
-
[4]
Pairing correlations We now examine the attractive regime,K >1, where the system should exhibit a superconducting instability withp-wave pairing [31]. Although true long-range order is suppressed in one dimension due to strong phase fluc- tuations, bosonization predicts an algebraic decay of the pair correlator,ρ 2 ∝ |x|−2/K , consistent with quasi-long- ...
-
[5]
An important property ofs(q) for experiments, which can be described by the Luttinger theory, is its small momen- tum behavior
Static Structure Factor In this section, we consider the static structure factor, which is an experimentally accessible quantity defined via the density-density correlationsg 2 as [60] g2(i−j) = ⟨ninj⟩ n2 0 − δij n0 ,(54) s(q) = 1 +n 0 L−1X j=0 [g2(j)−1]e −iqj .(55) Here, the momentum takes valuesq= 2πn/Lwithn∈N. An important property ofs(q) for experimen...
-
[6]
D. A. Mazziotti, Structure of fermionic density matrices: Completen-representability conditions, Phys. Rev. Lett. 108, 263002 (2012)
2012
-
[7]
Ground State Lattice Energy To demonstrate the utility of the constructive bosonization approach taken here, we show that con- tinuum expressions for the 1-RDM and 2-RDM can be leveraged (through the interaction cutoffϵ) to provide access to even short-range observables in the context of a microscopic model. We compute the two-body lattice ground state en...
-
[8]
+e ikF x′ 2 Ψ† +(x′ 2) ih e−ikF x′ 1 Ψ† −(x′
Show all 85 references
-
[9]
In order to satisfy the condition⟨0|Ψ †(x′ 2)Ψ†(x′ 1)Ψ(x1)Ψ(x2)|0⟩ ̸= 0, if a right or left-moving fermion is created, one must also be destroyed
+e ikF x′ 1 Ψ† +(x′ 1) i × eikF x1 Ψ−(x1) +e −ikF x1 Ψ+(x1) eikF x2 Ψ−(x2) +e −ikF x2 Ψ+(x2) .(A1) Carrying out the multiplication, we obtain sixteen total terms; however, the majority will vanish. In order to satisfy the condition⟨0|Ψ †(x′ 2)Ψ†(x′ 1)Ψ(x1)Ψ(x2)|0⟩ ̸= 0, if a r...
-
[10]
+iϵ) |γ2 |sin π L ((x2 −x 1) +iϵ) |γ2 |sin π L ((x′ 2 −x 1) +iϵ) |γ2 |sin π L ((x′ 1 −x 2) +iϵ) |γ2 (A7) ⟨Ψ† α(x′ 2)Ψ† β(x′ 1)Ψβ(x1)Ψα(x2)⟩= 1 4L2 1 sin (π L (x′ 2 −x 1)) sin (π L (x′ 1 −x 2)) × |sin ( π L ((x′ 2 −x ′
-
[11]
Combining all non-zero terms (e.g.⟨Ψ † R(x′ 2)Ψ† R(x′ 1)ΨR(x1)ΨR(x2)⟩and⟨Ψ † L(x′ 2)Ψ† L(x′ 1)ΨL(x1)ΨL(x2)⟩), we obtain a final expression for the 2-RDM, Eq
+iϵ))| λ|sin ( π L ((x2 −x 1) +iϵ))| λ |sin ( π L ((x′ 2 −x 2) +iϵ))| λ|sin ( π L ((x′ 1 −x 1) +iϵ))| λ × |sin iπ L ϵ|2γ2 |sin ( π L ((x′ 2 −x 1) +iϵ))| γ2 |sin ( π L ((x′ 1 −x 2) +iϵ))| γ2 (A8) ⟨Ψ† α(x′ 2)Ψ† β(x′ 1)Ψα(x1)Ψβ(x2)⟩=−⟨Ψ † α(x′ 2)Ψ† β(x′ 1)Ψβ(x2)Ψα(x1)⟩(A9) where ...
-
[12]
hϵhϵ,0 hϵ,−hϵ,+ γ2 −1 # + cos (2kF r−) 2π2h0,−
Finally, to connect with the set of orthogonal hyperplanes we mentioned earlier in section IV, we definer ± = (r′ ±r)/2 = (x ′ 2 −x ′ 1 ±x 2 ∓x 1)/2. The appearance of at least one of the functionsh 0,− andh 0,+ in the denominator in each of the three terms brings our attentio...
-
[13]
Wick’s theorem We simplify the expression by introducing the relative coordinatesr=x 2 −x 1,r ′ =x ′ 2 −x ′ 1, and center-of-mass coordinatesR= x1+x2 2 ,R ′ = x′ 1+x′ 2 2 , with ∆R=R ′ −R. The two-body density matrix becomes: ⟨Ψ†(x′ 2)Ψ†(x′ 1)Ψ(x1)Ψ(x2)⟩= hγ2 ϵ hλ ϵ,0 π2hγ2 ϵ,...
-
[14]
Similarly, we can define the limits forX,Y, andW
Calculating the limitx ′ 1 →x 1,x ′ 2 →x 2 andx 2 →x 1 Starting from ρ(x′ 2, x′ 1;x 1, x2) =⟨Ψ †(x′ 2)Ψ†(x′ 1)Ψ(x1)Ψ(x2)⟩ = |hϵ|γ2 |hϵ,0 |λ 2π2h0,+h0,−|hϵ,+hϵ,−|γ2 cos (kF (x′ 1 +x ′ 2 −x 1 −x 2))[h0,0 |hϵ,0 |δ +h 0,+|hϵ,+|δ −h 0,−|hϵ,−|δ] + |hϵ,0 |λ |hϵ,+|λ ⟨Ψ†(x′ 2)Ψ(x2)⟩⟨Ψ†...
-
[15]
+iϵ) δ sin π L ((x2 −x 1) +iϵ) δ − sin π L ((x′ 1 −x 2) +iϵ) δ sin π L ((x′ 2 −x 1) +iϵ) δi =− L2 π2 δ−1 δ 2 π L sin π L ((x2 −x 1) +iϵ) δ sin π L ((x′ 2 −x ′
-
[16]
+iϵ) δ−2 sin 2π L (x′ 2 −x ′ 1) − L2 π2 δ−1 δ 2 π L sin π L ((x′ 2 −x 1) +iϵ) δ sin π L ((x′ 1 −x 2) +iϵ) δ−2 sin 2π L (x′ 1 −x 2) .(C11) The derivative of the denominator is ∂ ∂x′ 1 sin (π L (x′ 1 −x 1)) sin (π L (x′ 2 −x 2)) = π L cos (π L (x′ 1 −x 1)) sin (π L (x′ 2 −x 2))....
-
[17]
+iϵ] λ sin π L [(x2 −x 1) +iϵ] λ sin iπ L ϵ 2γ2 × ( sin[kF (x′ 2 −x 2)] sin[kF (x′ 1 −x 1)] sin π L [(x′ 2 −x 2) +iϵ] −γ2 L2 sin π L (x′ 2 −x 2) sin π L (x′ 1 −x 1) sin π L [(x′ 1 −x 2) +iϵ] λ sin π L [(x′ 2 −x 1) +iϵ] λ × 1 sin π L [(x′ 1 −x 1) +iϵ] γ2 − sin[kF (x′ 2 −x 1)] s...
-
[18]
(C21) Finally, in limitx ′ 1 →x 1,x ′ 2 →x 2 andx 2 →x 1 we have lim x2→x1 lim x′ 2→x2 lim x′ 1→x1 Ψ†(x′
Ψ(x1) Ψ(x2) =F 2(X2 −Y 2) +W 2 = sin π L [(x2 −x 1) +iϵ] 2λ−2γ2 2L 2 ( − sin π L [(x2 −x 1) +iϵ] 2δ − sin iπ L ϵ 2δ sin2 π L (x2 −x 1) + δ 2 sin π L [(x2 −x 1) +iϵ] 2δ−4h 2 sin π L [(x2 −x 1) +iϵ] 2 cos 2π L (x2 −x 1) −sin 2 2π L (x2 −x 1) i) +n 2 0 − sin iπ L ϵ 2δ sin2 kF (x2...
-
[19]
Ψ(x1) Ψ(x2) = lim x2→x1 [F2(X2 −Y 2) +W 2] =F 3(X3 −Y 3) +W 3 = 0. (C22)
-
[20]
A. J. Coleman, Structure of Fermion Density Matrices, Rev. Mod. Phys.35, 668 (1963)
1963
-
[21]
Altman, E
E. Altman, E. Demler, and M. D. Lukin, Probing many- body states of ultracold atoms via noise correlations, Phys. Rev. A70, 013603 (2004)
2004
-
[22]
M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, 23 L. Pollet, I. Bloch, and C. Gross, Spin- and density- resolved microscopy of antiferromagnetic correlations in fermi-hubbard chains, Science353, 1257 (2016)
2016
-
[23]
T. L. Yang, P. Griˇ sins, Y. T. Chang, Z. H. Zhao, C. Y. Shih, T. Giamarchi, and R. G. Hulet, Measurement of the dynamical structure factor of a 1d interacting fermi gas, Phys. Rev. Lett.121, 103001 (2018)
2018
-
[24]
E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D. Drummond, P. Hannaford, and C. J. Vale, Universal be- havior of pair correlations in a strongly interacting fermi gas, Phys. Rev. Lett.105, 070402 (2010)
2010
-
[25]
T. Liu, L. Xu, J. Liu, and Y. Wang, Entanglement wit- ness for indistinguishable electrons using solid-state spec- troscopy, Phys. Rev. X15, 011056 (2025)
2025
-
[26]
D. A. Mazziotti, Significant conditions for the two- electron reduced density matrix from the constructive solution ofnrepresentability, Phys. Rev. A85, 062507 (2012)
2012
-
[27]
D. A. Mazziotti, Contracted schr¨ odinger equation: De- termining quantum energies and two-particle density ma- trices without wave functions, Phys. Rev. A57, 4219 (1998)
1998
-
[28]
Liebert, A
J. Liebert, A. O. Schouten, I. Avdic, C. Schilling, and D. A. Mazziotti, Refining ensemble N-representability of one-body density matrices from partial information, New J. Phys.27, 124511 (2025)
2025
-
[29]
Bulgac, M
A. Bulgac, M. Kafker, and I. Abdurrahman, Measures of complexity and entanglement in many-fermion systems, Phys. Rev. C107, 044318 (2023)
2023
-
[30]
Naldesi, A
P. Naldesi, A. Elben, A. Minguzzi, D. Cl´ ement, P. Zoller, and B. Vermersch, Fermionic correlation func- tions from randomized measurements in programmable atomic quantum devices, Phys. Rev. Lett.131, 060601 (2023)
2023
-
[31]
Polkovnikov, E
A. Polkovnikov, E. Altman, and E. Demler, Interfer- ence between independent fluctuating condensates, Proc. Natl. Acad. Sci. U.S.A.103, 6125 (2006)
2006
-
[32]
Gritsev, E
V. Gritsev, E. Altman, E. Demler, and A. Polkovnikov, Full quantum distribution of contrast in interference ex- periments between interacting one-dimensional Bose liq- uids, Nat. Phys.2, 705 (2006)
2006
-
[33]
Moitra and R
S. Moitra and R. Sensarma, Building entanglement entropy out of correlation functions for interacting fermions, Phys. Rev. B108, 174309 (2023)
2023
-
[34]
Rammelm¨ uller, W
L. Rammelm¨ uller, W. J. Porter, J. Braun, and J. E. Drut, Evolution from few- to many-body physics in one- dimensional fermi systems: One- and two-body density matrices and particle-partition entanglement, Phys. Rev. A96, 033635 (2017)
2017
-
[35]
D. L. B. Ferreira, T. O. Maciel, R. O. Vianna, and F. Iem- ini, Quantum correlations, entanglement spectrum, and coherence of the two-particle reduced density matrix in the extended Hubbard model, Phys. Rev. B105, 115145 (2022)
2022
-
[36]
O. S. Zozulya, M. Haque, K. Schoutens, and E. H. Rezayi, Bipartite entanglement entropy in fractional quantum Hall states, Phys. Rev. B76, 125310 (2007)
2007
-
[37]
O. S. Zozulya, M. Haque, and K. Schoutens, Particle partitioning entanglement in itinerant many-particle sys- tems, Phys. Rev. A78, 042326 (2008)
2008
-
[38]
Haque, O
M. Haque, O. Zozulya, and K. Schoutens, Entanglement Entropy in Fermionic Laughlin States, Phys. Rev. Lett. 98, 060401 (2007)
2007
-
[39]
Haque, O
M. Haque, O. S. Zozulya, and K. Schoutens, Entan- glement between particle partitions in itinerant many- particle states, J. Phys. A: Math. Theor.42, 504012 (2009)
2009
-
[40]
Barghathi, E
H. Barghathi, E. Casiano-Diaz, and A. Del Maestro, Par- ticle partition entanglement of one dimensional spinless fermions, J. Stat. Mech.: Theory Exp.2017(8), 083108
2017
-
[41]
Radhakrishnan, M
H. Radhakrishnan, M. Thamm, H. Barghathi, B. Rosenow, and A. Del Maestro, A scaling func- tion for the particle entanglement entropy of fermions, J. Stat. Mech.: Theory Exp.2023(8), 083101
2023
-
[42]
C. M. Herdman, P. N. Roy, R. G. Melko, and A. Del Maestro, Particle entanglement in continuum many-body systems via quantum Monte Carlo, Phys. Rev. B89, 140501(R) (2014)
2014
-
[43]
C. M. Herdman and A. Del Maestro, Particle partition entanglement of bosonic Luttinger liquids, Phys. Rev. B 91, 184507 (2015)
2015
-
[44]
Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Ann
U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (N.Y.) 326, 96 (2011)
2011
-
[45]
A. M. Tsvelik,Quantum Field Theory in Condensed Mat- ter Physics, 2nd ed. (Cambridge University Press, Cam- bridge, 2003)
2003
-
[46]
M. A. Cazalilla, Bosonizing one-dimensional cold atomic gases, J. Phys. B: At. Mol. Opt. Phys.37, S1 (2004)
2004
-
[47]
Cardy,Scaling and Renormalization in Statistical Physics(Cambridge University Press, 1996)
J. Cardy,Scaling and Renormalization in Statistical Physics(Cambridge University Press, 1996)
1996
-
[48]
J.-S. Caux, A. Lopez, and D. Suppa, Currents and cor- relations in Luttinger liquids and carbon nanotubes at finite temperature and size: A bosonization study, Nucl. Phys. B651, 413 (2003)
2003
-
[49]
F. D. M. Haldane, Effective Harmonic-Fluid Approach to Low-Energy Properties of One-Dimensional Quantum Fluids, Phys. Rev. Lett.47, 1840 (1981)
1981
-
[50]
Panel (b) shows agreement between the DMRG data and analytic result where only the coordinatex ′ 1 is free;ρ 2(L/2 + 5, x′ 1;L/2 + 5, L/2−3)
Dashed lines indicate the cuts that are plotted in the remaining panels. Panel (b) shows agreement between the DMRG data and analytic result where only the coordinatex ′ 1 is free;ρ 2(L/2 + 5, x′ 1;L/2 + 5, L/2−3). Panel (c) shows the analytic and numerical results for a circu...
-
[51]
Giamarchi,Quantum Physics in One Dimension (Clarendon Press, Oxford, U.K., 2004)
T. Giamarchi,Quantum Physics in One Dimension (Clarendon Press, Oxford, U.K., 2004)
2004
-
[52]
A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosonization and Strongly Correlated Systems(Cam- bridge University Press, 1998)
1998
-
[53]
von Delft and H
J. von Delft and H. Schoeller, Bosonization for beginners: Refermionization for experts, Ann. Phys. (Berlin)510, 225 (1998)
1998
-
[54]
I. E. Dzyaloshinskii and A. I. Larkin, Correlation func- tions for a one-dimensional Fermi system with long-range interaction (Tomonaga model), Sov. Phys. JETP38, 202 (1974)
1974
-
[55]
Luther and I
A. Luther and I. Peschel, Calculation of critical expo- nents in two dimensions from quantum field theory in one dimension, Phys. Rev. B12, 3908 (1975)
1975
-
[56]
M. A. Cazalilla, Effect of Suddenly Turning on Interac- tions in the Luttinger Model, Phys. Rev. Lett.97, 156403 (2006)
2006
-
[57]
Thamm, H
M. Thamm, H. Radhakrishnan, H. Barghathi, B. Rosenow, and A. Del Maestro, One-particle en- tanglement for one-dimensional spinless fermions after an interaction quantum quench, Phys. Rev. B106, 165116 (2022)
2022
-
[58]
Baldelli, H
N. Baldelli, H. Karlsson, B. Kloss, M. Fishman, and A. Wietek, Fragmented superconductivity in the hub- bard model as solitons in ginzburg–landau theory, npj Quantum Mater.10, 22 (2025)
2025
-
[59]
N. H. Aase and A. Sudbø, Dominant superconducting correlations in a luttinger liquid induced by spin fluctu- ations, Phys. Rev. B106, L241102 (2022). 24
2022
-
[60]
Mukhopadhyay, C
R. Mukhopadhyay, C. L. Kane, and T. C. Lubensky, Sliding luttinger liquid phases, Phys. Rev. B64, 045120 (2001)
2001
-
[61]
C. L. Kane, A. Stern, and B. I. Halperin, Pairing in lut- tinger liquids and quantum hall states, Phys. Rev. X7, 031009 (2017)
2017
-
[62]
Nayak, Density-wave states of nonzero angular mo- mentum, Phys
C. Nayak, Density-wave states of nonzero angular mo- mentum, Phys. Rev. B62, 4880 (2000)
2000
-
[63]
S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett.69, 2863 (1992)
1992
-
[64]
Del Maestro, H
A. Del Maestro, H. Barghathi, and B. Rosenow, Equiva- lence of spatial and particle entanglement growth after a quantum quench, Phys. Rev. B104, 195101 (2021)
2021
-
[65]
C. N. Yang, Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid He and of Supercon- ductors, Rev. Mod. Phys.34, 694 (1962)
1962
-
[66]
L¨ owdin, Quantum theory of many-particle systems
P.-O. L¨ owdin, Quantum theory of many-particle systems. i. physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction, Phys. Rev.97, 1474 (1955)
1955
-
[67]
B. C. Carlson and J. M. Keller, Eigenvalues of Density Matrices, Phys. Rev.121, 659 (1961)
1961
-
[68]
Ando, Properties of fermion density matrices, Rev
T. Ando, Properties of fermion density matrices, Rev. Mod. Phys.35, 690 (1963)
1963
-
[69]
Eggert, One-dimensional quantum wires: A pedestrian approach to bosonization (2009), arXiv:0708.0003 [cond- mat.str-el]
S. Eggert, One-dimensional quantum wires: A pedestrian approach to bosonization (2009), arXiv:0708.0003 [cond- mat.str-el]
2009 arXiv
-
[70]
[30], the second term in Eq
In Ref. [30], the second term in Eq. (7) has the wrong sign [27]
-
[71]
Lukyanov and V
S. Lukyanov and V. Terras, Long-distance asymptotics of spin-spin correlation functions for the XXZ spin chain, Nucl. Phys. B654, 323 (2003)
2003
-
[72]
Kutzelnigg and D
W. Kutzelnigg and D. Mukherjee, Cumulant expansion of the reduced density matrices, J. Chem. Phys.110, 2800 (1999)
1999
-
[73]
Voit, One-dimensional Fermi liquids, Rep
J. Voit, One-dimensional Fermi liquids, Rep. Prog. Phys. 58, 977 (1995)
1995
-
[74]
Garrod and M
C. Garrod and M. Rosina, Particle-hole matrix: Its con- nection with the symmetries and collective features of the ground state, J. Math. Phys.10, 1855 (1969)
1969
-
[75]
Des Cloizeaux, A soluble fermi-gas model
J. Des Cloizeaux, A soluble fermi-gas model. validity of transformations of the bogoliubov type, J. Math. Phys. 7, 2136 (1966)
1966
-
[76]
C. N. Yang and C. P. Yang, One-Dimensional Chain of Anisotropic Spin-Spin Interactions. I. Proof of Bethe’s Hypothesis for Ground State in a Finite System, Phys. Rev.150, 321 (1966)
1966
-
[77]
Fishman, S
M. Fishman, S. White, and E. M. Stoudenmire, The iten- sor software library for tensor network calculations, Sci- Post Phys. Codebases , 4 (2022)
2022
-
[78]
Fishman, S
M. Fishman, S. White, and E. M. Stoudenmire, Codebase release 0.3 for itensor, SciPost Phys. Codebases , 4 (2022)
2022
-
[79]
Giuliani and G
G. Giuliani and G. Vignale,Quantum Theory of the Elec- tron Liquid(Cambridge University Press, Cambridge, 2005)
2005
-
[80]
E. A. Carlen, E. H. Lieb, and R. Reuvers, Entropy and Entanglement Bounds for Reduced Density Matri- ces of Fermionic States, Commun. Math. Phys.344, 655 (2016)
2016
-
[81]
I. V. Protopopov, D. B. Gutman, and A. D. Mirlin, Many-particle correlations in a non-equilibrium luttinger liquid, J. Stat. Mech.: Theory Exp.2011(11), P11001
2011
-
[82]
M. Rosina, Transition amplitudes as ground-state varia- tional parameters, inReduced Density Matrices with Ap- plications to Physical and Chemical Systems, Queen’s Pa- pers in Pure and Applied Mathematics No. 11, edited by A. J. Coleman and R. M. Erdahl (Queen’s University, Ki...
1968
-
[83]
D. A. Mazziotti, Complete reconstruction of reduced den- sity matrices, Chem. Phys. Lett.326, 212 (2000)
2000
-
[84]
G. E. Massaccesi, O. B. O˜ na, L. Lain, A. Torre, J. E. Peralta, D. R. Alcoba, and G. E. Scuseria, Is the matrix completion of reduced density matrices unique?, J. Phys. Chem. Lett.17, 3430 (2026)
2026
-
[85]
Thamm, H
M. Thamm, H. Radhakrishnan, and A. Del Maestro, GitHub Repository, Zenodo doi:10.5281/zenodo.21384640 (2026)
2026 doi
Reviewed August 2, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.