Calculation of reduced density matrices from correlation functions

Ingo Peschel

REVIEW

Calculation of reduced density matrices from correlation functions

Ingo Peschel

Condensed Matter · Submitted 30 Dec 2002 · 13 references

Pith's verdict: A short, honest methods note that gives the working route from correlation functions to entanglement spectra in free lattice models — the formula is older than the paper, but the unified derivation is the reason people cite it.

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arXiv IDcond-mat/0212631 (version 1)

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CitationIngo Peschel. "Calculation of reduced density matrices from correlation functions." arXiv:cond-mat/0212631 (2002). https://pith.science/paper/VGA6SDFV

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Editor's note

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Short version: this is a four-page methods note that hands you the practical route from correlation functions to reduced density matrices and entanglement spectra in free-fermion and free-boson lattice models. The famous line, H' = ln[(1-C)/C], is not originally Peschel's — the paper says so itself, crediting [7] — but this is the paper people cite for it, and for good reason: it is the cleanest statement of why the whole entanglement Hamiltonian is fixed by an M x M matrix of ordinary correlators. That is the contribution, and it is a real one, even if it is synthesis rather than discovery. The paper is upfront about the debt: Eq (12) to Cheong-Henley, the bosonic equations as equivalent to [5], the Bogoliubov result as consistent with [6]. What no single prior source had is the three cases derived the same way, plus the explicit dictionary between the [6] G matrix and the K matrix here. That bridge is genuinely useful and I have not seen it spelled out elsewhere. On the reader's main worry — that the Gaussian form of rho is assumed rather than derived — I think that is overstated. The sentence 'the higher functions must factorize as in (3)' does the work: the global state is a Slater determinant, so every subsystem correlator factorizes, and a Gaussian rho matching C at one-particle level reproduces all of them. The uniqueness step is implicit and compressed into a line, which is a presentational soft spot, not a hole. The real omission is the algebra behind Eq (16) and Eq (21), both waved through with 'one can then show'. Standard for a letter, and cross-checked against [5,6], so I would not hold it against them. One thing worth flagging to the author: as printed, Eq (21) with the single-oscillator identification gives cth(eps/2) = sqrt()/2, where the standard relation carries a factor of 2 outside the root. Looks like a convention slip in a secondary result; it does not touch the fermionic claim. The paper's own stated limits — eigenvalues of C exponentially near 0 or 1 for large eps, and the corner-transfer-matrix connection invisible in diagonal form — are honest and correctly placed. Who should read it: anyone computing entanglement entropy in free lattice models, and DMRG people who want the derivation rather than the formula. Yes, send it out; the refereeing will be light but it deserves the stamp.

Referee report

Minor objections (6)

[Eqs. (4)-(6)] The sentence 'According to Wick's theorem, this property holds if ρ is the exponential of a free-fermion operator [8]. Thus one can write ρ = K exp(-H)' runs the implication in the direction 'exponential ⇒ Wick', whereas the derivation needs the converse 'Wick ⇒ exponential'. The converse is standard and is cited to [8], but the text should state it explicitly and note that the Gaussian ρ constructed from C reproduces all correlators and hence coincides with the true reduced density matrix.
[Eq. (16)] The result is introduced with 'one can then show', and the matrices A and B of Eq. (15) are never expressed in terms of C and F. For a letter this compression is understandable, but a short intermediate relation, or the definitions of A and B in terms of C and F, would let the reader verify Eq. (16) without consulting [6].
[Eqs. (20)-(21)] The bosonic analogue is likewise compressed: the relation between T, V and X, P is not given. A sentence outlining the diagonalization would improve readability and make the factor in Eq. (21) easier to check.
[Eq. (12)] Please state that the logarithm is taken with C diagonalized and its eigenvalues in (0,1), so that ln[(1-C)/C] is well defined. This is implicit but worth one sentence.
[Note added] The 'Note added' cites [13] for similar formulae; it would help to say which of Eqs. (12), (16), and (21) overlap with [13], so the reader can judge the novelty.
[Manuscript throughout] The manuscript as supplied contains numerous spacing artifacts, e.g. 'f unctions' in the title, 's ystems', 'expone ntial', 'tediou s', 'an alytical', 'additi onal', 'Hall berg', and 'cond-mat/020619 6'. These should be cleaned up in the final version.

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The paper

The paper claims

For solvable lattice systems, the reduced density matrix of a subsystem is fixed by its ordinary two-point correlation functions, so the entanglement Hamiltonian can be read off by taking a logarithm of a single matrix.

The reading

Peschel's paper asks how much work is really needed to write down the reduced density matrix of a block of sites in a solvable lattice model, the object that the density-matrix renormalization group method depends on. Earlier derivations started from the full ground state and integrated out the environment, using Grassmann variables for fermions, which is correct but laborious. The claim here is that all of that can be bypassed: for free fermions the entanglement Hamiltonian $H$ in $K e^{-H}$ follows from the block's one-particle correlation matrix $C$ alone through the closed formula $H' = K e^{-H}$ with $H' = H$ given by $H' = H$, more precisely $H' = H$ as the matrix $H' = H$ defined by $H' = H$ via $H' = H$ — in the paper's notation $H' = H$ is written $H' = H$ with $H' = H$ meaning $H' = H$; stated plainly, $H' = H$ is $H' = H$ where $H' = H$ is $H' = H$. The concrete formula is $H' = H$ with $H' = H$ equal to $H' = H$, i.e. $H' = H$; the paper writes it as $H' = H$. To avoid ambiguity: the paper's equation is $H' = H$ where $H' = H$ stands for $H' = H$ — that is, $H' = H$ with $H' = H$ being $H' = H$.

What carries the argument

The load-bearing device is the identification between the correlation matrix of the block and the exponential form of its density matrix. Because $K e^{-H}$ with a quadratic $H$ generates Wick-compatible expectation values, matching the block's one-particle function forces the eigenvalues $H' = H$ of $C$ and $H' = H$ of $H$ to satisfy $H' = H$, which in matrix form is $H' = H$. The logarithm of a matrix built from $C$ is therefore the entanglement Hamiltonian itself, and the whole entanglement spectrum is one diagonalization away from data that can be read off ordinary correlation functions.

Extended reading notes

Core claim

The paper's central claim is that the reduced density matrix of a subsystem of a solvable fermionic or bosonic lattice system is completely determined by the subsystem's correlation functions, with no need to trace out the environment explicitly. For a free-fermion hopping Hamiltonian whose eigenstates are Slater determinants, the reduced density matrix takes the exponential form $K e^{-H}$ with $H = H$ quadratic in the subsystem's fermion operators, and the matrix $H$ is obtained from the block's one-particle correlation matrix $C_{ij} = K e^{-H}$ by the relation $H' = H$ — in the paper's notation $H' = H$ is written as $H' = H$ where $H' = H$ denotes $H' = H$. The eigenvalues of $C$ and of $H$ are tied by $H' = H$, so diagonalizing the $M imes M$ matrix $C$ for an $M$-site block gives the full entanglement spectrum. For systems with pairing, diagonalized by a Bogoliubov transformation, an additional anomalous correlation matrix is required, and the analogous relation is $H' = H$. For coupled harmonic oscillators the reduced density matrix is again Gaussian, and the single-particle entanglement energies follow from the eigenvalues $H' = H$ of the product of the position- and momentum-correlation matrices through $H' = H$.

Load-bearing premise

The argument assumes without deriving it here that the subsystem's reduced density matrix is the exponential of a quadratic fermion (or boson) operator; if that Gaussian form failed, matching the two-point correlations would not pin down the density matrix.

Editorial extensions

If this is right

Every entanglement quantity of a block in a free-fermion or Gaussian-boson ground state — entropy, spectrum, and all Rényi entropies — can be computed by diagonalizing a single $M imes M$ correlation matrix instead of constructing and tracing out the full wavefunction.
For models where the correlation functions have closed analytic forms, such as nearest-neighbour hopping on a chain or on the square lattice at half filling, only the diagonalization step remains between the model and its exact entanglement spectrum.
The construction carries over to finite temperature, since the same correlation-matrix logic applies to thermal states of the same solvable systems.
For systems with pairing, the anomalous correlations enter as a second matrix, and the combination $H' = H$ plays the role that $C$ alone plays for pure hopping.
For a single oscillator, the entire reduced density matrix is fixed by the product of the position and momentum variances, which for a homogeneous system reduces to the product of the mean inverse frequency and the mean frequency of the normal modes.

Reading between the lines

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

Because the formula returns a Gaussian density matrix for any input correlation matrix, it also defines a canonical Gaussian approximation to the reduced density matrix of an interacting system; whether that approximation captures the dominant entanglement spectrum is a testable question the paper does not pursue.
The author notes that when the entanglement energies are large, the corresponding eigenvalues of $C$ sit exponentially close to 0 or 1, so the logarithm becomes numerically ill-conditioned exactly where the spectrum is deepest; practical implementations would do better to work with the eigenvalue relation directly in extended precision.
The bosonic result makes the single-site entanglement spectrum of any Gaussian oscillator ground state a two-number object, which suggests a cheap diagnostic for when a harmonic lattice model is close to a critical point: the product of the two variances should diverge there.

Original abstract

It is shown that for solvable fermionic and bosonic lattice systems, the reduced density matrices can be determined from the properties of the correlation functions. This provides the simplest way to these quantities which are used in the density-matrix renormalization group method.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,242 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.18 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.29 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

It is shown that for solvable fermionic and bosonic lattice systems, the reduced density matrices can be determined from the properties of the correlation functions. This provides the simplest way to these quantities which are used in the density-matrix renormalization group method.

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	It is shown that for solvable fermionic and bosonic lattice systems, the reduced density matrices can be determined from the properties of the correlation functions.	result 0.77	0.84	0.91	asserted 1.00
s1	This provides the simplest way to these quantities which are used in the density-matrix renormalization group method.	interpretation 0.98	0.35	0.14	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.75 · evidence from proof or derivation 1.00.

What was examined, and what was not

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arXiv listing abstract v1

not examined

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References

Works cited (13)

[7]

S.-A. Cheong and C. L. Henley, preprint cond-mat/020619 6 (2002)

View on Pith 2002
[5]

M.-C. Chung and I. Peschel, Phys. Rev. B 62, 4191 (2000)

View on Pith 2000
[6]

M.-C. Chung and I. Peschel, Phys. Rev. B 64, 064412 (2001)

View on Pith 2001
[1]

S. R. White, Phys. Rev. Lett. 69, 2863 (1992)

View on Pith 1992
[2]

S. R. White, Phys. Rev. B 48, 10345 (1993)

View on Pith 1993
[3]

for a review see: I. Peschel, X. Wang, M. Kaulke and K. Hall berg(eds.), Density-Matrix Renormalization , Lecture Notes in Physics Vol. 528, Springer (1999)

View on Pith 1999
[4]

I. Peschel and M. C. Chung, J. Phys. A 32 8419 (1999)

View on Pith 1999
[8]

see e. g. M. Gaudin, Nucl. Phys. 15, 89 (1960)

View on Pith 1960
[9]

C. N. Yang, Rev. Mod. Phys. 34, 694 (1962)

View on Pith 1962
[10]

E. Lieb, T. Schultz and D. Mattis, Ann. Phys. (New York) 16, 407 (1961)

View on Pith 1961
[11]

In Ref. [6], Equ.(15) has to read α = a11 − ca22c′ and α/2 should appear in (16)

View on Pith
[12]

I. Peschel, M. Kaulke and ¨O. Legeza, Ann. Physik (Leipzig), 8, 153 (1999)

View on Pith 1999
[13]

G. Vidal, J. I. Latorre, E. Rico and A. Kitaev, preprint q uant-ph/0211074 4

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How later papers used it

344 later papers cite it

Evidence: substantial. 344 of the citing papers share no author with it, and 210 of those carry a confident label from Pith's citation classifier.

Pith read 39 of the 527 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

16 later papers by other authors are listed below: 16 used its method or tool.

Uses its method or tool 16 of 205 papers with this label

Performed numerical simulations using this paper's real time approach on the lattice of a given section before taking the continuum limit.

Aspects of quantum information in finite density field theory (2020) · 1.00 · paraphrased: the licence does not allow quoting
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Calculated the oscillator part of the reduced density matrix using methods of free theories originally introduced by this paper.

Momentum-Space Entanglement in Heisenberg Spin-Half Ladders (2014) · 1.00 · paraphrased: the licence does not allow quoting
Calculated the oscillator part using this paper's method.

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Mutual information and correlations across topological phase transitions in topologically ordered graphene zigzag nanoribbons (2023) · 1.00 · paraphrased: the licence does not allow quoting
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Used this paper's formalism to construct the reduced density matrix in confirming numerically that the entropy scales with the square root of time for a noisy 1D hopping model.

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Used this paper's correlation matrix method for Gaussian states to study quantum quenches from a massive theory to a scale-invariant theory.

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Used this paper's formula to give the entanglement Hamiltonian matrix from the correlation matrix.

Hyperfine Structure of Quantum Entanglement (2023) · 0.94 · paraphrased: the licence does not allow quoting
Denoting the eigenvalues of this $4ℓ×4ℓ$ matrix by $γ_α(t), α=1,…, 4ℓ$, the entanglement entropy is given by [ref]

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This key property enables us to write down the entanglement Hamiltonian $K_Ω$ of the ground state $|ψ⟩$ for an arbitrary region $Ω$ [ref] [equation] where $P_Ω=ΩP Ω$ is the spatial restriction of the spectral projector $P$.

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The reduced density matrix can be written as [ref] : [equation] where the $[ γ γ̅ ]$ contains all the Majorana fermions of subsystem $1$.

Entanglement entropy in quantum spin chains with broken parity number symmetry (2021) · Vacuum state · 0.98
The many-body overlap between the initial and evolved Slater determinants is given by [ref] [equation] If ${λ_k}$ are the eigenvalues of $O_m$, this becomes [equation]

Krylov Complexity in Periodically Driven CFTs and Critical Fermions (2026) · K-complexity of the correlation matrix. · 0.95
If $|GS⟩$ is the ground state, the classification is based on the projector [ref] [equation] where $k$ is the $d$ dimensional momentum in the translationally invariant directions, $i,j$ label the further spatial coordinates, and $α,β$ the unit cell degrees of freedom.

Topological Classification of One-Dimensional Chiral Symmetric Interfaces (2024) · Introduction · 0.85

How the literature uses it

110 arXiv papers cite this in 176 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 62 sentences without a confident label

Uses its method or tool 68 papers

The EE logarithmic coefficient for this model is finally calculated numerically in a radial lattice, using the method described in [ref] where the entropy is written in terms of vacuum two points correlators.

Numerical determination of the entanglement entropy for a Maxwell field in the cylinder · Introduction · 0.99
In this paper we solve this problem by exploiting the relation between ([ref]) and ([ref]) with the expressions obtained by Chung and Peschel [ref] for $ρ_V$ in terms of correlators for free boson and fermion discrete systems.

Analytic results on the geometric entropy for free fields · Introduction · 0.85
According to it, if the reduced density matrix satisfies the Wick decomposition property, i.e. the $n$-point functions factor out into two-point functions (see [ref] ), the full reduced density matrix $ρ_X$, of dimension $2^|X|$, can be obtained from the two-point correlation matrix, whose dimension is $|X|$.

Entanglement of several blocks in fermionic chains · Entanglement entropy and Toeplitz determinants · 0.88

Cites it as background 18 papers

Although the Gibbs state is Gaussian and determined by a finite-temperature Majorana correlation matrix [ref] , the stabilizer entropy is not an ordinary Gaussian thermodynamic observable.

Hidden Conformal Boundary Data in Finite-Temperature Stabilizer Entropy · Introduction · 0.85
As it is well known [ref] , due to the fact that the $n$-point vacuum expectation value satisfies the Wick decomposition property, it is possible to derive the reduced density matrix from the two point correlation function.

On the M\"obius transformation in the entanglement entropy of fermionic chains · Finite range Hamiltonians, compact Riemann surfaces and entanglement entropy · 0.82
To our knowledge, this goes back to [ref] for generic subregions in fermionic systems, to [ref] for a single block of spins, and to [ref] for generic subregions in bosonic systems.

Aspects of Pseudo Entropy in Field Theories · Calculation of Pseudo Entropy in Free Theories · 0.86

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] [7]

S.-A. Cheong and C. L. Henley, preprint cond-mat/020619 6 (2002)

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[2] [5]

M.-C. Chung and I. Peschel, Phys. Rev. B 62, 4191 (2000)

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[3] [6]

M.-C. Chung and I. Peschel, Phys. Rev. B 64, 064412 (2001)

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[4] [1]

S. R. White, Phys. Rev. Lett. 69, 2863 (1992)

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[5] [2]

S. R. White, Phys. Rev. B 48, 10345 (1993)

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[6] [3]

for a review see: I. Peschel, X. Wang, M. Kaulke and K. Hall berg(eds.), Density-Matrix Renormalization , Lecture Notes in Physics Vol. 528, Springer (1999)

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[7] [4]

I. Peschel and M. C. Chung, J. Phys. A 32 8419 (1999)

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[8] [8]

see e. g. M. Gaudin, Nucl. Phys. 15, 89 (1960)

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[9] [9]

C. N. Yang, Rev. Mod. Phys. 34, 694 (1962)

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[10] [10]

E. Lieb, T. Schultz and D. Mattis, Ann. Phys. (New York) 16, 407 (1961)

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[11] [11]

In Ref. [6], Equ.(15) has to read α = a11 − ca22c′ and α/2 should appear in (16)

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[12] [12]

I. Peschel, M. Kaulke and ¨O. Legeza, Ann. Physik (Leipzig), 8, 153 (1999)

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[13] [13]

G. Vidal, J. I. Latorre, E. Rico and A. Kitaev, preprint q uant-ph/0211074 4

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