REVIEW
Increasing of entanglement entropy from pure to random quantum critical chains
Strongly Correlated Electrons · Submitted 22 Feb 2006 · 28 references
Pith's verdict: Extends SDRG entanglement to general N and gives a provocative N>41 counterexample, but the parafermionic fixed-point flow is asserted and written inconsistently.
T0 · AI review
Why Pith says this
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Referee report
The paper
The paper claims
For N>41, adding disorder increases entanglement in quantum chains.
The reading
What carries the argument
The machinery is the strong-disorder renormalization group applied to Hamiltonian (4), whose coefficients $\alpha_n$ control the symmetry class. The key step is the frozen-cluster wavefunction (8): a field-decimated cluster of $s$ spins is frozen in the state $|\psi\rangle = \frac{1}{\sqrt{N}}\sum_{q=0}^{N-1}|q\rangle^{\otimes s}$, so any segment boundary crossing that cluster contributes $\log N$ to the entanglement. Self-duality fixes the field-decimation probability at $p=1/2$, and the $N$-independent fixed-point distribution gives $\langle D\rangle_L = \frac{1}{3}\ln L$ crossing clusters, producing the prefactor $\log N/6$. For the parafermionic chain, the coefficients $\{\alpha_n\}$ are claimed to renormalize quickly to $\delta_{1,n}$, so the same fixed point governs the $Z_N$ clock model.
Extended reading notes
Core claim
The central claim is Eq. (11): for the family of random chains (4) with $\alpha_n \le 1$, the entanglement entropy of a segment of length $L$ scales as $S_{\rm dis}^N(L) \sim \frac{\log N}{6} \ln L$ at the infinite-randomness fixed point. Since the pure $Z_N$ parafermionic chain has central charge $c_N = 2(N-1)/(N+2)$, its entropy scales as $S_{\rm pure}^N(L) \sim \frac{2(N-1)}{3(N+2)}\ln L$. Comparing the two shows the ratio is $N$-dependent, so no universal model-independent correction exists, and for $N>41$ the disordered prefactor is larger, giving $S_{\rm dis}^N(L) > S_{\rm pure}^N(L)$ and contradicting the conjectured universal loss of entanglement under disorder.
Load-bearing premise
The calculation assumes that the coefficients $\alpha_n$ of the $Z_N$ parafermionic chain renormalize quickly to $\delta_{1,n}$ under the SDRG, so that the chain reaches the same infinite-randomness fixed point as the $Z_N$ clock model; if that flow is slow or nonuniversal, the frozen-cluster wavefunction (8) and the comparison with the pure central charge do not follow.
Editorial extensions
If this is right
- If Eq. (11) holds, the entanglement prefactor at the random fixed point is $N$-dependent, so entanglement can distinguish random chains with different $N$ even though other scaling functions and exponents are $N$-independent.
- The ratio of disordered to pure entropy prefactors is not universal; therefore an 'effective central charge' defined from entanglement is not simply the pure central charge times a fixed constant.
- The random Potts and random $Z_N$ clock chains share the same prefactor $(\log N)/6$ despite different symmetries, exactly analogous to distinct CFTs with equal central charge.
- For $N>41$, the disordered chain has larger entropy than the pure one, so a generalized $c$-theorem stating that disorder always reduces entanglement does not hold for these models.
- The loss of unitarity under disorder is suggested as the reason the monotone decrease can fail, since a random-bond Potts model is known to develop negative-norm operators at the random fixed point.
Reading between the lines
- One testable extension is to compute the free-energy scaling at the IRFP for these chains: the paper notes such a comparison would clarify whether the prefactor behaves as a true effective central charge, but does not perform it.
- A direct numerical iteration of the SDRG recursion (5)-(6) starting from the parafermionic coefficients would test how quickly $\alpha_n$ approaches $\delta_{1,n}$; if that flow is slow, the $\log N/6$ prefactor may only emerge at very large length scales.
- For $N$ just above 41, finite-size numerics on random $Z_N$ chains could detect the crossing by comparing the logarithmic prefactors at accessible $L$; the paper gives no numerical check of Eq. (13).
- The threshold itself depends on the pure CFT central charge $c_N$; similar thresholds would appear for any pure chain whose central charge is small enough, suggesting that disorder-driven entanglement growth is not unique to the $Z_N$ family.
Original abstract
It is known that the entropy of a block of spins of size $L$ embedded in an infinite pure critical spin chain diverges as the logarithm of $L$ with a prefactor fixed by the central charge of the corresponding conformal field theory. For a class of strongly random spin chains, it has been shown that the correspondent block entropy still remains universal and diverges logarithmically with an "effective" central charge. By computing the entanglement entropy for a family of models which includes the $N$-states random Potts chain and the $Z_N$ clock model, we give some definitive answer to some recent conjectures about the behaviour of the effective central charge. In particular, we show that the ratio between the entanglement entropy in the pure and in the disordered system is model dependent and we provide a series of critical models where the entanglement entropy grows from the pure to the random case.
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.
The abstract, sentence by sentence
It is known that the entropy of a block of spins of size $L$ embedded in an infinite pure critical spin chain diverges as the logarithm of $L$ with a prefactor fixed by the central charge of the corresponding conformal field theory. For a class of strongly random spin chains, it has been shown that the correspondent block entropy still remains universal and diverges logarithmically with an "effective" central charge. By computing the entanglement entropy for a family of models which includes the $N$-states random Potts chain and the $Z_N$ clock model, we give some definitive answer to some recent conjectures about the behaviour of the effective central charge. In particular, we show that the ratio between the entanglement entropy in the pure and in the disordered system is model dependent and we provide a series of critical models where the entanglement entropy grows from the pure to the random case.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | It is known that the entropy of a block of spins of size $L$ embedded in an infinite pure critical spin chain diverges as the logarithm of $L$ with a prefactor fixed by the central charge of the corresponding conformal field theory. | background 1.00 | 0.03 | 0.03 | asserted 1.00 |
| s1 | For a class of strongly random spin chains, it has been shown that the correspondent block entropy still remains universal and diverges logarithmically with an "effective" central charge. | background 1.00 | 0.05 | 0.07 | asserted 1.00 |
| s2 | By computing the entanglement entropy for a family of models which includes the $N$-states random Potts chain and the $Z_N$ clock model, we give some definitive answer to some recent conjectures about the behaviour of the effective central charge. | contribution 0.94 | 0.95 | 0.14 | asserted 0.99 |
| s3 | In particular, we show that the ratio between the entanglement entropy in the pure and in the disordered system is model dependent and we provide a series of critical models where the entanglement entropy grows from the pure to the random case. | result 0.99 | 0.96 | 0.83 | asserted 1.00 |
What was examined, and what was not
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Figures
Showing 1 of the paper's 2 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:cond-mat/0602527 by the authors.
References
Discussion
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