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Entanglement in Quantum Critical Phenomena
Open accessAbstract
Entanglement, one of the most intriguing features of quantum theory and a main resource in quantum information science, is expected to play a crucial role also in the study of quantum phase transitions, where it is responsible for the appearance of long-range correlations. We investigate, through a microscopic calculation, the scaling properties of entanglement in spin chain systems, both near and at a quantum critical point. Our results establish a precise connection between concepts of quantum information, condensed matter physics, and quantum field theory, by showing that the behavior of critical entanglement in spin systems is analogous to that of entropy in conformal field theories. We explore some of the implications of this connection.
Overview
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TL;DR
The paper computes block entanglement in XXZ and XY spin-1/2 chains near and at quantum critical points, finding logarithmic scaling at criticality and saturation in noncritical regimes. It connects critical spin-chain entanglement to conformal field theory entropy and argues entanglement decreases under renormalization group flow.
Key findings
- Noncritical entanglement either vanishes or grows monotonically with block size \(L\) and saturates at a length scale \(L_0\); critical entanglement diverges logarithmically with \(L\).
- For critical regimes, \(S_L = \frac{c+\bar c}{6}\log_2 L + k\), matching conformal field theory. Critical Ising/XY (\(\gamma \neq 0\)) has \(c=\bar c=1/2\), giving \(S_L \sim \log_2 L/6\); critical XX/XXZ has \(c=\bar c=1\), giving \(S_L \sim \log_2 L/3\).
- The paper reports numerical XXZ ground states for chains up to \(N=20\) spins and exact infinite-chain XY results, with Fig. 1 showing agreement between numerics and conformal predictions.
- It argues that critical entanglement has unboundedly growing support in the reduced density matrix, explaining why DMRG with a limited number of retained eigenvectors fails to reproduce quantum critical behavior.
Methods
- Entanglement is quantified by the von Neumann entropy \(S_L = -\mathrm{tr}\,\rho_L \log_2 \rho_L\) for a block of \(L\) contiguous spins in the ground state.
- XXZ model: Bethe ansatz plus numerical diagonalization for chains up to \(N=20\) spins. XY model: exact solution for an infinite chain using Majorana operators, Gaussian states, and a correlation matrix; \(S_L\) is computed by diagonalizing \(\Gamma_L\) and summing binary entropies of fermionic modes.
- Scaling behavior is compared with \(1+1\)-dimensional conformal field theory entropy formulas and with central charges of the relevant universality classes.
Limitations
- Stated: finite-\(N\) corrections to Eq. (6) are not discussed; the XXZ numerical analysis uses chains up to \(N=20\) spins.
- Stated: DMRG schemes retaining only a limited number of eigenvectors are expected to fail for critical \(S_L\) because the relevant support grows with \(L\).
- Evident from the text: the work is theoretical/numerical for specific spin-1/2 chains, with no experimental validation reported.
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In the paper
“Such a computation determines the scaling laws of entanglement in one-dimensional spin systems, while explicitly uncovering an accurate correspondence between critical entanglement in condensed matter systems and the entropy of the vacuum in conformal field theories.”
§ Entanglement in Quantum Critical Phenomena > Abstract
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Rests on an assumption: The block entanglement entropy S_L as a function of block size L captures the large-scale behavior of quantum correlations at a critical regime.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Indeed, more generally we encounter that the support or number of relevant eigenvectors of L unboundedly grows with L.”
§ Entanglement in Quantum Critical Phenomena > Critical regime.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Correspondingly, already for a chain of N 20 spins it is possible to observe a distinct, characteristic behavior of S L depending on whether the values ; in Eq. ( 2) belong or not to a critical regime.”
§ Entanglement in Quantum Critical Phenomena > Abstract
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For any L, the reduced density matrix L is effectively supported on just a small, bounded subspace of the Hilbert space of the L spins.”
§ Entanglement in Quantum Critical Phenomena > P H Y S I C A L R E V I E W L E T T E R S
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In critical chains, on the contrary, the entropy S L diverges logarithmically with L.”
- Also stated in: § P H Y S I C A L R E V I E W L E T T E R S · plain
“Instead, the entanglement of a block with a chain in a critical model displays a logarithmic divergence for large L: S L log 2 L=6 (stars) for the critical Ising chain, H XY a 1; 1; S L log 2 L=3 (triangles) for the critical XX chain with no magnetic field, H XY a 1; 0;”
§ Entanglement in Quantum Critical Phenomena > Critical regime.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § P H Y S I C A L R E V I E W L E T T E R S · plain
None of the sources cited for this claim have been checked yet.
In the paper
“On the other hand, entanglement is appointed to play a central role in the study of strongly correlated quantum systems [7][8][9], since a highly entangled ground state is at the heart of a large variety of collective quantum phenomena.”
§ Entanglement in Quantum Critical Phenomena > Abstract
What the sources say
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“If, as is the case in DMRG schemes, only a limited number of eigenvectors can be retained, then a local computation of L for sufficiently large L is bound to fail from the outset (arguably, even in an approximate sense).”
§ Entanglement in Quantum Critical Phenomena > Critical regime.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“Entanglement, one of the most intriguing features of quantum theory and a main resource in quantum information science, is expected to play a crucial role also in the study of quantum phase transitions, where it is responsible for the appearance of long-range correlations.”
- Also stated in: § Abstract · hedged
“Ground-state entanglement is, most promisingly, also a key factor to the understanding of quantum phase transitions, where it is responsible for the appearance of long-range correlations at zero temperature [12].”
§ Entanglement in Quantum Critical Phenomena > Abstract
What the sources say
- Not checked yetcites [12] source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § Abstract · hedged
In the paper
“As derived in Ref. [25], in a 1 1 conformal field theory the entropy of a subregion of length L reads S L c c c 6 log 2 L k; (12”
§ Entanglement in Quantum Critical Phenomena > Critical regime.
What the sources say
- Not checked yetcites [25] source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The general picture emerging from the above computations is that there is a clear distinction between noncritical and critical entanglement, corresponding to two forms of structurally inequivalent ground states.”
§ Entanglement in Quantum Critical Phenomena > P H Y S I C A L R E V I E W L E T T E R S
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes
- It may state two separate claims as one. 4
- The quote only partly states this claim. 1
- Probably not a substantial claim. 3
In the paper
“Noncritical entanglement is characterized by a saturation of S L as a function of the block size L: noncritical Ising chain (empty squares), H XY a 1:1; 1; noncritical XXZ chain (filled squares), H XXZ 2:5; 0.”
§ Entanglement in Quantum Critical Phenomena > P H Y S I C A L R E V I E W L E T T E R S
What the sources say
- Not checked yetcites [13]
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The calculation of S L , as sketched next, also uses the fact that the ground state j g i of the chain and the density matrices L for blocks of spins are Gaussian states that can be completely characterized by means of certain correlation matrix of second moments.”
§ Entanglement in Quantum Critical Phenomena > Abstract
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“Their analysis, focused on single-spin entropies [12] and on two-spin quantum correlations [12,14], suggestively shows that these one-and two-spin entanglement measures display a peak either near or at the critical point.”
§ Entanglement in Quantum Critical Phenomena > Abstract
Strength:hedged· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“The XY model, Eq. ( 3), is an exactly solvable model (see for instance [13]) and this allows us to consider an infinite chain, N ! 1.”
§ Entanglement in Quantum Critical Phenomena > Abstract
What the sources say
- Not checked yetcites [13]
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“It is noteworthy, then, that entanglement in spin systems decreases both under local operations and classical communication and along renormalization group trajectories.”
§ Entanglement in Quantum Critical Phenomena > P H Y S I C A L R E V I E W L E T T E R S
What the sources say
- Not checked yetcites [27] source
- Not checked yetcites [30]
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“DMRG techniques have reportedly failed to reproduce quantum critical behavior [21] and we may, in view of the divergent character of S L , be in a position to understand why.”
§ Entanglement in Quantum Critical Phenomena > Critical regime.
What the sources say
- Not checked yetcites [21]
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
In the paper
“These models consider a variety of spin-spin interactions between first neighbors as well as the effect of an external magnetic field along the z direction, and are used to described a large range of one-dimensional quantum systems [13].”
§ Entanglement in Quantum Critical Phenomena > Abstract
What the sources say
- Not checked yetcites [13]
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
The paper relies on this without support.
The block entanglement entropy S_L as a function of block size L captures the large-scale behavior of quantum correlations at a critical regime.
“The motivation behind the present approach is straightforward: by considering the entanglement S L of a spin block as a function of its size L, and by characterizing it for large L, one expects to capture the large-scale behavior of quantum correlations at a critical regime.”
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In the citing paper
“However, it is well known that even small disorder can lead to novel quantum phases such as Bose [182] or Fermi [183] glasses (see also the section on disorder in Ref. [184] and references therein).”
From Can one trust quantum simulators? (2012) · cites this paper as [184]
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Fiola et al. are cited as reference 26 but listed at 27
MinorThe text associates Fiola et al. with reference [26], which lists Holzhey, Larsen, and Wilczek. Fiola et al. are listed at reference 27.
“Fiola et al. [26]”
“26. C. Holzhey, F. Larsen, and F. Wilczek, Nucl. Phys. B424, 443 (1994).”
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Metadata
- Authors
- Vidal, G., Latorre, J. I., Rico, E., & Kitaev, A.
- Affiliations
- California Institute of Technology (US)
- Universitat de Barcelona (ES)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2003-06-02
- Volume
- 90
- Issue
- 22
- Pages
- 227902-227902
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W1990516747
- Semantic Scholar
- 4dcee3be43c2d9b785f36939fd4cf4462639b375
- Cited by
- 2,793
- References
- 30
- Citations by year
- 2026: 80 · 2025: 150 · 2024: 129 · 2023: 152 · 2022: 173 · 2021: 133 · 2020: 144 · 2019: 148 · 2018: 122 · 2017: 135 · 2016: 130 · 2015: 130 · 2014: 137 · 2013: 102 · 2012: 97
- FWCI
- 59.61
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- Dipòsit Digital de la Universitat de Barcelona (Universitat de Barcelona) (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CaltechAUTHORS (California Institute of Technology) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- RECERCAT (Consorci de Serveis Universitaris de Catalunya) (submittedVersion, public-domain)
- arXiv (Cornell University)
- Funders
- European Commission
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- quantum entanglement, quantum critical point, quantum phase transitions, spin chains, entanglement scaling, long-range correlations, conformal field theory, entanglement entropy, quantum information
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