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.
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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.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
0 of 22 cited sources checked · 0 supported · 0 flagged
- Findingown data
The block entanglement entropy S_L in spin chains near and at a quantum critical point follows scaling laws.
“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.”
Quote in the paperSubstantial claim- 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.
- Findingnone given
The support or number of relevant eigenvectors of rho_L unboundedly grows with L.
“Indeed, more generally we encounter that the support or number of relevant eigenvectors of L unboundedly grows with L.”
Quote in the paperSubstantial claim - Findingreasoning
Already for a chain of N=20 spins it is possible to observe a distinct, characteristic behavior of S_L depending on whether the parameters belong or not to a critical regime.
“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.”
Quote in the paperSubstantial claim - Findingreasoning
For any L, the reduced density matrix rho_L is effectively supported on just a small, bounded subspace of the Hilbert space of the L spins.
“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.”
Quote in the paperSubstantial claim - Findingown data
In critical chains, the entropy S_L diverges logarithmically with L.
“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;”
Quote in the paperSubstantial claim - 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
- Premisecitation
A highly entangled ground state is at the heart of a large variety of collective quantum phenomena.
“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.”
Quote in the paperSubstantial claim0 of 3 sources checked - Interpretationreasoning
If only a limited number of eigenvectors can be retained, then a local computation of rho_L for sufficiently large L is bound to fail.
“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).”
Quote in the paperSubstantial claim - Premisecitation
Entanglement is expected to play a crucial role in the study of quantum phase transitions, where it is responsible for the appearance of long-range correlations at zero temperature.
“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].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [12] source
Not checked in the free run.
- Also stated in: § Abstract · hedged
- Premisecitation
In a 1+1 conformal field theory, the entropy of a subregion of length L reads S_L = c/6 log2 L + k, with coefficient given by the central charges.
“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”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [25] source
Not checked in the free run.
- Interpretationown data
There is a clear distinction between noncritical and critical entanglement, corresponding to two forms of structurally inequivalent ground states.
“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.”
Quote in the paperSubstantial claim - Interpretationown data
For critical regimes, the block entanglement entropy S_L in spin chains coincides with the entropy of a subregion in conformal field theory.
“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.”
- Also stated in: § Critical regime. · boosted
“An exciting, far-reaching feature of the above results is that, for critical regimes, S L fully coincides with entropy computations performed in conformal field theory [22].”
- Also stated in: § Critical regime. · boosted
“In other words, starting from nonrelativistic spin chain models and by performing a microscopic analysis of a relevant quantity in quantum information, we have obtained a scaling law for entanglement that is in full agreement with previous findings in the context of, say, black-hole thermodynamics in 1 1 dimensions [25,26].”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [22] source
Not checked in the free run.
- cites [25] source
Not checked in the free run.
- cites [26]
Not checked in the free run.
- 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.
- Also stated in: § Critical regime. · boosted
- Premisecitation
Srednicki obtained that in 2+1 (3+1) conformal field theories, the entropy of a region R is proportional to the size of its boundary.
“Srednicki [23] has obtained the behavior of entropy in 2 1 (3 1) conformal field theories.For a region R in 2 (respectively 3) spatial dimensions, the entropy of R is proportional to the size R of its boundary, S R R: (13”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [23] source
Not checked in the free run.
- Interpretationcitation
The scaling law for entanglement obtained from spin chains is in full agreement with previous findings in black-hole thermodynamics in 1+1 dimensions.
“In other words, starting from nonrelativistic spin chain models and by performing a microscopic analysis of a relevant quantity in quantum information, we have obtained a scaling law for entanglement that is in full agreement with previous findings in the context of, say, black-hole thermodynamics in 1 1 dimensions [25,26].”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [25] source
Not checked in the free run.
- cites [26]
Not checked in the free run.
- Interpretationreasoning
Noncritical ground-state entanglement corresponds to a weak, semilocal form of entanglement driven by a length scale L0 due to a mass gap.
“Noncritical ground-state entanglement corresponds, thus, to a weak, semilocal form of entanglement driven by the appearance of a length scale L 0 , due, e.g., to a mass gap in the Hamiltonian.”
Quote in the paperSubstantial claim - Interpretationreasoning
The dependence of S_L at criticality is only on the central charges of the relevant theory.
“In the present case, the dependence of S L is only on the central charges of the relevant theory.”
Quote in the paperSubstantial claim - Interpretationcitation
Quantum correlations do not increase under renormalization group transformations.
“Furthermore, such a connection can be exploited to unveil a fundamental law for entanglement in spin systems -namely, that quantum correlations do not increase under renormalization group transformations.”
- 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
“The translation of this idea to the quantum information setting is that entanglement decreases along renormalization group flows.”
Quote in the paperSubstantial claim0 of 3 sources checked - 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
- Premisereasoning
In the XXZ model, the nonanalyticity of the ground-state energy characterizing a phase transition already occurs for a finite chain due to level crossing.
“We recall that in the XXZ model, and due to level crossing, the nonanalyticity of the groundstate energy characterizing a phase transition already occurs for a finite chain.”
Quote in the paperSubstantial claim - Premisenone given
Critical XX and XXZ models are described by a free bosonic field theory with central charges c_b = cbar_b = 1.
“whereas the critical regimes of the XX and XXZ models are described by a free bosonic field theory, c b c c b 1 (see Fig. 1).”
Quote in the paperSubstantial claim - Premisenone given
Critical Ising and XY (gamma != 0) models correspond to a free fermionic field theory with central charges c_f = cbar_f = 1/2.
“Thus, critical Ising and XY (8 Þ 0) models correspond to a free fermionic field theory, with central charges c f c c f 1=2,”
Quote in the paperSubstantial claim - Premisecitation
The c theorem states that central charges associated to the ultraviolet and infrared endpoints of renormalization group flows obey C_UV > C_IR for unitary theories.
“The c theorem states that the central charges associated to the ultraviolet and infrared end points of renormalization group flows, labeled by C UV and C IR , obey the inequality C UV > C IR for unitary theories.”
Quote in the paperSubstantial claim - Interpretationreasoning
Critical ground-state entanglement corresponds to a stronger form of entanglement that embraces the system at all length scales simultaneously.
“Critical ground-state entanglement corresponds to a stronger form of entanglement, one that embraces the system at all length scales simultaneously.”
Quote in the paperSubstantial claim
- May combine two claims. 4
- The quote only partly states this claim. 1
- Classified as probably not a substantial claim. 3
May combine two claims. The quote only partly states this claim.
FindingcitationIn noncritical chains, S_L either vanishes for all L or grows monotonically until it reaches a saturation value at a block size L0.
“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.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [13]
Not checked in the free run.
May combine two claims.
Premisenone givenThe ground state of the XY chain and density matrices for blocks of spins are Gaussian states completely characterized by a correlation matrix of second moments.
“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.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationOsterloh et al. and Osborne and Nielsen found that one- and two-spin entanglement measures in the XY model display a peak either near or at the critical point.
“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.”
Quote in the paperNot a substantial claim0 of 2 sources checkedMay combine two claims.
PremisecitationThe XY model is exactly solvable and allows consideration of an infinite chain.
“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.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [13]
Not checked in the free run.
May combine two claims.
InterpretationcitationEntanglement in spin systems decreases both under local operations and classical communication and along renormalization group trajectories.
“It is noteworthy, then, that entanglement in spin systems decreases both under local operations and classical communication and along renormalization group trajectories.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [27] source
Not checked in the free run.
- cites [30]
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Classified as probably not a substantial claim.
PremisecitationDMRG techniques have reportedly failed to reproduce quantum critical behavior.
“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.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [21]
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe XXZ and XY models are used to describe a large range of one-dimensional quantum systems.
“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].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [13]
Not checked in the free run.
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.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
It is well known that even small disorder can lead to novel quantum phases such as Bose or Fermi glasses.
“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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Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
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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- 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 › Computer Science › Artificial Intelligence › 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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