Time-dependence of correlation functions following a quantum quench

John Cardy; Pasquale Calabrese

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Time-dependence of correlation functions following a quantum quench

Pasquale Calabrese, John Cardy

Statistical Mechanics · Submitted 11 Jan 2006 · 14 references

Pith's verdict: A genuinely foundational Letter: the quench-to-boundary-CFT mapping is the real contribution, the equal-time formulas hold up, and the soft spots are all in the deferred detail rather than the load-bearing claims.

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arXiv IDcond-mat/0601225 (version 2)

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CitationJohn Cardy, Pasquale Calabrese. "Time-dependence of correlation functions following a quantum quench." arXiv:cond-mat/0601225 (2006). https://pith.science/paper/SRZLBDGG

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Editor's note

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The thing to know: this is the paper that turned quench dynamics into a boundary-critical-problem statement, and the statement is right. Prepare a state in the ground state of H0, evolve with a critical H, and the correlation functions are those of a d+1-dimensional slab with conformally invariant boundary conditions and an extrapolation length tau0. For d=1 that gives you Eqs. (5), (9), (10) essentially for free from the standard strip-to-half-plane map. The exponential relaxation of one-point functions with rate set by the bulk scaling dimension x, and the light-cone saturation of two-point functions at t = r/2v, are the results people now quote without thinking. The credit is earned: the derivation is three lines of standard BCFT, but nobody had put those three lines together, and the quasiparticle picture in the introduction is what made the whole thing legible to the cold-atom community. The free-boson integral Eq. (14) is a real independent check, not decoration, and the Barouch-McCoy comparison is a genuine external one. One correction to the reader's report: the classifier says no limitations were stated, but the authors flag the euclidean-to-real-time continuation explicitly in the text, and they are right to. For the equal-time formulas the continuation is safe, as the stress-test notes, since the mapped points sit on the imaginary axis and never hit a singularity for r > 0. So that assumption is acknowledged and, for the load-bearing results, benign. The soft spots are all downstream. The t^{-3/2} approach to the asymptote, the lattice dispersion details, and the stationary-phase argument for the group-velocity picture are deferred to ref [10], which is listed as in preparation, so a reader cannot check them here. The claim that the vertex operator e^{iq phi} reproduces the CFT formulas from Eq. (14) is asserted rather than shown. The Ising check against Ref. [4] is for a time-averaged correlator, which is a weaker test than the instantaneous Eq. (10). And the connected correlator in the ordered phase, where x_b = 0, is noted as subleading but not computed. None of these threatens the central claim. This is for anyone working on non-equilibrium dynamics in extended quantum systems, and it assumes fluency in BCFT, which limits the audience but not the correctness. Send it to referees; the questions worth pressing are the deferred [10] material and the ordered-phase connected correlator.

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Major objections (2)

[Introduction and Eq. (9)] The introduction states that 'the connected correlations do not change significantly from their initial values until time t ∼ |r|/2v,' but Eq. (9) gives, for t<r/2, a correlator proportional to e^{-π(x-xb)t/τ0} times its t=0 value. When xb > x — for instance the Ising free-boundary case quoted later, with x=1/8 and xb=1/2 — this factor grows with t, so the correlator changes substantially well before the light cone. Please reconcile the verbal light-cone picture with Eq. (9), or state the regime in which the correlator is approximately constant.
[Free-boson check, after Eq. (14)] The statement that 'Taking Φ to be the conformal field e^{iqφ} then reproduces all the CFT results before' is asserted rather than shown, and the quantitative asymptotics of Eq. (14) (including the t^{-3/2} correction and the gapped-dispersion oscillations) are deferred to Ref. [10], which is listed as in preparation. Since the abstract's claim that the results 'are checked against those available from solvable models' rests on these checks, please outline at least the leading asymptotic evaluation of Eq. (14) and the vertex-operator identification here, or state explicitly which checks are deferred.

Minor objections (8)

[d>1 discussion] The d=3 formula '⟨φ(r,t)φ(0,t)⟩ ∼ r^{-1}e^{-π(t-r/2v)/τ0} for t < r/2v' has the exponent sign such that the expression grows with r in the stated regime; presumably the intended form is e^{-π(r/2v-t)/τ0}. Please check.
[d>1 discussion] The phrase 'the energy density parameter ⟨φ²⟩' in d>1 should be distinguished from the conserved T_tt discussed in d=1, since the d=1 text emphasizes that the local energy density does not decay; as written the two statements invite confusion.
[Free-boson check, after Eq. (14)] For the CFT dispersion Ω=v|k| the group velocity Ω'_k is constant, so there is no stationary point; the light-cone behaviour of Eq. (14) comes from the k=0 kink rather than from stationary phase. Please clarify that the stationary-phase argument applies to the general nonlinear dispersion.
[Ising-XY comparison] The comparison with Ref. [4] is for the time-averaged correlation function, which is a weaker test than the instantaneous Eq. (10); a sentence stating this would make the strength of the check clear.
[Eq. (2)] The contour is specified as τ1 = -ε-it and τ2 = ε-it, but the operator is then inserted at τ = ε+it; please make the sign convention consistent.
[Free-boson check] The notation 'Ω_0 > 0' appears to mean the gap m0 of H0, but Ω_{0k} is already used for the initial dispersion; please use a distinct symbol.
[Other CFT results, Eq. (11)-(12)] Eqs. (11) and (12) involve operators at two different real times, so they rely on a stronger analytic continuation than the equal-time formulas; a sentence noting this would be helpful.
[PACS numbers] The first PACS code, 73.43.Nq, appears unrelated to the subject matter; consider a more appropriate code.

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The paper

The paper claims

A quench's aftermath can be read off from a boundary critical problem in one higher dimension, giving explicit decay laws and a light-cone time for correlations.

The reading

Take a quantum system in its ground state, then suddenly change the hamiltonian so it evolves under a new one. The paper argues that the resulting time-dependent correlation functions are not a model-by-model mystery: they are the correlation functions of a classical critical system in one extra dimension, living in a slab whose two faces are set by the initial state. When the new hamiltonian sits at a quantum critical point with linear quasiparticle dispersion, the slab problem becomes a boundary conformal field theory, and everything can be computed. The payoff is explicit formulas: one-point functions decay exponentially at a rate proportional to the operator's scaling dimension, and two-point correlations stay frozen until a sharp time set by separation divided by twice the quasiparticle speed, then saturate to a separation-dependent value. The authors check these against exactly solvable lattice models and explain them with a picture of entangled quasiparticles flying apart at a classical speed.

What carries the argument

The load-bearing device is the identification of the real-time expectation value with a euclidean path integral on a slab of width $2t_0$, where $t_0$ is the extrapolation length of boundary critical phenomena — the distance the actual initial state sits from the renormalisation-group-invariant boundary state. For $d=1$ critical quenches the slab is mapped to the upper half-plane by $w = (2t_0/1000000000)1000000000 z$, turning the problem into standard boundary conformal field theory: one-point functions come from the half-plane amplitude $A_b (2,1000000000 z)^{-x}$, two-point functions from the universal half-plane form with its undetermined function $F(1000000000)$, and the limits of $F$ are fixed by bulk-boundary short-distance expansions. The second device is the quasiparticle picture, which explains the formulas and generalises past CFT: quasiparticles entangled over the initial correlation length fly apart at the group velocity, so correlations at separation $r$ change only once $t$ reaches $r/2v$.

Extended reading notes

Core claim

The central claim is that a quench — preparation in the ground state of $H_0$, then unitary evolution under a different $H$ — maps onto a $d+1$-dimensional slab geometry whose boundary conditions encode the initial state, so that the renormalisation-group theory of boundary critical phenomena controls the late-time, long-distance behaviour. In $d=1$ with $H$ at a critical point described by a conformal field theory, the slab correlators follow from the conformal map $w=(2t_0/1000000000...)$ — concretely, from $w = (2t_0/1000000000)$-style strip-to-half-plane mapping with an extrapolation length $t_0$ of order the inverse gap of $H_0$. The resulting predictions are that a primary scalar of bulk dimension $x$ decays as $[1000000000/(4t_0)/1000000000(1000000000 t/2t_0)]^x$, that the two-point function is essentially unchanged for $t<r/2v$ and saturates to $e^{-1000000000 x r/2t_0}$ for $t>r/2v$, and that the local energy density does not decay at all because it is conserved. The authors present these as universal in their exponents and in the light-cone time, with the overall scale $t_0$ non-universal but cancelling in ratios.

Load-bearing premise

The authors state it themselves: the whole real-time prediction rests on the assumption that the euclidean boundary-conformal-field-theory answer, valid at large imaginary times, can simply be analytically continued to large real times.

Editorial extensions

If this is right

Ratios of relaxation rates for different local observables are universal and equal ratios of their bulk scaling dimensions, giving a parameter-free prediction for quench experiments near a critical point.
Connected correlations after a critical quench do not decay to zero but saturate at an exponentially small value in the separation, because the excess energy does not relax.
Locally conserved densities, notably the energy density, are exceptions: they approach a constant with an oscillatory power-law correction set by the slowest quasiparticles at the zone boundary.
For gapped or lattice dispersion relations the sharp light cone is smeared, the first signal arrives at $r/2v_m$ with $v_m$ the maximum group velocity, and spatial oscillations appear when that velocity occurs at nonzero momentum.
In $d=3$ above the upper critical dimension, mean-field slab results give $r^{-1}e^{-1000000000(t-r/2v)/t_0}$ before saturation.

Reading between the lines

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

Because only the exponent $x$ and the light-cone time enter, cold-atom experiments that measure two observables at once could test the theory without knowing $t_0$ at all, by checking that decay rates scale as the ratio of the two scaling dimensions.
The construction suggests a practical diagnostic: any quench experiment whose one-point relaxation is not exponential, or whose correlation front is not at $r/2v$, is either away from the critical point or dominated by a conserved quantity rather than a primary field.
The same slab logic should apply to Renyi and entanglement entropies of subsystems after a quench, since those are also boundary-condition-dominated quantities, which would tie the observed linear entanglement growth to the same quasiparticle flight time.
If the analytic continuation from imaginary to real time fails for non-primary operators, the natural place to look for the failure is a quench to a theory with a dangerously irrelevant coupling, where the euclidean boundary fixed point is reached only asymptotically.

Original abstract

We show that the time-dependence of correlation functions in an extended quantum system in d dimensions, which is prepared in the ground state of some hamiltonian and then evolves without dissipation according to some other hamiltonian, may be extracted using methods of boundary critical phenomena in d+1 dimensions. For d=1 particularly powerful results are available using conformal field theory. These are checked against those available from solvable models. They may be explained in terms of a picture, valid more generally, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate classically through the system.

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We show that the time-dependence of correlation functions in an extended quantum system in d dimensions, which is prepared in the ground state of some hamiltonian and then evolves without dissipation according to some other hamiltonian, may be extracted using methods of boundary critical phenomena in d+1 dimensions. For d=1 particularly powerful results are available using conformal field theory. These are checked against those available from solvable models. They may be explained in terms of a picture, valid more generally, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate classically through the system.

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	We show that the time-dependence of correlation functions in an extended quantum system in d dimensions, which is prepared in the ground state of some hamiltonian and then evolves without dissipation according to some other hamiltonian, may be extracted using methods of boundary critical phenomena in d+1 dimensions.	contribution 0.86	0.88	0.86	hedged possibility 0.70
s1	For d=1 particularly powerful results are available using conformal field theory.	background 0.58	0.13	0.05	asserted 0.93
s2	These are checked against those available from solvable models.	method 0.79	0.74	0.05	asserted 0.91
s3	They may be explained in terms of a picture, valid more generally, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate classically through the system.	interpretation 1.00	0.46	0.21	hedged possibility 0.98

What kind of paper the classifier reports this is: theoretical result 0.55 · evidence from proof or derivation 0.91.

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arXiv listing abstract v2

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References

Works cited (14)

[10]

P. Calabrese and J. Cardy, in preparation

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[4]

K. Sengupta et al., Phys. Rev. A 69, 053616 (2004)

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F. Igloi and H. Rieger, Phys. Rev. Lett. 85, 3233 (2000)

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M. Greiner et al., Nature (London) 415, 39 (2002); C. Orzel et al., Science 291, 2386 (2001)

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R. W. Cherng and L. S. Levitov, cond-mat/0512689

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H. W. Diehl, in Phase Transitions and Critical Phenom- ena, v. 10, ed C. Domb and J. L. Lebowitz (Academic, London, 1986); Int. J. Mod. Phys. B 11, 3503 (1997)

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[7]

J. L. Cardy, Nucl. Phys. B 240, 514 (1984)

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[8]

J. L. Cardy, Boundary Conformal Field Theory, in Ency- clopedia of Mathematical Physics , ed J.-P. Francoise, G. Naber, and S. Tsun Tsou, (Elsevier, Amsterdam, 2006)

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[9]

P. Calabrese and J. Cardy, J. Stat. Mech. P04010 (2005) [cond-mat/0503393]

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H. Bl¨ ote et al., Phys. Rev. Lett. 56, 742 (1986)

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M. Krech et al., Phys. Rev. E 52, 1345, (1995)

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R. Klimpel and S. Dietrich, Phys. Rev. B 60, 16977 (1999)

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M. Krech, Phys. Rev. E 56 1642 (1997), and references therein

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hep-th 2010-05 conditional novelty 5.0 of 10

A probe Dp-brane rotating in the internal sphere of AdS×S develops a worldvolume horizon with Hawking temperature proportional to the angular velocity, making the dual defect CFT behave thermally while the bulk stays ...
Entanglement Entropy from a Holographic Viewpoint
gr-qc 2012-04 conditional novelty 2.0 of 10

A survey of the Ryu-Takayanagi proposal that entanglement entropy in a conformal field theory equals the area of a minimal surface in its dual anti-de Sitter gravity, with applications to black holes and boundary CFT.
A holographic view on physics out of equilibrium
hep-th 2010-06 accept novelty 2.0 of 10

Holography recasts non-equilibrium dynamics of strongly coupled quantum field theories as classical evolution of black hole geometries, organized by how far the system sits from equilibrium in amplitude and wavelength.
Entanglement entropy and conformal field theory
cond-mat.stat-mech 2009-05 accept novelty 2.0 of 10

Conformal field theory fixes the universal scaling of entanglement entropy in one-dimensional critical systems, with the central charge setting the logarithmic coefficient.
Holographic systems far from equilibrium: a review
hep-th 2018-10 accept novelty 1.0 of 10

Holography can follow real-time evolution of inhomogeneous, anisotropic strongly coupled systems by solving bulk gravity PDEs, and this review maps the resulting steady-state and quench phenomenology.

How later papers used it

331 later papers cite it

Evidence: substantial. 319 of the citing papers share no author with it, and 25 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 478 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

1 paper by the same authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 12 papers with this label · 1 by the same authors

By the same authors

Used the path integral approach in imaginary time developed in this paper to map quantities to correlation functions in a strip geometry with boundary conditions corresponding to conformally invariant boundary states.

Quenches in initially coupled Tomonaga-Luttinger Liquids: a conformal field theory approach (2021) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

119 arXiv papers cite this in 177 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 · 89 sentences without a confident label

Contests it 1 paper

In the limit when the duration of the quench $δt→0$ and $d2≤Δ<d$ the response of the system becomes singular, which is at odds with the predictions of sudden quench approximation [ref] .

Universality of fast quenches from the conformal perturbation theory · Introduction · contest 0.82

Builds on it 1 paper

To obtain the conformal dynamics on the strip, we generalize the Calabrese–Cardy ansatz for critical initial states to the non-Hermitian setting [ref] .

Biorthogonal Conformal Dynamics in Non-Hermitian Quantum Quenches · Complex extrapolation parameters · 0.99

Uses its method or tool 5 papers

Within CFT we use the Calabrese-Cardy representation of such a state as [ref] | Ψ_0 ⟩∝e^- τ_0 H_0 |B ⟩⟩, where $|B ⟩⟩$ is a (unnormalized) conformally invariant boundary state.

Entanglement entropy growth in stochastic conformal field theory and the KPZ class · Introduction · 0.90
Following the setup of [ref] , the global quench is effectively modeled as a BCFT calculation.

Entanglement Tsunami in (1+1)-Dimensions · Global Quench · 0.86
The strategy of [ref] to compute correlators after this process is to treat them as path integrals on a bounded complex surface in complex spacetime (real space times complex time), conformally map this surface to the complex upper half plane (UHP) and compute the correlators in the UHP using boundary conformal field theory (BCFT), also known as surface critical behavior [ref] .

Mutual information after a local quench in conformal field theory · Introduction · 0.93

Uses its data or values 1 paper

From Ref. [ref] one has $Δ_b=0$.

Entanglement entropy growth in stochastic conformal field theory and the KPZ class · Introduction · 0.93

Compares with it 5 papers

[In [ref] , it was shown that UV modes of a $1+1$ dimensional CFT thermalize faster than the IR modes irrespective of the strength of the coupling constant.] While a standard perturbation technique in QCD predicts that the thermalization process should be bottom-up [ref] , these papers get an opposite behavior.

Holographic Thermalization with Weyl Corrections · Introduction · 0.80
This is contrasted with the finite behavior in the case of an instantaneous quench [ref] .

On fast quenches and spinning correlators · Introduction · 1.00
The physics is similar to that seen in homogenous quantum quenches in quantum field theories [ref] , but the explicit entanglement between subsystems brings in new features.

Quenched coupling, entangled equilibria, and correlated composite operators: a tale of two O(N) models · Introduction · 0.89

Cites it as support 1 paper

As argued by the authors of [ref] , the EE after quench is well approximated by the universal parts.

Quantum Local Quench, AdS/BCFT and Yo-Yo String · Entanglement entropy after the quench · 0.96

Cites it as background 65 papers

In the quantum field theory context, the conformal symmetry is strong enough to fix the dynamics of entanglement for certain subregions in two dimensional conformal field theories (CFT) [ref] .

On the Time Scaling of Entanglement in Integrable Scale-Invariant Theories · Introduction · 0.90
Up to now, only few results [ref] are known beyond non-interacting Hamiltonians.

Quenching the magnetic flux in 1d fermionic ring: Loschmidt echo and edge singularity · Introduction · 0.89
Recall the representation of gapped states with small coherence length introduced by Calabrese and Cardy [ref] .

Entanglement entropy growth in stochastic conformal field theory and the KPZ class · Ic- Twist operators & entanglement entropy in stochastic CFT · 0.95

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.

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