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Form factors in equilibrium and non-equilibrium mixed states of the Ising model
Strongly Correlated Electrons · Submitted 2 May 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We verify that this is in agreement with a non-equilibrium generalization of the KMS relations, and we conjecture that the leading large-distance behavior of order and disorder non-equilibrium correlation functions contains oscillations in the log of the distance between the fields.
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The abstract, sentence by sentence
Using the "Liouville space'' (the space of operators) of the massive Ising model of quantum field theory, there is a natural definition of form factors in any mixed state. These generalize the usual form factors, and are building blocks for mixed-state correlation functions. We study the cases of mixed states that are diagonal in the asymptotic particle basis, and obtain exact expressions for all mixed-state form factors of order and disorder fields. We use novel techniques based on deriving and solving a system of nonlinear functional differential equations. We then write down the full form factor expansion for mixed-state correlation functions of these fields. Under weak analytic conditions on the eigenvalues of the density matrix, this is an exact large-distance expansion. The form factors agree with the known finite-temperature form factors when the mixed state is specialized to a thermal Gibbs ensemble. Our results can be used to analyze correlation functions in generalized Gibbs ensembles (which occur after quantum quenches). Applying this to the density matrix for non-equilibrium steady states with energy flows, we observe that non-equilibrium form factors have branch cuts in rapidity space. We verify that this is in agreement with a non-equilibrium generalization of the KMS relations, and we conjecture that the leading large-distance behavior of order and disorder non-equilibrium correlation functions contains oscillations in the log of the distance between the fields.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Using the "Liouville space'' (the space of operators) of the massive Ising model of quantum field theory, there is a natural definition of form factors in any mixed state. | contribution 0.65 | 0.38 | 0.07 | asserted 1.00 |
| s1 | These generalize the usual form factors, and are building blocks for mixed-state correlation functions. | interpretation 0.45 | 0.36 | 0.08 | asserted 1.00 |
| s2 | We study the cases of mixed states that are diagonal in the asymptotic particle basis, and obtain exact expressions for all mixed-state form factors of order and disorder fields. | contribution 0.60 | 0.96 | 0.17 | asserted 1.00 |
| s3 | We use novel techniques based on deriving and solving a system of nonlinear functional differential equations. | method 0.99 | 0.95 | 0.03 | asserted 1.00 |
| s4 | We then write down the full form factor expansion for mixed-state correlation functions of these fields. | contribution 0.64 | 0.96 | 0.12 | asserted 1.00 |
| s5 | Under weak analytic conditions on the eigenvalues of the density matrix, this is an exact large-distance expansion. | result 0.67 | 0.84 | 0.10 | asserted 0.99 |
| s6 | The form factors agree with the known finite-temperature form factors when the mixed state is specialized to a thermal Gibbs ensemble. | result 0.91 | 0.77 | 0.09 | asserted 1.00 |
| s7 | Our results can be used to analyze correlation functions in generalized Gibbs ensembles (which occur after quantum quenches). | interpretation 0.66 | 0.40 | 0.10 | hedged possibility 0.72 |
| s8 | Applying this to the density matrix for non-equilibrium steady states with energy flows, we observe that non-equilibrium form factors have branch cuts in rapidity space. | result 0.98 | 0.95 | 0.17 | asserted 0.99 |
| s9 | We verify that this is in agreement with a non-equilibrium generalization of the KMS relations, and we conjecture that the leading large-distance behavior of order and disorder non-equilibrium correlation functions contains oscillations in the log of the distance between the fields. | result 0.68 | 0.86 | 0.42 | speculative 0.88 |
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The paper
Original abstract
Using the "Liouville space'' (the space of operators) of the massive Ising model of quantum field theory, there is a natural definition of form factors in any mixed state. These generalize the usual form factors, and are building blocks for mixed-state correlation functions. We study the cases of mixed states that are diagonal in the asymptotic particle basis, and obtain exact expressions for all mixed-state form factors of order and disorder fields. We use novel techniques based on deriving and solving a system of nonlinear functional differential equations. We then write down the full form factor expansion for mixed-state correlation functions of these fields. Under weak analytic conditions on the eigenvalues of the density matrix, this is an exact large-distance expansion. The form factors agree with the known finite-temperature form factors when the mixed state is specialized to a thermal Gibbs ensemble. Our results can be used to analyze correlation functions in generalized Gibbs ensembles (which occur after quantum quenches). Applying this to the density matrix for non-equilibrium steady states with energy flows, we observe that non-equilibrium form factors have branch cuts in rapidity space. We verify that this is in agreement with a non-equilibrium generalization of the KMS relations, and we conjecture that the leading large-distance behavior of order and disorder non-equilibrium correlation functions contains oscillations in the log of the distance between the fields.
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How the literature uses it
5 arXiv papers cite this in 17 sentences
Papers citing it · 7 sentences without a confident label
Contests it 1 paper
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Arguments and counter-examples against the LM series for two-point functions were presented in [ref] , whereas the result for the one-point function was still believed to be true [ref] .
LeClair-Mussardo series for two-point functions in Integrable QFT · Introduction · contest 0.71
Builds on it 1 paper
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Generalising the program used in [ref] , we set up the Liouville space and defined the mixed-state form factor in the Dirac theory.
Mixed-state form factors of $U(1)$ twist fields in the Dirac theory · Conclusion · 0.87
Uses its method or tool 1 paper
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Now, our problem has been reduced to a question of computing the matrix element of a pure exponential on the right-hand side of ([ref]), which can be performed using Bogoliubov transformation, similarly to [ref] .
Mixed-state form factors of $U(1)$ twist fields in the Dirac theory · General solution as integral-operator kernel · 0.91
Compares with it 1 paper
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The results of [ref] only concern the large time limit, nevertheless it is remarkable that the integral series takes the same form as suggested by the works [ref] .
LeClair-Mussardo series for two-point functions in Integrable QFT · Introduction · 0.97
Cites it as background 2 papers
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In [ref] it was suggested that finite temperature correlations should be described by dressed form factors that already take into account certain effects of the background.
LeClair-Mussardo series for two-point functions in Integrable QFT · Earlier results for two-point functions in iQFT · 0.84
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We note that an alternative approach to thermal correlations was developed by Doyon [ref] , however, at present it seems to be confined to the Ising model.
Finite temperature one-point functions in non-diagonal integrable field theories: the sine-Gordon model · Introduction · 0.85
What it cites · 81 sentences without a confident label
Builds on 4 papers
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Generalizing the thermal (or finite-temperature) form factors of [ref] , we have defined mixed-state form factors and evaluated them explicitly in the Ising model, and we have shown how to obtain a full large-distance expansion for mixed-state two-point correlation functions of order and disorder fields from these form factors.
Finite-Temperature Form Factors: a Review · Work done · 1.00
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Generalizing the thermal (or finite-temperature) form factors of [ref] , we have defined mixed-state form factors and evaluated them explicitly in the Ising model, and we have shown how to obtain a full large-distance expansion for mixed-state two-point correlation functions of order and disorder fields from these form factors.
Finite-temperature form factors in the free Majorana theory · Work done · 1.00
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Our form factor expansion can be trivially re-expressed as the determinant of a matrix linear integral operator, generalizing for instance the ideas of [ref] (as was done in [ref] for the thermal case).
From Form Factors to Correlation Functions: The Ising Model · Work done · 0.93
Compares with 4 papers
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In non-equilibrium steady states our expansion requires further regularization which we do not study in detail here, but the leading large-distance behavior is in agreement with the results of [ref] , and we conjecture the presence of logarithmic oscillating subleading factors.
Out of equilibrium correlations in the XY chain · Introduction · 0.99
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It can be used to analyze their large-distance asymptotic expansion: for instance, we find agreement with leading large-distance results of [ref] , and we compute new subleading terms.
Quantum Quench in the Transverse Field Ising Chain II: Stationary State Properties · Introduction · 0.98
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The approach of the order parameter (one-point function), in the ordered phase, to its steady average value after the quench was studied using form factors in [ref] , reproducing the results of [ref] .
Dynamics in the Ising field theory after a quantum quench · Quantum quenches · 0.84
Cites as background 32 papers
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A particular quench corresponding to an abrupt change of the external magnetic field from $h_0$ to $h$ was considered in [ref] .
Quantum Quench in the Transverse Field Ising Chain · Quantum quenches · 0.93
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A particular quench corresponding to an abrupt change of the external magnetic field from $h_0$ to $h$ was considered in [ref] .
Quantum Quench in the Transverse Field Ising chain I: Time evolution of order parameter correlators · Quantum quenches · 0.93
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A particular quench corresponding to an abrupt change of the external magnetic field from $h_0$ to $h$ was considered in [ref] .
Quantum Quench in the Transverse Field Ising Chain II: Stationary State Properties · Quantum quenches · 0.93
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