PHYSICAL REVIEW E 90, 022107 (2014)
Fluctuation theorems and the generalized Gibbs ensemble in integrable systems James M. Hickey and Sam Genway School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom (Received 18 March 2014; published 8 August 2014) We derive fluctuation relations for a many-body quantum system prepared in a generalized Gibbs ensemble subject to a general nonequilibrium protocol. By considering isolated integrable systems, we find generalizations to the Tasaki-Crooks and Jarzynski relations. Our approach is illustrated by studying the one-dimensional quantum Ising model subject to a sudden change in the transverse field, where we find that the statistics of the work done and irreversible entropy show signatures of quantum criticality. We discuss these fluctuation relations in the context of thermalization. DOI: 10.1103/PhysRevE.90.022107
PACS number(s): 05.40.−a, 02.30.Ik, 05.70.Ln
I. INTRODUCTION
The last decade has seen an explosion of interest, both experimental and theoretical, in the exploration of out-ofequilibrium quantum systems. A particular motivation has been the advance in ultracold-atom experiments, which allow closed quantum systems to be realized, manipulated, and probed [1–4]. Driven by this progress, theoretical investigations have revealed a number of interesting phenomena involving nonequilibrium dynamics, fluctuations, and thermalization [5–13]. In particular, much interest has arisen in the nonequilibrium fluctuations of a system when parametrically changing the Hamiltonian over time. When applying such nonequilibrium protocols to systems prepared initially in a thermal state, these fluctuations have been shown to obey Tasaki-Crooks and Jarzynski relations [14–19]. A much-studied nonequilibrium protocol is the global quantum quench, whereby the system is prepared in a ground state |0 of a specified Hamiltonian H (λ0 ) which depends parametrically on an experimentally tunable global parameter λ0 . This parameter is changed abruptly such that λ0 → λt , and the state |0 is allowed to evolve coherently under the Hamiltonian: |ψt = e−iH (λt )t |0. It has been shown that after a long time t, small subsystems within the closed system can be described by statistical ensembles. Generically, the reduced density matrices of subsystems evolve towards thermal states [20–25]. However, for integrable systems where there are additional local conserved quantities, subsystems are known to often approach the so-called generalized Gibbs ensemble (GGE) [26–32]. The GGE is specified by an extensive number of conserved charges Ik on the global system and associated Lagrange multipliers μk . In this work we consider one of the GGE subsystems of an integrable system in isolation and ask about fluctuation phenomena. By studying isolated integrable systems and using a two-point measurement scheme [33,34], we derive generalizations to the Tasaki-Crooks and Jarzynski relations for systems initially in a GGE. We emphasize the mathematical analogy with traditional fluctuation theorems by using terms such as “generalized work done.” We illustrate our results by applying them to the transverse-field Ising model (TFIM) [35] and calculate generalized work, irreversible entropy, and joint probability distributions associated with quench protocols. Finally it must be noted that recent work has shown that in certain instances the GGE is not necessarily the statistical 1539-3755/2014/90(2)/022107(5)
description of the stationary state properties of an integrable model [36–38]. The fluctuation theorems derived in this work may be used as a check of whether a system has converged to a GGE.
II. GENERALIZED WORK AND IRREVERSIBLE ENTROPY
We will study systems with Hamiltonian H (λ0 ) initially in states described by the GGE ρGGE (λ0 ) =
e−
k
Z
μk Ik
,
(1)
where Ik are conserved quantities with [H (λ0 ),Ik ] = [Ik ,Ik ] = 0 ∀ k,k , μk are associated Lagrange multipliers, and the normalization is defined by Z = Tr(e− k μk Ik ). For times t 0 up to the time t = τ , work is done on the system under a protocol where λ0 → λτ . The final Hamiltonian H (λτ ) has a set of conserved quantities we will denote Jk . We consider a continuous class of operators Ik (λ) such that Ik (λ0 ) = Ik and Ik (λτ ) = Jk ; in essence each quantity Jk of H (λτ ) is the continuation of the conserved quantity Ik of H (λ0 ) with the same k. We first consider performing a measurement on the initial set of conserved quantities {Ik }, which results in eigenvalues {0k }; this collapses the state of the system into an eigenstate of the set {Ik }. We then evolve the system under unitary dynamics U (τ,0) describing the protocol λ0 → λτ before measuring the eigenvalues {τk } associated with the final set of conserved quantities {Jk }, at time τ . We will use two complete basis sets |m,{0k } and |n,{τk } where the indices m(n) distinguish between states with the same set of eigenvalues {0k }({τk }) associated with {Ik } ({Jk }). According to this measurement scheme, the probability density, i.e., the probability for the generalized work done for the forward protocol, is given by PF (W ) =
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m,n,{0k },{τk }
2 e− k μk 0 k n, τ U (τ,0)m, 0k δ Z(λ0 ) k
k k × W− μ˜k τ − μk 0 .
(2)
k
©2014 American Physical Society
JAMES M. HICKEY AND SAM GENWAY
PHYSICAL REVIEW E 90, 022107 (2014)
Multiplying both sides by e−W and rearranging we find PF (W ) = eW −F , PB (−W )
(3)
where F = log Tr(e− k μk Ik )/Tr(e− k μ˜ k Jk ). This is a generalization of the Tasaki-Crooks fluctuation relation and encodes the relationship between the measurement distributions of the forward protocol, as described above, and the backward protocol. The generalized work done is the relative entropy between the forward and backward nonequilibrium process. Therefore it shares mathematical properties, such as concavity, with thermodynamic entropies. Integrable systems generically do not thermalize and so are not directly related to the thermodynamics of the system. The backward work distribution is obtained by preparing the system initially in the state ρGGE (λτ ) = k e−μ˜ k Jk /Z(λτ ) and evolving the system under the reverse protocol λτ → λ0 defined by U † (τ,0). Taking the Fourier transform of both sides of Eq. (3) allows this fluctuation theorem to be expressed in terms of characteristic functions [39–42]: χF (u,τ ) Z(λτ ) = , χB (−u + i,τ ) Z(λ0 ) where
(4)
χF (u,τ ) =
dW eiuW PF (W )
= Tr[U † (τ,0)eiu
k
μ˜ k Jk
U (τ,0)e−iu
k
μk Ik
ρGGE ]. (5)
Eq. (7) we obtain the standard Tasaki-Crooks and Jarzynski equalities [43]. For example, one could consider the standard nonequilibrium work protocol for a thermal state, where the system is initially thermalized at a temperature β and the final ˜ With such a scheme one state is at a different temperature β. would need to examine the relative entropies or the generalized work done rather than the standard thermodynamic work in order to test the generalized fluctuation relations presented here. The fluctuation relations (7) encompass all the information on the generalized work done in an integrable model: regardless of how far the system is driven from equilibrium, they are determined by the equilibrium properties of the system in a GGE. In the thermodynamic limit, the law of large numbers ensures that entropy of an isolated never decreases under a thermodynamic process. For a finite size system the second law of thermodynamics must be extended, and, in accordance with the Jarzynski equality, the expectation value of the work done is greater than or equal to the change in equilibrium free energy. This extends from the canonical ensemble to GGEs, with Eq. (6) implying that W F . For nonideal processes, the difference between these two processes is the average generalized irreversible work done: W = W irr + F . The change in entropy under this protocol has contributions associated with normalisation changes and irreversibility. In this instance, as the system is closed, the irreversible entropy production is simply the irreversible “work done” [15,33]:
This global quench protocol does not change the overall Hilbert space dimension of the problem. Evaluating the characteristic function at the point u = i we obtain χF (u = i) = e−W = e−F =
Z(λτ ) , Z(λ0 )
(6)
which is the extension of the Jarzynski equality [15] to the GGE. The above relationships simplify if three requirements are met: the first is that the Lagrange multipliers of the initial and final GGEs are equal, i.e., μk = μ˜k , ∀k; the second is that the initial [H (λ0 )] and final [H (λτ )] Hamiltonian be expressed as a tensor product of the Hilbert spaces spanned by the associated conserved quantities; the final requirement is that conserved charges Jk and Ik live in the same subspace ∀k and the nonequilibrium protocol U (τ,0) does not connect the different subspaces spanned by these conserved quantities. Then we can consider measurements solely on individual Jk and Ik leading to the simpler analogs of Eqs. (4) and (6): Zk (λτ ) χF (u,τ ) = , χF (i) = e−W = e−Fi , χB (−u + i,τ ) Zk (λ0 )
S irr = W − F = W irr .
Until now we have studied a scheme where we measure all of the conserved quantities {Ik } and {Jk } under changing λ0 → λτ . This measurement scheme could be a challenge to implement in practice. However, we can also consider a simpler scheme where we examine the joint probability density for measuring one conserved quantity from each set Ik and Jk at the beginning and end of the nonequilibrium protocol, respectively. The outcomes of these measurements are eigenvalues 0k and τk , where in general we consider k = k. Taking two complete basis sets as above, |m,0k and |n,τk , where m(n) now distinguishes between states with only one common eigenvalue 0k (τk ), we find a generic two-point joint probability density [33,34]: PFTP (W ) =
m,0k ρGGE (λ0 )m,0k
m,n,0k ,τk
2
× n,τk U (τ,0)m,0k δ W − τk − 0k . (9)
(7)
where Zk is the normalization associated with the kth conserved charge and Fk = − log Zk (λτ )/Zk (λ0 ). Later we will find that this simplification can arise when studying a quench protocol, i.e., U (τ,0) = 1 as τ → 0+ . We note that if the initial state is a thermal state, there is only one Lagrange multiplier, i.e., μk = μ˜k = β, and the conserved quantities of interest (the Hamiltonians) live within the same Hilbert space. In this instance the above requirements hold necessarily, and from
(8)
The superscript TP indicates we are considering a two-point measurement joint distribution associated with two conserved quantities. Although similar to our previous considerations, here the “generalized work done” has no special dependence on the chemical potentials and we consider it to be the most general “work” distribution, which is directly related to the GGE form for integrable systems.
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PHYSICAL REVIEW E 90, 022107 (2014)
III. EXAMPLE: TRANSVERSE-FIELD ISING MODEL
Writing Nk (λ0 ) = Zk (λ0 )Z−k (λ0 ) we may express the trace using the fermionic number states |nk ,n−k of the initial Hamiltonian H (λ0 ) to find 1 nk ,n−k |eiu(μ˜ k Jk +μ˜ −k J−k ) χF (u) = Nk (λ0 ) n =0,1
We now apply the theoretical framework described previously to the concrete example of the one-dimensional transverse-field Ising model (TFIM). The TFIM is described by the Hamiltonian z σiz σi+1 −λ σix . (10) H =− i
×e
i
In the N-spin model with periodic boundary conditions which we will study, the operators σix,z are the Pauli operators at site i and the parameter λ denotes the transverse field strength. This is a paradigmatic model of a quantum phase transition: in the thermodynamic limit the ground state changes singularly at λ = ±1. For |λ| < 1 the ground state is ferromagnetic in nature while when |λ| > 1 it is paramagnetic [35]. This Hamiltonian may be diagonalized exactly using textbook techniques employing Jordan-Wigner fermions and a Bogoliubov transformation with a set of Bogoluibov angles ϕk , which reduce the Hamiltonian to † H = k (λ)(γk γk − 1/2) = Hk (λ0 ). (11) k
±k
k
|nk ,n−k .
(12)
These matrix elements are computable analytically, and the analytic form of the forward characteristic function is χF (u) =
1 [e−2(1+iu)μk (e2iuμ˜ k cos2 αk + sin2 αk ) Nk (λ0 ) + 2e−(1+iu)μk eiuμ˜ k + (cos2 αk + e2iuμ˜ k sin2 αk )], (13)
where αk = α˜ k − αk and we have used the fact μk = μ−k and μ˜ k = μ˜ −k . Similarly, the backward characteristic function may be evaluated straightforwardly: χB (v) =
†
Here γk (γk ) is the fermionic quasiparticle annihilation (cre ation) operator and k (λ) = 2 (λ − cos k)2 + sin2 k. The wave vectors k lie in the range [−π,π ] and are given by k = π n/N where n = −N + 1, −N + 3, . . . ,N − 1. (To be concrete, we focus on the case where N is even.) In this model the local conserved quantities of the model are simply linear combinations of the occupation numbers [29] of each † fermionic mode γk γk . Thus one can prepare a subsystem in † a state ρGGE (λ0 ) ∝ e− k μk γk γk via a quantum quench from a state |ψ(λt 0 ) with the Lagrange multipliers μk fixed † † by the expectation values γk γk = Tr[ρGGE (λ0 )γk γk ]. The † eigenvalues of these fermionic occupation operators γk γk are labeled nk = 0,1; we note also this quench preserves integrability. Having prepared our system of interest in a GGE, we will consider a protocol implemented via instantaneous switch from λ0 to λ†τ , where thefinal Hamiltonian is H (λτ ) = k k (λτ )(γ˜k γ˜k − 1/2) = k Hk (λτ ). Here we note that the pre- and postquench fermionic operators are necessarily different but related to each other through the difference in Bogoliubov angles ϕk which diagonalize the Hamiltonian. Furthermore, the chemical potentials can be calculated for the initial and final GGEs [7,26] by μk (μ˜ k ) = − log tan2 αk (− log tan2 α˜ k ) where αk (α˜k ) are the difference in Bogoliubov angles which diagonalize H (λt 0 ) and H (λ0 )[H (λτ )]. In the TFIM, the fermionic occupations form a set of good quantum numbers, and the quench protocol implies U (τ,0) = 1 as τ → 0+ . The final fermionic operators † after the quench given by Jk = γ˜k γ˜k are linear combinations † of γ±k and γ±k . They therefore remain within the ±k subspace of the original conserved quantities, and under the quench protocol satisfy the three requirements for the simplified Eq. (7) to hold. Therefore we first consider the case of measuring just J±k and I±k . Using the results above, Eq. (5) reduces to χF (u) = Tr(eiu(μ˜ k Jk +μ˜ −k J−k ) e−(1+iu)(μk Ik +μ−k I−k ) )/Zk (λ0 )Z−k (λ0 ).
−(1+iu)(μk Ik +μ−k I−k )
1 [e−2(1+iv)μ˜ k (e2ivμk cos2 αk + sin2 αk ) Nk (λτ ) + 2e−(1+iv)μ˜ k eivμk + (cos2 αk + e2ivμk sin2 αk )]. (14)
Setting v = −u + i we find the ratio of the forward and backward characteristic functions equals Nk (λτ )/Nk (λ0 ) = Zk (λτ )Z−k (λτ )/Zk (λ0 )Z−k (λ0 ), which is expected from the form of Eq. (7). Evaluating the derivative ∂iu χF (u)|u=0 = W we find the irreversible work done under this nonequilibrium protocol is given by W = [2μ˜ k e−2μk cos2 αk − 2μk e−2μk + 2μ˜k e−μk − 2μk e−μk + 2μ˜ k sin2 αk ],
(15)
and hence the irreversible entropy is given by S irr = k (λτ ) log N + W . This quantity is plotted in Fig. 1 as a Nk (λ0 ) function of the prequench field λ0 for different k modes and system size N with fixed λt 0 = 0. Focusing on the irreversible entropy associated with measuring on the k mode with the smallest magnitude k = π/N , we find a peak around the quantum critical point λ0 = 1. This peak sharpens with increasing N and becomes singular in the thermodynamic limit. The interpretation of this is simple: if we prepare the system in a GGE at the quantum critical point via a quench numerous excitations are produced due to vanishing energy gap. The existence of these excitations leads to the production of irreversible entropy on driving the system away from criticality, i.e., the irreversible entropy is associated with the initial state and is not intrinsically linked to the nonequilibrium protocol [44]. Another interesting regime is where the initial and final Lagrange multipliers are equal, but λ0 = λτ . In the TFIM these conditions, combined with the spectral properties of {Ik } and {Jk }, imply the ratio of the forwards- and backwards-protocol characteristic functions is unity and the irreversible entropy is always zero, irrespective of any singular features of the model. This is in stark contrast to the case of a thermal state whereupon approaching the quantum critical point the irreversible entropy
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PHYSICAL REVIEW E 90, 022107 (2014)
λτ=1.5
k=0.1, λ0 = 0.1
3
1
k = 0.1 k = 0.5 k = 1.5 W
ΔS
TP
irr
2
λτ = 0.75 λτ = 0.95 λτ = 1.0 λτ = 1.1
0.8
1
0.6 0.4 0.2
0 0
0.5
1 λ0
1.5
2
0 0
λτ=1.5 N = 10 N = 100 N = 500 N = 2500
irr
ΔS
3
1
1.5 k’
2
2.5
3
FIG. 2. (Color online) The work done as defined by the two-point measurement on modes k = 0.1 and k as a function of k . The initial GGE is prepared via quench in the transverse field from 0 → 0.1, while the GGE compared with at the end of the protocol is prepared via quenches from 0 → λτ . When the final GGE is in the ferromagnetic phase excitations are added to each k mode with the largest change in occupation occurring at k = 0.
5 4
0.5
2 1 0 0
0.5
1 λ0
1.5
2
FIG. 1. (Color online) Top: The irreversible entropy associated with the mode k is plotted as a function of the value λ0 . This grows and sharpens about the quantum critical point as k → 0, marking the irreversibility associated with criticality. Bottom: The irreversible entropy of the longest wavelength mode, k = π/N , grows and sharpens about the quantum critical point with increasing system size.
in the thermodynamic limit diverges due to the closing of the gap in the spectrum [35,43]. Further evidence of the quantum critical point is provided from a general two-point measurement: we now consider † measuring the Ik = γk γk followed by a measurement of † another mode’s occupation Jk = γ˜k γ˜k , where k = k. Using the two-point measurement scheme the joint probability density [see Eq. (9)] and characteristic function are accessible analytically. From this we find the “generalized work done” during such a protocol is †
†
W TP = Tr[(γ˜k γ˜k − γk γk )ρGGE (λ0 )] = +
1 1 + e μk
1 −2μk (e cos2 αk + e−μk + sin2 αk ). (16) Nk
Fixing λ0 = 0.1, λt 0 = 0 and k = 0.1, we plot the work done as a function of k in Fig. 2, for various λτ . Effectively the “work done” is the difference in occupation of the fermionic modes. The nonequilibrium protocol adds excitations to the system leading to the difference in occupation being positive, and increases as the protocol approaches the critical point λτ → 1. At the critical point, the difference in occupation at k = 0 equals 1/2. If the nonequilibrium protocol crosses the critical point, many excitations are produced, and this leads
the occupation difference to approach unity as k → 0. The fact that the largest change in occupation occurs at k = 0 is consistent with the gap closing around this wave vector at criticality.
IV. CONCLUSIONS
We derived appropriate fluctuation theorems for integrable systems whose equilibrium states are described be GGEs, exemplified with the archetypal example of the TFIM. We find a peak in the irreversible entropy when quenching around the critical point which sharpens as the wavelength of the mode and system size increases. This we can understand in terms of the closing of a gap at the equilibrium critical point. Correspondingly features of this singularity manifest within a simpler two-point measurement scheme. These generalized fluctuation theorems provide a nonequilibrium test of whether a system follows a GGE; this is of timely interest as several recent studies highlight that a GGE description may be inappropriate in certain cases. Of particular interest in future work will be the case of systems which exhibit prethermalization where the system relaxes close to a GGE on short time scales, but eventually reaches thermal equilibrium on a longer time scale [4,45–47]. In such systems it will be of interest to study how fluctuation properties are affected by the time scale on which the initial state is prepared.
ACKNOWLEDGMENTS
We thank Matteo Marcuzzi for fruitful discussions and comments on the manuscript. This work was supported by the European Union Research Scholarship provided by the University of Nottingham and the Leverhulme Trust Grant No. F/00114/BG.
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022107-5
Fluctuation theorems and the generalized Gibbs ensemble in integrable systems James M. Hickey and Sam Genway School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom (Received 18 March 2014; published 8 August 2014) We derive fluctuation relations for a many-body quantum system prepared in a generalized Gibbs ensemble subject to a general nonequilibrium protocol. By considering isolated integrable systems, we find generalizations to the Tasaki-Crooks and Jarzynski relations. Our approach is illustrated by studying the one-dimensional quantum Ising model subject to a sudden change in the transverse field, where we find that the statistics of the work done and irreversible entropy show signatures of quantum criticality. We discuss these fluctuation relations in the context of thermalization. DOI: 10.1103/PhysRevE.90.022107
PACS number(s): 05.40.−a, 02.30.Ik, 05.70.Ln
I. INTRODUCTION
The last decade has seen an explosion of interest, both experimental and theoretical, in the exploration of out-ofequilibrium quantum systems. A particular motivation has been the advance in ultracold-atom experiments, which allow closed quantum systems to be realized, manipulated, and probed [1–4]. Driven by this progress, theoretical investigations have revealed a number of interesting phenomena involving nonequilibrium dynamics, fluctuations, and thermalization [5–13]. In particular, much interest has arisen in the nonequilibrium fluctuations of a system when parametrically changing the Hamiltonian over time. When applying such nonequilibrium protocols to systems prepared initially in a thermal state, these fluctuations have been shown to obey Tasaki-Crooks and Jarzynski relations [14–19]. A much-studied nonequilibrium protocol is the global quantum quench, whereby the system is prepared in a ground state |0 of a specified Hamiltonian H (λ0 ) which depends parametrically on an experimentally tunable global parameter λ0 . This parameter is changed abruptly such that λ0 → λt , and the state |0 is allowed to evolve coherently under the Hamiltonian: |ψt = e−iH (λt )t |0. It has been shown that after a long time t, small subsystems within the closed system can be described by statistical ensembles. Generically, the reduced density matrices of subsystems evolve towards thermal states [20–25]. However, for integrable systems where there are additional local conserved quantities, subsystems are known to often approach the so-called generalized Gibbs ensemble (GGE) [26–32]. The GGE is specified by an extensive number of conserved charges Ik on the global system and associated Lagrange multipliers μk . In this work we consider one of the GGE subsystems of an integrable system in isolation and ask about fluctuation phenomena. By studying isolated integrable systems and using a two-point measurement scheme [33,34], we derive generalizations to the Tasaki-Crooks and Jarzynski relations for systems initially in a GGE. We emphasize the mathematical analogy with traditional fluctuation theorems by using terms such as “generalized work done.” We illustrate our results by applying them to the transverse-field Ising model (TFIM) [35] and calculate generalized work, irreversible entropy, and joint probability distributions associated with quench protocols. Finally it must be noted that recent work has shown that in certain instances the GGE is not necessarily the statistical 1539-3755/2014/90(2)/022107(5)
description of the stationary state properties of an integrable model [36–38]. The fluctuation theorems derived in this work may be used as a check of whether a system has converged to a GGE.
II. GENERALIZED WORK AND IRREVERSIBLE ENTROPY
We will study systems with Hamiltonian H (λ0 ) initially in states described by the GGE ρGGE (λ0 ) =
e−
k
Z
μk Ik
,
(1)
where Ik are conserved quantities with [H (λ0 ),Ik ] = [Ik ,Ik ] = 0 ∀ k,k , μk are associated Lagrange multipliers, and the normalization is defined by Z = Tr(e− k μk Ik ). For times t 0 up to the time t = τ , work is done on the system under a protocol where λ0 → λτ . The final Hamiltonian H (λτ ) has a set of conserved quantities we will denote Jk . We consider a continuous class of operators Ik (λ) such that Ik (λ0 ) = Ik and Ik (λτ ) = Jk ; in essence each quantity Jk of H (λτ ) is the continuation of the conserved quantity Ik of H (λ0 ) with the same k. We first consider performing a measurement on the initial set of conserved quantities {Ik }, which results in eigenvalues {0k }; this collapses the state of the system into an eigenstate of the set {Ik }. We then evolve the system under unitary dynamics U (τ,0) describing the protocol λ0 → λτ before measuring the eigenvalues {τk } associated with the final set of conserved quantities {Jk }, at time τ . We will use two complete basis sets |m,{0k } and |n,{τk } where the indices m(n) distinguish between states with the same set of eigenvalues {0k }({τk }) associated with {Ik } ({Jk }). According to this measurement scheme, the probability density, i.e., the probability for the generalized work done for the forward protocol, is given by PF (W ) =
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m,n,{0k },{τk }
2 e− k μk 0 k n, τ U (τ,0)m, 0k δ Z(λ0 ) k
k k × W− μ˜k τ − μk 0 .
(2)
k
©2014 American Physical Society
JAMES M. HICKEY AND SAM GENWAY
PHYSICAL REVIEW E 90, 022107 (2014)
Multiplying both sides by e−W and rearranging we find PF (W ) = eW −F , PB (−W )
(3)
where F = log Tr(e− k μk Ik )/Tr(e− k μ˜ k Jk ). This is a generalization of the Tasaki-Crooks fluctuation relation and encodes the relationship between the measurement distributions of the forward protocol, as described above, and the backward protocol. The generalized work done is the relative entropy between the forward and backward nonequilibrium process. Therefore it shares mathematical properties, such as concavity, with thermodynamic entropies. Integrable systems generically do not thermalize and so are not directly related to the thermodynamics of the system. The backward work distribution is obtained by preparing the system initially in the state ρGGE (λτ ) = k e−μ˜ k Jk /Z(λτ ) and evolving the system under the reverse protocol λτ → λ0 defined by U † (τ,0). Taking the Fourier transform of both sides of Eq. (3) allows this fluctuation theorem to be expressed in terms of characteristic functions [39–42]: χF (u,τ ) Z(λτ ) = , χB (−u + i,τ ) Z(λ0 ) where
(4)
χF (u,τ ) =
dW eiuW PF (W )
= Tr[U † (τ,0)eiu
k
μ˜ k Jk
U (τ,0)e−iu
k
μk Ik
ρGGE ]. (5)
Eq. (7) we obtain the standard Tasaki-Crooks and Jarzynski equalities [43]. For example, one could consider the standard nonequilibrium work protocol for a thermal state, where the system is initially thermalized at a temperature β and the final ˜ With such a scheme one state is at a different temperature β. would need to examine the relative entropies or the generalized work done rather than the standard thermodynamic work in order to test the generalized fluctuation relations presented here. The fluctuation relations (7) encompass all the information on the generalized work done in an integrable model: regardless of how far the system is driven from equilibrium, they are determined by the equilibrium properties of the system in a GGE. In the thermodynamic limit, the law of large numbers ensures that entropy of an isolated never decreases under a thermodynamic process. For a finite size system the second law of thermodynamics must be extended, and, in accordance with the Jarzynski equality, the expectation value of the work done is greater than or equal to the change in equilibrium free energy. This extends from the canonical ensemble to GGEs, with Eq. (6) implying that W F . For nonideal processes, the difference between these two processes is the average generalized irreversible work done: W = W irr + F . The change in entropy under this protocol has contributions associated with normalisation changes and irreversibility. In this instance, as the system is closed, the irreversible entropy production is simply the irreversible “work done” [15,33]:
This global quench protocol does not change the overall Hilbert space dimension of the problem. Evaluating the characteristic function at the point u = i we obtain χF (u = i) = e−W = e−F =
Z(λτ ) , Z(λ0 )
(6)
which is the extension of the Jarzynski equality [15] to the GGE. The above relationships simplify if three requirements are met: the first is that the Lagrange multipliers of the initial and final GGEs are equal, i.e., μk = μ˜k , ∀k; the second is that the initial [H (λ0 )] and final [H (λτ )] Hamiltonian be expressed as a tensor product of the Hilbert spaces spanned by the associated conserved quantities; the final requirement is that conserved charges Jk and Ik live in the same subspace ∀k and the nonequilibrium protocol U (τ,0) does not connect the different subspaces spanned by these conserved quantities. Then we can consider measurements solely on individual Jk and Ik leading to the simpler analogs of Eqs. (4) and (6): Zk (λτ ) χF (u,τ ) = , χF (i) = e−W = e−Fi , χB (−u + i,τ ) Zk (λ0 )
S irr = W − F = W irr .
Until now we have studied a scheme where we measure all of the conserved quantities {Ik } and {Jk } under changing λ0 → λτ . This measurement scheme could be a challenge to implement in practice. However, we can also consider a simpler scheme where we examine the joint probability density for measuring one conserved quantity from each set Ik and Jk at the beginning and end of the nonequilibrium protocol, respectively. The outcomes of these measurements are eigenvalues 0k and τk , where in general we consider k = k. Taking two complete basis sets as above, |m,0k and |n,τk , where m(n) now distinguishes between states with only one common eigenvalue 0k (τk ), we find a generic two-point joint probability density [33,34]: PFTP (W ) =
m,0k ρGGE (λ0 )m,0k
m,n,0k ,τk
2
× n,τk U (τ,0)m,0k δ W − τk − 0k . (9)
(7)
where Zk is the normalization associated with the kth conserved charge and Fk = − log Zk (λτ )/Zk (λ0 ). Later we will find that this simplification can arise when studying a quench protocol, i.e., U (τ,0) = 1 as τ → 0+ . We note that if the initial state is a thermal state, there is only one Lagrange multiplier, i.e., μk = μ˜k = β, and the conserved quantities of interest (the Hamiltonians) live within the same Hilbert space. In this instance the above requirements hold necessarily, and from
(8)
The superscript TP indicates we are considering a two-point measurement joint distribution associated with two conserved quantities. Although similar to our previous considerations, here the “generalized work done” has no special dependence on the chemical potentials and we consider it to be the most general “work” distribution, which is directly related to the GGE form for integrable systems.
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III. EXAMPLE: TRANSVERSE-FIELD ISING MODEL
Writing Nk (λ0 ) = Zk (λ0 )Z−k (λ0 ) we may express the trace using the fermionic number states |nk ,n−k of the initial Hamiltonian H (λ0 ) to find 1 nk ,n−k |eiu(μ˜ k Jk +μ˜ −k J−k ) χF (u) = Nk (λ0 ) n =0,1
We now apply the theoretical framework described previously to the concrete example of the one-dimensional transverse-field Ising model (TFIM). The TFIM is described by the Hamiltonian z σiz σi+1 −λ σix . (10) H =− i
×e
i
In the N-spin model with periodic boundary conditions which we will study, the operators σix,z are the Pauli operators at site i and the parameter λ denotes the transverse field strength. This is a paradigmatic model of a quantum phase transition: in the thermodynamic limit the ground state changes singularly at λ = ±1. For |λ| < 1 the ground state is ferromagnetic in nature while when |λ| > 1 it is paramagnetic [35]. This Hamiltonian may be diagonalized exactly using textbook techniques employing Jordan-Wigner fermions and a Bogoliubov transformation with a set of Bogoluibov angles ϕk , which reduce the Hamiltonian to † H = k (λ)(γk γk − 1/2) = Hk (λ0 ). (11) k
±k
k
|nk ,n−k .
(12)
These matrix elements are computable analytically, and the analytic form of the forward characteristic function is χF (u) =
1 [e−2(1+iu)μk (e2iuμ˜ k cos2 αk + sin2 αk ) Nk (λ0 ) + 2e−(1+iu)μk eiuμ˜ k + (cos2 αk + e2iuμ˜ k sin2 αk )], (13)
where αk = α˜ k − αk and we have used the fact μk = μ−k and μ˜ k = μ˜ −k . Similarly, the backward characteristic function may be evaluated straightforwardly: χB (v) =
†
Here γk (γk ) is the fermionic quasiparticle annihilation (cre ation) operator and k (λ) = 2 (λ − cos k)2 + sin2 k. The wave vectors k lie in the range [−π,π ] and are given by k = π n/N where n = −N + 1, −N + 3, . . . ,N − 1. (To be concrete, we focus on the case where N is even.) In this model the local conserved quantities of the model are simply linear combinations of the occupation numbers [29] of each † fermionic mode γk γk . Thus one can prepare a subsystem in † a state ρGGE (λ0 ) ∝ e− k μk γk γk via a quantum quench from a state |ψ(λt 0 ) with the Lagrange multipliers μk fixed † † by the expectation values γk γk = Tr[ρGGE (λ0 )γk γk ]. The † eigenvalues of these fermionic occupation operators γk γk are labeled nk = 0,1; we note also this quench preserves integrability. Having prepared our system of interest in a GGE, we will consider a protocol implemented via instantaneous switch from λ0 to λ†τ , where thefinal Hamiltonian is H (λτ ) = k k (λτ )(γ˜k γ˜k − 1/2) = k Hk (λτ ). Here we note that the pre- and postquench fermionic operators are necessarily different but related to each other through the difference in Bogoliubov angles ϕk which diagonalize the Hamiltonian. Furthermore, the chemical potentials can be calculated for the initial and final GGEs [7,26] by μk (μ˜ k ) = − log tan2 αk (− log tan2 α˜ k ) where αk (α˜k ) are the difference in Bogoliubov angles which diagonalize H (λt 0 ) and H (λ0 )[H (λτ )]. In the TFIM, the fermionic occupations form a set of good quantum numbers, and the quench protocol implies U (τ,0) = 1 as τ → 0+ . The final fermionic operators † after the quench given by Jk = γ˜k γ˜k are linear combinations † of γ±k and γ±k . They therefore remain within the ±k subspace of the original conserved quantities, and under the quench protocol satisfy the three requirements for the simplified Eq. (7) to hold. Therefore we first consider the case of measuring just J±k and I±k . Using the results above, Eq. (5) reduces to χF (u) = Tr(eiu(μ˜ k Jk +μ˜ −k J−k ) e−(1+iu)(μk Ik +μ−k I−k ) )/Zk (λ0 )Z−k (λ0 ).
−(1+iu)(μk Ik +μ−k I−k )
1 [e−2(1+iv)μ˜ k (e2ivμk cos2 αk + sin2 αk ) Nk (λτ ) + 2e−(1+iv)μ˜ k eivμk + (cos2 αk + e2ivμk sin2 αk )]. (14)
Setting v = −u + i we find the ratio of the forward and backward characteristic functions equals Nk (λτ )/Nk (λ0 ) = Zk (λτ )Z−k (λτ )/Zk (λ0 )Z−k (λ0 ), which is expected from the form of Eq. (7). Evaluating the derivative ∂iu χF (u)|u=0 = W we find the irreversible work done under this nonequilibrium protocol is given by W = [2μ˜ k e−2μk cos2 αk − 2μk e−2μk + 2μ˜k e−μk − 2μk e−μk + 2μ˜ k sin2 αk ],
(15)
and hence the irreversible entropy is given by S irr = k (λτ ) log N + W . This quantity is plotted in Fig. 1 as a Nk (λ0 ) function of the prequench field λ0 for different k modes and system size N with fixed λt 0 = 0. Focusing on the irreversible entropy associated with measuring on the k mode with the smallest magnitude k = π/N , we find a peak around the quantum critical point λ0 = 1. This peak sharpens with increasing N and becomes singular in the thermodynamic limit. The interpretation of this is simple: if we prepare the system in a GGE at the quantum critical point via a quench numerous excitations are produced due to vanishing energy gap. The existence of these excitations leads to the production of irreversible entropy on driving the system away from criticality, i.e., the irreversible entropy is associated with the initial state and is not intrinsically linked to the nonequilibrium protocol [44]. Another interesting regime is where the initial and final Lagrange multipliers are equal, but λ0 = λτ . In the TFIM these conditions, combined with the spectral properties of {Ik } and {Jk }, imply the ratio of the forwards- and backwards-protocol characteristic functions is unity and the irreversible entropy is always zero, irrespective of any singular features of the model. This is in stark contrast to the case of a thermal state whereupon approaching the quantum critical point the irreversible entropy
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PHYSICAL REVIEW E 90, 022107 (2014)
λτ=1.5
k=0.1, λ0 = 0.1
3
1
k = 0.1 k = 0.5 k = 1.5 W
ΔS
TP
irr
2
λτ = 0.75 λτ = 0.95 λτ = 1.0 λτ = 1.1
0.8
1
0.6 0.4 0.2
0 0
0.5
1 λ0
1.5
2
0 0
λτ=1.5 N = 10 N = 100 N = 500 N = 2500
irr
ΔS
3
1
1.5 k’
2
2.5
3
FIG. 2. (Color online) The work done as defined by the two-point measurement on modes k = 0.1 and k as a function of k . The initial GGE is prepared via quench in the transverse field from 0 → 0.1, while the GGE compared with at the end of the protocol is prepared via quenches from 0 → λτ . When the final GGE is in the ferromagnetic phase excitations are added to each k mode with the largest change in occupation occurring at k = 0.
5 4
0.5
2 1 0 0
0.5
1 λ0
1.5
2
FIG. 1. (Color online) Top: The irreversible entropy associated with the mode k is plotted as a function of the value λ0 . This grows and sharpens about the quantum critical point as k → 0, marking the irreversibility associated with criticality. Bottom: The irreversible entropy of the longest wavelength mode, k = π/N , grows and sharpens about the quantum critical point with increasing system size.
in the thermodynamic limit diverges due to the closing of the gap in the spectrum [35,43]. Further evidence of the quantum critical point is provided from a general two-point measurement: we now consider † measuring the Ik = γk γk followed by a measurement of † another mode’s occupation Jk = γ˜k γ˜k , where k = k. Using the two-point measurement scheme the joint probability density [see Eq. (9)] and characteristic function are accessible analytically. From this we find the “generalized work done” during such a protocol is †
†
W TP = Tr[(γ˜k γ˜k − γk γk )ρGGE (λ0 )] = +
1 1 + e μk
1 −2μk (e cos2 αk + e−μk + sin2 αk ). (16) Nk
Fixing λ0 = 0.1, λt 0 = 0 and k = 0.1, we plot the work done as a function of k in Fig. 2, for various λτ . Effectively the “work done” is the difference in occupation of the fermionic modes. The nonequilibrium protocol adds excitations to the system leading to the difference in occupation being positive, and increases as the protocol approaches the critical point λτ → 1. At the critical point, the difference in occupation at k = 0 equals 1/2. If the nonequilibrium protocol crosses the critical point, many excitations are produced, and this leads
the occupation difference to approach unity as k → 0. The fact that the largest change in occupation occurs at k = 0 is consistent with the gap closing around this wave vector at criticality.
IV. CONCLUSIONS
We derived appropriate fluctuation theorems for integrable systems whose equilibrium states are described be GGEs, exemplified with the archetypal example of the TFIM. We find a peak in the irreversible entropy when quenching around the critical point which sharpens as the wavelength of the mode and system size increases. This we can understand in terms of the closing of a gap at the equilibrium critical point. Correspondingly features of this singularity manifest within a simpler two-point measurement scheme. These generalized fluctuation theorems provide a nonequilibrium test of whether a system follows a GGE; this is of timely interest as several recent studies highlight that a GGE description may be inappropriate in certain cases. Of particular interest in future work will be the case of systems which exhibit prethermalization where the system relaxes close to a GGE on short time scales, but eventually reaches thermal equilibrium on a longer time scale [4,45–47]. In such systems it will be of interest to study how fluctuation properties are affected by the time scale on which the initial state is prepared.
ACKNOWLEDGMENTS
We thank Matteo Marcuzzi for fruitful discussions and comments on the manuscript. This work was supported by the European Union Research Scholarship provided by the University of Nottingham and the Leverhulme Trust Grant No. F/00114/BG.
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