PRL 111, 100401 (2013)
week ending 6 SEPTEMBER 2013
PHYSICAL REVIEW LETTERS
Infinite-Time Average of Local Fields in an Integrable Quantum Field Theory After a Quantum Quench G. Mussardo SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136 Trieste, Italy International Centre for Theoretical Physics (ICTP), I-34151 Trieste, Italy (Received 28 April 2013; revised manuscript received 5 August 2013; published 3 September 2013) The infinite-time average of the expectation values of local fields of any interacting quantum theory after a global quench process are key quantities for matching theoretical and experimental results. For quantum integrable field theories, we show that they can be obtained by an ensemble average that employs a particular limit of the form factors of local fields and quantities extracted by the generalized Bethe ansatz. DOI: 10.1103/PhysRevLett.111.100401
PACS numbers: 05.30.Ch, 05.30.Jp, 11.10.Gh, 11.10.Kk
The aim of this Letter is to set up a statistical ensemble formula for explicitly computing the infinite-time average of the expectation value (EV) of local fields in a ð1 þ 1Þ-dimensional quantum integrable field theory (QIFT) [1] after a quantum quench, i.e., after an abrupt change of the parameters of the Hamiltonian. The subject of quantum quenches has recently attracted a lot of attention, from both experimental and theoretical points of view; see, for instance, Refs. [2–11]. QIFTs are special continuum models of quantum many-body systems: they are special for the presence of an infinite number of conservation charges Qn that strongly constrain their scattering processes and their dynamics (see, e.g., [12] and references therein). In a situation of out-ofequilibrium dynamics, one expects that the asymptotic infinite-time regime of these theories will violate the ergodicity property and therefore its properties could not be recovered by the usual Gibbs ensemble based only on the Hamiltonian. Indeed, it has been advocated in [8] that to describe the stationarity properties of these integrable systems one has to consider a generalized Gibbs ensemble, i.e., an ensemble that not only employs the Hamiltonian but also all the other conserved charges. Such an hypothesis has been shown to be valid in a series of examples, among which are those studied in [9–11], and has acquired by now a well-established level of consensus. Yet, despite important advances on many topics, an explicit formula for computing the infinite-time average of the EV of local fields in QIFT has thus far been elusive. This formula is put forward and proved in this Letter: it consists of the following identity, hOiDA ¼ hOiGGEA ;
(1)
where the two quantities of this equation are defined hereafter. Let Oðx; tÞ be a local field of this QIFT and h c 0 jOðtÞj c 0 i its expectation value on a macroscopic state j c 0 i, not an eigenstate of the Hamiltonian. As shown in [4], this state encodes all the information about the quench process. Being a macroscopic state, j c 0 i is necessarily 0031-9007=13=111(10)=100401(5)
made of an infinite superposition of multiparticle states [9], but the statistical nature of this state is more interesting than that and will be discussed in more detail later. Let us now define the dynamical average (DA) of the field Oðx; tÞ on j c 0 i as the infinite-time average after the quench at t ¼ 0: 1 Zt dth c 0 jOð0; tÞj c 0 i: (2) hOiDA lim t!1 t 0 In infinite volume, or with periodic boundary conditions on a finite interval L, this average is independent of x by the translation invariance of the theory. The compact definition of the generalized Gibbs ensemble average (GGEA) entering Eq. (1) is given by n 1 X 1 Z1 Y di ~ conn ; hOiGGEA fði Þ hQ jOð0; 0Þji n¼0 n! 1 i¼1 2 (3) ~ j1 ; . . . ; n i (hQ j hn ; . . . ; 1 j) denotes the where ji asymptotic multiparticle states of the QIFT expressed in terms of the rapidities i , with relativistic dispersion relation EðÞ ¼ m cosh, pðÞ ¼ m sinh. The GGEA employs the connected diagonal form factor (FF) of the operator O, which are finite functions of the rapidities defined as ~ conn FO hQ jOji 2n;conn ð1 ; . . . n Þ ~ Q i þ i ¼ FPð lim h0jOj; Q iÞ; i !0
(4)
where Q ðn ; . . . ; 1 Þ and FP in front of the expression means taking its finite part, i.e., omitting all the terms of the form i =j and 1=pi , where p is a positive integer, in taking the limit Q ! 0 in the matrix element given above. Equation (3) also employs the filling factor fðÞ of the one-particle state fði Þ ¼ ½eði Þ Sð0Þ1 ;
(5)
where SðÞ is the exact two-body S matrix of the model while the pseudoenergy ðÞ is the solution of the
100401-1
Ó 2013 American Physical Society
PRL 111, 100401 (2013)
generalized Bethe ansatz equation based on all the conserved charges of the theory [13]; see Eq. (23) below. It is worth making a series of comments. (a) The final formula (3) of the generalized Gibbs ensemble average may be regarded as a generalization of the so-called LeClair-Mussardo formula [14], previously established in the context of pure thermal equilibrium. The main difference between the two is that, while the LeClair-Mussardo formula employs the thermodynamic Bethe ansatz [15,16], expression (3) instead employs the generalized Bethe ansatz [9,13], i.e., the formalism that takes into account all the conserved charges of the initial state used in the quench process. (b) Equation (3) consists of a well-defined and, usually, fast convergent series [17]. Particularly useful is the fast convergence of the series, because it permits us to compute the infinite-time averaged EV, for all intents and purposes, by employing just the first few terms, thus saving a lot of analytic and numerical effort. (c) It is also interesting to mention that, restoring in the QIFT an @ dependence, the limit @ ! 0 of the formula (3) solves a long-standing problem of purely mathematical physics, i.e., how to determine the infinite-time averages in purely classical relativistic integrable models when one is in the presence of the so-called infinite-gap solutions [18]. Before embarking on the proof of the identity (1), it is useful to spell out its content by means of the simplest QIFT, i.e., the free theory. Although elementary, the important pedagogical value of this example is to show clearly the necessity to employ in the identity (1) the generalized Gibbs ensemble average (3) [19]. In the infinite volume, the solution of the free equation of motion ðh þ m2 Þðx; tÞ ¼ 0 is Z 1 d ½AðÞei½EðÞtpðÞx þ c:c:; (6) ðx; tÞ ¼ 1 2 with ½AðÞ; Ay ð0 Þ ¼ 2ð 0 Þ. Such a dynamic is supported by the infinite number of nonlocal conserved quantities given by all the mode number occupations 1 jAðÞj2 ; 8 : NðÞ ¼ (7) 2 As shown in the Supplemental Material [20], one can also find the infinite set of local conserved charges Qn made up of two sets Qþ n and Qn , the first is even under the Z2 space-parity, the second odd. Qþ 0 and Q0 are, respectively, the energy and momentum of the field, while the others, up to normalization, can be written as Z d 2nþ1 jAðÞj2 q Q (8) n ¼m n ðÞ; 2
2nþ1 Q n j1 ; . . . ; k i ¼ m
(9) The multiparticle states are common eigenvectors of all these conserved quantities, with eigenvalues
q n ði Þ j1 ; . . . ; k i:
(10)
It is rather evident that the knowledge of the mode occupation jAðÞj2 fixes all the local charges but, under general mathematical and physical conditions, it is also true in 2 reverse that the Q n ’s fix the jAðÞj . Being linearly related one to the other, the two types of conservation laws are then essentially interchangeable. It must be stressed that Eqs. (7), (8), and (10) hold exactly the same also in interacting QIFT (as the SinhGordon model, for instance), where the only thing to do is to substitute, in Eqs. (7) and (8), jAðÞj2 ! jZðÞj2 , where ZðÞ and Zy ðÞ satisfy the Faddev-Zamolodchikov algebra involving the exact S matrix: Zð1 ÞZy ð2 Þ ¼ Sð1 2 ÞZy ð2 ÞZð1 Þ þ 2ð1 2 Þ: In the free theory, the exact solution (6) of the equation of motion allows us to easily compute the DA of any local function F½ðx; tÞ, defined with a proper normal ordering of the operators. Since we are interested in field configurations with finite energy density, our theory has to be defined on a circle of length L and then send L ! 1 so that E^ ¼ E=L is always finite, even in this limit. The momenta of the particles will be quantized in units of 2=L, which become dense when L ! 1. The initial state j c 0 i fixes the modes AðÞ and Ay ðÞ through the condition h c 0 jðx; 0Þj c 0 i ¼ c 0 ðxÞ, where c 0 ðxÞ is a real R periodic function c 0 ðxÞ ¼ ^ c 0 ðx þ LÞ, such that ð1=2LÞ L0 ½ð@x c 0 Þ2 þ m2 c 20 dx ¼ E. Let us now consider in the free theory a series of quenches, whose initial states have in common only the same energy density E=L. Since the energy is a very degenerate observable, each of these quenches corresponds to initial states having different EV of all other conserved charges. The typical outputs for an observable as :2 : are shown in Fig. 1: the strong dependence on the initial data is evident from the large spread of these DAs. These features are easily explained. We focus attention on the DA of this 2
8
6
4
2
2000
q n ðÞ ¼ sinh½ð2n þ 1Þ:
X k i¼1
where qþ n ðÞ ¼ cosh½ð2n þ 1Þ;
week ending 6 SEPTEMBER 2013
PHYSICAL REVIEW LETTERS
4000
6000
8000
10000
t
2 ðtÞ ð1=tÞ Rt dth c 0 j2 ðtÞj c 0 i as a FIG. 1 (color online). 0 function of time t, for different initial states j c 0 i with the same EV of the energy density. The DAs, as defined in Eq. (2), are the asymptotic values of these curves.
100401-2
PRL 111, 100401 (2013)
week ending 6 SEPTEMBER 2013
PHYSICAL REVIEW LETTERS
infinite set of operators (easily computable by a phase-stationary argument): Z d jAðÞj2 b; h:2 :iDA ¼ (11) 2 h:2n :iDA ¼ ð2n 1Þ!!bn :
(12)
Since they explicitly depend on the initial condition through the jAðÞj2 ’s, these DA can never collapse on a single value or be derived by a Gibbs ensemble average involving only the Hamiltonian h:2n :iDA Z1 Trð:2n :eH Þ;
(13)
even if all initial states share the same energy. Notice, however, that the DA (11) and (12) can be put precisely in the form of the generalized Gibbs ensemble average (3), since the pseudoenergy and the nonzero connected FF of these operators are in this case ðÞ ¼ log½1 þ jAðÞj2 ; hQ m j:2n ð0Þ:j~m iconn ¼ 2n ð2nÞ!n;m (see the Supplemental Material [20]). In summary, we have verified that the identity (1) holds in the integrable free theory. Moreover, all the DA of free theory can be recovered expanding in the GGEA generating function 2 Z d hexp½iiGGEA ¼ exp jAðÞj2 : (14) 2 2
As it stands, however, this expression is highly problematic: all terms of the sum in the numerator as well as those present in Z are in fact divergent. In the latter, the divergencies come from the normalization of the eigenstates, Q h0m ; . . . 01 j1 . . . n i ¼ i ð0 Þ, which gives rise to ½ð0Þn when 0i ¼ . In the former, the divergencies come from the FF h0m . . . 01 jOð0Þjj1 . . . n i, once evaluated at 0i ¼ i . These divergencies are an unavoidable consequence of the kinematical pole structure of the FFs [21]. The first cure of the divergencies is to define the theory on a finite interval L. In this case, for large but finite L, the rapidities of the n-particle states entering (15) are solutions of the Bethe equation X 2Ni mL sinhi þ &ði k Þ ¼ ; i ¼ 1;. .. ;n; (17) L ki where &ðÞ ¼ i logSðÞ is the phase shift and fNi g is a sequence of increasing integers. Let us denote the finite volume eigenstates associated with the integers fNi g as j1 ; . . . ; n iL : the corresponding density of states is given by the Jacobian Jð1 ; . . . ; n Þ ¼ detJjk , with Jjk ¼ ð@J j =@k Þ. The functions J i are given by the derivative of the right-hand ide. of (17), X (18) J i ð1 ; . . . ; n Þ ¼ mL coshi þ ’ði k Þ; ki
with the kernel ’ðÞ ¼ iðd=dÞ logSðÞ. There is now a relation between the diagonal form factors in finite volume and the infinite-volume connected form factors defined in Eq. (4) [22]: h1 ; . . . ; n jOj1 ; . . . ; n iL
Let us now proceed to the general proof of the identity (1) for an interacting QIFT, with a two-body S matrix SðÞ assumed to be of fermionic type, i.e., Sð0Þ ¼ 1. Expanding the initial state j c 0 i on the basis of the multiparticle states, which are common eigenvectors of H and all higher charges, its general form is 1 n X 1ZY di Kn ð1 ;. .. ;n Þj1 ; .. .; n i: (15) n¼0 n! i¼1 2 P Posing En ¼ m ni¼1 coshi , at any later time t the EV of a local observable Oðx; tÞ on j c 0 i is jc 0i ¼
1 n Y m X h c 0 jOðtÞj c 0 i 1 Z Y di d0j ¼ Z1 hc 0jc 0i m;n¼0 n!m! i¼1 j¼1 2 2 ðf0 gÞK ðfgÞ eitðEm En Þ Km n
h0m . . . 01 jOð0Þjj1 . . . n i (Z ¼ h c 0 j c 0 i). Taking the dynamical average, one inevitably ends up with the so-called diagonal ensemble n X 1 Z Y di hOiDA ¼ Z1 jKn ð1 ; . . . ; n j2 n! 2 n¼0 i¼1 hn . . . 1 jOð0Þjj1 . . . n i:
(16)
¼
1 J n ð1 ; . . . ; n Þ X O F2l;conn ð ÞJ nl ðfþ g; f gÞ; S fþ g
(19)
f g
where the sum runs on all possible bipartite partitions of the set of rapidities f1 ; . . . ; n g in two disjoint sets made by l and n l rapidities, and J nl ðfþ g; f gÞ ¼ detJþ is the restricted determinant of the submatrix J þ corresponding to the particles in the set fþ g in the presence of those in f g. Notice that the relation (19) involves the kernel ’ðÞ of the Bethe ansatz equations (18), as explicitly shown in the Supplemental Material [20]. Let us now focus our attention on the initial state: if the state j c 0 i is statistically characterized by the EV of all its conserved charges h c 0 jQ n j c 0 i ¼ LQn , it can be shown (see the Supplemental Material [20]) that the quantities jKn ð1 ; . . . ; n Þj2 factorize in terms of a function jKðÞj2 , n Y jKn ð1 ; . . . ; n Þj2 ¼ jKði Þj2 ; (20) i¼1
and moreover KðÞ can always be expressed in terms of an infinite set of variables f n g, conjugated to the conserved charges Q n as
100401-3
PRL 111, 100401 (2013)
PHYSICAL REVIEW LETTERS
jKðÞj2 ¼ e0 ðÞ ; 0 ðÞ ¼
1 X
(21)
þ ½þ n qn ðÞ þ n qn ðÞ;
n¼0
q n ðÞ
given in (9). with the functions With all the information collected above, let us now come back to the dynamical average (16): with the regularization given by the finite interval L, its right-hand side can be written as hOiDA ¼ lim
L!1
TrðeH OÞL ðTreH ÞL
;
(22)
where H is the generalized Hamiltonian that includes all the conserved charges H ¼
1 X
þ ½þ n Qn þ n Qn :
n¼0
One can now easily repeat the argument given in [14,22] and show that the right-hand side precisely coincides with the generalized Gibbs ensemble average (3), where the function ðÞ satisfies the nonlinear integral equation of the generalized Bethe ansatz [13]: ðÞ ¼
1 X
þ ½þ n q ðÞ þ n q ðÞ
n¼0
Z d0 0 ’ð 0 Þ logð1 þ eð Þ Þ: 2
(23)
This concludes the general proof of the identity (1). A pragmatic approach to find the functions ðÞ and 0 ðÞ given the initial state j c 0 i has recently been proposed in [23]. Applications to quench processes in the sinhGordon model (at both the quantum and the classical level) are presented in [18]. It must be stressed that the same formalism can be applied to compute the infinitetime average of local fields in the Lieb-Liniger model, a system which has recently attracted a lot of interest for the on-going experiments in cold-atom physics: indeed, as shown in [24], to recover the Lieb-Liniger results one can take advantage of the fact that the Lieb-Liniger model may be reached by taking the nonrelativistic limit of the sinh-Gordon model, whose form factors are all known. An example relative to quench processes in the Lieb-Liniger system is presented in the Supplemental Material [20]. I would like to thank P. Assis and, in particular, A. De Luca for discussions. This work is supported by the IRSES grants QICFT. Note added.—Recently there has been another proposal [25] to compute the EV, based purely on the Bethe ansatz and checked for the free case of the quantum Ising model. Although very similar to the one presented here, it remains to see how it applies to the interactive case.
week ending 6 SEPTEMBER 2013
[1] For simplicity, we consider here QIFTs with only one type of particle and with a unique ground state. The generalization to theories with many types of particles is straightforward; less trivial is the extension to theories with many vacua, a case that will be considered somewhere else. [2] T. Kinoshita, T. Wenger, and D. S. Weiss, Nature (London) 440, 900 (2006); M. Greiner, O. Mandel, T. W. Ha¨nsch, and I. Bloch, Nature (London) 419, 51 (2002); S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Nature (London) 449, 324 (2007). [3] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991); M. Srednicki, Phys. Rev. E 50, 888 (1994). [4] P. Calabrese and J. Cardy, J. Stat. Mech. (2007) P06008. [5] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011), and references therein. [6] A. Iucci and M. A. Cazalilla, Phys. Rev. A 80, 063619 (2009); T. Barthel and U. Schollwo¨ck, Phys. Rev. Lett. 100, 100601 (2008); S. R. Manmana, S. Wessel, R. M. Noack, and A. Muramatsu, Phys. Rev. B 79, 155104 (2009); D. Rossini, A. Silva, G. Mussardo, and G. Santoro, Phys. Rev. Lett. 102, 127204 (2009); D. Rossini, S. Suzuki, G. Mussardo, G. E. Santoro, and A. Silva, Phys. Rev. B 82, 144302 (2010). [7] J. Berges, S. Borsanyi, and C. Wetterich, Phys. Rev. Lett. 93, 142002 (2004); M. Kollar, F. A. Wolf, and M. Eckstein, Phys. Rev. B 84, 054304 (2011); M. C. Banu˜ls, J. I. Cirac, and M. B. Hastings, Phys. Rev. Lett. 106, 050405 (2011); G. Biroli, C. Kollath, and A. M. La¨uchli, Phys. Rev. Lett. 105, 250401 (2010); G. P. Brandino, A. De Luca, R. M. Konik, and G. Mussardo, Phys. Rev. B 85, 214435 (2012). [8] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007); M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008). [9] D. Fioretto and G. Mussardo, New J. Phys. 12, 055015 (2010). [10] P. Calabrese, F. H. L. Essler, and M. Fagotti, Phys. Rev. Lett. 106, 227203 (2011); J. Stat. Mech. (2012) P07016; J. Stat. Mech. (2012) P07022. [11] V. Gurarie, J. Stat. Mech. (2013) P02014; S. Sotiriadis, D. Fioretto, and G. Mussardo, J. Stat. Mech. (2012) P02017; T. Caneva, E. Canovi, D. Rossini, G. E. Santoro, and A. Silva, J. Stat. Mech. (2011) P07015. [12] G. Mussardo, Statistical Field Theory (Oxford University Press, Oxford, England, 2010). [13] J. Mossel and J. S. Caux, J. Phys. A 45, 255001 (2012). [14] A. LeClair and G. Mussardo, Nucl. Phys. B552, 624 (1999). [15] C. N. Yang and C. P. Yang, J. Math. Phys. (N.Y.) 10, 1115 (1969). [16] Al. B. Zamolodchikov, Nucl. Phys. B342, 695 (1990). [17] The fast convergence of the series can be argued on a phase space argument, see J. Cardy and G. Mussardo, Nucl. Phys. B410, 451 (1993), and has been observed in many examples involving operators with smooth behavior of their form factors at large value of the rapidities. [18] G. Mussardo, P. E. Assis, and A. De Luca (to be published). [19] To appreciate the pedagogical value of this example, notice that the main difference between a free and an interacting QIFT is only at the technical level: while the free theory is solved by the Fourier transform, an interacting QIFT
100401-4
PRL 111, 100401 (2013)
PHYSICAL REVIEW LETTERS
needs the inverse scattering transform, i.e., the Bethe ansatz techniques. [20] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.111.100401 for infinitetime averages in Lieb-Liniger model. [21] F. A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory (World Scientific, Singapore, 1992).
week ending 6 SEPTEMBER 2013
[22] B. Pozsgay and G. Takacs, Nucl. Phys. B788, 209 (2008); B. Pozsgay, J. Stat. Mech. (2011) P01011. [23] J. S. Caux and R. M. Konik, Phys. Rev. Lett. 109, 175301 (2012). [24] M. Kormos, G. Mussardo, and A. Trombettoni, Phys. Rev. A 81, 043606 (2010). [25] J. S. Caux and F. Essler, Phys. Rev. Lett. 110, 257203 (2013).
100401-5
week ending 6 SEPTEMBER 2013
PHYSICAL REVIEW LETTERS
Infinite-Time Average of Local Fields in an Integrable Quantum Field Theory After a Quantum Quench G. Mussardo SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136 Trieste, Italy International Centre for Theoretical Physics (ICTP), I-34151 Trieste, Italy (Received 28 April 2013; revised manuscript received 5 August 2013; published 3 September 2013) The infinite-time average of the expectation values of local fields of any interacting quantum theory after a global quench process are key quantities for matching theoretical and experimental results. For quantum integrable field theories, we show that they can be obtained by an ensemble average that employs a particular limit of the form factors of local fields and quantities extracted by the generalized Bethe ansatz. DOI: 10.1103/PhysRevLett.111.100401
PACS numbers: 05.30.Ch, 05.30.Jp, 11.10.Gh, 11.10.Kk
The aim of this Letter is to set up a statistical ensemble formula for explicitly computing the infinite-time average of the expectation value (EV) of local fields in a ð1 þ 1Þ-dimensional quantum integrable field theory (QIFT) [1] after a quantum quench, i.e., after an abrupt change of the parameters of the Hamiltonian. The subject of quantum quenches has recently attracted a lot of attention, from both experimental and theoretical points of view; see, for instance, Refs. [2–11]. QIFTs are special continuum models of quantum many-body systems: they are special for the presence of an infinite number of conservation charges Qn that strongly constrain their scattering processes and their dynamics (see, e.g., [12] and references therein). In a situation of out-ofequilibrium dynamics, one expects that the asymptotic infinite-time regime of these theories will violate the ergodicity property and therefore its properties could not be recovered by the usual Gibbs ensemble based only on the Hamiltonian. Indeed, it has been advocated in [8] that to describe the stationarity properties of these integrable systems one has to consider a generalized Gibbs ensemble, i.e., an ensemble that not only employs the Hamiltonian but also all the other conserved charges. Such an hypothesis has been shown to be valid in a series of examples, among which are those studied in [9–11], and has acquired by now a well-established level of consensus. Yet, despite important advances on many topics, an explicit formula for computing the infinite-time average of the EV of local fields in QIFT has thus far been elusive. This formula is put forward and proved in this Letter: it consists of the following identity, hOiDA ¼ hOiGGEA ;
(1)
where the two quantities of this equation are defined hereafter. Let Oðx; tÞ be a local field of this QIFT and h c 0 jOðtÞj c 0 i its expectation value on a macroscopic state j c 0 i, not an eigenstate of the Hamiltonian. As shown in [4], this state encodes all the information about the quench process. Being a macroscopic state, j c 0 i is necessarily 0031-9007=13=111(10)=100401(5)
made of an infinite superposition of multiparticle states [9], but the statistical nature of this state is more interesting than that and will be discussed in more detail later. Let us now define the dynamical average (DA) of the field Oðx; tÞ on j c 0 i as the infinite-time average after the quench at t ¼ 0: 1 Zt dth c 0 jOð0; tÞj c 0 i: (2) hOiDA lim t!1 t 0 In infinite volume, or with periodic boundary conditions on a finite interval L, this average is independent of x by the translation invariance of the theory. The compact definition of the generalized Gibbs ensemble average (GGEA) entering Eq. (1) is given by n 1 X 1 Z1 Y di ~ conn ; hOiGGEA fði Þ hQ jOð0; 0Þji n¼0 n! 1 i¼1 2 (3) ~ j1 ; . . . ; n i (hQ j hn ; . . . ; 1 j) denotes the where ji asymptotic multiparticle states of the QIFT expressed in terms of the rapidities i , with relativistic dispersion relation EðÞ ¼ m cosh, pðÞ ¼ m sinh. The GGEA employs the connected diagonal form factor (FF) of the operator O, which are finite functions of the rapidities defined as ~ conn FO hQ jOji 2n;conn ð1 ; . . . n Þ ~ Q i þ i ¼ FPð lim h0jOj; Q iÞ; i !0
(4)
where Q ðn ; . . . ; 1 Þ and FP in front of the expression means taking its finite part, i.e., omitting all the terms of the form i =j and 1=pi , where p is a positive integer, in taking the limit Q ! 0 in the matrix element given above. Equation (3) also employs the filling factor fðÞ of the one-particle state fði Þ ¼ ½eði Þ Sð0Þ1 ;
(5)
where SðÞ is the exact two-body S matrix of the model while the pseudoenergy ðÞ is the solution of the
100401-1
Ó 2013 American Physical Society
PRL 111, 100401 (2013)
generalized Bethe ansatz equation based on all the conserved charges of the theory [13]; see Eq. (23) below. It is worth making a series of comments. (a) The final formula (3) of the generalized Gibbs ensemble average may be regarded as a generalization of the so-called LeClair-Mussardo formula [14], previously established in the context of pure thermal equilibrium. The main difference between the two is that, while the LeClair-Mussardo formula employs the thermodynamic Bethe ansatz [15,16], expression (3) instead employs the generalized Bethe ansatz [9,13], i.e., the formalism that takes into account all the conserved charges of the initial state used in the quench process. (b) Equation (3) consists of a well-defined and, usually, fast convergent series [17]. Particularly useful is the fast convergence of the series, because it permits us to compute the infinite-time averaged EV, for all intents and purposes, by employing just the first few terms, thus saving a lot of analytic and numerical effort. (c) It is also interesting to mention that, restoring in the QIFT an @ dependence, the limit @ ! 0 of the formula (3) solves a long-standing problem of purely mathematical physics, i.e., how to determine the infinite-time averages in purely classical relativistic integrable models when one is in the presence of the so-called infinite-gap solutions [18]. Before embarking on the proof of the identity (1), it is useful to spell out its content by means of the simplest QIFT, i.e., the free theory. Although elementary, the important pedagogical value of this example is to show clearly the necessity to employ in the identity (1) the generalized Gibbs ensemble average (3) [19]. In the infinite volume, the solution of the free equation of motion ðh þ m2 Þðx; tÞ ¼ 0 is Z 1 d ½AðÞei½EðÞtpðÞx þ c:c:; (6) ðx; tÞ ¼ 1 2 with ½AðÞ; Ay ð0 Þ ¼ 2ð 0 Þ. Such a dynamic is supported by the infinite number of nonlocal conserved quantities given by all the mode number occupations 1 jAðÞj2 ; 8 : NðÞ ¼ (7) 2 As shown in the Supplemental Material [20], one can also find the infinite set of local conserved charges Qn made up of two sets Qþ n and Qn , the first is even under the Z2 space-parity, the second odd. Qþ 0 and Q0 are, respectively, the energy and momentum of the field, while the others, up to normalization, can be written as Z d 2nþ1 jAðÞj2 q Q (8) n ¼m n ðÞ; 2
2nþ1 Q n j1 ; . . . ; k i ¼ m
(9) The multiparticle states are common eigenvectors of all these conserved quantities, with eigenvalues
q n ði Þ j1 ; . . . ; k i:
(10)
It is rather evident that the knowledge of the mode occupation jAðÞj2 fixes all the local charges but, under general mathematical and physical conditions, it is also true in 2 reverse that the Q n ’s fix the jAðÞj . Being linearly related one to the other, the two types of conservation laws are then essentially interchangeable. It must be stressed that Eqs. (7), (8), and (10) hold exactly the same also in interacting QIFT (as the SinhGordon model, for instance), where the only thing to do is to substitute, in Eqs. (7) and (8), jAðÞj2 ! jZðÞj2 , where ZðÞ and Zy ðÞ satisfy the Faddev-Zamolodchikov algebra involving the exact S matrix: Zð1 ÞZy ð2 Þ ¼ Sð1 2 ÞZy ð2 ÞZð1 Þ þ 2ð1 2 Þ: In the free theory, the exact solution (6) of the equation of motion allows us to easily compute the DA of any local function F½ðx; tÞ, defined with a proper normal ordering of the operators. Since we are interested in field configurations with finite energy density, our theory has to be defined on a circle of length L and then send L ! 1 so that E^ ¼ E=L is always finite, even in this limit. The momenta of the particles will be quantized in units of 2=L, which become dense when L ! 1. The initial state j c 0 i fixes the modes AðÞ and Ay ðÞ through the condition h c 0 jðx; 0Þj c 0 i ¼ c 0 ðxÞ, where c 0 ðxÞ is a real R periodic function c 0 ðxÞ ¼ ^ c 0 ðx þ LÞ, such that ð1=2LÞ L0 ½ð@x c 0 Þ2 þ m2 c 20 dx ¼ E. Let us now consider in the free theory a series of quenches, whose initial states have in common only the same energy density E=L. Since the energy is a very degenerate observable, each of these quenches corresponds to initial states having different EV of all other conserved charges. The typical outputs for an observable as :2 : are shown in Fig. 1: the strong dependence on the initial data is evident from the large spread of these DAs. These features are easily explained. We focus attention on the DA of this 2
8
6
4
2
2000
q n ðÞ ¼ sinh½ð2n þ 1Þ:
X k i¼1
where qþ n ðÞ ¼ cosh½ð2n þ 1Þ;
week ending 6 SEPTEMBER 2013
PHYSICAL REVIEW LETTERS
4000
6000
8000
10000
t
2 ðtÞ ð1=tÞ Rt dth c 0 j2 ðtÞj c 0 i as a FIG. 1 (color online). 0 function of time t, for different initial states j c 0 i with the same EV of the energy density. The DAs, as defined in Eq. (2), are the asymptotic values of these curves.
100401-2
PRL 111, 100401 (2013)
week ending 6 SEPTEMBER 2013
PHYSICAL REVIEW LETTERS
infinite set of operators (easily computable by a phase-stationary argument): Z d jAðÞj2 b; h:2 :iDA ¼ (11) 2 h:2n :iDA ¼ ð2n 1Þ!!bn :
(12)
Since they explicitly depend on the initial condition through the jAðÞj2 ’s, these DA can never collapse on a single value or be derived by a Gibbs ensemble average involving only the Hamiltonian h:2n :iDA Z1 Trð:2n :eH Þ;
(13)
even if all initial states share the same energy. Notice, however, that the DA (11) and (12) can be put precisely in the form of the generalized Gibbs ensemble average (3), since the pseudoenergy and the nonzero connected FF of these operators are in this case ðÞ ¼ log½1 þ jAðÞj2 ; hQ m j:2n ð0Þ:j~m iconn ¼ 2n ð2nÞ!n;m (see the Supplemental Material [20]). In summary, we have verified that the identity (1) holds in the integrable free theory. Moreover, all the DA of free theory can be recovered expanding in the GGEA generating function 2 Z d hexp½iiGGEA ¼ exp jAðÞj2 : (14) 2 2
As it stands, however, this expression is highly problematic: all terms of the sum in the numerator as well as those present in Z are in fact divergent. In the latter, the divergencies come from the normalization of the eigenstates, Q h0m ; . . . 01 j1 . . . n i ¼ i ð0 Þ, which gives rise to ½ð0Þn when 0i ¼ . In the former, the divergencies come from the FF h0m . . . 01 jOð0Þjj1 . . . n i, once evaluated at 0i ¼ i . These divergencies are an unavoidable consequence of the kinematical pole structure of the FFs [21]. The first cure of the divergencies is to define the theory on a finite interval L. In this case, for large but finite L, the rapidities of the n-particle states entering (15) are solutions of the Bethe equation X 2Ni mL sinhi þ &ði k Þ ¼ ; i ¼ 1;. .. ;n; (17) L ki where &ðÞ ¼ i logSðÞ is the phase shift and fNi g is a sequence of increasing integers. Let us denote the finite volume eigenstates associated with the integers fNi g as j1 ; . . . ; n iL : the corresponding density of states is given by the Jacobian Jð1 ; . . . ; n Þ ¼ detJjk , with Jjk ¼ ð@J j =@k Þ. The functions J i are given by the derivative of the right-hand ide. of (17), X (18) J i ð1 ; . . . ; n Þ ¼ mL coshi þ ’ði k Þ; ki
with the kernel ’ðÞ ¼ iðd=dÞ logSðÞ. There is now a relation between the diagonal form factors in finite volume and the infinite-volume connected form factors defined in Eq. (4) [22]: h1 ; . . . ; n jOj1 ; . . . ; n iL
Let us now proceed to the general proof of the identity (1) for an interacting QIFT, with a two-body S matrix SðÞ assumed to be of fermionic type, i.e., Sð0Þ ¼ 1. Expanding the initial state j c 0 i on the basis of the multiparticle states, which are common eigenvectors of H and all higher charges, its general form is 1 n X 1ZY di Kn ð1 ;. .. ;n Þj1 ; .. .; n i: (15) n¼0 n! i¼1 2 P Posing En ¼ m ni¼1 coshi , at any later time t the EV of a local observable Oðx; tÞ on j c 0 i is jc 0i ¼
1 n Y m X h c 0 jOðtÞj c 0 i 1 Z Y di d0j ¼ Z1 hc 0jc 0i m;n¼0 n!m! i¼1 j¼1 2 2 ðf0 gÞK ðfgÞ eitðEm En Þ Km n
h0m . . . 01 jOð0Þjj1 . . . n i (Z ¼ h c 0 j c 0 i). Taking the dynamical average, one inevitably ends up with the so-called diagonal ensemble n X 1 Z Y di hOiDA ¼ Z1 jKn ð1 ; . . . ; n j2 n! 2 n¼0 i¼1 hn . . . 1 jOð0Þjj1 . . . n i:
(16)
¼
1 J n ð1 ; . . . ; n Þ X O F2l;conn ð ÞJ nl ðfþ g; f gÞ; S fþ g
(19)
f g
where the sum runs on all possible bipartite partitions of the set of rapidities f1 ; . . . ; n g in two disjoint sets made by l and n l rapidities, and J nl ðfþ g; f gÞ ¼ detJþ is the restricted determinant of the submatrix J þ corresponding to the particles in the set fþ g in the presence of those in f g. Notice that the relation (19) involves the kernel ’ðÞ of the Bethe ansatz equations (18), as explicitly shown in the Supplemental Material [20]. Let us now focus our attention on the initial state: if the state j c 0 i is statistically characterized by the EV of all its conserved charges h c 0 jQ n j c 0 i ¼ LQn , it can be shown (see the Supplemental Material [20]) that the quantities jKn ð1 ; . . . ; n Þj2 factorize in terms of a function jKðÞj2 , n Y jKn ð1 ; . . . ; n Þj2 ¼ jKði Þj2 ; (20) i¼1
and moreover KðÞ can always be expressed in terms of an infinite set of variables f n g, conjugated to the conserved charges Q n as
100401-3
PRL 111, 100401 (2013)
PHYSICAL REVIEW LETTERS
jKðÞj2 ¼ e0 ðÞ ; 0 ðÞ ¼
1 X
(21)
þ ½þ n qn ðÞ þ n qn ðÞ;
n¼0
q n ðÞ
given in (9). with the functions With all the information collected above, let us now come back to the dynamical average (16): with the regularization given by the finite interval L, its right-hand side can be written as hOiDA ¼ lim
L!1
TrðeH OÞL ðTreH ÞL
;
(22)
where H is the generalized Hamiltonian that includes all the conserved charges H ¼
1 X
þ ½þ n Qn þ n Qn :
n¼0
One can now easily repeat the argument given in [14,22] and show that the right-hand side precisely coincides with the generalized Gibbs ensemble average (3), where the function ðÞ satisfies the nonlinear integral equation of the generalized Bethe ansatz [13]: ðÞ ¼
1 X
þ ½þ n q ðÞ þ n q ðÞ
n¼0
Z d0 0 ’ð 0 Þ logð1 þ eð Þ Þ: 2
(23)
This concludes the general proof of the identity (1). A pragmatic approach to find the functions ðÞ and 0 ðÞ given the initial state j c 0 i has recently been proposed in [23]. Applications to quench processes in the sinhGordon model (at both the quantum and the classical level) are presented in [18]. It must be stressed that the same formalism can be applied to compute the infinitetime average of local fields in the Lieb-Liniger model, a system which has recently attracted a lot of interest for the on-going experiments in cold-atom physics: indeed, as shown in [24], to recover the Lieb-Liniger results one can take advantage of the fact that the Lieb-Liniger model may be reached by taking the nonrelativistic limit of the sinh-Gordon model, whose form factors are all known. An example relative to quench processes in the Lieb-Liniger system is presented in the Supplemental Material [20]. I would like to thank P. Assis and, in particular, A. De Luca for discussions. This work is supported by the IRSES grants QICFT. Note added.—Recently there has been another proposal [25] to compute the EV, based purely on the Bethe ansatz and checked for the free case of the quantum Ising model. Although very similar to the one presented here, it remains to see how it applies to the interactive case.
week ending 6 SEPTEMBER 2013
[1] For simplicity, we consider here QIFTs with only one type of particle and with a unique ground state. The generalization to theories with many types of particles is straightforward; less trivial is the extension to theories with many vacua, a case that will be considered somewhere else. [2] T. Kinoshita, T. Wenger, and D. S. Weiss, Nature (London) 440, 900 (2006); M. Greiner, O. Mandel, T. W. Ha¨nsch, and I. Bloch, Nature (London) 419, 51 (2002); S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Nature (London) 449, 324 (2007). [3] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991); M. Srednicki, Phys. Rev. E 50, 888 (1994). [4] P. Calabrese and J. Cardy, J. Stat. Mech. (2007) P06008. [5] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011), and references therein. [6] A. Iucci and M. A. Cazalilla, Phys. Rev. A 80, 063619 (2009); T. Barthel and U. Schollwo¨ck, Phys. Rev. Lett. 100, 100601 (2008); S. R. Manmana, S. Wessel, R. M. Noack, and A. Muramatsu, Phys. Rev. B 79, 155104 (2009); D. Rossini, A. Silva, G. Mussardo, and G. Santoro, Phys. Rev. Lett. 102, 127204 (2009); D. Rossini, S. Suzuki, G. Mussardo, G. E. Santoro, and A. Silva, Phys. Rev. B 82, 144302 (2010). [7] J. Berges, S. Borsanyi, and C. Wetterich, Phys. Rev. Lett. 93, 142002 (2004); M. Kollar, F. A. Wolf, and M. Eckstein, Phys. Rev. B 84, 054304 (2011); M. C. Banu˜ls, J. I. Cirac, and M. B. Hastings, Phys. Rev. Lett. 106, 050405 (2011); G. Biroli, C. Kollath, and A. M. La¨uchli, Phys. Rev. Lett. 105, 250401 (2010); G. P. Brandino, A. De Luca, R. M. Konik, and G. Mussardo, Phys. Rev. B 85, 214435 (2012). [8] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007); M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008). [9] D. Fioretto and G. Mussardo, New J. Phys. 12, 055015 (2010). [10] P. Calabrese, F. H. L. Essler, and M. Fagotti, Phys. Rev. Lett. 106, 227203 (2011); J. Stat. Mech. (2012) P07016; J. Stat. Mech. (2012) P07022. [11] V. Gurarie, J. Stat. Mech. (2013) P02014; S. Sotiriadis, D. Fioretto, and G. Mussardo, J. Stat. Mech. (2012) P02017; T. Caneva, E. Canovi, D. Rossini, G. E. Santoro, and A. Silva, J. Stat. Mech. (2011) P07015. [12] G. Mussardo, Statistical Field Theory (Oxford University Press, Oxford, England, 2010). [13] J. Mossel and J. S. Caux, J. Phys. A 45, 255001 (2012). [14] A. LeClair and G. Mussardo, Nucl. Phys. B552, 624 (1999). [15] C. N. Yang and C. P. Yang, J. Math. Phys. (N.Y.) 10, 1115 (1969). [16] Al. B. Zamolodchikov, Nucl. Phys. B342, 695 (1990). [17] The fast convergence of the series can be argued on a phase space argument, see J. Cardy and G. Mussardo, Nucl. Phys. B410, 451 (1993), and has been observed in many examples involving operators with smooth behavior of their form factors at large value of the rapidities. [18] G. Mussardo, P. E. Assis, and A. De Luca (to be published). [19] To appreciate the pedagogical value of this example, notice that the main difference between a free and an interacting QIFT is only at the technical level: while the free theory is solved by the Fourier transform, an interacting QIFT
100401-4
PRL 111, 100401 (2013)
PHYSICAL REVIEW LETTERS
needs the inverse scattering transform, i.e., the Bethe ansatz techniques. [20] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.111.100401 for infinitetime averages in Lieb-Liniger model. [21] F. A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory (World Scientific, Singapore, 1992).
week ending 6 SEPTEMBER 2013
[22] B. Pozsgay and G. Takacs, Nucl. Phys. B788, 209 (2008); B. Pozsgay, J. Stat. Mech. (2011) P01011. [23] J. S. Caux and R. M. Konik, Phys. Rev. Lett. 109, 175301 (2012). [24] M. Kormos, G. Mussardo, and A. Trombettoni, Phys. Rev. A 81, 043606 (2010). [25] J. S. Caux and F. Essler, Phys. Rev. Lett. 110, 257203 (2013).
100401-5
