A remark on the subleading order in the asymptotics of the nonequilibrium emptiness formation probability
Abstract
We study the asymptotic behavior of the emptiness formation probability for large spin strings in a translation invariant quasifree nonequilibrium steady state of the isotropic XY chain. Besides the overall exponential decay, we prove that, out of equilibrium, the exponent of the subleading power law contribution to the asymptotics is nonvanishing and strictly positive due to the singularities in the density of the steady state.
Keywords Nonequilibrium steady state, emptiness formation probability, Toeplitz theory
Mathematics Subject Classifications (2000) 46L60, 47B35, 82C10, 82C23
1 Introduction
In this note, I propose to enlarge upon the study started in Aschbacher [6] of the asymptotic behavior of a special and important correlator, the socalled emptiness formation probability (EFP). Written down in the framework of a spin system over the twosided discrete line, the EFP observable is given by
(1) 
where denotes the projection at site onto the spin down configuration of the spin in the direction. For a given state of the spin system, the probability that a ferromagnetic string of length is formed in this state is thus expressed by
(2) 
Due to the existence of the JordanWigner transformation which maps, in a certain sense, spins onto fermions, the EFP has been heavily studied for states of the XY chain whose formal Hamiltonian is given in Remark 3 below. As a matter of fact, this model becomes a gas of independent fermions under the JordanWigner transformation, and it is thus ideally suited for rigorous analysis.^{3}^{3}3Lowdimensional magnetic systems are also heavily studied experimentally, see for example Sologubenko et al. [17].
The large behavior of the EFP in the XY chain has already been analyzed for the cases where the state is a ground state or a thermal equilibrium state at positive temperature. In both cases, the EFP can be written as the determinant of the section of a Toeplitz operator with scalar symbol. Since the higher order asymptotics of a Toeplitz determinant is highly sensitive to the regularity of the symbol of the Toeplitz operator, the asymptotic behavior of the ground state EFP is qualitatively different in the socalled critical and noncritical regimes corresponding to certain values of the anisotropy and the exterior magnetic field of the XY chain.^{4}^{4}4I.e., in (15) below, the parameters and , respectively. It has been found in Shiroishi et al. [16] that the EFP decays like a Gaussian in one of the critical regimes.^{5}^{5}5With some additional explicit numerical prefactor and some power law prefactor, see Shiroishi et al. [16] and references therein. In a second critical regime and in all noncritical regimes, the EFP decays exponentially.^{6}^{6}6In contrast to the noncritical regime, there is an additional power law prefactor in the second critical regime whose exponent differs from the one in the first critical regime, see Abanov and Franchini [1, 13]. These results have been derived by using wellknown theorems of Szegő, Widom, and FisherHartwig, and the yet unproven BasorTracy conjecture and some of its extensions, see Widom [19] and Böttcher and Silbermann [11, 12]. On the other hand, in thermal equilibrium at positive temperature, the EFP can again be shown to decay exponentially by using a theorem of Szegő, see for example Shiroishi et al. [16] and Franchini and Abanov [13].
In contrast, for the case where is the nonequilibrium steady state (NESS) constructed in Aschbacher and Pillet [9],^{7}^{7}7And in Araki and Ho [5] for . the EFP can still be written as a Toeplitz determinant, but now, the symbol is, in general, no longer scalar. Due to the lack of control of higher order determinant asymptotics in Toeplitz theory with nontrivial irregular block symbols, I started off by studying bounds on the leading asymptotic order for a class of general block Toeplitz determinants in Aschbacher [6]. It turned out that suitable basic spectral information on the density of the state is sufficient to derive a bound on the rate of the exponential decay of the EFP in general translation invariant fermionic quasifree states. This bound proved to be exact not only for the decay rates of the EFP in the ground states and the equilibrium states at positive temperature treated in Abanov and Franchini [1, 13] and Shiroishi et al. [16], but it will also do so for the nonequilibrium situation treated here exhibiting the socalled left moverright mover structure already found in Aschbacher [7] and Aschbacher and Barbaroux [8] for nonequilibrium expectations of different correlation observables.^{8}^{8}8This has already been noted in Aschbacher [6]. Hence, given this exponential decay in leading order which parallels qualitatively the behavior in thermal equilibrium at positive temperature, one may wonder whether there is some characteristic signature of the nonequilibrium left at some lower order of the EFP asymptotics. It turns out, and this is the main result of this note, that, in contradistinction to the leading order contribution, the subleading power law contribution to the large asymptotics of the EFP in FisherHartwig theory is sensitive to the singularity of the symbol of the underlying Toeplitz operator, and it has a strictly positive exponent if and only if the system is truly out of equilibrium. This may be interpreted as the manifestation, in subleading order, of the longrange nature of the underlying formal effective Hamiltonian of the NESS.^{9}^{9}9See Remark 3 in Aschbacher and Pillet [9]. This effective Hamiltonian is to be understood on a formal level only. It has been shown by Matsui and Ogata [14] that there exists no dynamics on the Pauli spin algebra w.r.t. which this NESS is a KMS state. This connection is not made precise here, though, but it is left to be studied in greater detail elsewhere.
2 Nonequilibrium setting
In this section, I shortly summarize the setting for the system out of equilibrium used in Aschbacher and Pillet [9]. In contradistinction to the presentation there, I will skip the formulation of the twosided XY chain as a spin system and rather focus directly on the underlying dynamical system structure in terms of Bogoliubov automorphisms on a selfdual CAR algebra as introduced by Araki [4].^{10}^{10}10For an introduction to the algebraic approach to open quantum systems, see also for example Aschbacher et al. [10].
For some , the nonequilibrium configuration is set up by cutting the finite piece
(3) 
out of the twosided discrete line . This piece will play the role of the confined sample, whereas the remaining parts,
(4)  
(5) 
will act as infinitely extended thermal reservoirs at different temperatures to which the sample will be suitably coupled.
We first specify the observables contained in the system to be considered.
Definition 1 (Observables)
Let denote the fermionic Fock space built over the oneparticle Hilbert space of wave functions on the discrete line,
(6) 
With the help of the usual creation and annihilation operators for any ,^{11}^{11}11The bounded operators on the Hilbert space are denoted by . the complex linear mapping is defined, for , by
(7) 
Moreover, the antiunitary involution is given, for all , by
(8) 
The observables are described by the selfdual CAR algebra over with involution generated by the operators for all . I denote this algebra by .^{12}^{12}12The concept of a selfdual CAR algebra has been introduced and developed by Araki [2, 3]. Here, it is just a convenient way of working with the linear combination (7).
The time evolution is generated as follows.
Definition 2 (Dynamics)
Let , and let be the translation operator defined by for all and all . The coupled and the decoupled oneparticle Hamiltonians are defined by
(9)  
(10) 
respectively, where the decoupling operators have the form
(11)  
(12) 
and the projection is given by for all .^{13}^{13}13I write for the finite rank operators on the Hilbert space . Moreover, for denotes the Kronecker function. Finally, for an operator , the real part is . For all , the coupled and the decoupled time evolutions are the Bogoliubov automorphisms defined on the generators with by
(13)  
(14) 
Remark 3
As mentioned above, this model has its origin in the XY spin chain whose formal Hamiltonian is given by
(15) 
where denotes the anisotropy, the external magnetic field, and the Pauli basis of reads
(16) 
The Hamiltonian from (9) corresponds to the isotropic XY chain, i.e. to the case where .
The left and right reservoirs carry the inverse temperatures and , respectively. Pour fixer les idées, we assume w.l.o.g. that they satisfy
(17) 
We next specify the state in which the system is prepared initially. It consists of a KMS state at the corresponding temperature for each reservoir, and, w.l.o.g., of the chaotic state for the sample. For the definition of quasifree states, see Appendix A.
Definition 4 (Initial state)
The initial state is the quasifree state specified by the density of the form
(18) 
where the operators are defined by
(19) 
and is given by
(20) 
Here, for , I used the definitions and , where is the natural injection defined, for any , by if , and zero otherwise.
The following definition is due to Ruelle [15].
Definition 5 (Ness)
A NESS associated with the dynamical system having the initial state is a weak limit point for of the net
(21) 
In the model specified by the Definitions 1, 2, and 4, we get the following NESS using the scattering approach of Ruelle [15].
Theorem 6 (Xy Ness)
There exists a unique quasifree NESS w.r.t. the initial state and the coupled dynamics whose density reads
(22) 
where the operators act in momentum space as multiplication by
(23) 
and the functions are defined by
(24) 
Here, we set and , and the sign function is defined by if , and if .
Proof. See Aschbacher and Pillet [9].
The main object of our study is the following.
Definition 7 (Ness Efp)
Let . The EFP observable is defined by
(25) 
where, for all , the form factors are given by
(26)  
(27) 
and the initial form factors look like
(28) 
Moreover, the expectation value of the EFP observable in the NESS is denoted by^{14}^{14}14As for the name EFP, note that , and that, for , we have
(29) 
The next assertion states the main structural property of the EFP correlation function. For the basic facts of Toeplitz theory, see Appendix B.2.
Proposition 8 (EFP determinantal structure)
The NESS EFP is given by the determinant of the finite section of the Toeplitz operator ,
(30) 
Proof. Proceeding as in Aschbacher and Barbaroux [8], we have that, on one hand, the skewsymmetric EFP correlation matrix , defined, for , by
(31) 
where for are the form factors from Definition 7, relates to the EFP as
(32) 
and, on the other hand, that it has the Toeplitz structure
(33) 
Here, is the block symbol of the Toeplitz operator which I computed in Aschbacher [6] to be of the form , where is the density of the NESS in momentum space and . Theorem 6 then implies that, in the present case, the symbol has the form
(34) 
Hence, there exists an with s.t., using Lemma 19, we can reduce the block Toeplitz Pfaffian to a scalar Toeplitz determinant,
(35)  
where I used the fact that .^{15}^{15}15 stands for the orthogonal matrices in . This is the assertion.
3 Subleading order in the NESS EFP asymptotics
Due to Proposition 8, the study of the large behavior of the EFP correlation function boils down to the analysis of a large truncated Toeplitz operator whose symbol is scalar and has the form given in Theorem 6,
(36) 
see Figure 1.
In a true nonequilibrium situation, i.e. for , the r.h.s. of (36) is no longer continuous. Hence, as described in the introduction, we want to study the asymptotic behavior of the EFP NESS with the help of the socalled FisherHartwig theory whose main content is summarized in Theorem 26 of Appendix B.3. We first introduce the socalled pure jump symbols. For notation and definitions, see Appendix B.1.
Definition 9 (Pure jump)
Let the argument function be defined by and for all . For and , the pure jump symbol is defined, for all , by
(37) 
Remark 10
Note that has at most one jump discontinuity at the point , namely
(38) 
Moreover, the socalled jump phases are defined as follows.
Definition 11 (Jump phases)
Let with , and let for . The numbers for , called the pure jump phases, are defined by
(39) 
Next, we define a regularized symbol which will be extracted from below.
Definition 12 (Regularized symbol)
Let and . The regularized symbol is defined by
(40) 
where, for , the function has the form
(41) 
see Figure 2.
Using the pure jump phases and the regularized symbol, we can recast into the following form.
Lemma 13 (Restricted FisherHartwig form)
The NESS EFP symbol has the following properties.



The jump phases of at the points and are given by
(42) (43) 
The symbol can be written as
(44)
Proof. The assertions (a) and (b) immediately follow from the form of the symbol given in (36). Moreover, using the choice
(45) 
where the argument function is given in Definition 9, we get the pure jump phases (42) and (43) in assertion (c). As for assertion (d), writing, for ,
(46)  
(47) 
with the sign function defined after (24), we get equality (44) in involving the regularized symbol .
In order to be able to apply the FisherHartwig theory to the symbol , we have to make sure that the regularized symbol is indeed sufficiently regular.
Lemma 14 (FisherHartwig regularity)
The regularized symbol has the following properties.

for all


Proof. It follows from (40) that for all which implies assertions (a) and (b). As for assertion (c), we use the fact given in Lemma 27 (b) of Appendix B.3 that . This allows us to bound the integrand in (65) in the Definition 21 of the Besov space as
(48)  
Since, due to Lemma 27 (a)–(c), is continuous and differentiable at all but finitely many points having a bounded derivative , we have , and, thus, it follows from the fundamental theorem of calculus that with Lipschitz constant . Using the Lipschitz continuity on the r.h.s. of (48), we get
(49) 
Hence, we arrive at assertion (c).
We are now ready to formulate the main result of this note.
Theorem 15 (NESS EFP asymptotics)
For , the NESS EFP behaves asymptotically as
(50) 
where the base of the exponential factor is given by
(51) 
satisfying for all inverse temperatures in the range . Furthermore, the exponent of the power law factor has the form
(52) 
Thus, if and only if .
Proof. Due to (44) of Lemma 13, the nonequilibrium symbol has the form , where and, w.r.t. to the form (72) of the restricted FisherHartwig symbol, we have , and from (42) and (43) in Lemma 13. Hence, assumptions (a) and (b) of Theorem 26 from Appendix B.3 are satisfied. Moreover, due to Lemma 14, assumptions (c)–(d) of Theorem 26 are also satisfied by the regularized symbol. Then, (50) follows from (73) in Theorem 26, and it remains to derive the exponential, the power law, and the constant factors in (50) with the help of (74)–(76). As for , we get (51) from (74) and (40) in Definition 12. Moreover, plugging (42) and (43) into (75), we find (52). Finally, using (76), the last factor on the r.h.s. of (50) has the form
(53) 
where , , and are given in (77), (78), and (80), respectively.
Remark 16
In Aschbacher [6], I derived a bound on the decay rate of the exponential decay for the NESS EFP in the more general anisotropic XY chain. As noted there and discussed in the present introduction, Theorem 15 yields that this bound is exact for the special isotropic case at hand.^{16}^{16}16This can also be seen by directly using Szegő’s first limit theorem, see for example Böttcher and Silbermann [11, p.139].
Appendix A Fermionic quasifree states
Let be the selfdual CAR algebra from Definition 1. We denote by the set of states on the algebra .^{17}^{17}17I.e. the normalized positive linear functionals on .
Definition 17 (Density)
The density of a state is defined to be the operator with and satisfying, for all ,
(54) 
A special class of states are the important fermionic quasifree states.
Definition 18 (Quasifree state)
A state is called quasifree if it vanishes on the odd polynomials in the generators, and if it is a Pfaffian on the even polynomials in the generators, i.e. if, for all and for any , we have
(55) 
where the skewsymmetric matrix is defined, for , by
(56) 
Here, the Pfaffian is defined, on all skewsymmetric matrices ,^{18}^{18}18 is the transposition of the matrix . by
(57) 
where the sum is running over all pairings of the set , i.e. over all the permutations in the permutation group of elements which satisfy and , see Figure 3. The set of quasifree states is denoted by .
The following lemma has been used in Section 2.
Lemma 19 (Pfaffian)
The Pfaffian has the following properties.

Let with . Then,
(58) 
Let . Then,
(59)
Proof. See for example Stembridge [18].
Appendix B Toeplitz theory
b.1 Function classes
Let stand for the unit circle. We denote by the continuous functions, by the times continuously differentiable functions, and by with the usual Lebesgue spaces. Moreover, and stand for the absolutely continuous and the Lipschitz continuous functions on . Finally, we need the following function class.
Definition 20 (Piecewise continuous)
The set of piecewise continuous functions is defined by
(60) 
For and any of the form with , we use the notation
(61) 
Moreover, the set of jumps of is defined by
(62) 
Finally, the set of piecewise continuous functions with finitely many jumps is defined by
(63) 
In order to be able to make use of the FisherHartwig theory from Appendix B.3, we also need to introduce the following function class.
Definition 21 (Besov space)
Let and . The operator is defined, on all , and for all , by
(64) 
Moreover, for any , we recursively set . For and , the Besov class is defined by
(65) 
where is s.t. .^{19}^{19}19Note that the definition does not depend on the choice of such an .
Finally, we need the following definition.
Definition 22 (Index)
Let with for all , and let with be s.t. . The index (or winding number) of is defined by
(66) 
b.2 Toeplitz operators
For , we denote by the space of all squaresummable valued sequences.^{20}^{20}20W.r.t. the Euclidean norm on . Moreover, we set
(67) 
We then have the following classical result.
Theorem 23 (Toeplitz)
Let . The linear operator defined on all with maximal domain by
(68) 
is a bounded operator on if and only if there exists an s.t., for all , it holds
(69) 
Proof. See Böttcher and Silbermann [11, p.186].
We then make the following definition.
Definition 24 (Symbol)
Under the assumptions of Theorem 23, we write the Toeplitz operator as . It has the matrix form
(70) 
The function is called the symbol of . If , the symbol and the Toeplitz operator are called scalar, whereas for they are called block.
Finally, the Toeplitz operators are truncated as follows.
Definition 25 (Finite section)
Let . The projection