A remark on the subleading order in the asymptotics of the nonequilibrium emptiness formation probability
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
In this note, I propose to enlarge upon the study started in Aschbacher  of the asymptotic behavior of a special and important correlator, the so-called emptiness formation probability (EFP). Written down in the framework of a spin system over the two-sided discrete line, the EFP observable is given by
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
Due to the existence of the Jordan-Wigner 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 Jordan-Wigner transformation, and it is thus ideally suited for rigorous analysis.333Low-dimensional magnetic systems are also heavily studied experimentally, see for example Sologubenko et al. .
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 so-called critical and noncritical regimes corresponding to certain values of the anisotropy and the exterior magnetic field of the XY chain.444I.e., in (15) below, the parameters and , respectively. It has been found in Shiroishi et al.  that the EFP decays like a Gaussian in one of the critical regimes.555With some additional explicit numerical prefactor and some power law prefactor, see Shiroishi et al.  and references therein. In a second critical regime and in all noncritical regimes, the EFP decays exponentially.666In 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 well-known theorems of Szegő, Widom, and Fisher-Hartwig, and the yet unproven Basor-Tracy conjecture and some of its extensions, see Widom  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.  and Franchini and Abanov .
In contrast, for the case where is the nonequilibrium steady state (NESS) constructed in Aschbacher and Pillet ,777And in Araki and Ho  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 . 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. , but it will also do so for the nonequilibrium situation treated here exhibiting the so-called left mover-right mover structure already found in Aschbacher  and Aschbacher and Barbaroux  for nonequilibrium expectations of different correlation observables.888This has already been noted in Aschbacher . 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 Fisher-Hartwig 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 long-range nature of the underlying formal effective Hamiltonian of the NESS.999See Remark 3 in Aschbacher and Pillet . This effective Hamiltonian is to be understood on a formal level only. It has been shown by Matsui and Ogata  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 . In contradistinction to the presentation there, I will skip the formulation of the two-sided 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 .101010For an introduction to the algebraic approach to open quantum systems, see also for example Aschbacher et al. .
For some , the nonequilibrium configuration is set up by cutting the finite piece
out of the two-sided discrete line . This piece will play the role of the confined sample, whereas the remaining parts,
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 one-particle Hilbert space of wave functions on the discrete line,
With the help of the usual creation and annihilation operators for any ,111111The bounded operators on the Hilbert space are denoted by . the complex linear mapping is defined, for , by
Moreover, the antiunitary involution is given, for all , by
The observables are described by the selfdual CAR algebra over with involution generated by the operators for all . I denote this algebra by .121212The 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 one-particle Hamiltonians are defined by
respectively, where the decoupling operators have the form
and the projection is given by for all .131313I 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
As mentioned above, this model has its origin in the XY spin chain whose formal Hamiltonian is given by
where denotes the anisotropy, the external magnetic field, and the Pauli basis of reads
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
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
where the operators are defined by
and is given by
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 .
Definition 5 (Ness)
A NESS associated with the -dynamical system having the initial state is a weak- limit point for of the net
Theorem 6 (Xy Ness)
There exists a unique quasifree NESS w.r.t. the initial state and the coupled dynamics whose density reads
where the operators act in momentum space as multiplication by
and the functions are defined by
Here, we set and , and the sign function is defined by if , and if .
Proof. See Aschbacher and Pillet .
The main object of our study is the following.
Definition 7 (Ness Efp)
Let . The EFP observable is defined by
where, for all , the form factors are given by
and the initial form factors look like
Moreover, the expectation value of the EFP observable in the NESS is denoted by141414As for the name EFP, note that , and that, for , we have
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 ,
Proof. Proceeding as in Aschbacher and Barbaroux , we have that, on one hand, the skew-symmetric EFP correlation matrix , defined, for , by
where for are the form factors from Definition 7, relates to the EFP as
and, on the other hand, that it has the Toeplitz structure
Here, is the block symbol of the Toeplitz operator which I computed in Aschbacher  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
Hence, there exists an with s.t., using Lemma 19, we can reduce the block Toeplitz Pfaffian to a scalar Toeplitz determinant,
where I used the fact that .151515 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,
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 so-called Fisher-Hartwig theory whose main content is summarized in Theorem 26 of Appendix B.3. We first introduce the so-called 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
Note that has at most one jump discontinuity at the point , namely
Moreover, the so-called 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
Next, we define a regularized symbol which will be extracted from below.
Definition 12 (Regularized symbol)
Let and . The regularized symbol is defined by
where, for , the function has the form
see Figure 2.
Using the pure jump phases and the regularized symbol, we can recast into the following form.
Lemma 13 (Restricted Fisher-Hartwig form)
The NESS EFP symbol has the following properties.
The jump phases of at the points and are given by
The symbol can be written as
Proof. The assertions (a) and (b) immediately follow from the form of the symbol given in (36). Moreover, using the choice
In order to be able to apply the Fisher-Hartwig theory to the symbol , we have to make sure that the regularized symbol is indeed sufficiently regular.
Lemma 14 (Fisher-Hartwig regularity)
The regularized symbol has the following properties.
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
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
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
where the base of the exponential factor is given by
satisfying for all inverse temperatures in the range . Furthermore, the exponent of the power law factor has the form
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 Fisher-Hartwig 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
In Aschbacher , 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.161616This 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 .171717I.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 ,
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
where the skew-symmetric matrix is defined, for , by
Here, the Pfaffian is defined, on all skew-symmetric matrices ,181818 is the transposition of the matrix . by
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,
Let . Then,
Proof. See for example Stembridge .
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
For and any of the form with , we use the notation
Moreover, the set of jumps of is defined by
Finally, the set of piecewise continuous functions with finitely many jumps is defined by
In order to be able to make use of the Fisher-Hartwig 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
Moreover, for any , we recursively set . For and , the Besov class is defined by
where is s.t. .191919Note 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
b.2 Toeplitz operators
For , we denote by the space of all square-summable -valued sequences.202020W.r.t. the Euclidean norm on . Moreover, we set
We then have the following classical result.
Theorem 23 (Toeplitz)
Let . The linear operator defined on all with maximal domain by
is a bounded operator on if and only if there exists an s.t., for all , it holds
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
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