The quantum Ising model: finite sums and hyperbolic functions

We derive exact closed-form expressions for several sums leading to hyperbolic functions and discuss their applicability for studies of finite-size Ising spin chains. We show how they immediately lead to closed-form expressions for both fidelity susceptibility characterizing the quantum critical point and the coefficients of the counterdiabatic Hamiltonian enabling arbitrarily quick adiabatic driving of the system. Our results generalize and extend the sums presented in the popular Gradshteyn and Ryzhik Table of Integrals, Series, and Products.


Results
There are four interestingly looking sums in Chapter 1.382 of ref. which are valid for integer n ≥ 1 (we assume such n everywhere in this article). To explain where these sums come from, we consider polynomials  which can be also written as  If we now substitute into Eqs (5-8) and differentiate them with respect to x, we will get Eqs (1-4) after simple algebra. As will be explained in the Discussion section, the parameter g (9) is the magnetic field in the quantum Ising model. The sums (1)-(4) can be generalized in the following way ( )   2 . The sums (10)- (17) have been conjectured by applying the techniques described in Appendix A of ref. 16. Two of them, (10) and (14), have been already proven in ref. 16. We will outline below the steps allowing for an inductive proof of all the sums. This way we will generalize the former results and simplify a bit the proof.
To compactify notation, we define two functions where the momenta k are chosen such that ∑ k replaces ∑ j in Eqs (10-17). Next, we show that . These recurrence relations can be derived by substituting Second, we check that our inductive conjecture holds for m = 1. This is done by substituting Eqs (21,22) into Eqs (18,19) taken at m = 1 and then comparing the resulting expressions to Eqs (1-4).

Discussion
The results obtained in the previous section provide useful analytical tools for the studies of the quantum Ising model. The Hamiltonian of this model reads where N is the number of spins, σ i denote the Pauli matrices acting on the i-th spin, g is a magnetic field, and periodic boundary conditions are assumed σ σ = + i N i . This exactly solvable model has many interesting properties 9,10 . We will mention two of them that are relevant for the present discussion. Forgetting about the irrelevant sign of k and the 0 and π momentum modes, the summations over these four choices of momenta correspond to the four summations over j seen in Eqs (10)(11)(12)(13)(14)(15)(16)(17).
What's more, the denominator of all the sums that we have computed, ( / ) + ( / ) k x sin 2 sinh 2 2 2 , can be mapped to the form familiar from the Ising model computations, Note that we have already used this mapping at the beginning of the Results section (9). Since both the eigenvalues and eigenstates of quantum Ising model depend on the factor (30)-see e.g. ref. 9-it is not surprising that the sums involving it appear in the studies of this model.
Finally, we note that the sums involving positive integer powers of the factor (30) in the denominator can be easily generated through the differentiation of the sums (10)-(17) with respect to x. We are now ready to show two applications for the results of the previous section.
Fidelity approach to quantum phase transitions. The location of the critical point and the critical exponent quantifying power-law divergence of the correlation length around the critical point can be extracted from the overlap between ground states corresponding to different external fields [19][20][21][22][23][24] . Such an overlap is called fidelity 19 . It defines the so-called fidelity susceptibility χ through a Taylor expansion in the field shift δ 25 where g is a ground state of a Hamiltonian depending on an external field g. It can be easily shown that 19 in the quantum Ising model (25); the restriction to π < < k 0 in the sum is applied to momenta from Eqs (26)(27)(28)(29).
Therefore, to compute the exact closed-form expression for fidelity susceptibility χ ( ) g one needs to (i) determine the parity of the ground state and choose the appropriate momenta k from the list (26)- (29); (ii) choose the relevant sum from Eqs (14)(15)(16)(17), set m = 1 to obtain ( ) k sin 2 in the numerator of the summand, and compute derivative of the resulting expression with respect to x to square the summand's denominator; (iii) substitute (31) into the resulting expression. One gets in the end The quantum Ising model has the critical point at g c = 1 separating the ferromagnetic phase (0 ≤ g < 1) from the paramagnetic phase (g > 1). It is thus expected that its ground states are most sensitive to the change of the magnetic field near the critical point. Therefore, fidelity (32) should have minimum near the critical point, and so fidelity susceptibility should have maximum there. One can use Eq. (34) to easily analytically compute the location of the maximum of fidelity susceptibility and quantify how fast it approaches g c = 1 as the size of the system is increased. One can also study with the help of Eq. (34) the dependence of fidelity susceptibility on the system size and the distance from the critical point, which yields further insights into the quantum criticality of the Ising model (see ref. 26 for the details).
Counterdiabatic driving. Perfectly adiabatic evolutions of a quantum system subjected to arbitrary time variation of an external field can be engineered by a proper modification of a Hamiltonian 11,[27][28][29] .
For the quantum Ising chain this can be obtained after adding to the Hamiltonian (25) describes interactions of m + 1 adjacent spins. This so-called counterdiabatic Hamiltonian was computed for even-sized chains evolving in the positive parity subspace in ref. 30. We have checked that it can be written in such a form for systems of an arbitrary size evolving in a subspace of arbitrary parity. It should be perhaps explained that the fairly complicated counterdiabatic Hamiltonian is obtained by "reverse-engineering". It is written down first in the momentum space, where it takes a simple form, and then it is transformed to the real space form. It is thus assumed that the system evolves in the subspace of the Hilbert space of definite parity. Such an assumption can be made because parity is conserved during time evolutions with the counterdiabatic Hamiltonian: Using mapping (9) as well as Eqs (14)(15)(16)(17)(26)(27)(28)(29), one immediately finds that for any system size N > 1 and = , , , − where the + (-) sign should be used for evolutions taking place in the positive (negative) parity subspaces of the Hilbert space. The closed-form expression for Eq. (35) in even-sized systems evolving in the positive parity subspace was previously reported in ref. 16. Summarizing, we have derived several expressions for the sums leading to hyperbolic functions. They generalize and extend the results listed in the popular Gradshteyn and Ryzhik book 15 . We expect that they will be useful in different studies and illustrated their applicability to the quantum Ising model, which has provided the primary motivation for this work.