An exact description of integrable spin chains at finite temperature is provided using an elementary algebraic approach in the complete Hilbert space of the system. We focus on spin chain models that admit a description in terms of free fermions, including paradigmatic examples such as the one-dimensional transverse-field quantum Ising and XY models. The exact partition function is derived and compared with the ubiquitous approximation in which only the positive parity sector of the energy spectrum is considered. Errors stemming from this approximation are identified in the neighborhood of the critical point at low temperatures. We further provide the full counting statistics of a wide class of observables at thermal equilibrium and characterize in detail the thermal distribution of the kink number and transverse magnetization in the transverse-field quantum Ising chain.
An exact description of integrable spin chains at finite temperature is provided using an elementary algebraic approach in the complete Hilbert space of the system. We focus on spin chain models that admit a description in terms of free fermions, including paradigmatic examples such as the one-dimensional transverse-field quantum Ising and XY models. The exact partition function is derived and compared with the ubiquitous approximation in which only the positive parity sector of the energy spectrum is considered. Errors stemming from this approximation are identified in the neighborhood of the critical point at low temperatures. We further provide the full counting statistics of an arbitrary observable at thermal equilibrium and characterize in detail the thermal distribution of the kink number and transverse magnetization in the transverse-field quantum Ising chain.
Abstract We derive the exact expression for the Uhlmann fidelity between arbitrary thermal Gibbs states of the quantum XY model in a transverse field with finite system size. Using it, we conduct a thorough analysis of the fidelity susceptibility of thermal states for the Ising model in a transverse field. We compare the exact results with a common approximation that considers only the positive-parity subspace, which is shown to be valid only at high temperatures. The proper inclusion of the odd parity subspace leads to the enhancement of maximal fidelity susceptibility in the intermediate range of temperatures. We show that this enhancement persists in the thermodynamic limit and scales quadratically with the system size. The correct low-temperature behavior is captured by an approximation involving the two lowest many-body energy eigenstates, from which simple expressions are obtained for the thermal susceptibility and specific heat.
We propose to describe correlations in classical and quantum systems in terms of full counting statistics of a suitably chosen discrete observable. The method is illustrated with two exactly solvable examples: the classical one-dimensional Ising model and the quantum spin-1/2 XY chain. For the one-dimensional Ising model, our method results in a phase diagram with two phases distinguishable by the long-distance behavior of the Jordan-Wigner strings. For the anisotropic spin-1/2 XY chain in a transverse magnetic field, we compute the full counting statistics of the magnetization and use it to classify quantum phases of the chain. The method, in this case, reproduces the previously known phase diagram. We also discuss the relation between our approach and the Lee-Yang theory of zeros of the partition function.
We present a general study of a large family of exactly integrable quantum chains with multispin interactions. The exact integrability follows from the algebraic properties of the energy density operators defining the quantum chains. The Hamiltonians are characterized by a parameter $p=1,2,\ensuremath{\cdots}$ related to the number of interacting spins in the multispin interaction. In the general case, the quantum spins are of infinite dimension. In special cases, characterized by the parameter
N=2,3,..., the quantum chains describe the dynamics of
Z(N) quantum spin chains with open boundary conditions. The simplest case
p=1 corresponds to the free fermionic quantum Ising chain (
N=2) or the
Z(N) free parafermionic quantum chain. The eigenenergies of the quantum chains are given in terms of the roots of special polynomials, and for general values of
p the quantum chains are characterized by a free fermionic (
N=2) or free parafermionic (
N>2) eigenspectrum. The models have a special critical point when all coupling constants are equal. At this point, the ground-state energy is exactly calculated in the bulk limit, and our analytical and numerical analyses indicate that the models belong to universality classes of critical behavior with dynamical critical exponent
z=(p+1)/N and specific-heat exponent $\ensuremath{\alpha}=max{0,1\ensuremath{-}(p+1)/N}$.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.