The correlation functions of certain random antiferromagnetic spin-1∕2 critical chains

Abstract

We study the spin-spin correlations in two distinct random critical XX spin-1/2 chain models via exact diagonalization. For the well-known case of uncorrelated random coupling constants, we study the non-universal numerical prefactors and relate them to the corresponding Lyapunov exponent of the underlying single-parameter scaling theory. We have also obtained the functional form of the correct scaling variables important for describing even the strongest finite-size effects. Finally, with respect to the distribution of the correlations, we have numerically determined that they converge to a universal (disorder-independent) non-trivial and narrow distribution when properly rescaled by the spin-spin separation distance in units of the Lyapunov exponent. With respect to the less known case of correlated coupling constants, we have determined the corresponding exponents and shown that both typical and mean correlations functions decay algebraically with the distance. While the exponents of the transverse typical and mean correlations are nearly equal, implying a narrow distribution of transverse correlations, the longitudinal typical and mean correlations critical exponents are quite distinct implying much broader distributions. Further comparisons between this models are given.

Corrections to the singlet-length distribution and the entanglement entropy in random singlet phases

“…In the SDRG approach, their average value is directly proportional to the singlet-length distribution p s (l). In a recent exact diagonalization study of the correlation functions of the random XX chain, weak non-universal corrections have been seen in the transversal (α = z) correlation function 27 . It is an interesting question whether the corrections next to the leading term of the correlation function are compatible with the SDRG predictions of this work.…”

“…From the distribution of singlet lengths obtained numerically by the SDRG method we calculated the average entanglement entropy S ℓ of subsystems of size ℓ through Eq. (27). In order to eliminate the constant term in the asymptotic dependence of S ℓ on ℓ, see Eq.…”

Powerful method to evaluate the mass gaps of free-particle quantum critical systems

“…Reliable numerical results for quantum chains in the presence of quenched disorder are rare, specially for those exhibiting infinite-randomness criticality. The use of standard methods such as matrix diagonalization suffers from numerical instabilities even for moderate lattice sizes [7]. In this section, we apply our method to the random transverse-field Ising (RTFI) chain in order to illustrate its effectiveness and practicality.…”

“…This is a problem in the cases where we need larger systems L ∼ 10 4 or L ∼ 10 5 , or even for small chains in the case of quench disordered quantum chains since a large number of disorder configurations (typically 10 5 ) are necessary for achieving good statistics. A more severe constraint arises in quenched disordered systems since the diagonalization procedure suffers from numerical instabilities even for fairly small chains (L ∼ 100) when the associ-ated dynamical critical exponent is sufficiently large [7].…”

Exact bounds on the energy gap of transverse-field Ising chains by mapping to random walks

“…Due to this latter property, they are promising starting points of possible analytic treatments toward the determination of the finite-size scaling of the gap. We will demonstrate the power of these bounds by obtaining the finite-size scaling of the gap of the RTIC with coupling-field correlations, which is relevant in the context of adiabatic quantum computing and has been studied by several authors 27,[29][30][31][32][33] . The derivation rests on an exact relationship between the open RTIC and continuous-time random walks with an absorbing boundary.…”