In a disordered system one can either consider a microcanonical ensemble, where there is a precise constraint on the random variables, or a canonical ensemble where the variables are chosen according to a distribution without constraints. We address the question as to whether critical exponents in these two cases can differ through a detailed study of the random transverse-field Ising chain. We find that the exponents are the same in both ensembles, though some critical amplitudes vanish in the microcanonical ensemble for correlations which span the whole system and are particularly sensitive to the constraint. This can appear as a different exponent. We expect that this apparent dependence of exponents on ensemble is related to the integrability of the model, and would not occur in nonintegrable models.
In a disordered system one can either consider a microcanonical ensemble, where there is a precise constraint on the random variables, or a canonical ensemble where the variables are chosen according to a distribution without constraints. We address the question as to whether critical exponents in these two cases can differ through a detailed study of the random transverse-field Ising chain. We find that the exponents are the same in both ensembles, though some critical amplitudes vanish in the microcanonical ensemble for correlations which span the whole system and are particularly sensitive to the constraint. This can appear as a different exponent. We expect that this apparent dependence of exponents on ensemble is related to the integrability of the model, and would not occur in nonintegrable models.
The random transverse-field Ising chain is the simplest disordered model presenting a quantum phase transition at T=0. We compare analytically its finite-size scaling properties in two different ensembles for the disorder: (i) the canonical ensemble, where the disorder variables are independent and (ii) the microcanonical ensemble, where there exists a global constraint on the disorder variables. The observables under study are the surface magnetization, the correlation of the two surface magnetizations, the gap and the end-to-end spin-spin correlation C(L) for a chain of length L. At criticality, each observable decays typically as ${e}^{\ensuremath{-}w\sqrt{L}}$ in both ensembles, but the probability distributions of the rescaled variable w are different in the two ensembles, in particular, in their asymptotic behaviors. As a consequence, the dependence on L of averaged observables differs in the two ensembles. For instance, the correlation C(L) decays algebraically as 1/L in the canonical ensemble, but subexponentially as ${e}^{\ensuremath{-}{\mathrm{cL}}^{1/3}}$ in the microcanonical ensemble. Off criticality, probability distributions of rescaled variables are governed by the critical exponent $\ensuremath{\nu}=2$ in both ensembles, but the following observables are governed by the exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=1$ in the microcanonical ensemble, instead of the exponent $\ensuremath{\nu}=2$ in the canonical ensemble: (a) In the disordered phase, the averaged surface magnetization, the averaged correlation of the two surface magnetizations and the averaged end-to-end spin-spin correlation; (b) in the ordered phase, the averaged gap. In conclusion, the measure of the rare events that dominate various averaged observables can be very sensitive to the microcanonical constraint.
Ensemble equivalence — the expectation that microcanonical, canonical, and grand-canonical descriptions of a system agree in the thermodynamic limit — is a cornerstone of statistical mechanics, yet it can fail for observables that probe exponentially rare events. Recent work on a simplified quantum error correction (QEC) model reports that critical exponents governing fidelity and magic at the error-correction threshold depend on the noise ensemble chosen, and that even the existence of the transition can be ensemble-dependent [1,2]. Here we develop a minimal, fully analytic treatment of this phenomenon using a random-unitary encoding model subject to depolarizing noise. We derive explicit expressions for the decoding fidelity in grand-canonical (independent per-qubit errors) and canonical (fixed error number) ensembles, and exhibit concrete numerical instances of the discrepancy: for a distance-3 code with 100 physical qubits and error rate p = 0.01, the grand-canonical fidelity is 0.735762 while the canonical fidelity is exactly 1. We show that the discrepancy survives the vanishing of relative fluctuations (σ/μ → 0) because fidelity is a large-deviation tail quantity, and we compute the associated entropy and capacity budgets. We further derive the statistical cost of resolving ensemble-dependent exponents: distinguishing two candidate exponents at distance Δp from threshold requires a sample count scaling as Δp⁻⁴ and system sizes scaling as Δp⁻², a compound burden that we quantify under stated assumptions. We relate the mechanism to the known ensemble dependence of the random transverse-field Ising chain [3], discuss saturation of the information-theoretic bound of Feldman et al. as reported in [1,2], and outline falsifiable predictions. The results clarify when ensemble choice is physically consequential rather than a matter of convenience in QEC threshold physics. ## 1. Introduction Quantum error correction protects quantum information from decoherence by en Full text and updates: papers.qnfo.org/papers/ensemble-dependence-of-critical-exponents-at-quantum-error-correction-thresholds/
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian:
We study the dynamical properties of the random transverse-field Ising chain at criticality using a mapping to free fermions with which we can obtain numerically exact results for system sizes L as large as 256. The probability distribution of the local imaginary time correlation function $S(\ensuremath{\tau})$ is investigated and found to be simply a function of $\ensuremath{\alpha}\ensuremath{\equiv}\ensuremath{-}\mathrm{ln}S(\ensuremath{\tau})/\mathrm{ln}\ensuremath{\tau}.$ This scaling behavior implies that the typical correlation function decays algebraically, ${S}_{\mathrm{typ}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{typ}}},$ where the exponent ${\ensuremath{\alpha}}_{\mathrm{typ}}$ is determined from $P(\ensuremath{\alpha}),$ the distribution of $\ensuremath{\alpha}.$ The precise value for ${\ensuremath{\alpha}}_{\mathrm{typ}}$ depends on exactly how the ``typical'' correlation function is defined. The form of $P(\ensuremath{\alpha})$ for small $\ensuremath{\alpha}$ gives a contribution to the average correlation function ${S}_{\mathrm{av}}(\ensuremath{\tau})$ namely, ${S}_{\mathrm{av}}(\ensuremath{\tau})\ensuremath{\sim}(\mathrm{ln}\ensuremath{\tau}{)}^{\ensuremath{-}{2x}_{m}},$ where xm is the bulk magnetization exponent, which was reported recently in Europhys. Lett. 39, 135 (1997). These results represent a type of ``multiscaling'' different from the well-known ``multifractal'' behavior.
We consider disordered ladders of the transverse-field Ising model and study their critical properties and entanglement entropy for varying width, $w\ensuremath{\le}20$, by numerical application of the strong disorder renormalization group method. We demonstrate that the critical properties of the ladders for any finite w are controlled by the infinite disorder fixed point of the random chain and the correction to scaling exponents contain information about the two-dimensional model. We calculate sample dependent pseudocritical points and study the shift of the mean values as well as scaling of the width of the distributions and show that both are characterized by the same exponent, $\ensuremath{\nu}(2d)$. We also study scaling of the critical magnetization, investigate critical dynamical scaling as well as the behavior of the critical entanglement entropy. Analyzing the w dependence of the results we have obtained accurate estimates for the critical exponents of the two-dimensional model: $\ensuremath{\nu}(2d)=1.25(8)$, x(2d)=0.996(15), and $\ensuremath{\psi}(2d)=0.51(3)$.
A renormalization-group analysis of the spin-1/2 transverse field Ising model with quenched randomness is presented; it become exact asymptotically near the zero temperature ferromagnetic phase transition. The spontaneous magnetization is found to vanish with an exponent \ensuremath{\beta}=1/2(3- \ensuremath{\surd}5 ), while in the disordered phase the typical and average spin correlations are found to decay with different correlation lengths, which diverge with exponents \ensuremath{\nu}\ifmmode \tilde{}\else \~{}\fi{}=1 and \ensuremath{\nu}=2, respectively.