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Vanishing critical magnetization in the quantum Ising model

Jakob E. Björnberg

Published Aug 16, 2016• 8 citations
OpenAlex

Abstract

Adapting the recent argument of Aizenman, Duminil-Copin and Sidoravicius for the classical Ising model, it is shown here that the magnetization in the transverse-field Ising model vanishes at the critical point. The proof applies to the ground state in dimension $d\geq2$ and to positive-temperature states in dimension $d\geq 3$, and relies on graphical representations as well as an infrared bound.

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Adapting the recent argument of Aizenman, Duminil-Copin and Sidoravicius for the classical Ising model, it is shown here that the magnetization in the transverse-field Ising model vanishes at the critical point. The proof applies to the ground state in dimension d≥2d\geq2d≥2 and to positive-temperature states in dimension d≥3d\geq 3d≥3, and relies on graphical representations as well as an infrared bound.
We prove an infrared bound for the transverse field Ising model. This bound is stronger than the previously known infrared bound for the model, and allows us to investigate mean-field behaviour. As an application we show that the critical exponent γ\gammaγ for the susceptibility attains its mean-field value γ=1\gamma=1γ=1 in dimension at least 4 (positive temperature), respectively 3 (ground state), with logarithmic corrections in the boundary cases.
A zero-temperature real-space renormalization-group method is presented and applied to the quantum Ising model with a transverse field in one dimension. The transition between the low-field and high-field regimes is studied. Magnetization components, spin correlation functions, and critical exponents are derived and checked against the exact results. It is shown that increasing the size of the blocks in the iterative procedure yields more accurate results, especially for the critical "magnetic" exponents near the transition.
The critical behavior of an Ising model of classical spins in a transverse field HHH is derived for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$ using an $\ensuremath{\epsilon}$ expansion based on the Wilson renormalization group. To first order in $\ensuremath{\epsilon}$ the model exhibits a line of critical points Tc(H){T}_{c}(H)Tc​(H) with the same fixed point and critical exponents as in previous studies of the Wilson effective Hamiltonian.
Based on the obtained exact results we systematically study the quench dynamics of a one-dimensional spin-1/2 transverse field Ising model with zero- and finite-temperature initial states. We focus on the magnetization of the system after a sudden change of the external field and a coherent time-evolution process. With a zero-temperature initial state, the quench magnetic susceptibility as a function of the initial field strength exhibits strongly similar scaling behaviors to those of the static magnetic susceptibility, and the quench magnetic susceptibility as a function of the final field strength shows a discontinuity at the quantum critical point. This discontinuity remains robust and always occurs at the quantum critical point even for the case of finite-temperature initial systems, which indicates a great advantage of employing quench dynamics to study quantum phase transitions.
We consider the effect of a random longitudinal field on the Ising model in a transverse magnetic field. For spatial dimension d>2d>2d>2, there is at low strength of randomness and transverse field a phase with true long-range order that is destroyed at higher values of the randomness or transverse field. The properties of the quantum phase transition at zero temperature are controlled by a fixed point with no quantum fluctuations. This fixed point also controls the classical finite-temperature phase transition in this model. Many critical properties of the quantum transition are therefore identical to those of the classical transition. In particular, we argue that the dynamical scaling is activated, i.e., the logarithm of the diverging time scale rises as a power of the diverging length scale.
The equivalence of the critical behavior of the ddd-dimensional Ising model at a finite temperature to that of a ddd- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.