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arXiv:2003.05804 · Physical Review Research

Ising model in a light-induced quantized transverse field

Jonas Rohn, Max Hörmann, Claudiu Genes, Kai Phillip Schmidt

Published May 6, 2020• 40 citations
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Abstract

This paper investigates the effect of light-matter interactions on correlated quantum matter. The non-local bosonic light field of an optical resonator drastically changes the phase transition in an Ising chain, with a tuning parameter that resembles a non-classical transverse magnetic field with quantized values
DOI
10.1103/physrevresearch.2.023131

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We study the quantum phase transition in a spin chain with variable Ising interactions and position-dependent coupling to a resonator field. Such a complicated model, usually not present in natural physical systems, can be simulated by an array of qubits based on man-made devices and exhibits interesting behavior. We show that, when the coupling between the qubit and field is strong enough, a superradiant phase transition occurs, and it is possible to pick a particular field mode to undergo this phase transition by properly modulating the strength of the Ising interaction. We also study the impact of the resonator field on the magnetic properties of the spin chain and find a rich set of phases characterized by distinctive qubit correlation functions.
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.
In this thesis I have successfully recreated the results that describe the behavior of a transverse-field Ising chain. There is a phase transition when the parameter that controls the influence of the transverse field is one (that is: the factor in front of the transverse-field term and the interaction term are equally big). However, I could not determine exactly what the order parameter should be.\nThen for the transverse-field Ising ladder I have used perturbation theory in the parameter that controls the interaction between the chains. I have found that the phase transition doesn't change position, but that the ground state is a different state compared to the transverse-field Ising chain, when the transverse-field term is smaller than the interaction.
The paradigmatic example of a continuous quantum phase transition is the transverse field Ising ferromagnet. In contrast to classical critical systems, whose properties depend only on symmetry and the dimension of space, the nature of a quantum phase transition also depends on the dynamics. In the transverse field Ising model, the order parameter is not conserved, and increasing the transverse field enhances quantum fluctuations until they become strong enough to restore the symmetry of the ground state. Ising pseudospins can represent the order parameter of any system with a twofold degenerate broken-symmetry phase, including electronic nematic order associated with spontaneous point-group symmetry breaking. Here, we show for the representative example of orbital-nematic ordering of a non-Kramers doublet that an orthogonal strain or a perpendicular magnetic field plays the role of the transverse field, thereby providing a practical route for tuning appropriate materials to a quantum critical point. While the transverse fields are conjugate to seemingly unrelated order parameters, their nontrivial commutation relations with the nematic order parameter, which can be represented by a Berry-phase term in an effective field theory, intrinsically intertwine the different order parameters.
We gain quantitative insights on effects of light-matter interactions on correlated quantum matter by quantum Monte Carlo simulations. We introduce a wormhole algorithm for the paradigmatic Dicke-Ising model which combines the light-matter interaction of the Dicke model with Ising interactions. The quantum phase diagrams for ferro- and antiferromagnetic interactions on the chain and the square lattice are determined. The occurring superradiant phase transitions are in the same universality class as the Dicke model, leading to a well-known peculiar finite-size scaling that we elucidate in terms of scaling above the upper critical dimension. For the ferromagnetic case, the transition between the normal and the superradiant phase is of second order with Dicke criticality (first order) for large (small) longitudinal fields separated by a multicritical point. For antiferromagnetic interactions, we establish the light-matter analog of a lattice supersolid with off-diagonal superradiant and diagonal magnetic order and determine the nature of all transition lines. Our achievements allow to quantitatively enter the domain of mesoscopic light-matter systems, which is an increasingly prominent topic at the interface of quantum optics and condensed matter physics.
The transverse field Ising chain model is ideally suited for testing the fundamental ideas of quantum phase transitions because its well-known T ¼ 0 ground state can be extrapolated to finite temperatures.Nonetheless, the lack of appropriate model materials hindered the past effort to test the theoretical predictions.Here, we map the evolution of quantum fluctuations in the transverse field Ising chain based on nuclear magnetic resonance measurements of CoNb 2 O 6 , and we demonstrate the finite-temperature effects on quantum criticality for the first time.From the temperature dependence of the 93 Nb longitudinal relaxation rate 1=T 1 , we identify the renormalized classical, quantum critical, and quantum disordered scaling regimes in the temperature (T) vs transverse magnetic field (h ⊥ ) phase diagram.Precisely at the critical field h c ⊥ ¼ 5.25 Æ 0.15 T, we observe a power-law behavior, 1=T 1 ∼ T -3=4 , as predicted by quantum critical scaling.Our parameter-free comparison between the data and theory reveals that quantum fluctuations persist up to as high as T ∼ 0.4J, where the intrachain exchange interaction J is the only energy scale of the problem.