Equilibrium order parameter

In the previous part of this thesis the dynamical quantum phase transition in the Ising model has been analyzed in terms of the Loschmidt amplitude and topological quantum numbers. Although in principle measurable quantities in experiments local observables and correlation functions are more easily accessible. In the following, the dynamics of the local order parameter of the continuous equilibrium phase transition will be analyzed.

4.2 Dynamical quantum phase transitions in the Ising model 105 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 L o n g it u d in a l m a g n et iz a tio n | ρx ( t ) | time (t−tϕ)/t∗ -1 0 1 -1 0 1 0 2 4 t/ t ∗ -1 0 1 -1 0 1 0 1 2 | gf − 1 | t gi= 0.1→gf = 2.5 gi= 0.8→gf = 1.6 gi= 0.1→gf = 1.9 gi= 0.4→gf = 1.6 sx(t) sy(t) t/ t ∗ sx(t) sy(t) | gf − 1 | t

Figure 4.11: Dynamics of the longitudinal magnetization after quenches across the quan- tum critical point starting in the ferromagnetic phase. The lower plot shows the damped oscillatory decay of the longitudinal magnetization for different quenches from the ferro- magnetic to the disordered phase. The period of the oscillation is entirely determined by the nonequilibrium time scalet∗_nas can be clearly seen from this graph. Note that the zeros of the order parameter are not precisely located at t = t∗_n. The curves have been shifted

by a magnetic field dependent constant tϕ. The dynamics of the normalized local spin

expectation values sx(t) and sy(t) in the xy-plane are illustrated in the upper two graphs. For quenches within one phase the ferromagnetic order is so rigid that the change in the external magnetic field is not able to induce a significant displacement in the xy-plane. In the opposite case of quenches across the quantum critical point as shown in the upper right plot the magnetization starts to precess with a frequency Ω∗ =π/t∗.

The longitudinal magnetization ρx(t) is the local observable that measures the ferro- magnetic order in the Ising ordering direction. Its equilibrium and already known nonequi- librium properties have been summarized already in Sec. 4.2.1. Formally, the longitudinal magnetization after the quench is given by

ρx(t) = 1 2hψ0(t)|σ x l|ψ0(t)i. (4.97) As the operatorσx

l connects states with different fermionic parity, compare the introduction to Sec. 4.2, its expectation value cannot simply evaluated in terms of the Jordan-Wigner

fermions. This technical detail can be circumvented by alternatively analyzing the spin- spin correlation function

ρxx(t, r) = 1 4hψ0(t)|σ x lσ x l+r|ψ0(t)i (4.98)

from which the value of the order parameter can be deduced in the limit of large distances

r_{→ ∞} [118]:

ρ2_x(t) = lim

r→∞ρxx(t, r). (4.99)

This is called the Cluster decomposition. Most importantly, the fermion parity is conserved by evaluating ρxx(t, r) such that the Jordan-Wigner representation of the Ising model can be used. Expressing the Pauli matrices in terms of Jordan-Wigner fermions it is possible to write ρxx(t, r) [109]

ρxx(t, r) =hψ0|Bl(t)Al+1(t)Bl+1(t). . . Bl+r−1(t)Al+r(t)|ψ0i (4.100)

with the (up to a prefactor) Majorana operators Al=c

†

l +cl, Bl=c †

l −cl. (4.101)

Each of the Al(t) and Bl(t) are linear combinations of fermionic operators and |ψ0i is the

ground state of a quadratic operator allowing to use Wicks theorem [181]. As a consequence the spin-spin correlation function can be related to a Pfaffian [147] that can be evaluated numerically.

For quenches within one phase the longitudinal magnetization decays exponentially in time [22, 23], see Eq. (4.32). On the other hand, for large transverse magnetic fields where the coupling between the neighboring spins can be neglected one may expect that the spin

expectation values precess in the xy-plane. The transition from exponential decay to a

damped Larmor precession happens precisely at that point where the final magnetic field gf crosses the quantum critical point gf = 1. At this value the external force becomes strong enough to overcome the rigidity of the ferromagnetic order.

In Fig. 4.11 numerical data for the longitudinal magnetization is shown for quenches from the ordered into the disordered phase. The upper three-dimensional plots exemplify the different qualitative behavior for quenches within the ferromagnetic phase and across the quantum critical point. In equilibrium the magnetization in y-direction

ρy(t) = 1

2hψ0(t)|σ y

l|ψ0(t)i (4.102)

vanishes as a consequence of time-reversal symmetry which is, however, broken in the nonequilibrium scenario. Instead it acquires a nonzero value as one can directly deduce from the equation of motion of the order parameter ∂tρx(t) = −gfρy(t). To analyze the precession in the xy-plane it is suitable to introduce normalized functions sx(t) = N−1_(t)ρ

x(t) and sy(t) = N−1(t)ρy(t) on the unit circle with N(t) = q

ρ2

4.2 Dynamical quantum phase transitions in the Ising model 107

this way the exponential decay is eliminated making the precession visible in Fig. 4.11. As one can see in Fig. (4.11) quenches within the ferromagnetic phase do not lead to a significant dynamics in the spin orientation in the xy-plane. This is a consequence of the rigidity of the ferromagnetic order. The external magnetic field does not provide enough energy to overcome its stability.

In contrast, for quenches across the quantum critical point the spin expectation values in thexy-plane start to precess as is expected for large transverse magnetic fields. Remark- ably, the frequency Ω∗ of this precession is not set by the mass of the final Hamiltonian or the Larmor frequency. Instead Ω∗ =π/t∗ is solely given by the emergent nonequilibrium time scale t∗ that marks the periodicity with which the real-time nonanalyticities in the Loschmidt amplitude appear, see Sec. 4.2.4. This finding is purely numerical and up to the present moment there is no analytic proof for a connection of the dynamics between the Loschmidt amplitude and the longitudinal magnetization. As the data shows, however, there is a deep relationship between the dynamical quantum phase transition and physical quantities. But note that there are other observables such as the transverse magnetization ρz(t) that do not show an influence of the dynamical quantum phase transition. From a physical point of view one may expect that those quantities that are especially sensitive to the initial state are most likely influenced by the nonanalyticities in the Loschmidt am- plitude and thus the dynamical quantum phase transition as discussed in more detail in Sec 4.1.3.

Chapter 5

Periodically driven many-body

quantum systems

One important aspect in the study of nonequilibrium systems is the possibility to generate dynamically new states with properties not accessible through equilibrium thermodynam- ics. For quenches and ramps as investigated in Sec. 2 systems can be trapped dynamically in intermediate states with long lifetimes that are associated with a regime termed prether- malization [13, 120]. The dynamics of integrable systems is restricted by the presence of conservation laws leading to the expectation that they do not thermalize in the long-time limit [137]. Instead, the asymptotic steady state is supposed to be described by generalized Gibbs ensembles that bear some similarity to canonical states but require the maximiza- tion of the entropy constrained by more constants of motion than just energy [145]. As shown for a restricted class of systems there is a deep connection between prethermalization plateaus, generalized Gibbs ensembles, and the proximity to integrable theories [95].

Another possibility for the dynamical generation of new quantum states is provided by a periodic driving. Periodically time-dependent electric fields can lead to a dynamical localization of noninteracting particles on a chain [37]. This takes over to interacting parti- cles on a lattice where it has been demonstrated that the periodic driving can yield effective renormalized microscopic parameters for time-averaged quantities allowing to manipulate and design model Hamiltonians dynamically [38, 134, 175]. As shown below in Sec. 5.1 fast periodic driving can generate nonequilibrium steady states showing time-translational invariance as equilibrium states but with violated fluctuation-dissipation theorem and prop- erties not accessible within equilibrium thermodynamics [71].

Through a periodic driving energy is continuously pumped into the system. For local perturbations the increase in energy can be typically redistributed among the system’s internal degrees of freedom provided the whole system is infinitely large signaling the im- portance of the thermodynamic limit, see for example Sec. 5.1. If the periodic perturbation acts globally the increase in energy can lead to a substantial heating of the system provided there is no external dissipation mechanism included in the description. This is especially important for effective low-energy models such as the Luttinger liquid, see Sec. 5.3 below, as the continuous heating will drive the model description beyond its limits of validity at

some point in time. For the periodically driven Luttinger liquid the absence of dissipation mechanisms manifests in a parametric instability with exponentially growing perturbations. This sets an intrinsic time scale beyond which other internal or external perturbations have to be included for a realistic description.

Compared to quenches or ramps periodically driven systems, especially interacting ones, pose new challenges in terms of methodology. Most theories for interacting many-body systems are based on low-energy approximations whose extension including the periodic driving beyond the linear response regime is not straightforward. Concerning strongly- correlated systems only a restricted class has been addressed. As for the quench or ramp case one can distinguish local periodic perturbations that only act on a small subsystem and global ones. The former has been studied in a number of works for periodically driven Anderson impurity models [126, 113, 1, 114, 124, 69] and related Kondo models [87, 86, 60, 61, 152] using different approaches. The latter for Bose-Hubbard [38, 134], Fermi- Hubbard [82, 42, 41, 175, 9], and Falicov-Kimball models [50, 49, 173, 174].

Despite of these fundamental questions it is important and instructive to study those particular models that exhibit exact and nonperturbative solutions. Below, three different examples will be investigated such as a Kondo model with periodically switched exchange interaction in Sec. 5.1, transport through a noninteracting quantum dot in presence of a periodic bias voltage in Sec. 5.2, and a Luttinger liquid with periodically modulated interaction strength in Sec. 5.3.