The above properties describe the single particle characteristics of the AFAI, and lead to its defining characteristic as a many body system: robust quantized charge pumping that persists in a **non**-adiabatic driving regime. This behavior is in con- trast to that of the one-dimensional “Thouless pump,” where the pumped charge is quantized only in the limit of infinitely slow driving [105]. The setup that realizes **non**-adiabatic quantized pumping is illustrated in Fig. 8.1. We consider a strip of AFAI in which all of the states close to one edge are populated with fermions. In this situation, the total current flowing through the strip is quantized as an integer times the inverse driving period, hIi = eW/T. Here, W is the bulk winding number char- acterizing the parent anomalous Floquet topological phase. The current averaged over many driving periods is hIi. The quantized charge pumping is a direct result of the edge structure defined above: when the fermion occupation is of the form shown in Fig. 8.1, the edge states on one side are completely filled, while the localization of the bulk states prevents current from flowing in the direction perpendicular to the edge. In Sec. 8.4, we derive the relation between the quantization of the charge current and the bulk topological invariant, and discuss temporal fluctuations about the quantized value.

Show more
182 Read more

VII. CONCLUSION
We have introduced a model of **driven**-dissipative two- level **systems** with coherent and dissipative branching and coagulation dynamics, which features a unique absorbing state throughout the entire parameter regime. By mapping the dis- sipative Heisenberg-Langevin equations to a nonequilibrium path integral for the density of the excited atomic levels, we have shown that this model undergoes a phase transition from the absorbing state towards an active, finite excitation density state for sufficiently strong branching rates. In the classical limit, i.e., in the limit of weak coherent branching, the system corresponds to a classical contact process and the absorbing state phase transition belongs to the universality class of directed percolation. On the other hand, in the **quantum** limit, i.e., in the limit of vanishing incoherent branching, the phase transition is drastically modified and becomes a discontinuous nonequilibrium first-order transition. These two regimes are separated by a bicritical point, which features a continuous absorbing state phase transition, which resembles the tricritical DP class. The dynamics at this point represents the **quantum** analog of the classical contact process. Performing a functional renormalization group analysis, we have analyzed the critical scaling behavior and characterized the universality class of this **quantum** contact process below its upper critical dimension d c = 3. By showing that the critical scaling regime of the bicritical point is extended in parameter space, we have demonstrated that the **quantum** contact universality class can be explored experimentally for reasonably large system sizes and with moderate parameter fine tuning. The experimental realization of the **quantum** contact process with ensembles of laser-excited Rydberg atoms opens the door for the exploration of novel **quantum** and classical nonequilibrium phase transi- tions in the framework of current cold-atom experiments.

Show more
31 Read more

derlying theory, and to this day thermodynamics remains one of the most successful and complete physical theories.
The next step taken was to connect and derive these thermody- namic quantities from microscopic descriptions of the system, a field that is known as statistical mechanics. But as already mentioned, it is not practical nor even theoretically possible to know at a microscopic level the simultaneous position and velocities of the ⇠ 10 23 particles involved. For this reason, one introduces the concept of a statistical en- semble, a large collection of virtual independent copies of the system in various states. It is thus a probability distribution over all states of the system. Statistical **equilibrium** denotes the point not at which the particles have stopped moving, but at which the ensembles do not change any longer. For an isolated system, there are three **equilibrium** ensembles which are usually considered. The simplest is called the mi- crocanonical ensemble and it describes a system with a precisely fixed energy and particle number. It is motivated by the equal a priori prob- ability postulate, which states that in the microcanonical ensemble, the probability for every possible microscopic configuration, often called a microstate, is the same. It takes the value of p i = 1/W , where W is the total number of available microstates. The famous Boltzmann equation, S = k B log W, carved on Boltzmann’s gravestone in Vienna, relates the thermodynamic property entropy with the microscopic de- scription of the system. The logarithm naturally arises, as for two combined **systems**, the number of possible microstates should mul- tiply, while the entropy as an extensive quantity should be additive, S = k B log W 1 W 2 = S 1 + S 2 .

Show more
202 Read more

change” (or “quasi-static change” for the classical **systems**.) However, this idea no longer holds when λ is tuned toward the transition point, because the relaxation time will eventu- ally diverge at the critical point, as discussed in Sec. 1.1. The first attempt to get around this problem is the Kibble-Zurek (KZ) arguments, [ 14 , 15 ] which originally focused on quan- titatively relating defect formation (e.g., the typical defect size and the density of defects) to the rate of change (the quench velocity) of a parameter of the system (such as the tem- perature, external fields, etc.). The KZ mechanism and extensions of it have successfully been used to describe out-of-**equilibrium** physics at both classical and **quantum** phase tran- sitions, for a general review, see Ref. [ 16 ]. In Sec. 1.2.1 we outline the general ideas and basic scalings associated with KZ. In Sec. 1.2.2, based on KZ argument, we derive the gen- eralized dynamic finite-size scaling, and also the **non**-**equilibrium** version of the dual scaling behavior analogous to the **equilibrium** case Eq. (1.8). This **non**-**equilibrium** dual scaling behavior will be tested and verified in various **systems** throughout this dissertation.

Show more
221 Read more

The off-diagonal elements can also be used as a diagnostic of interesting behaviour. While the diagonal elements of an operator describe the aver- age values of quantities, the off-diagonal elements are associated with the relaxation to these averages. As such, the behaviour of off-diagonal elements in the ergodic phase preceding the MBL transition has received recent in- terest. It has been found in numerical studies that the onset of anomalous subdiffusive transport is accompanied by a change in the distributions of off-diagonal matrix elements, which develop long tails, violating Berry’s conjecture that the fluctuations should be Gaussian [63, 67]. This behaviour is demonstrated in Fig. 1.2 [63], where the distributions are consistently Gaus- sian for all system sizes with weak **disorder**, but develop long tails when the **disorder** is stronger. The matrix elements of spin operators in the one- dimensional disordered XXZ model were also shown to satisfy a modified version of the ETH, which includes a power-law correction to the exponential decay of the off-diagonal elements with system size,

Show more
121 Read more

E-mail: imp24@cam.ac.uk
Abstract. We review the practical conditions required to achieve a **non**-**equilibrium** BEC **driven** by **quantum** dynamics in a system comprising a microcavity field mode and a distribution of localised two-level **systems** **driven** to a step-like population inversion profile. A candidate system based on eight 3.8nm layers of In 0

The phase **transitions** between charge-ordered and - disor- dered metallic phases can display **quantum** critical phenomena in close analogy with the heavy-fermion **systems** 16–19 with the critical charge rather than the spin fluctuations driving the CO transition. Such type of fluctuations may be at the origin of both the anomalous properties in the metallic state and Cooper-pair formation. Indeed, **non**-Fermi-liquid behavior as well as **non**-BCS superconductivity have both been predicted and observed in quarter-filled organic materials of the α, β
, and θ -(ET) 2 X type. 63–67 Such heavy-fermion behavior arising from molecular π electrons instead of the d or f electrons, as occurs in the rare earths, may indeed find a natural explanation based on the properties of matter expected near a QCP.

Show more
13 Read more

Figure E.1 depicts the analytically calculated topological phase diagram for an RSW wire as a function of the **disorder** strength and µ for various magnetic field strengths. The transition between a RSW wire and a pair of oppositely polarized PW wires can be seen as increasing magnetic field polarizes the system. The topological order is less robust against **disorder** for higher magnetic fields, because the coherence length becomes longer with increasing B. This is the reason why the spin polarized regimes where PW model applies is typically less robust than the lower field regimes where both spin species exist as seen in Fig. E.1(a) and E.1(c) or (d). In order to complete the discussion, we also present an analytical plot (Figure E.2) for an RSW wire for which B is greater than the subband spacing but less than the bandwidth. While this regime is experimentally very hard to achieve, it is useful for comparing the PW and the RSW regimes. The vertical blue line denotes the bottom of the higher energy spin band beyond which both spin species exist. We note that the critical **disorder** strength increases with the chemical potential, hence spin-polarized regime, which appear at lower chemical potential values, is less robust against **disorder**.

Show more
124 Read more

Discussion
**Driven** **non**-**equilibrium** **systems** are often composed of a driving process and a relaxing process. The later is char- acterized by **transitions** from “higher” states to “lower” states, and is often a sample space reducing process. SSR processes with simple driving processes have been shown to be analytically solvable. They exhibit **non**-Gaussian statistics that is often encountered in **driven** complex **systems**. In particular SSR processes offer an alternative route to understand the origin of power-laws. Here we showed that SSR processes exhibit a much wider range of statistical diversity if the driving process becomes **non**-trivial. Assuming that driving rates may vary with the current state of the system, we demonstrated that practically any distribution function can be naturally associated with state-dependent driving processes. The functional form of the driving function can be extremely simple. Constant driving leads to exact power-laws, a linear driving function λ(x) gives exponential or Gamma distribu- tions, a quadratic function yields the normal distribution. Also the Weibull and Gompertz distributions arise as a consequence from relatively simple driving functions. It is well known how noise and drift parameters can be defined in standard stochastic processes to derive specific stationary distributions. In this sense, note that Eq. ( 4 ) is also the solution of a general family of stochastic differential equation 15 , where the drift and noise terms are

Show more
3 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
(Received 9 July 2015; revised manuscript received 2 October 2015; published 24 November 2015) We study electrical transport in a strongly coupled strange metal in two spatial dimensions at finite temperature and charge density, holographically dual to the Einstein-Maxwell theory in an asymptotically four-dimensional anti – de Sitter space spacetime, with arbitrary spatial inhomogeneity, up to mild assumptions including emergent isotropy. In condensed matter, these are candidate models for exotic strange metals without long-lived quasiparticles. We prove that the electrical conductivity is bounded from below by a universal minimal conductance: the **quantum** critical conductivity of a clean, charge-neutral plasma. Beyond nonperturbatively justifying mean-field approximations to **disorder**, our work demon- strates the practicality of new hydrodynamic insight into holographic transport.

Show more
3 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Dated: November 26, 2015)
We study electrical transport in a strongly coupled strange metal in two spatial dimensions at finite temperature and charge density, holographically dual to Einstein-Maxwell theory in an asymp- totically AdS 4 spacetime, with arbitrary spatial inhomogeneity, up to mild assumptions including emergent isotropy. In condensed matter, these are candidate models for exotic strange metals with- out long-lived quasiparticles. We prove that the electrical conductivity is bounded from below by a universal minimal conductance: the **quantum** critical conductivity of a clean, charge-neutral plasma.

Show more
In his pioneering paper, Hertz 8 studied the paramagnet- to-ferromagnet **quantum** phase transition of itinerant fermions that occurs by varying the exchange coupling between electron spins. He derived an effective action for dynamical fluctu- ations of the bosonic-order parameter. Later, Millis 9 used this approach to calculate temperature dependencies of the correlation length, susceptibility, and specific heat. In the past decade, several authors 10–15 carried out diagrammatic calculations that extend beyond Hertz-Millis theory. They showed that the free energy contains a nonanalytic dependence on the order parameter and its gradients. It was argued 10 that nonanalytic terms occur due to additional soft particle-hole modes that couple to the order-parameter fluctuations. These nonanalytic corrections can render the **transitions** weakly first order at low temperatures and can lead to the instability of **quantum** critical points for the formation of new phases.

Show more
14 Read more

G rec = g rec ( 1 − δ νν 0 ) δ q x ,0 δ k,k 0 , and take a constant density of states ρ 0 for photons with energies
~ ω & E gap .
We now describe the processes resulting from the coupling to the electromagnetic environment in the **driven** system that we consider. The most dominant of these involve **transitions** from Floquet states of predominantly conduction band character to final states of predominantly valence band character, and follow directly from processes present in the **non**-**driven** case. Due to the band inversion described in detail in Sec. 2.2, the − Floquet band has conduction band character for momenta | k | < k R . Therefore, spontaneous **transitions** from the − to the + Floquet band are active for states in this momentum range. Note that these Floquet-Umklapp processes increase the total electronic quasi-energy, and play an important role in determining the density of excitations in the steady state of the system (see Sec. 2.3). The rates of these processes may be controlled to some extent by placing the system in a cavity or photonic crystal, which modifies the photon density of states. In addition, spontaneous **transitions** from the + to the − Floquet band are allowed in the momentum region |k | > k R , where the + Floquet band has predominantly conduction band character. These processes help to reduce the total electronic quasi-energy, but will play a minor role near the steady state where the + Floquet band is mostly empty.

Show more
224 Read more

of **transitions**, where each node represents a state and each link a transition rate between two different states. In this chapter we will try to understand how topological features of the underlying transition network influence the value of the entropy production. Interestingly, for **systems** **driven** out of **equilibrium** both by asymmetric transition rates and an external probability current, close to **equilibrium**, the stationary entropy production is composed by two contributions. The first one is related to the Joule’s law for the heat dissipated by a classical electrical circuit properly introduced, whereas the second contribution has a Gaussian distribution with an extensive mean and a finite variance that does not depend on the microscopic details of the system.

Show more
155 Read more

Nevertheless, by elaborating a synthesis of these various studies, we concluded that the full functionality of the rst scheme based on a narrow bandbass emission spectrum is somehow model dependent, in the sense that it strongly relies on the existence of at photonic bands. In particular, the ability of stabilizing the many-body ground-state in some class of topologically protected **systems** appears to be somehow accidentally related to the existence of iso-energetic Landau levels/quasi-hole excitations, as well as vanishing matrix elements toward higher energy states away from the Laughlin sub-manifold. As a counterpart, in the Bose-Hubbard model the observation of competiting **quantum** eects between localization and delocalization is compromised with such scheme, since its eciency does not extend to the regime where the hopping is comparable to the interaction strength. In view of broader applications, this restrictive functionality led to us to conceive a more advanced and fully novel model based on tailored **non**-Markovian reservoirs with broad bandpass spectra (Chapter 3), designed in such a way to mimick the eect of a tunable articial chemical potential combined with a zero temperature. As a rst step, a **non**-Markovian model featuring a square-shaped spectrum was developed, and the possibility of reproducing the zero temperature **equilibrium** phenomenology of the Bose-Hubbard model in a wide range of parameters was predicted. In particular, our numerical study conrmed the existence of Mott regions with arbitrary integer densities featuring strong robustness against tunneling and losses processes. The system can then undergo a transition toward a superuid-like state either by changing the articial chemical potential or increasing the tunneling. Still, our analysis pointed out exotic behaviours leading to the generation of a weak but **non**- vanishing entropy in some regions of the phase diagram, and those deviations from the **equilibrium** physics were related to the existence of **non**-**equilibrium** channels allowing for the dynamical creation of doublon excitations. The implementation of additional frequency- dependent losses was nally considered in order to circumvent this eect, and it was then showed that the ground-state was successfully recovered for any choice of parameters. This last scheme, which should be accessible with current technology, is to our knowledge the simplest existing experimental proposal of a reliable and exible **quantum** simulator of zero temperature physics in strongly correlated photonic platforms. Since it is only based on generic relaxation mechanisms in the energy landscape, we believe in the universality of our proposal, in the sense that we do not expect neither its eciency to be limited to the Bose-Hubbard model nor that restrictive constraints on the nature of the aimed many-body system need being imposed.

Show more
148 Read more

Chapter 3
Exploration of the stability of
many-body localization in d > 1
Recent works on many-body localization [120, 71, 70] have started to re-examine the case for the existence and stability of the MBL state, going beyond the perturbative arguments of Basko et al [29] (see also Ref. [103]). For instance, Imbrie has given a mathematical proof for the existence of MBL in spin chains with short range interactions [120]. On the other hand, other analyses have pointed to **non**-perturbative effects that can destabilize MBL under certain conditions [71, 70]. In particular, De Roeck and Huveneers [70] have argued that the MBL state is unstable in two or higher dimensions or in **systems** with interactions that decay sub-exponentially with distance. Their instability mechanism hinges upon finite ergodic “bubbles” of weak **disorder** that occur naturally inside an insulator (see Fig. 3.1). According to this narrative, such rare regions may trigger a “thermalization avalanche”— the avalanche commences by thermalizing the immediate surroundings of the bubble, thus creating a larger and more potent bubble which reinforces the process. Because this argument has far reaching implications such as the absence of MBL in two dimensional **systems**, it is important to test the crucial assumptions underlying this conclusion.

Show more
173 Read more

5. Conclusion
Here, we summarize our main ﬁndings related to the dynamics of spin–boson **systems**, and more precisely to the dy- namics of **driven** and dissipative **quantum** impurity **systems**.
When studying the real time dynamics of spin observables, we have made connections with classical Bloch equations and a statistical approach described by an effective temperature. At the same time, the stochastic dynamics of the **quantum** trajectories can produce novel phenomena in terms of many-body physics, such as localization and dissipation-induced **quantum** phase **transitions**, synchronization revivals, and a **quantum** dynamo effect when probing the topology of a **driven** spin-1/2 particle on the Bloch sphere. It is important to note that one is able to measure Berry phase properties of a spin-1/2 particle with the stochastic Schrödinger equation approach, even though paths are encoded in classical blip and sojourn variables. Here, the Berry phase or Chern number is directly related to spin observables. Then, we have shown how the stochastic aspect can be generalized to dissipative spin chains, making some links towards Dicke-type models and mean-ﬁeld approaches.

Show more
36 Read more

These are all results related to purely longitudinal interaction. However, a cou- pling along the transverse direction might as well realize an interesting thermal improvement, since it couples to the tunnelling term that causes **transitions** (the ˆ σ x operator in the Landau-Zener Hamiltonian). Notice that a transverse interaction might be relevant for Landau-Zener processes happening in AQC-QA. Although the longitudinal coupling is dominant for a single qubit in the D-Wave machine, when the Landau-Zener physics emerges from the two lowest-lying eigenstates of a complex multi-qubit Hamiltonian, one would expect that an appropriate model to describe the dissipation should include couplings to the transverse directions. At zero tem- perature, the dissipative Landau-Zener model has been exactly solved for generic coupling directions [90, 91], finding an analytic formula for the final ground state probability that generalizes the famous Landau-Zener formula for the coherent case. Very interestingly, if the coupling has some transverse component, in some cases the final ground state probability can be larger than its coherent counterpart, real- izing again a dissipative enhancement in the spirit of Ref. [33]. The same problem has also been tackled numerically at finite temperature and for an ohmic spectral function [95], utilizing Bloch-Redfield QME (with RWA) and QUAPI: the study, however, does not investigate the **non**-adiabatic region, where we do find the most interesting physics.

Show more
128 Read more

2. Uniqueness of KMS state at high temperatures
It is well-known that, at sufficiently high temperatures, there are no phase-**transitions**, and one expects that **equilibrium** states are unique. This claim is backed by various mathematical results, such as analyticity of the free energy at high temperatures. In this section, we show that, for a large class of **quantum** lattice **systems**, assuming that the temperature is high enough, only a single state satisfies the KMS condition that characterizes thermal

16 Read more

a δ -function two body interaction), but it is strong enough and **non** negligible. The cloud is split by a laser beam in two counter-propagating clouds with opposite momentum.
The two clouds then climb the harmonic potential up to the maximum value allowed by energy conservation, sub- sequently move back toward the centre of the trap where they interact; after this interaction the clouds climb again the potential and the process is repeated many times until the system becomes stationary (when it does). The system is recorded for many of these oscillations (obviously, since the measures are destructive, in the experiment this proce- dure is repeated many times, see the original reference for all details and for a suggestive graphical representation of the experiment). It has also been argued that the time evo- lution is essentially unitary during the whole probed time window. The results of this experiment are considered milestones. Indeed, it has been shown that the momen- tum distribution function attains for large time a stationary distribution for arbitrary space dimensionality. The details of the stationary values do depend on spatial dimension- ality. In one dimension, the system relaxes slowly in time to a **non**-thermal distribution, while in two and three di- mensions, **systems** relax very quickly and thermalise. It has been suggested that the one-dimensional case is spe- cial because the system is almost integrable, as we will discuss in the following.

Show more