The Complex Systems Research Group deals with the cooperative behavior, ordering, equilibrium and far from equilibrium dynamics of interacting systems with many degrees of freedom. Our main research topics are the following.

**1. Disordered quantum magnets**

- application and development of the strong disorder renormalization group method
- critical behavior of long-range interacting systems
- ground state entanglement in inhomogeneous systems

**2. Ground state of interacting classical and quantum systems **

- periodic ordering at finite temperatures
- consequences of Galilei invariance on the ground state superfluidity and ordering

**3. Nonequilibrium relaxation of quantum systems **

- quench dynamics in closed systems
- dynamics of open systems in a fluctuating field

**4.** **Quantum Hall effect **

- study of anomalous quantum Hall effect and topologically protected edge states

**5. Dynamics in nonequilibrium models and networks **

- dynamics and phase transitions of stochastic many-particle systems (lattice gases, models of population dynamics)
- dynamical processes on complex networks

(* ***2016**)

The research activity of Complex Systems research group in 2017 covered various topics in the field of cooperative behavior, phase transitions and nonequilibrium dynamics of systems with many degrees of freedom.

**Entanglement entropy of composite quantum spin chains.** — The entanglement entropy in clean, as well as in random critical quantum spin chains is well known to have a logarithmic scaling with the size of the subsystem. We considered a composite, antiferromagnetic XX chain that consists of a clean and a random part, and found a double-logarithmic scaling of the half-chain entanglement entropy: S∼lnln(*L*). We also considered the case, when the disorder penetrates into the homogeneous part in such a way that its strength decays with the distance *l* as ∼*l*^{−κ}. For *κ*<1/2, the entropy scales logarithmically with a modified prefactor as *S(κ)≃(1−2κ)S(κ=0)*, while for *κ≥1/2*, we recover the double-logarithmic scaling. These results were explained by strong-disorder RG arguments. We also studied the half-chain entanglement entropy across a symmetric, extended random defect, where the strength of disorder decays algebraically on both sides of the interface. In the whole regime *κ*≥0, we found a logarithmic scaling of the entropy, but the variation of the prefactor with *κ* is non-monotonic and discontinuous at *κ*=1/2.

**Quantum relaxation of lattice bosons with cavity-induced interactions.** —The coupling of cold atoms to the radiation field within a high-finesse optical resonator induces long-range interactions which can compete with an underlying optical lattice. The interplay between short- and long-range interactions gives rise to new phases of matter including supersolidity (SS) and density waves (DW). We have shown that for hard-core bosons in one dimension, the ground-state phase diagram (see Fig. 1) and the quantum relaxation after sudden quenches can be calculated exactly in the thermodynamic limit. Remanent DW order is observed for quenches from a DW ground state into the superfluid (SF) phase below a dynamical transition line. After sufficiently strong SF to DW quenches beyond a static metastability line, DW order emerges on top of remanent SF order, giving rise to a dynamically generated supersolid state.

**Proof of phase transition in homogeneous systems of interacting bosons.** — Using the rigorous path integral formalism of Feynman and Kac, we proved London's eighty-year-old conjecture that during the superfluid transition in liquid helium, Bose-Einstein condensation (BEC) takes place. The result is obtained by proving first that, at low enough temperatures, macroscopic permutation cycles appear in the system, and then showing that this implies BEC. We found also that, in the limit of zero temperature, the infinite cycles cover the whole system, while BEC remains partial. Via the equivalence of 1/2 spins and hard-core bosons the method extends to lattice models. We showed that, at low enough temperatures, the spin-1/2 axially anisotropic Heisenberg models, including the isotropic ferro- and antiferromagnet and the XY model, undergo magnetic ordering.

* Figure 1. Phase diagram of the Bose-Hubbard model with cavity-induced infinite-range interactions. Τ, μ, ε, and x denote the tunneling constant, the chemical potential, the strength of cavity-induced interactions, and the density-wave order parameter (imbalance) , respectively. MI, SF, and DW stand for Mott insulator, superfluid, and density wave, respectively.*

**Critical behavior of the contact process near an extended defect.** — The contact process is a simple stochastic lattice model of epidemic spreading. We considered its inhomogeneous variant, where the deviation of the local control parameter from the bulk value tends to zero with the distance from the surface as *λ(l)−λ(∞)=Al ^{−s}*. In the marginal case,

The Complex Systems research group is interested mainly in the cooperative behavior, phase transitions, equilibrium and nonequilibrium dynamics of systems with many degrees of freedom. Our research activity in 2016 covered various interrelated topics of this broad area, which are subjects of intensive research currently.

The *non-equilibrium quench dynamics*, i.e., the time evolution of a closed or open quantum system after a sudden change of global or local parameters is in the focus of current research. Such phenomena are important because they are experimentally accessible in systems of cold atoms trapped in optical lattices. We studied these phenomena in integrable models under different circumstances such as in the presence of a global noise or quenched spatial disorder. The investigation of *quantum correlations*, especially the entanglement properties of the ground state of extended quantum systems with inhomogeneities or at first-order phase transitions also constituted an important part of our activity. An almost inevitable feature of real systems is the presence of *quenched disorder*, which has striking effects both on the static and dynamic properties especially at phase transition points. Another feature of many real systems, which affects the critical behavior, is that the interactions between particles are *long-ranged*, i.e. their strength decays as an inverse power of the distance. We were interested in the interplay of the above two features in theoretical models, which turned out to result in new types of critical behavior characterized by a Kosterlitz-Thouless type of essential singularity or by a *mixed-order* of the phase transition. The latter means that the transition resembles first-order ones in that the order parameter is discontinuous, but, at the same time, the correlations diverge, which is a property of second-order transitions. Mixed-order transitions occur typically in systems with long-range interactions. We have found that simple short-range models can also exhibit this phenomenon locally under a special geometry, namely, near junctions of chains. Besides, we have a contribution to the solution of the long-standing problem of *crystallographic phase retrieval* as well, by introducing a new method called volumic omit map. The results obtained in the above fields are described in detail in the following.

**Non-equilibrium quench dynamics of noisy quantum systems.** — We continued our work on the nonequilibrium dynamics of the quantum Ising chain in the presence of a stochastically varying global transverse field. The dynamics of entanglement entropy and magnetization can be understood by a semi-classical theory of the spreading of quasiparticle excitations, which are described by a continuous-time random quantum walk with stochastic transition amplitudes. In general, the stochastic noise gives rise to decoherence and a diffusive behavior of excitations. For the special case of a dichotomous noise, there can also be coherent modes, which give a ballistic contribution beside the diffusive one and result ultimately in a superdiffusive behavior. We extended these investigations in several directions. We considered a dichotomous noise which changes in discrete times steps with a frequency that decays algebraically in time as p(t)~t^{-κ}. We found that the excitations spread ballistically for κ > 1, while the spreading is anomalous with a κ-dependent dynamical exponent for κ < 1. We also considered different aperiodic modulations such as the Fibonacci sequence, where the spreading of excitations is found again to be anomalous with a non-universal dynamical exponent z. In the high frequency limit, for sequences with a non-positive wandering exponent, z is found to be close to 1, while, for a positive wandering exponent, it is considerably less than 1. We studied the effect of a point-like source of dichotomous noise on the dynamics of an excitation initially localized near the source. Here, the distributions of the position at different times still follow the ballistic scaling, which is characteristic of the noiseless quantum walk. However, a depletion zone with nearly zero probability develops around the origin, the width of which increases linearly in time.

**Critical quench dynamics of random quantum spin chains. - **By means of free-fermion techniques combined with multiple-precision arithmetic, we studied the time evolution of the average magnetization, m(t), of the random transverse-field Ising chain after global quenches. We observed different relaxation behaviors for quenches starting from different initial states to the critical point. Starting from a fully ordered initial state, the relaxation is logarithmically slow described by m(t) ∼ ln^{a}t, and in a finite sample of length L, the average magnetization saturates at a size-dependent plateau m_{p}(L) ∼ L^{-b}; here, the two exponents satisfy the relation b/a = ψ = 1/2. Starting from a fully disordered initial state, the magnetization stays at zero for a period of time until t = t_{d} with ln t_{d} ∼ L^{Ψ} and then starts to increase until it saturates to an asymptotic value m_{p}(L) ∼ L^{-b’}, with b’ ≈ 1.5. For both quenching protocols, finite-size scaling is satisfied in terms of the scaled variable ln t/L^{Ψ} . Furthermore, the distribution of long-time limiting values of the magnetization shows that the typical and the average values scale differently and the average is governed by rare events. The non-equilibrium dynamical behavior of the magnetization is explained through a semi-classical theory.

**Entanglement entropy of the Q ≥ 4 quantum Potts chain. -- **The entanglement entropy, S, is an indicator of quantum correlations in the ground state of a many-body quantum system. At a second-order quantum phase-transition point in one dimension, S generally has a logarithmic singularity. We considered quantum spin chains with a first-order quantum phase transition, the prototype being the Q-state quantum Potts chain for Q > 4 and calculated S across the transition point. According to numerical density matrix renormalization group results, at the first-order quantum phase transition point S shows a jump, which is expected to vanish for Q → 4^{+}. This jump was calculated in leading order as ∆S = lnQ[1−4/Q−2/(QlnQ)+ O(1/Q^{2})].

**Long-range random transverse-field Ising model in three dimensions. — **We considered quantum magnets with long-range interactions in the presence of quenched disorder. Such a system is realized by the compound LiHo_{1-x}Y_{x}F_{4}, in which a fraction x of the magnetic Ho atoms are replaced by non-magnetic Y atoms. A related but somewhat simplified model, which describes the low-energy properties is the three-dimensional random transverse-field Ising model with long-range ferromagnetic interactions, which decay as a power α of the distance. Using a variant of the strong-disorder renormalization group method we studied numerically the phase-transition point and the paramagnetic phase. The schematic flow diagram in terms of the dynamical exponent z and the ratio r of the frequency of coupling and field decimations can be seen in Fig. 1. We found that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but possibly not asymptotically exact. The distribution of the (sample dependent) pseudo-critical points is found to scale with 1/lnL, L being the linear size of the sample. Similarly, the critical magnetization scales with (lnL)^{χ}/L^{d} and the excitation energy behaves as L^{−α}. Using extreme-value statistics, we argued that, extrapolating from the ferromagnetic side, the magnetization approaches a finite limiting value and thus the transition is of mixed order.

**Figure 1.** RG flow diagram of the long-range random transverse-field Ising model. At r=0, the line of stable (a/z>1) and unstable (α/z<1) fixed points is separated by the critical fixed point indicated by the red dot.

**Random-bond Potts chain with long-range interactions**. — We studied phase transitions of the ferromagnetic q-state Potts chain with random nearest-neighbor couplings having a variance ∆^{2} and with homogeneous long-range interactions, which decay with the distance as a power r^{−(1+σ)}, σ > 0. In the large-q limit, the free energy of random samples of length L ≤ 2048 was calculated exactly by a combinatorial optimization algorithm. The phase transition reamins of first order for σ < σ_{c}(∆) ≤ 0.5, while the correlation length becomes divergent at the transition point for σ_{c}(∆) < σ < 1. In the latter regime, the average magnetization is continuous for small enough ∆, but, for larger ∆, it is discontinuous at the transition point, thus the phase transition is of mixed order. A schematic phase diagram showing the short-range (SR), long-range (LR), first-order (FO), second-order (SO), and mixed-order (MO) phases can be seen in Fig. 2.

**Figure 2.** Schematic phase diagram of the long-range random Potts model.

**Critical behavior of the contact process near multiple junctions.** — The contact process is a basic stochastic lattice model of epidemic spreading or population dynamics. It displays a nonequilibrium phase transition between a fluctuating active phase and an absorbing phase, which is continuous in any dimensions and falls into the robust universality class of directed percolation. Discontinuous transitions are rare in low dimensional fluctuating systems; for the particular case of one-dimensional systems with short-range interactions, they are conjectured to be impossible since fluctuations destabilize the ordered phase. We demonstrated by numerical simulations that a suitable topology of the underlying lattice is able to induce a discontinuous local transition even with a simple dynamics such as the contact process. We have considered, namely, a multiple junction of M semi-infinite chains. As opposed to the continuous transitions of the translationally invariant (M=2) and the semi-infinite (M=1) system, the local-order parameter is found to be discontinuous for M>2. The temporal correlation length diverges algebraically at the critical point, thus the transition is of mixed order. Interestingly, the corresponding exponents on the two sides of the transition are different. We proposed a scaling theory, which is compatible with the numerical results and explains the exponent asymmetry by the presence of an irrelevant local variable, which is harmless in the active phase, but becomes dangerous in the inactive phase. Quenched spatial disorder is found to make the transition continuous, in agreement with earlier renormalization group results.

**Phase problem of crystallography.** – In the framework of our study of the problem of crystallographic phase retrieval, we introduced a new method called &quo;volumic omit map&quo;, to accelerate slowly converging dual space iterative procedures. Alternating-projection-type dual-space algorithms have a clear construction, but are susceptible to stagnation and, thus, inefficient for solving the phase problem ab initio. To improve this behavior, new omit maps were introduced, which are real-space perturbations applied periodically during the iteration process. The omit maps are called volumic because they delete some predetermined subvolume of the unit cell without searching for atomic regions or analyzing the electron density in any other way. The basic algorithms of positivity, histogram matching and low-density elimination were tested by their solution statistics. It is concluded that, while all these algorithms based on weak constraints are practically useless in their pure forms, appropriate volumic omit maps can transform them to practically useful methods. In addition, the efficiency of the already useful reflector-type charge-flipping algorithm can be further improved. It is important that these results are obtained by using non-sharpened structure factors and without any weighting scheme of reciprocal-space perturbation.