Please find the program of the 11th Trieste-Ljubljana-Zagreb meeting

Simulation of the single-particle tunneling problem by means of the Suzuki-Trotter approximation (STA) is analyzed. The target system describes a particle hopping across a chain of sites with position-dependent potential profile. The latter is assumed to be smooth and posses several local minima separated by a potential barrier, arranging a tunneling problem between the localized states in different minima. The STA error is found to manifest itself in three ways: i) perturbative energy shifts, ii) nonperturbartive renormalization of the tunneling rates, and iii) perturbative leakage of total probability to other states. In the general case, the first type of error is the most essential, as the emerging detuning of the tunneling resonance has to be compared with exponentially small tunneling rates. In absence of detuning (e.g. if the resonance is protected by symmetry), STA is found to cause exponential enhancement of the tunneling rates. The last type of error classifies the overall defect in the wave function and thus delineates the region of sufficiently weak distortion of the wave function due to STA. The conducted analysis confirms the naive criteria of applicability max{T, P } ≪ 1/δt (with T, P being the typical scales of kinetic and potential terms, respectively), while also revealing the structure of error and its actual behavior with system parameters. Analysis of the case of large Trotter step is also performed, with the main result being the reconstruction of the low-energy spectrum due to coupling between states with energy difference close to 2π/δt. The connection of the obtained results with the rigorous upper error bounds on the STA error is discussed, with particular emphasis on the reasons for the fact that these rigorous bounds are not always saturated. We also point out that the proposed problem can be directly implemented on existing quantum devices [arXiv:2012.00921]. In particular, we give a detailed description of an experimental design that demonstrates the described physics. The talk is based on the recent paper [arXiv:2312.04735].

We study the out-of-time-order correlation function (OTOC) in a
lattice extension of the Sachdev-Ye-Kitaev (SYK) model with quadratic
perturbations. The results obtained are valid for arbitrary time
scales, both shorter and longer than the Ehrenfest time. We
demonstrate that the region of well-developed chaos is separated from
the weakly chaotic region by the “front region”, which moves
ballistically across the lattice. The front velocity is calculated for
various system’s parameters, for the first time for SYK-like models.

*Aleksey Lunkin, Landau Institute Moscow*

We consider black hole linear perturbation theory in a four-dimensional Schwarzschild (anti-)de Sitter background. We describe two methods that provide the quantization condition for the quasinormal mode frequencies of the perturbation field. The first consists of using the Nekrasov-Shatashvili functions, or, equivalently, the classical Virasoro conformal blocks, to obtain the connection coefficients for the differential equation encoding the spectral problem. The second method is based on a perturbative expansion of the local solutions of the differential equation, that involves multiple polylogarithmic functions. We conclude by showing how the two methods shed light on the mathematical structure of the quasinormal mode frequencies, and discussing how they can be generalized to problems in different backgrounds, emphasizing their effectiveness.

Paolo Arnaudo, SISSA

We study localization phenomena in an SNS junction with a disordered metallic wire as its normal part. Standard description of the Josephson effect in such systems is based on the Usadel equation. However, the classical approach remains valid only while the junction is shorter than the localization length in the wire. In the opposite limit, quantum effects become important and the Usadel description is no longer valid.

We develop a general theory of the Josephson effect taking into account all localization (quantum) contributions. Our theory is based on the nonlinear sigma model and can be applied to junctions of arbitrary length including very long junctions (exceeding the localization length) when a fully quantum description is required. Applied to the Josephson effect in this limit, the theory predicts three qualitatively different regimes depending on relation between the length of the junction, superconducting coherence length, and localization length. In all these regimes, we found analytical expressions for the current-phase relation and estimates for the critical current. In particular, we demonstrate that Ambegaokar-Baratoff relation can remain valid under certain conditions even in the strongly localized limit.

Pavel Ostrovsky, Max Planck Institute Stuttgart

We consider a harmonic chain of oscillators in the presence of a disordered magnetic field. The disorder, in general, causes localization of the normal modes, due to which a system becomes insulating. However, for this system, the localization length diverges as the normal mode frequency approaches zero. Therefore, the low-frequency modes contribute to the heat transmission and the heat current goes down as a power law with the system size. This power law is determined by the small frequency behavior of some Lyapunov exponents and the heat transmission in the thermodynamic limit. We find the behavior of these two quantities and show that the power law for the current is different for zero and nonzero expectation value of the magnetic field.

*Junaid Majeed-Bhat, FMF Ljubljana

Unusual day – Friday 2pm

In the talk, I briefly discuss the present-day view of the magnetic field evolution of neutron stars and its main observational appearance. Then, I focus on the possibility of finding magnetars in binary systems of a different kind. Possible candidates include accreting neutron stars (in particular, ultra-luminous X-ray sources), gamma-ray sources, and fast radio bursts. I present a scenario in which the existence of a highly magnetized neutron star in a high-mass X-ray binary is possible at an age ~few million years.

A celebrated one-parameter scaling of “Gang of four” was a guiding paradigm in the field of Anderson localization for decades. It was shown  to work very well in three dimensions and was  analytically proven in 2+\epsilon dimension using the renormalization group (RG) analysis of the non-linear supersymmetric sigma-model. However, its applicability to the Anderson localization problem on lattices of high dimensionality $d$ and for hierarchical graphs like Cayley tree and Random Regular Graph (RRG) has recently been put in question. The point is that the Anderson transition on high-dimensional lattices corresponds to small dimensional conductance where the saddle-point approximation which leads to a geometrical constraint Q^{2}=1 in the non-linear sigma model, is no longer parametrically justified. Numerical methods also have very limited access to this problem because of the small-size limitation. The situation is even more serious for hierarchical graphs. The point is that the enlargement of blocks which is in the core of the RG procedure, leads to increasing of the coordination number on such graphs, in contrast to the lattices of any dimension where it stays constant. Thus the two-parameters scaling naturally emerges for such graphs, the additional parameter being the coordination number.

In this work we make an analytical analysis of the RG on RRG on the basis of the numerical data for the  size-dependent fractal dimension $D_{1}(L)$ of eigenfunctions

which stands for a principal parameter of RG instead of the dimensionless conductance. The main observation is that the single-parameter scaling is established at large enough sizes by merging of the two-parameter flow trajectory with a “single parameter arc” that connects the critical points D=0 and D=1.

This single-parameter arc results from the continuous deformation of the single-parameter flow at finite-dimensions, while the two-parameter trajectories originate from the irrelevant parameters in d-dimensions. The role of the latter increase with increasing the lattice dimensionality d, being minimal at d=3 and d=4 but becoming numerically significant at d=5 and d=6.

These observations raise the question about existence of upper critical dimension d_{c} such that for 2<d<d_{c} the RG is qualitatively similar to the one in 2+\epsilon dimensions and for $d>d_{c}$ it is similar to that on RRG.

The talk will also be broadcast over Zoom via the following link

https://uni-lj-si.zoom.us/j/4933857795?pwd=TnkycEJsYnRuemdjR0l6czVnTHRLUT09

We discuss a unified and systematic symmetry classification of general (interacting) open quantum systems coupled to a Markovian environment, described by a Lindbladian superoperator. Our classification is based on the behavior of the matrix representation of the Lindbladian under antiunitary symmetries and unitary involutions. We find that Hermiticity preservation reduces the number of symmetry classes, while trace preservation and complete positivity do not, and that the set of admissible
classes depends on the presence of additional unitary symmetries: in their absence or in symmetry sectors containing steady states, Lindbladians belong to one of ten non-Hermitian symmetry classes; if however, there are additional symmetries and we consider non-steady-state sectors, they belong to a different set of 14 classes. In both cases, it does not include classes with Kramer’s degeneracy. Moreover, we show that the examples in each class display unique random-matrix correlations. To fully resolve all symmetries, we employ the combined analysis of bulk complex spacing ratios and the overlap of eigenvector pairs related by symmetry operations. To conclude, we discuss how the classification can be extended to PT-symmetric fermionic Hamiltonians, by mapping the requirements of Hermiticity preservation and PT symmetry into each other. As a paradigmatic case study, we consider a two-site non-Hermitian Sachdev-Ye-Kitaev model. By varying the parameters of this model, we can realize 14 different classes in our classification.

*Lucas de Barros Pacheco Seara de Sá, Technical University of Lisbon

We develop a theory of measurement-induced phase transitions (MIPT) for d-dimensional lattice free fermions subject to random projective measurements of local site occupation numbers. Our analytical approach is based on the Keldysh path-integral formalism and replica trick.

In the limit of rare measurements, \gamma << J (where \gamma is measurement rate per site and J is hopping constant), we derive a non-linear sigma model (NLSM) as an effective field theory of the problem. Its replica-symmetric sector is a U(2)/U(1) x U(1) NLSM describing diffusive behavior of average density fluctuations. The replica-asymmetric sector, which describes propagation of quantum information in a system, is a (d+1)-dimensional isotropic NLSM defined on SU(R) manifold with the replica limit R->1, establishing close relation between MIPT and Anderson transitions. On the Gaussian level, valid in the limit \gamma/J -> 0, this model predicts “critical” (i.e. logarithmic enhancement of area law) behavior for the entanglement entropy.

However, one-loop renormalization group analysis shows [1] that for d=1, the logarithmic growth saturates at a finite value ~(J / \gamma)^2 even for rare measurements, implying existence of a single area-law phase. The crossover between logarithmic growth and saturation, however, happens at an exponentially large scale, ln(l_corr)∼J/\gamma, thus making it easy to confuse with a transition in a finite-size system.
Furthermore, utilizing \epsilon-expansion, we demonstrate [2] that the “critical” phase is stabilized for d > 1 with a transition to the area-law phase at a finite value of \gamma / J.

The analytical calculations are supported and are in excellent agreement with extensive numerical simulations [1,2] for d=1 and d=2. For d=2 we determine numerically the position of the transition and estimate the value of correlation length critical exponent.

[1] https://arxiv.org/abs/2304.03138
[2] https://arxiv.org/abs/2309.12405