Том 194, № 2 (2018)

We present arguments supporting the use of the Bogoliubov method of quasiaverages for quantum systems. First, we elucidate how it can be used to study phase transitions with spontaneous symmetry breaking (SSB). For this, we consider the example of Bose–Einstein condensation in continuous systems. Analysis of different types of generalized condensations shows that the only physically reliable quantities are those defined by Bogoliubov quasiaverages. In this connection, we also solve the Lieb–Seiringer–Yngvason problem. Second, using the scaled Bogoliubov method of quasiaverages and considering the example of a structural quantum phase transition, we examine a relation between SSB and critical quantum fluctuations. We show that the quasiaverages again provide a tool suitable for describing the algebra of critical quantum fluctuation operators in both the commutative and noncommutative cases.

Using the projection operator method, we obtain approximate time-local and time-nonlocal master equations for the reduced statistical operator of a multilevel quantum system with a finite number N of quantum eigenstates coupled simultaneously to arbitrary classical fields and a dissipative environment. We show that the structure of the obtained equations is significantly simplified if the free Hamiltonian dynamics of the multilevel system under the action of external fields and also its Markovian and non-Markovian evolutions due to coupling to the environment are described via the representation of the multilevel system in terms of the SU(N) algebra, which allows realizing effective numerical algorithms for solving the obtained equations when studying real problems in various fields of theoretical and applied physics.

We study the low-temperature properties of the p-spin spin glass model in the spin-one (three-state) case for large values of p. We show that the one-step replica symmetry-breaking phase is unstable at a very low temperature, and we calculate the explicit boundary of the stability interval, the Gardner temperature, analytically for large values of p. This temperature for the spin-one model has the same form of dependence on p as in the case of Ising spins (two states). In the one-step replica symmetrybreaking state, a quadrupolar orientational glass coexists with the spin glass and also with a regular quadrupole ordering.

We study a new property of integrable systems, the existence of infinitely many local three-dimensional conservation laws for pairs of integrable two-dimensional hydrodynamic chains. We describe an effective algorithm for successively computing an infinite set of three-dimensional conservation laws for the Benney pair of commuting flows.

We study the optimization problem for remote one- and two-qubit state creation via a homogeneous communication line of spin-1/2 particles using local unitary transformations of the multiqubit sender and extended receiver. We show that the maximum length of a communication line used for the needed state creation (the critical length) increases as the dimensionality of the sender and extended receiver increases. We use the model with the sender and extended receiver comprising up to ten qubits for the one-qubit state creation and consider the creation of two particular states, the almost pure state and the maximally mixed state. Regarding the two-qubit state creation, we numerically study the dependence of the critical length on a particular triad of independent eigenvalues to be created using the model with a four-qubit sender without an extended receiver.