We present a general dynamic finite-size scaling theory for the quantum dynamics after an abrupt quench, at both continuous and first-order quantum transitions. For continuous transitions, the scaling laws are naturally ruled by the critical exponents and the renormalization-group dimension of the perturbation at the transition. In the case of first-order transitions, it is possible to recover a universal scaling behavior, which is controlled by the size behavior of the energy gap between the lowest-energy levels. We discuss these findings in the framework of the paradigmatic quantum Ising ring, and support the dynamic scaling laws by numerical evidence.
We study off-equilibrium behaviors at first-order transitions (FOTs) driven by a time dependence of the temperature across the transition point T(c), such as the linear behavior T(t)/T(c)=1±t/t(s) where t(s) is a time scale. In particular, we investigate the possibility of nontrivial off-equilibrium scaling behaviors in the regime of slow changes, corresponding to large t(s). We consider the two-dimensional Potts models, which provide an ideal theoretical laboratory to investigate issues related to FOTs driven by thermal fluctuations. We put forward general ansatzes for off-equilibrium scaling behaviors around the time t=0 corresponding to T(c). Then we present numerical results for the q=10 and 20 Potts models. We show that off-equilibrium scaling behaviors emerge at FOTs with relaxational dynamics, when appropriate boundary conditions are considered, such as mixed boundary conditions favoring different phases at the opposite sides of the system, which enforce an interface in the system.
This thesis reports the results obtained during my PhD research in the field of out of equilibrium quantum many-body systems. Chapter 1 consists in a brief introduction of the field and the introduction of concept that are useful for the following chapters. In Chapter 2 the statistics of the work as a tool for characterizing the dynamics of many-body quantum systems is introduced its general features discussed. Then, such a statistics is computed for generic time-dependent protocols (both global and local) in the quantum Ising chain and in the Gaussian field theory, showing, in particular, that in its low-energy part there are features that are independent of the details of the specific chosen protocol. Chapter 3 is devoted to the study of the dynamical phase transition in the
O(N) quantum vector model in the
N→∞ limit, whose critical properties in generic dimensions are characterized. Moreover, a strong connection between such a transition and the statistics of excitations produced in a double quench as a function of the waiting time is showed. The chapter ends by studying the fate of the dynamical transition and the its critical properties when a ramp of finite duration
τ is applied to the system instead of a sudden quench. In particular, we will show that when
τ→∞ the critical point tends to the equilibrium critical point (at zero temperature) in a power-law fashion and that for every finite
τ the critical properties are always the same (and different from the equilibrium critical properties). Finally in Chapter 4 we will discuss the emergence of a non adiabatic behavior in the dynamics of the order parameter for a low dimensional quantum system driven within a gapped phase by considering in detail the case of a quantum Ising chain subject to a linear variation in time of the transverse field, showing that, no matter how slowly the ramp is performed, such a change leads eventually to the disruption of the order.
Scaling behavior for first-order phase transitions can be derived alternatively but consistently from renormalization-group, phenomenological, or finite-size considerations. A general analysis of densities at a renormalization-group fixed point demonstrates that if the coexistence of
p distinct phases is possible, then
p distinct eigenvalue exponents must equal the spatial dimensionality. This basic eigenvalue (or scaling) exponent condition can also be derived phenomenologically by various arguments not depending on detailed renormalization-group considerations. A scaling description of first-order phase transitions is presented and extended to finite systems with linear dimensions
L, leading to a rounding proportional to ${L}^{\ensuremath{-}d}$, response-function maxima varying as ${L}^{d}\ensuremath{\propto}N$, and boundary-condition-dependent shifts which may be as large as $\ensuremath{\sim}{L}^{\ensuremath{-}1}$.
In this paper, I give an overview over a recently developed rigorous theory of finite-size scaling near first order transitions. Leaving out details of the mathematical proofs, the main emphasis is put on the underlying physical ideas and the discussion of the validity of the results for regions which are, in the sense of mathematical rigour, not covered by the original papers. I present both the finite-size scaling for cubic systems and for long cylinders, discussing also a recent controversy which stems from our work on the finite-size scaling of the mass gap in long cylinders.