We show that quantum fluctuations display a singularity at thermal critical points, involving the dynamical z exponent. Quantum fluctuations, captured by the quantum variance [Frérot et al., Phys. Rev. B 94, 075121 (2016)PRBMDO2469-995010.1103/PhysRevB.94.075121], can be expressed via purely static quantities; this in turn allows us to extract the z exponent related to the intrinsic Hamiltonian dynamics via equilibrium unbiased numerical calculations, without invoking any effective classical model for the critical dynamics. These findings illustrate that, unlike classical systems, in quantum systems static and dynamic properties remain inextricably linked even at finite-temperature transitions, provided that one focuses on static quantities that do not bear any classical analog-namely, on quantum fluctuations.
We revisit the two-dimensional quantum Ising model by computing renormalization group flows close to its quantum critical point. The low but finite temperature regime in the vicinity of the quantum critical point is squashed between two distinct non-Gaussian fixed points: the classical fixed point dominated by thermal fluctuations and the quantum critical fixed point dominated by zero-point quantum fluctuations. Truncating an exact flow equation for the effective action, we derive a set of renormalization group equations and analyze how the interplay of quantum and thermal fluctuations, both non-Gaussian in nature, influences the shape of the phase boundary and the region in the phase diagram where critical fluctuations occur. The solution of the flow equations makes this interplay transparent: we detect finite temperature crossovers by computing critical exponents and we confirm that the power law describing the finite temperature phase boundary as a function of control parameter is given by the correlation length exponent at zero temperature as predicted in an
ϵ-expansion with
ϵ=1 by Sachdev [Phys. Rev. B 55, 142 (1997)].
We study the critical dynamics of the quantum random-field Ising model using the nonperturbative functional renormalization group (NP-FRG). The static critical behavior is found to be controlled by the zero-temperature fixed point of the classical random-field Ising model, where both thermal and quantum fluctuations are dangerously irrelevant. Considering a family of quantum dynamical universality classes defined by a bare dynamical kernel
FΛ(ω)∼∣ω∣σ, we show how this fluctuationless fixed point nevertheless controls the quantum dynamics by computing the full Matsubara-frequency dependence of the running dynamical kernel
Fk(ω). This is essential at zero temperature: a naive treatment of the dynamical kernel flow leads to a divergence at a finite length scale, resulting in apparent localization. In contrast, keeping the full frequency dependence of the dynamical kernel and choosing a regulator adapted to its running scale yields a controlled flow. The resulting dynamics is of activated form, with a relaxation time given by
lnτ∼ξΨ. The exponent
Ψ is determined by the static RFIM fixed-point exponents and by
σ. At finite temperature, the flow crosses over to the classical thermally activated scaling of the random-field Ising model. These results provide a quantitative field-theoretic realization of the heuristic activation scenario proposed earlier for the quantum random-field model and establish a framework for analyzing the dynamics of other disordered quantum systems that may exhibit similar tentative localization-like singularities.
We face the problem of detecting and featuring footprints of quantum criticality in the finite-temperature behavior of quantum many-body systems. Our strategy is that of comparing the phase diagram of a system displaying a
T=0 quantum phase transition with that of its classical limit, in order to single out the genuinely quantum effects. To this aim, we consider the one-dimensional Ising model in a transverse field: while the quantum
S=1∕2 Ising chain is exactly solvable and extensively studied, results for the classical limit $(S\ensuremath{\rightarrow}\ensuremath{\infty})$ of such model are lacking, and we supply them here. They are obtained numerically, via the transfer-matrix method, and their asymptotic low-temperature behavior is also derived analytically by self-consistent spin-wave theory. We draw the classical phase diagram according to the same procedure followed in the quantum analysis, and the two phase diagrams are found unexpectedly similar: Three regimes are detected also in the classical case, each characterized by a functional dependence of the correlation length on temperature and field analogous to that of the quantum model. What discriminates the classical from the quantum case are the different values of the exponents entering such dependencies, a consequence of the different nature of zero-temperature quantum fluctuations with respect to the thermal ones.
The stochastic quantization of Nelson is extended to the finite-temperature equilibrium states and is applied to the study of the quantum effects in the critical dynamics of the time-dependent Ginzburg-Landau model in the large-
n limit. A temperature-dependent (non-Markovian) stochastic process for the order parameter is derived which contains the crossover from the ground-state dynamics dominated by the quantum fluctuations, to classical critical dynamics in the neighborhood of the finite-temperature critical points. The crossover is exhibited by means of the effective dynamical critical exponent.
We study the effect of quenched time-independent random fields coupled linearly to the order parameter on the dynamical critical behavior of spin systems. We assume that the dynamics is described by a Langevin equation without conservation of the order parameter. It is shown that the dominant fluctuations are those induced by the random fields and, therefore, thermal fluctuations are irrelevant. This allows us to establish a relation between this model and a quantum spin system in the presence of a quenched random field. Moreover, we find that only static exponents in
D dimensions are the same as those of the pure ($D\ensuremath{-}2$)-dimensional theory, but the dynamical exponent
z does not satisfy this relation. The quantum system in
D dimensions is studied through its (
D+1)-dimensional equivalent model where the quenched random fields are totally correlated in the additional imaginary-time ($\ensuremath{\tau}$) direction. The system is anisotropic, and there is a new exponent
zA associated with the scaling behavior in the $\ensuremath{\tau}$ direction. We find the relation
z=2zA to all orders in perturbation theory. For the zero-temperature quantum model we find that the static (zero-frequency) exponents are the same as those of the ($D\ensuremath{-}3$)-dimensional pure quantum model. At finite temperature, when the classical system is finite in the $\ensuremath{\tau}$ direction, we predict a crossover to
D-dimensional classical behavior in nonstatic response and correlation functions, with crossover exponent ${({z}_{A}{\ensuremath{\nu}}_{(D\ensuremath{-}2)})}^{\ensuremath{-}1}$, where ${\ensuremath{\nu}}_{(D\ensuremath{-}2)}$ is the exponent $\ensuremath{\nu}$ for the ($D\ensuremath{-}2$)-dimensional pure classical system. The static correlation functions do not have this cross-over behavior and are the same as those of the ($D\ensuremath{-}3$)-dimensional pure quantum model. The dimensional shift in static quantities for both quantum and classical dynamical systems is a consequence of a supersymmetry in the underlying field theory. The exponent ${z}_{A}=1+\ensuremath{\eta}+O(\frac{1}{{N}^{2}})$ in the large-
N limit or ${z}_{A}=1+c\ensuremath{\eta}+O({\ensuremath{\epsilon}}^{4})$ in the $\ensuremath{\epsilon}$ expansion where $c=1\ensuremath{-}\frac{3}{4}[\frac{(N+2)}{{(N+8)}^{2}}]\ensuremath{\epsilon}$ and $\ensuremath{\eta}$ is the same as in the ($D\ensuremath{-}2$)-dimensional pure classical system. We also study the above classical random-field Ising problem using the interface approach, but are unable to draw any definite conclusion about the dynamics at the lower critical dimension
D=3.
We explore the robustness of universal dynamical scaling behavior in a quantum system near criticality with respect to initialization in a large class of states with finite energy. By focusing on a homogeneous
XY quantum spin chain in a transverse field, we characterize the nonequilibrium response under adiabatic and sudden quench processes originating from a pure as well as a mixed excited initial state, and involving either a regular quantum critical or a multicritical point. We find that the critical exponents of the ground-state quantum phase transition can be encoded in the dynamical scaling exponents despite the finite energy of the initial state. In particular, we identify conditions on the initial distribution of quasiparticle excitation that ensure that Kibble-Zurek scaling persists. The emergence of effective thermal equilibrium behavior following a sudden quench toward criticality is also investigated, with focus on the long-time expectation value of the quasiparticle number operator. Despite the integrability of the
XY model, this observable is found to behave thermally in quenches to a regular quantum critical point, provided that the system is initially prepared at sufficiently high temperature. However, a similar thermalization behavior fails to occur in quenches toward a multicritical point. We argue that the observed lack of thermalization originates in this case in the asymmetry of the impulse region that is also responsible for anomalous multicritical dynamical scaling.