We consider translationally invariant quantum spin-1/2 chains with local interactions and a discrete symmetry that is spontaneously broken at zero temperature. We envision experimenters switching off the couplings between two parts of the system and preparing them in independent equilibrium states. One side of the chain is prepared in a disordered phase, and the other in a symmetry-breaking ground state. When the couplings are switched back on, time evolution ensues. We argue that in integrable systems the front separating the ordered region recedes at the maximal velocity of quasiparticle excitations over the ground state. We infer that, generically, the order parameters should vary on a subdiffusive scale of order t^{1/3}, where t is time, and their fluctuations should exhibit the same scaling. This interfacial region exhibits full range correlations, indicating that it cannot be decomposed into nearly uncorrelated subsystems. Using the transverse-field Ising chain as a case study, we demonstrate that all order parameters follow the same universal scaling functions. Through an analysis of the skew information, we uncover that the breakdown of cluster decomposition has a quantum contribution: each subsystem within the interfacial region, with extent comparable to the region, exists in a macroscopic quantum state.
Using a renormalization group approach, we solve the time evolution of random Ising spin chains with generic interactions starting from initial states of arbitrary energy. As a function of the Hamiltonian parameters, the system is tuned through a dynamical transition, similar to the ground-state critical point, at which the local spin correlations establish true long-range temporal order. In the state with a dominant transverse field, a spin that starts in an up state loses its orientation with time, while in the ``ordered'' state it never does. As in ground-state quantum phase transitions, the dynamical transition has unique signatures in the entanglement properties of the system. When the system is initialized in a product state, the entanglement entropy grows as
log(t) in the two ``phases,'' while at the critical point it grows as ${\mathrm{log}}^{\ensuremath{\alpha}}(t)$, with $\ensuremath{\alpha}$ a universal number. This universal entanglement growth requires generic (``integrability breaking'') interactions to be added to the pure transverse field Ising model.
We study the dynamics of the quantum Ising chain following a zero-temperature quench of the transverse field strength. Focusing on the behavior of two-point spin correlation functions, we show that the correlators of the order parameter display an effective asymptotic thermal behavior; i.e., they decay exponentially to zero with a phase coherence rate and a correlation length dictated by the equilibrium law with an effective temperature set by the energy of the initial state. On the contrary, the two-point correlation functions of the transverse magnetization or the density-of-kinks operator decay as a power law and do not exhibit thermal behavior. We argue that the different behavior is linked to the locality of the corresponding operator with respect to the quasiparticles of the model: nonlocal operators, such as the order parameter, behave thermally, while local ones do not. We study which features of the two-point correlators are a consequence of the integrability of the model by analyzing their robustness with respect to a sufficiently strong integrability-breaking term.
Quantum phase transitions occur strictly at zero temperature and are accessed by varying a physical parameter such as a magnetic field or chemical potential. While these transitions themselves are experimentally inaccessible, the existence of a zero temperature quantum critical point has remarkable effects on the dynamical properties of interacting quantum many-body systems at finite temperatures. In this thesis, we study dynamics in four strongly interacting many-body systems near quantum critical points in the context of NMR relaxation, quantum quenches, many-body quantum chaos, and charge diffusion. In chapter 1, we begin by revisiting the issue of NMR relaxation in the quantum disordered phase of Ising spin chains. We find that the imaginary part of the spin susceptibility at low frequences is dominated by rare quantum processes which leads to an NMR relaxation rate that scales as
1/T1∼exp(−T2Δ) near the quantum critical point rather than the previously conjectured
1/T1∼exp(−TΔ). These results resolve certain discrepancies between the energy scales measured with different experimental probes in the quantum disordered paramagnetic phase of the Ising chain system
CoNb2O6. In chapter 2, we study the non-equilibrium dynamics of Sachdev-Ye-Kitaev (SYK) models with
q body interactions after a quantum quench, considering a quench protocol which drives the system from an initial state with long-lived quasiparticle excitations to a final state lacking any long-lived quasiparticle excitations. For
q=4, we find that the timescale for the system to locally thermalize
τeq∼kBTℏ and for
q→∞ we find that the system thermalizes instantaneously. In chapter 3, we study the real time dynamics of quantum electrodynamics in 2+1 dimensions with
Nf flavors of fermions. We focus on thermalization and the onset of many-body quantum chaos and compute the quantum Lyapunov exponent in a
Nf1 expansion. We briefly comment on chaos, locality, and gauge invariance. Finally, in Chapter 4 we consider whether the butterfly velocity - a speed at which quantum information propagates - may provide a fundamental bound on diffusion constants in dirty incoherent metals by computing the charge diffusion constant and the butterfly velocity in charge-neutral holographic matter with long wavelength ``hydrodynamic" disorder. In this limit, we find that the butterfly velocity does not set a sharp lower bound for the charge diffusion constant.
In this work, we present numerical evidence for the spontaneous breaking of a continuous U ( 1 ) symmetry in a nearest-neighbor interacting spin-1 chain at a quantum critical point separating two XY quasi-long-range ordered phases distinguished by a spontaneously broken Z 2 symmetry. Remarkably, the continuous symmetry breaking emerges precisely at the critical point of the discrete order parameter, suggesting a mechanism beyond currently established scenarios. At criticality, the XY correlations develop true long-range order, accompanied by a finite perpendicular magnetization, a zero-frequency Bragg peak in the transverse dynamical structure factor, and sharp gapless collective excitations. From complementary static and dynamical observables, we quantitatively determine the critical exponents, obtaining a dynamical exponent z = 1.50 ± 0.04 and an anomalous dimension η = 1.04 ± 0.03 . Remarkably, the value of z coincides with the one-dimensional Kardar-Parisi-Zhang exponent despite the system being an equilibrium quantum many-body system. We further show that the noninteracting continuum limit is equivalent to the recently introduced transverse quantum fluid, displaying their off-diagonal long-ranged order in one dimension. Complementing the numerical study, we derive the renormalization-group flow equations of the continuum theory to second order in the ɛ expansion. We identify an interacting fixed point whose critical behavior differs from the Ising universality class already at two-loop order, although the perturbative exponents remain far from the numerical values.
We consider the unitary time evolution of an interface in the regime of spontaneously broken symmetry of the two-dimensional quantum Ising model. The interface is induced by an initial condition interpolating between the two degenerate ground states in one of the spatial dimensions. The interpolation is left generic in order to investigate the dependence of the late time dynamics on the initial condition. Exploiting the basis of asymptotic quasiparticle states of the bulk theory, the order parameter is analytically determined at large times in the rough phase. The mechanism allowing the breakdown of this phase as the distance from criticality increases emerges from the theory.
In this thesis we consider isolated quantum systems influenced by disorder and external driving, and explore aspects of localisation and ergodicity-breaking in three different models. We first consider the XYZ spin chain in a disordered magnetic field; this system breaks the U(1) symmetry of the XXZ model and therefore violates the conservation of the z component of the total magnetisation. We identify a quantum phase transition between an ergodic phase and a many-body localised phase at a finite disorder strength that is strongly dependent on the size of the U(1) symmetry-breaking term. We also investigate the breakdown of thermalisation by studying the matrix elements of local operators in the basis of the system's eigenstates, and we find suggestive evidence of subdiffusive energy transport at disorder strengths preceding the localisation transition. We then consider a two-level time-dependent quantum system that is influenced by a fluctuating classical noise term. We show that the noise does not fully randomise the quantum state if the fluctuations of every noise parameter coupled to the system are perfectly correlated; the state of the system instead becomes distributed over a finite subset of the full state space. Lastly we consider a one-dimensional Anderson-localised system that is affected by randomly fluctuating onsite potentials. We explore how the spatiotemporal correlations between these onsite potentials affect the delocalisation of a particle initially localised on a single site.