Suppose a quantum system starts to evolve under a Hamiltonian from some initial state. When, for the first time, will an observable attain a preassigned value? To answer this question, one method often adopted is to make instantaneous measurements periodically and note down the serial number for which the desired result is obtained for the first time. We apply this protocol to an interacting spin system at zero temperature and show analytically that the response of this system shows a nonanalyticity as a function of the parameter of the Hamiltonian and the time interval of measurement. In contrast to quantum phase transitions, this type of phase transition is not a property of the ground state and arises from the Hamiltonian dynamics and quantum-mechanical nature of the measurement. The specific system studied is the transverse Ising chain, and the measurement performed is whether the total transverse magnetic moment (per site) is not equal to 1. The results for some other types of measurement are also discussed.
Suppose a quantum system starts to evolve under a Hamiltonian from some initial state. When, for the first time, will an observable attain a preassigned value? To answer this question, one method often adopted is to make instantaneous measurements periodically and note down the serial number for which the desired result is obtained for the first time. We apply this protocol to an interacting spin system at zero temperature and show analytically that the response of this system shows a nonanalyticity as a function of the parameter of the Hamiltonian and the time interval of measurement. In contrast to quantum phase transitions, this type of phase transition is not a property of the ground state and arises from the Hamiltonian dynamics and quantum-mechanical nature of the measurement. The specific system studied is the transverse Ising chain, and the measurement performed is whether the total transverse magnetic moment (per site) is not equal to 1. The results for some other types of measurement are also discussed.
Measurement-induced phase transitions are often studied in random quantum circuits, with local measurements performed with a certain probability. We present here a model where a global measurement is performed with certainty at every time-step of the measurement protocol. Each time step, therefore, consists of evolution under the transverse Ising Hamiltonian for a time τ, followed by a measurement that provides a ``yes/no'' answer to the question, ``Are all spins up?''. The survival probability after n time-steps is defined as the probability that the answer is ``no'' in all the n time-steps. For various τ values, we compute the survival probability, entanglement in bipartition, and the generalized geometric measure, a genuine multiparty entanglement, for a chain of size L∼26, and identify a transition at τc∼0.2 for field strength h=1/2. We then analytically derive a recursion relation that enables us to calculate the survival probability for system sizes up to 1000, which provides evidence of a scaling τc∼1/L. The transition at finite τc for L∼28 seems therefore to recede to τc=0 in the thermodynamic limit. Additionally, at large time-steps, survival probability decays logarithmically only when the ground state of the Hamiltonian is paramagnetic. Such decay is not present when the ground state is ferromagnetic.
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.
Phase transitions occur when a macroscopic number of local degrees of freedom coherently change their behavior. In ground states of quantum many-body systems, phase transitions due to quantum fluctuations are observed as non-analytic behaviors of order parameters, such as magnetization, as functions of a conjugate parameter, such as the magnetic field. However, as soon as thermal fluctuations are present, these effects are believed to disappear for local observables. We show that this is not necessarily the case: order parameters may still show non-analytic behaviors within their dynamics. With the example of the Ising model and using methods based on hydrodynamic fluctuations, we evaluate the exact order-parameter correlation time, in space-time directions of all velocities, in equilibrium states at nonzero temperature. We reveal non-analytic behaviors of spin correlation times as functions of the magnetic field, velocity, and temperature. As a function of the magnetic field, they occur at values that continuously approach that of the zero-temperature equilibrium transition point as the velocity is decreased and reach it within the light cone, where we obtain a new, temperature-independent logarithmic divergence characterizing the collective dynamics. Thus, collective effects induced by quantum fluctuations persist within the dynamics of local observables.
Abstract Dynamical phase transition in quantum many body systems is usually studied by taking it in the ground state and then quenching a parameter to a new value. We investigate here the dynamics when one performs the time evolution of a generic state and observe that the rate function related to the Loschmidt echo shows non-analytic behavior of two types, one related, and the other unrelated, to the appearance of a quantum phase transition. Specifically, we consider a quantum Ising chain in an initial configuration which is a generic superposition of the eigenstates, and follow its dynamics under the transverse Ising Hamiltonian with a constant field. Depending on the the configuration of the initial state, some singularities appear in the rate function which do not correspond to the equilibrium phase transition of the Hamiltonian. However another class of singularity is found having a connection with the quantum critical point. Some features of the singularities of the rate function have been derived analytically and it is observed that the occupancy of quasiparticle eigenstates plays the key role in triggering the non-analyticities.
We study the dynamics of entanglement in the infinite asymmetric XY spin chain, in an applied transverse field. The system is prepared in a thermal equilibrium state (ground state at zero temperature) at the initial instant, and it starts evolving after the transverse field is completely turned off. We investigate the evolved state of the chain at a given fixed time, and show that the nearest-neighbor entanglement in the chain exhibits a critical behavior (which we call a dynamical phase transition), controlled by the initial value of the transverse field. The character of the dynamical phase transition is qualitatively different for short and long evolution times. We also find a nonmonotonic behavior of entanglement with respect to the temperature of the initial equilibrium state. Interestingly, the region of the initial field for which we obtain a nonmonotonicity of entanglement (with respect to temperature) is directly related to the position and character of the dynamical phase transition in the model.
Short-time dynamics of many-body systems may exhibit nonanalytical behavior of the systems' properties at particular times, thus dubbed dynamical quantum phase transition. Simulations showed that in the presence of disorder, new critical times appear in the quench evolution of the Ising model. We study the physics behind these new critical times. We discuss the spectral features of the Ising model responsible for the disorder-induced phase transitions. We found the critical value of the disorder sufficient to induce the dynamical phase transition as a function of the number of spins. Most importantly, we argue that this dynamical phase transition while nontopological lacks a local order parameter.