Abstract We study gapless quantum spin chains with spin 1/2 and 1: the Fredkin and Motzkin models. Their entangled groundstates are known exactly but not their excitation spectra. We first express the groundstates in the continuum which allows for the calculation of spin and entanglement properties in a unified fashion. Doing so, we uncover an emergent conformal-type symmetry, thus consolidating the connection to a widely studied family of Lifshitz quantum critical points in 2d. We then obtain the low lying excited states via large-scale DMRG simulations and find that the dynamical exponent is z = 3.2 in both cases. Other excited states show a different z , indicating that these models have multiple dynamics. Moreover, we modify the spin-1/2 model by adding a ferromagnetic Heisenberg term, which changes the entire spectrum. We track the resulting non-trivial evolution of the dynamical exponents using DMRG. Finally, we exploit an exact map from the quantum Hamiltonian to the non-equilibrium dynamics of a classical spin chain to shed light on the quantum dynamics.
We study a quantum spin-1/2 chain that is dual to the canonical problem of non-equilibrium Kawasaki dynamics of a classical Ising chain coupled to a thermal bath. The Hamiltonian is obtained for the general disordered case with non-uniform Ising couplings. The quantum spin chain (dubbed Ising-Kawasaki) is stoquastic, and depends on the Ising couplings normalized by the bath's temperature. We give its exact ground states. Proceeding with uniform couplings, we study the one- and two-magnon excitations. Solutions for the latter are derived via a Bethe Ansatz scheme. In the antiferromagnetic regime, the two-magnon branch states show intricate behavior, especially regarding their hybridization with the continuum. We find that that the gapless chain hosts multiple dynamics at low energy as seen through the presence of multiple dynamical critical exponents. Finally, we analyze the full energy level spacing distribution as a function of the Ising coupling. We conclude that the system is non-integrable for generic parameters, or equivalently, that the corresponding non-equilibrium classical dynamics are ergodic.
We investigate the critical behavior of a spin chain coupled to bosonic baths characterized by a spectral density proportional to ${\ensuremath{\omega}}^{s}$, with
s>1. Varying
s changes the effective dimension
deff=d+z of the system, where
z is the dynamical critical exponent and the number of spatial dimensions
d is set to one. We consider two extreme cases of clock models, namely Ising-like and
U(1)-symmetric ones, and find the critical exponents using Monte Carlo methods. The dynamical critical exponent and the anomalous scaling dimension $\ensuremath{\eta}$ are independent of the order parameter symmetry for all values of
s. The dynamical critical exponent varies continuously from $z\ensuremath{\approx}2$ for
s=1 to
z=1 for
s=2, and the anomalous scaling dimension evolves correspondingly from $\ensuremath{\eta}\ensuremath{\gtrsim}0$ to $\ensuremath{\eta}=1/4$. The latter exponent values are readily understood from the effective dimensionality of the system, being ${d}_{\mathrm{eff}}\ensuremath{\approx}3$ for
s=1, while for
s=2 the anomalous dimension takes the well-known exact value for the two-dimensional Ising and
XY models, since then
deff=2. However, a noteworthy feature is that
z approaches unity and $\ensuremath{\eta}$ approaches
1/4 for values of
s<2, while naive scaling would predict the dissipation to become irrelevant for
s=2. Instead, we find that $z=1,\ensuremath{\eta}=1/4$ for $s\ensuremath{\approx}1.75$ for both Ising-like and
U(1) order parameter symmetry. These results lead us to conjecture that for all site-dissipative
Zq chains, these two exponents are related by the scaling relation $z=\mathrm{max}{(2\ensuremath{-}\ensuremath{\eta})/s,1}$. We also connect our results to quantum criticality in nondissipative spin chains with long-range spatial interactions.
In this Thesis we discussed several issues related to the dynamics of isolated quantum many-body systems. The brief overview of Chapter 1 highlighted a series of open fundamental theoretical questions which are crucial for the investigation of nonequilibrium phenomena. Among them, we mainly focused on the emergence of thermal behaviors and on the characterization of dynamical phase transitions and of the associated universal dynamics. To this end, we analyzed some paradigmatic models, such as the Fermi-Hubbard model, the quantum Ising chain, and the O(N) model. -Chapter 2: Absence of thermalization in a Fermi liquid. In this Chapter we report two results that question the common expectation of thermalization in a Landau-Fermi liquid after an interaction quench. We first consider the perturbative expansion in the interaction strength of the long-wavelength structure factor S(q) and show that it does not satisfy the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q) has an analytical component ∼ const. + O(q^2) for q → 0, compatible with thermalization, but it retains also a nonanalytic term ∼ |q| characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity turns in a power-law decay of the density-density correlations, in contrast with the exponential decay associated to thermalization. We next analyze the low-density case, where one can obtain results valid at any order in interaction but at leading in the density, and find that in the steady state the momentum distribution jump at the Fermi surface is strictly finite, though smaller than at equilibrium. -Chapter 3: Nonadiabatic stationary behavior in a driven Ising chain. In this Chapter we discuss the emergence of a nonadiabatic behavior in the dynamics of the order parameter in a low-dimensional system subject to a linear ramp of one of its parameters within a gapped phase, which should be the most favorable situation for an adiabatic evolution. We study in details the dynamics of the spontaneous magnetization m_x(t) in a quantum Ising chain after a linear variation in time of the transverse field within the ordered phase. In particular, focusing on the value of the order parameter at the end of the ramp m_x(τ), we show that the smaller the switching rate of the transverse field is the closer m_x(τ) gets to its ground state value. Nonetheless, this apparently small correction to adiabaticity eventually leads to the disruption of the order exponentially fast in the subsequent time evolution, no matter how slowly the ramp is performed. -Chapter 4: Aging and coarsening in isolated quantum systems after a quench. This Chapter analyzes the nonequilibrium dynamics of an isolated quantum system after a sudden quench to the dynamical critical point, where the emergence of scaling and universal exponents is expected, due to the absence of time and length scales. We explore these features for a bosonic interacting scalar field theory with O(N) symmetry in the large-N limit, where the model is exactly solvable and the exponents and scaling forms of the relevant two-time correlation functions can be analytically calculated. Moreover, we provide evidence of the emergence of a dynamical scaling behavior also for quenches below the dynamical critical point, associated with coarsening. We find that the latter case is characterized by the same scaling functions as those describing the critical case, yet with different exponents. -Chapter 5: Dynamical transitions and statistics of excitations. In this Chapter we study the dynamics of the O(N) model (introduced in the previous Chapter) resulting from a different protocol: a linear ramp of one of its parameters. We find that the presence of a dynamical phase transition, as well as its critical properties, are robust against the change of the protocol. We show that a characterization based on the critical dimensions and exponents would suggest that the dynamical phase transition is analogous to the equilibrium thermal one. However, its nonequilibrium nature becomes evident analyzing the statistics of excitations produced in the ramp process. In particular, the critical properties are encoded in the fluctuations in the number of excitations, which display qualitatively different behaviors depending on the ramp being performed above, at, or below the dynamical critical point. These behaviors bear no dependence on the duration of the protocol.
The excitation spectrum of the one-dimensional spin-orbital model in a magnetic field is studied, using a recently developed dynamical density-matrix renormalization-group technique. The method is applied to chains with up to 80 sites, and examined for test cases such as the one-dimensional spin-1/2 Heisenberg antiferromagnet whose excitation spectrum is known exactly from the Bethe ansatz. In the spin-orbital chain, the characteristic dynamical response depends strongly on the model parameters and the applied magnetic field. The coupling between the spin and orbital degrees of freedom is found to influence the incommensuration at finite magnetizations. In the regions of the phase diagram with only massive spin and orbital excitations, a finite field is required to overcome the spin gap. An orbital mode is found to become massless in this partially spin-polarized regime, indicating a strong coupling between the two degrees of freedom. In the critical region with three elementary gapless excitations, a prominent particle-hole excitation is observed at higher energies, promoted by the biquadratic term in the model Hamiltonian of the spin-orbital chain.
In this Letter we present, in a number conserving framework, a model of interacting fermions in a two-wire geometry supporting nonlocal zero-energy Majorana-like edge excitations. The model has an exactly solvable line, on varying the density of fermions, described by a topologically nontrivial ground state wave function. Away from the exactly solvable line we study the system by means of the numerical density matrix renormalization group. We characterize its topological properties through the explicit calculation of a degenerate entanglement spectrum and of the braiding operators which are exponentially localized at the edges. Furthermore, we establish the presence of a gap in its single particle spectrum while the Hamiltonian is gapless, and compute the correlations between the edge modes as well as the superfluid correlations. The topological phase covers a sizable portion of the phase diagram, the solvable line being one of its boundaries.