Strong zero modes (SZMs) are conserved operators localized at the edges of certain quantum spin chains, which give rise to long coherence times of edge spins. Here we define and analyze analogous operators in one-dimensional classical stochastic systems. For concreteness, we focus on chains with single occupancy and nearest-neighbor transitions, in particular particle hopping and pair creation and annihilation. For integrable choices of parameters we find the exact form of the SZM operators. Being in general nondiagonal in the classical basis, the dynamical consequences of stochastic SZMs are very different from those of their quantum counterparts. We show that the presence of a stochastic SZM is manifested through a class of exact relations between time-correlation functions, absent in the same system with periodic boundaries.
Strong zero modes (SZMs) are conserved operators localized at the edges of certain quantum spin chains, which give rise to long coherence times of edge spins. Here we define and analyze analogous operators in one-dimensional classical stochastic systems. For concreteness, we focus on chains with single occupancy and nearest-neighbor transitions, in particular particle hopping and pair creation and annihilation. For integrable choices of parameters we find the exact form of the SZM operators. Being in general nondiagonal in the classical basis, the dynamical consequences of stochastic SZMs are very different from those of their quantum counterparts. We show that the presence of a stochastic SZM is manifested through a class of exact relations between time-correlation functions, absent in the same system with periodic boundaries.
Strong zero modes (SZMs) are edge-localized operators that commute with the Hamiltonian up to corrections exponentially small in system size, yielding anomalously long edge coherence times. In some settings, notably certain integrable models, this commutator can be made to vanish exactly at finite size, defining an exact SZM (ESZM). Existing ESZM constructions in the integrable setting, however, have proceeded model by model and have not been unified into a common framework. Here, I show that ESZMs arise generically in a broad family of integrable spin models with anisotropic interactions. Their existence follows from two algebraic properties of the underlying R- and K-matrices -- quasi-periodicity in the spectral parameter and tracelessness, respectively -- providing a uniform, model-independent mechanism. The framework recovers the known ESZM in XXZ chain and its higher-spin generalizations as special cases and predicts ESZMs in previously unrecognized models.
The work presented in this thesis is divided into two main parts: strong zero modes in quantum spin chains with open boundaries and large deviations in classical spin systems. The results concerning the strong zero modes are presented in chapters 2 and 3, while chapter 4 contains the results about large deviations in systems of many, but independent, classical degrees of freedom. These two distinct classes of problems in spin chains are both about aspects of slow relaxation to equilibrium in many-body systems. Nowadays slow dynamics and non-ergodicity in many-body quantum systems are receiving much attention due to their notable technological potential for storing and processing, for very long times, the quantum information encoded in boundary degrees of freedom. In the first part of the work presented in this thesis we mainly focus on quantum spin chains with open boundaries, which can display coherence times for edge spins, that diverge with the system size, as a consequence of the emergence, in the thermodynamic limit, of conserved local quantities, called strong zero modes. First, we discuss the fate of these coherence times when dynamics becomes dissipative. Understanding the behavior of these long correlation times in open quantum systems, undergoing Markovian (memoryless) dynamics, would allow for the possibility of exploiting the protected information encoded in the boundary spins for applications in quantum devices in realistic non-equilibrium settings. Second, we study one possible simple generalization of the transverse field Ising model to a class of ladder spin systems with plaquette interactions in the presence of a transverse field. Also for these systems one can explicitly construct the strong zero mode operators and understand the behavior of the consequential long coherence times for the edge degrees of freedom under isolated (unitary) dynamics. This is possible because the models we consider, while appearing complex superficially, can be brought to simple forms by exploiting an extensive number of conserved quantities they possess. The results presented in chapters 2 and 3 are closely related, even if they focus on different models and investigate different aspects of their dynamics. Nevertheless, the underlying concept they are based on is the same one, that is, an eigenstate phase transition for the strong zero modes that has interesting consequences for the full energy spectrum. In the second part of the thesis we focus on a different type of phase transitions. Here we address the issue of dynamical large deviation phase transitions from a disordered phase to an ordered one, in classical systems which consist in a collection of independent Ising spins evolving stochastically. Surprisingly, by reweighing trajectories accordingly to a certain type of time-integrated observable, even a system of independently evolving spins can display some critical behavior, at the fluctuation level, in trajectory space in the large size and long time limit. By making use of the tilted generator techniques we study the statistics of two particular time-integrated observables. Their associated tilted generators have an exact Kramers-Wannier duality and they look like, up to a sign and an additive constant, some specific well known quantum Hamiltonians. In this sense, the dynamical large deviation phase transitions we observe in these classical stochastic systems translate in ground state quantum phase transitions taking place at the self dual point of the tilted generators we study.
Strong Zero Modes (SZMs) are (approximately) conserved quantities that result in (approximate) double degeneracies in the entire spectra of certain Hamiltonians, with the Majorana zero mode of the transverse-field Ising chain being a primary example. In this work, we discover via a systematic search that many examples of SZMs can be understood as symmetries in the commutant algebra framework, which reveals novel algebraic structures hidden in Hamiltonians with well-known SZMs, including the transverse-field Ising chain. Our findings unify the understanding of different examples of SZMs in the literature, demystify their connections to ground state phases of matter, and reveal novel symmetries in simple models, such as exact quasilocal U(1) symmetries that sometimes accompany the SZMs such as in the spin-1/2 XY model for certain parameter values. Moreover, while analytically tractable SZMs have mostly been demonstrated only for non-interacting or integrable models, the algebraic structures revealed in this work can be exploited to construct integrability-breaking interactions that exactly preserve these SZMs. Such non-integrable models are expected to show more clear dynamical signatures of SZMs without the interference of other conserved quantities that appear in integrable models, and we discuss many examples, including those of novel hydrodynamic modes associated with such symmetries for some special parameter values. We also show that while this commutant understanding extends to the non-interacting limit of the celebrated Fendley SZM in the spin-1/2 XYZ chain, the SZM in the interacting case cannot be understood in this framework. This suggests that there are two types of SZMs -- those that survive integrability breaking and those that do not. We close by using this commutant understanding to construct an alternate proof of the Fendley SZM, which might be of independent interest.
I will discuss how edge spins in certain one-dimensional spin chains retain memory of their initial state for very long times, even with interactions and at infinite temperature. These long coherence times do not require disorder or integrability. I will show how this is a consequence of an "almost" strong edge zero mode that almost commutes with the Hamiltonian, and explain how to compute this operator explicitly with computer-aided algebra. I will examine the effect of this almost strong zero mode on the behaviour of a wide variety of one-dimensional systems analytically, from the ground state to infinite temperature, and from finite size to the thermodynamic limit. I will also examine the effect of the almost strong zero mode on the eigenstate spectrum and the matrix elements of the edge spin, and show that despite the long coherence time of the edge spin, the almost strong zero mode does not induce degeneracies in the entire spectrum, unlike its counterpart exact strong zero mode in non-interacting systems. At special values of the couplings, resonances dramatically limit the coherence time. I will explain how these resonances are related to physical processes which flip the edge spin for no change in the energy of the system, and calculate the coupling strengths for which they appear. I will propose an experimental test of the almost strong zero mode and the long coherence time of the edge spin in a trapped ion chain which is feasible to implement with current technology. Lastly, I will explain how these almost strong edge zero modes in a system with symmetry-protected topological order can be used to form a boundary qubit with both long decoherence and dephasing times.
We derive exact strong zero mode (ESZM) operators for integrable spin-S chains with open boundary conditions and a boundary field. Their locality properties are generally weaker than in the previously known cases, but they still imply infinite coherence times in the vicinity of the edges. We explain how such integrable chains possess multiple ground states describing a first-order quantum phase transition, and that the odd number of such states for integer S makes the weaker locality properties necessary. We make contact with more traditional approaches by showing how the ESZM for S=1/2 acts on energy eigenstates given by solutions of the Bethe equations.
It is a classic result that certain interacting integrable spin chains host robust edge modes known as strong zero modes (SZMs). In this Letter, we extend this result to the Floquet setting of local quantum circuits, focusing on a prototypical model providing an integrable Trotterization for the evolution of the XXZ Heisenberg spin chain. By exploiting the algebraic structures of integrability, we show that an exact SZM operator can be constructed for these integrable quantum circuits in certain regions of parameter space. Our construction, which recovers a well-known result by Paul Fendley in the continuous-time limit, relies on a set of commuting transfer matrices known from integrability, and allows us to easily prove important properties of the SZM, including normalizabilty. Our approach is different from previous methods and could be of independent interest even in the Hamiltonian setting. Our predictions, which are corroborated by numerical simulations of infinite-temperature autocorrelation functions, are potentially interesting for implementations of the XXZ quantum circuit on available quantum platforms.