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Register a new userUltra-slow dynamics in a translationally invariant spin model for multiplication and factorization
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We construct a model of short-range interacting Ising spins on a translationally invariant two-dimensional lattice that mimics a reversible circuit that multiplies or factorizes integers, depending on the choice of boundary conditions. We prove that, for open boundary conditions, the model exhibits no finite-temperature phase transition. Yet we find that it displays glassy dynamics with astronomically slow relaxation times, numerically consistent with a double exponential dependence on the inverse temperature. The slowness of the dynamics arises due to errors that occur during thermal annealing that cost little energy but flip an extensive number of spins. We argue that the energy barrier that needs to be overcome in order to heal such defects scales linearly with the correlation length, which diverges exponentially with inverse temperature, thus yielding the double exponential behavior of the relaxation time.
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We introduce a new class of open, translationally invariant spin chains with long-range interactions depending on both spin permutation and (polarized) spin reversal operators, which includes the Haldane-Shastry chain as a particular degenerate case. The new class is characterized by the fact that the Hamiltonian is invariant under twisted translations, combining an ordinary translation with a spin flip at one end of the chain. It includes a remarkable model with elliptic spin-spin interactions, smoothly interpolating between the XXX Heisenberg model with anti-periodic boundary conditions and a new open chain with sites uniformly spaced on a half-circle and interactions inversely proportional to the square of the distance between the spins. We are able to compute in closed form the partition function of the latter chain, thereby obtaining a complete description of its spectrum in terms of a pair of independent su(1|1) and ${rm su}(m/2)$ motifs when the number $m$ of internal degrees of freedom is even. This implies that the even $m$ model is invariant under the direct sum of the Yangians $Y$(gl(1|1)) and $Y$(gl$(0|m/2)$). We also analyze several statistical properties of the new chains spectrum. In particular, we show that it is highly degenerate, which strongly suggests the existence of an underlying (twisted) Yangian symmetry also for odd $m$.
We numerically construct translationally invariant quasi-conserved operators with maximum range M which best-commute with a non-integrable quantum spin chain Hamiltonian, up to M = 12. In the large coupling limit, we find that the residual norm of the commutator of the quasi-conserved operator decays exponentially with its maximum range M at small M, and turns into a slower decay at larger M. This quasi-conserved operator can be understood as a dressed total spin-z operator, by comparing with the perturbative Schrieffer-Wolff construction developed to high order reaching essentially the same maximum range. We also examine the operator inverse participation ratio of the operator, which suggests its localization in the operator Hilbert space. The operator also shows almost exponentially decaying profile at short distance, while the long-distance behavior is not clear due to limitations of our numerical calculation. Further dynamical simulation confirms that the prethermalization-equilibrated values are described by a generalized Gibbs ensemble that includes such quasi-conserved operator.
Investigating translationally invariant qudit spin chains with a low local dimension, we ask what is the best possible tradeoff between the scaling of the entanglement entropy of a large block and the inverse-polynomial scaling of the spectral gap. Restricting ourselves to Hamiltonians with a rewriting interaction, we find the pair-flip model, a family of spin chains with nearest neighbor, translationally invariant, frustration-free interactions, with a very entangled ground state and an inverse-polynomial spectral gap. For a ground state in a particular invariant subspace, the entanglement entropy across a middle cut scales as $log n$ for qubits (it is equivalent to the XXX model), while for qutrits and higher, it scales as $sqrt{n}$. Moreover, we conjecture that this particular ground state can be made unique by adding a small translationally-invariant perturbation that favors neighboring letter pairs, adding a small amount of frustration, while retaining the entropy scaling.
The PH-Pfaffian topological order has been proposed as a candidate order for the $ u=5/2$ quantum Hall effect. The PH-Pfaffian liquid is known to be the ground state in several coupled wire and coupled stripe constructions. No translationally and rotationally invariant models with the PH-Pfaffian ground state have been identified so far. By employing anyon condensation on top of a topological order, allowed in an isotropic system, we argue that the PH-Pfaffian order is possible in the presence of rotational and translational symmetries.
Asymptotic Kardar-Parisi-Zhang (KPZ) properties are investigated in the totally asymmetric simple exclusion process (TASEP) with a localized geometric defect. In particular, we focus on the universal nature of nonequilibrium steady states of the modified TASEP. Since the original TASEP belongs to the KPZ universality class, it is mathematically and physically a quite interesting question whether the localized columnar defect, the slow bond (SB), is really always relevant to the KPZ universality or not. However, it is numerically controversial to address the possibility of the non-queued SB phase in the weak-strength SB limit. Based on the detailed statistical analysis of KPZ-type growing interfaces, we present a comprehensive view of the non-queue SB phase, compared to finite-size crossover effects that reported in our earlier work [Soh {it et al.}, Phys. Rev. E {bf 95}, 042123 (2017)]. Moreover, we employ two types of passive tracer dynamics as the probe of the SB dynamics. Finally, we provide intuitive arguments for additional clues to resolve the controversy of the SB problem.
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