In his paper “Constraints on order and disorder parameters in quantum spin chains,” Michael Levin derived general constraints on order and disorder parameters in Ising symmetric quantum spin chains. Levin’s main result in his paper was a theorem showing that in a circular spin chain, any local Hamiltonian that has a non-degenerate ground state and is translationally invariant and Ising symmetric must at least have either a nonzero order parameter or a nonzero disorder parameter. In the process of proving the theorem, he also proved a lemma that made a general statement about correlation and symmetry defect properties of any state in a circular spin chain. These properties namely had to do with notions of order and disorder that are weaker than long-range order and disorder. In this thesis, I prove an extension of the lemma to any simultaneous eigenstate of an onsite representation of an arbitrary finite group. Based on this generalization, we discuss some possible implications regarding how Levin’s theorem could be generalized to arbitrary finite symmetries as well. It is important to note that my generalization only applies to representations with one-dimensional invariant subspaces. While this condition is satisfied by any representation of an abelian group, it only holds for a subset of non-abelian group representations.
The author considers one-dimensional n-state quantum chains having as symmetry an Abelian group A of order n (generalised Ising chain). These Hamiltonians depend on 2(n-1) coupling constants. For special values of the coupling constants the symmetry of the quantum chains is larger than A corresponding to a (in general non-Abelian) group G with m elements (m)n). For each element of G he finds explicitly a boundary condition which leaves the Hamiltonian translationally invariant. This problem is relevant in finite-size scaling studies at the critical point of a second-order phase transition.
We study the eigenstate phases of disordered spin chains with on-site finite non-Abelian symmetry. We develop a general formalism based on standard group theory to construct local spin Hamiltonians invariant under any on-site symmetry. We then specialize to the case of the simplest non-Abelian group,
S3, and numerically study a particular two-parameter spin-1 Hamiltonian. We observe a thermal phase and a many-body localized phase with a spontaneous symmetry breaking (SSB) from
S3 to
Z3 in our model Hamiltonian. We diagnose these phases using full entanglement distributions and level statistics. We also use a spin-glass diagnostic specialized to detect spontaneous breaking of the
S3 symmetry down to
Z3. Our observed phases are consistent with the possibilities outlined by A. C. Potter and R. Vasseur [Phys. Rev. B 94, 224206 (2016)], namely thermal/ergodic and many-body-localized spin-glass (MBL-SG) phases. We also speculate on the nature of an intermediate region between the thermal and MBL + SSB regions where full
S3 symmetry exists.
We prove a trade-off theorem for order and disorder parameters in one-dimensional quantum spin systems with quenched disorder. For a disordered ensemble with exact Ising symmetry and average translation symmetry, any gapped ensemble must have one and only one of the following: an
O(1) order parameter or an
O(1) disorder parameter with even parity, both of the Edwards-Anderson type. The result extends to nearly gapped ensembles that accommodate Griffiths-type rare-region effects. These results offer a powerful and rigorous framework to understand the disorder effects beyond perturbative approaches. As applications, we (1) establish the existence of string order parameters for SPT phases; (2) derive a Lieb-Schultz-Mattis-type constraint for disordered ensembles, which requires a nearly gapped ensemble to spontaneously break the symmetry; and (3) discuss similar trade-off relations for disordered fermion chains, leading to an improved understanding of certain "intrinsically disordered" topological phases.
We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and therefore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with an number of examples. Keywords: quantum spin chains, Heisenberg model, spectral gap, symmetr...
Let
(Ω,¶) be a standard probability space and let
ϑ:Ω→Ω be a measure preserving ergodic homeomorphism. Let
A be a
C∗-algebra with a unit and let
AZ be the quasi-local algebra associated to the spin chain with one-site algebra
A. Equip
AZ with the group action of translation by
k-units,
τk∈Aut(AZ) for
k∈Z. We study the problem of finding a disordered matrix product state decomposition for disordered states
ψ(ω) on
AZ with the covariance symmetry condition
ψ(ω)∘τk=ψ(ϑkω). This can be seen as an ergodic generalization of the results of Fannes, Nachtergaele, and Werner [31]. To reify our structure theory, we present a disordered state
νω obtained by sampling the AKLT model [2] in parameter space. We go on to show that
νω has a nearest-neighbor parent Hamiltonian, its bulk spectral gap closes, but it has almost surely exponentially decaying correlations, and finally, that
νω is time-reversal invariant with a Tasaki index of
−1 almost surely.