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arXiv:hep-th/9202082

The Pokrovski-Talapov Phase Transitions and Quantum Groups

Haye Hinrichsen B, V. Rittenberg

Published Feb 24, 1992• 2 citations
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Abstract

We show that the XY quantum chain in a magnetic field is invariant under a two parameter deformation of the $SU(1/1)$ superalgebra. One is led to an extension of the braid group and the Hecke algebras which reduce to the known ones when the two parameter coincide. The physical significance of the two parameters is discussed. When both are equal to one, one gets a Pokrovski-Talapov phase transition. We also show that the representation theory of the quantum superalgebras indicates how to take the appropriate thermodynamical limits.

DOI
10.48550/arxiv.hep-th/9202082

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We show that the XY quantum chain in a magnetic field is invariant under a two parameter deformation of the SU(1/1)SU(1/1)SU(1/1) superalgebra. One is led to an extension of the braid group and the Hecke algebras which reduce to the known ones when the two parameter coincide. The physical significance of the two parameters is discussed. When both are equal to one, one gets a Pokrovski-Talapov phase transition. We also show that the representation theory of the quantum superalgebras indicates how to take the appropriate thermodynamical limits.
With appropriate boundary conditions the anisotropic XY chain in a magnetic field is known to be invariant under quantum group transformations. We generalize this model defining a class of integrable chains with several fermionic degrees of freedom per site. In order to maintain the quantum group symmetry a general condition on the parameters of these systems is derived. It is shown that the corresponding quantum algebra is a multi-parameter deformation of the Clifford algebra. Discussing a special physical example we observe a new type of zero mode.
The structure and representation theory of a two-parameter generalization of Hecke algebras, denoted HL(q,η), which plays an important role in the construction of the L-site XY chain in a magnetic field is investigated. In particular, the structure of the XY chain representation of the algebra, on C2⊗L, and hence the structure of the centralizer algebra EndHL(q,η)(C2⊗L) for the XY chain representation, is analyzed. It is proved that the full algebra never reduces to the ordinary (one-parameter) Hecke algebra HL(q) in any specialization of the parameters, but contains the Hecke algebra as a quotient in a certain specialization of one of the parameters. This corrects an earlier claim of Hinrichsen and Rittenberg [Phys. Lett. B 275, 350 (1992)]. Bounds are established on the generic dimension of the centralizer algebra 4 ≤ dim(EndHL(q,η)(C⊗L2)) ≤ 8 (the upper bound is realized in every known case). Hence, it is argued that the structure of the centralizer algebra is independent of L, which is consistent with a result of Hinrichsen and Rittenberg, but that it is isomorphic to two commuting copies (rather than one copy) of the Pauli matrices. This contrasts sharply with the dimension of EndHL(q)(C⊗L2), which grows without bound as L increases.
We use simple diagrammatic techniques to analyse the ordinary representation theory of the Hecke algebras , and to construct modules (resp. representations) which are generically simple (irreducible) and well defined in every specialization of q, including roots of unity. We determine several physically important properties of these modules, generalizing properties of the Temperley - Lieb algebra and its diagrams which have proved useful for lattice models. We show how these results can be used to locate energy level crossings in invariant quantum spin chains, and locate a new crossing of the thermodynamic limit spin chain at as an example.
A C∗C^*C∗-Algebraic Approach to Parametrized Quantum Spin Systems and Their Phases in One Spatial Dimension
This thesis investigates parametrized quantum spin systems in the thermodynamic limit from a C∗C^*C∗-algebraic point of view. Our main physical result is the construction of a phase invariant for one-dimensional quantum spin chains parametrized by a topological space XXX. This invariant is constructed using C∗C^*C∗-algebraic techniques and takes values in degree one Čech cohomology H1(X;PU(H))H^1(X;\mathbb{P}\mathrm{U}(\mathscr{H}))H1(X;PU(H)), where PU(H)\mathbb{P}\mathrm{U}(\mathscr{H})PU(H) is the projective unitary group of an infinite-dimensional Hilbert space H\mathscr{H}H, endowed with the strong operator topology. Using Dixmier-Douady theory [Bry93, DD63, HJJS08] one may equivalently view this as an element of H3(X;Z)H^3(X;\mathbb{Z})H3(X;Z). An exactly solvable model of a one-dimensional spin system parametrized by the 3-sphere X=S3X = \mathbb{S}^3X=S3 is presented and it is shown that its invariant is nontrivial [WQB+22]. We also prove several mathematical results on topological aspects of the pure state space P(A)\mathscr{P}(\mathfrak{A})P(A) of a C∗C^*C∗-algebra A\mathfrak{A}A. We prove that P(A)\mathscr{P}(\mathfrak{A})P(A), endowed with the weak* topology, has trivial fundamental group for every UHF algebra A\mathfrak{A}A [BHM+23]. We review the results of [SMQ+22] that show how the outputs of the GNS representation and Kadison transitivity theorem can be understood to depend continuously on their inputs. These results are expanded upon with the construction of a distinguished Čech class in H1(P(A);U(1))H^1(\mathscr{P}(\mathfrak{A});\mathrm{U}(1))H1(P(A);U(1)) that generalizes the principal U(1)\mathrm{U}(1)U(1)-bundle SH→PH\mathbb{S} \mathscr{H} \rightarrow \mathbb{P} \mathscr{H}SH→PH in a representation independent way, where SH\mathbb{S} \mathscr{H}SH and PH\mathbb{P} \mathscr{H}PH are the unit sphere and projective Hilbert space of H\mathscr{H}H. Finally, we prove a selection theorem in the vein of [SMQ+22] for the weak* topology on P(A)\mathscr{P}(\mathfrak{A})P(A).