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Feng Xu

Publications and source records attributed to Feng Xu.

268 records · Page 15Linked to original sources
arXiv2004

Experimental Simulation of a Pairing Hamiltonian on an NMR Quantum Computer

We have developed a concrete quantum simulation scheme and experimentally simulated a pairing model on an NMR quantum computer. The design of our experiment includes choosing an appropriate initial state in order to make our scheme scalable in near future, and the accomplishment of our experiment makes use of twice Fourier transforms so that our method is applicable to other physical models. Our results show that the experimental simulation can give the spectrum of the simulated Hamiltonian. Consequently, the potential power of a quantum computer on the simulation of complex physical systems is verified.

quant-ph↗
arXiv2004

A way to solve BCS-type pairing model

We propose a way to solve BCS-type pairing model by to exactly solve its spin-analogy in the subspace. The advantages of our method are to avoid to directly deal with the approximate procedure and to transfer an exponentially complicated problem to a polynomial problem. Moreover, it builds the theoretical foundation of quantum simulation of pairing model and can be used to check the precision of quantum simulation. It is also helpful to understand the many-body quantum theory.

cond-mat.supr-con↗
arXiv2004

Intersections of finite families of finite index subfactors

We prove that finiteness of the index of the intersection of a finite set of finite index subalgebras in a von Neumann algebra (with small centre) is equivalent to the finite dimensionality of the algebra generated by the conditional expectations onto the subalgebras.

math.OA↗
arXiv2004

Topological Sectors and a Dichotomy in Conformal Field Theory

Let A be a local conformal net of factors on the circle with the split property. We provide a topological construction of soliton representations of the tensor product of n copies of A, that restrict to true representations of subnet inviant under cyclic permutations (cyclic orbifold). We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the the cyclic orbifold has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the mu-index of A finite then A turns out to be strongly additive. By [KLM], if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is non-degenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.

math.OA↗
arXiv2004

Solitons in Affine and Permutation Orbifolds

We consider properties of solitons in general orbifolds in the algebraic quantum field theory framework and constructions of solitons in affine and permutation orbifolds. Under general conditions we show that our construction gives all the twisted representations of the fixed point subnet. This allows us to prove a number of conjectures: in the affine orbifold case we clarify the issue of ``fixed point resolutions''; in the permutation orbifold case we determine all irreducible representations of the orbifold, and we also determine the fusion rules in a nontrivial case, which imply an integral property of chiral data for any completely rational conformal net.

math.OA↗
arXiv2003

Strong additivity and conformal nets

We show that the fixed point subnet of a strongly additive conformal net under the action of a compact group is strongly additive. Using the idea of the proof we define the notion of strong additivity for a pair of conformal nets and we show that a key result about the induction of the pair which we proved previously under the finite index assumption can be generalized to strongly additive pairs of conformal nets. These results are applied to classify conformal nets of central charge $c=1$ which are not necessarily rational and satisfy a spectrum condition.

math.QA↗
arXiv2001

On the equivalence of certain coset conformal field theories

We demonstrate the equivalence of Kazama-Suzuki cosets $G(m,n,k)$ and $G(k,n,m)$ based on complex Grassmannians by proving that the corresponding conformal precosheaves are isomorphic. We also determine all the irreducible representations of the conformal precosheaves.

math.QA↗
arXiv2000

Algebraic orbifold conformal field theories

We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also show that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time.

math.QA↗
arXiv2000

On a Conjecture of Kac-Wakimoto

We prove a conjecture about mininmal index of certain representations of Coset Algebraic Conformal Field Theories under general conditions as formulated previously by us. As a by-product, the Kac-Wakimoto Conjecture (KWC) which is related to the asympotics of the coset characters is true under the same conditions. The same idea in the proof also proves a recent conjecture related to subfactors from conformal inclusions.

math.RT↗
arXiv2000

3-manifold invariants from cosets

We construct unitary modular categories for a general class of coset conformal field theories based on our previous study of these theories in the algebraic quantum field theory framework using subfactor theory. We also consider the calculations of the corresponding 3-manifold invariants. It is shown that under certain index conditions the link invaraints colored by the representations of coset factorize into the products of the the link invaraints colored by the representations of the two groups in the coset. But the 3-manifold invariants do not behave so simply in general due to the nontrivial branching and selection rules of the coset. Examples in the parafermion cosets and diagonal cosets show that 3-manifold invariants of the coset may be finer than the products of the 3-manifold invariants associated with the two groups in the coset, and these two invariants do not seem to be simply related in some cases, for an example, in the cases when there are issues of ``fixed point resolutions''. In the later case our framework provides a representation theory understanding of the underlying unitary modular categories which has not been obtained by other methods.

math.GT↗
arXiv2000

Algebraic Coset Conformal field theories

All unitary Rational Conformal Field Theories (RCFT) are conjectured to be related to unitary coset Conformal Field Theories, i.e., gauged Wess-Zumino-Witten (WZW) models with compact gauge groups. In this paper we use subfactor theory and ideas of algebraic quantum field theory to approach coset Conformal Field Theories. Two conjectures are formulated and their consequences are discussed. Some results are presented which prove the conjectures in special cases. In particular, one of the results states that a class of representations of coset $W_N$ ($N\geq 3$) algebras with critical parameters are irreducible, and under the natural compositions (Connes' fusion), they generate a finite dimensional fusion ring whose structure constants are completely determined, thus proving a long-standing conjecture about the representations of these algebras.

math.OA↗
arXiv1999

Algebraic Conformal Field Theories II

Some mathematical questions relating to Coset Conformal Field Theories (CFT) are considered in the framework of Algebraic Quantum Field Theory as developed previously by us. We consider the issue of fixed point resolution in the diagonal coset of type A, and show how to decompose reducible representations into irreducibles. We show the corresponding coset CFT gives rise to a unitary tensor modular category in the sense of Turaev, and therefore may be used to construct 3-manifold invariants. We also show that Kac-Wakimoto Hypothesis (KWH) and Kac-Wakimoto Conjecture (KWC) are equivalent under general conditions which can be checked in examples, a result which seems to be hard to prove by purely representation considerations. Examples are also presented.

math.OA↗
arXiv1997

Applications of braided endomorphisms from conformal inclusions

We give three applications of general theory about braided endomorphisms from conformal inclusions developed previously by us. The first is an example of subfactors associated with conformal inclusion whose dual fusion ring is non-commutative. In the second application we show that the Kac-Wakimoto hypothesis about certain relations between branching rules and S-matrices, which has existed for almost a decade, is not true in at least three examples. Finally we show that the fusion rings of subfactors associated with the conformal inclusions $SU(n)_{(n+2)}\subset SU(n(n+1)/2)$ and $SU(n+2)_n\subset SU((n+1)(n+2)/2)$ are canonically isomorphic using a version of level-rank duality.

q-alg↗
arXiv1997

A note on quivers with symmetries

We show that the bases of irreducible integrable highest weight module of a non-symmetric Kac-Moody algebra, which is associated to a quiver with a nontrivial admissible automorphism, can be naturally identified with a set of certain invariant Langrangian irreducible subvarieties of certain varieties associated with the quiver defined by Nakajima. In the case of non-symmetric affine or finite Kac-Moody algebras, the bases can be naturally identified with a set of certain invariant Langrangian irreducible subvarieties of a particular deformation of singularities of the moduli space of instantons over A-L-E spaces. The motivation of this paper comes from string/string duality and the paper is ended with questions and speculations related to string/string duality.

q-alg↗
arXiv1997

Jones-Wassermann Subfactors for Disconnected Intervals

We show that the Jones-Wassermann subfactors for disconnected intervals, which are constructed from the representations of loop groups of type $A$, are finite-depth subfactors. The index value and the dual principal graphs of these subfactors are completely determined. The square root of the index value in the case of two disjoint intervals for vacuum representation is the same as the Quantum 3-manifold invariant of type $A$ evaluated on $S^1\times S^2$.

q-alg↗