Search

Quantum phase transitions between different topologically ordered phases exhibit rich structures and are generically challenging to study in microscopic lattice models. In this work, we propose a tensor-network solvable model that allows us to tune between different symmetry enriched topological (SET) phases. Concretely, we consider a decorated two-dimensional toric code model for which the ground state can be expressed as a two-dimensional tensor-network state with bond dimension $D=3$ and two tunable parameters. We find that the time-reversal (TR) symmetric system exhibits three distinct phases (i) an SET toric code phase in which anyons transform non-trivially under TR, (ii) a toric code phase in which TR does not fractionalize, and (iii) a topologically trivial phase that is adiabatically connected to a product state. We characterize the different phases using the topological entanglement entropy and a membrane order parameter that distinguishes the two SET phases. Along the phase boundary between the SET toric code phase and the toric code phase, the model has an enhanced $U(1)$ symmetry and the ground state is a quantum critical loop gas wavefunction whose squared norm is equivalent to the partition function of the classical $O(2)$ model. By duality transformations, this tensor-network solvable model can also be used to describe transitions between SET double-semion phases and between $\mathbb{Z}_2\times\mathbb{Z}_2^T$ symmetry protected topological phases in two dimensions., Lukas Haller and Wen-Tao Xu contributed equally to this work

The most important feature of two-dimensional quantum Rokhsar-Kivelson (RK) type models is that their ground state wavefunction norms can be mapped into the partition functions of two-dimensional statistical models so that the quantum phase transitions become the thermal phase transitions of the corresponding statistical models. For a generic quantum critical point, we generalize the framework of RK wavefunctions by introducing the concept of the thermofield double (TFD) state, which is a purification of the equilibrium density operator. Moreover, by expressing the TFD state in terms of the projected entangled pair state, its $N$-order of R\'{e}nyi entropy results in a three-dimensional statistical model in Euclidian spacetime, describing the generic quantum phase transitions. Using the toric code model with two parallel magnetic fields as an example, we explain these ideas and derive the partition function of the three-dimensional $Z_2$ lattice gauge-Higgs model, where the phase transitions are characterized by the three-dimensional universality classes., Comment: 6 pages, 3 figures. some corrections are made

Xin-chang Zhang, Hao Chen, Wen-tao Xu, Yang-yang Song, Ya-hui Gu, Guang-xia Ni The Second Clinical College, Nanjing University of Chinese Medicine, Nanjing, Jiangsu Province, China Purpose: Fibromyalgia (FM) can cause chronic widespread pain and seriously affect the quality of patient lives. Acupuncture therapy is widely used for pain management. However, the effect of acupuncture on FM is still uncertain. The aim of this review was to determine the effect and safety of acupuncture therapy on the pain intensity and quality of life in patients with FM.Materials and methods: We searched PubMed, the Cochrane Library, Embase, the China National Knowledge Infrastructure, the Chinese Science and Technology Periodical Database, and the Chinese Biomedical Literature Database to collect randomized controlled trials (RCTs) of acupuncture for FM published before May 2018. A meta-analysis was performed according to the Cochrane systematic review method by using RevMan 5.3 software, and GRADE was used to evaluate the quality of the evidence.Results: We identified 12 RCTs that compared acupuncture therapy to sham acupuncture or conventional medication. Meta-analysis showed that acupuncture was significantly better than sham acupuncture for relieving pain (MD =-1.04, 95% CI [-1.70, –0.38], P=0.002, I2=78%) and improving the quality of life (MD =-13.39, 95% CI [-21.69, –5.10], P=0.002, I2=82%), with low- to moderate-quality evidence in the short term. At follow-up in the long term, the effect of acupuncture was also superior to that of sham acupuncture. No serious adverse events were found during acupuncture.Conclusion: Acupuncture therapy is an effective and safe treatment for patients with FM, and this treatment can be recommended for the management of FM. Keywords: acupuncture, fibromyalgia, pain, quality of life, meta-analysis

We consider a generic Fibonacci topological wave function on a square lattice, and the norm of this wave function can be mapped into the partition function of two-coupled $\phi ^{2}$-state Potts models with $\phi =(\sqrt{5}+1)/2$ as the golden ratio. A global phase diagram is thus established to display non-abelian topological phase transitions. The Fibonacci topological phase corresponds to an emergent new phase of the two-coupled Potts models, and continuously change into two non-topological phases separately, which are dual each other and divided by a first-order phase transition line. Under the self-duality, the Fibonacci topological state enters into the first-order transition state at a quantum tricritical point, where two continuous quantum phase transitions bifurcate. All the topological phase transitions are driven by condensation of anyonic bosons consisting of Fibonacci anyon and its conjugate. However, a fractional supersymmetry is displayed at the quantum tricritical point, characterized by a coset conformal field theory., Comment: 5.5 pages, 4 figures, and 8.5 page supplementary

We construct a general wave function with the topological order by introducing the $\mathbb{Z}_{2}$ gauge degrees of freedom, characterizing both the toric code state and double semion state. Via calculating the correlation length defined from the one-dimensional quantum transfer operator of the wave function norm, we can map out the complete phase diagram in terms of the parameter $\lambda $ and identify three different quantum critical points (QCPs) at $\lambda =0$, $\pm 1.73$. The first one separates the toric code phase and double semion phase, while later two describe the topological phase transitions from the toric code phase or double semion phase to the symmetry breaking phase, respectively. When mapping to the exactly solved statistical models, the norm of the tensor network wave function is transformed into the partition function of the eight-vertex model. Actually such a quantum-classical mapping can not reveal the rich structures of low-energy excitations at these three QCPs. So we further demonstrate that the full eigenvalue spectra of the transfer operators with/without the flux insertions can describe the complete quantum criticalities, which are characterized by the two-dimensional compactified free boson conformal field theories (CFTs) with the compactified radii $R=\sqrt{6}$ for the QCPs at $\lambda =\pm\sqrt{3}$ and $R=\sqrt{8/3}$ for the QCP at $\lambda =0$. For the QCP at $\lambda =0$, there are no anyon condensation, and the emerged matrix product operator symmetries result in a rich structure of the low-energy excitations, distinct from those of both toric code and double semion phases. Finally, we discuss the possible relation between our conformal quantum criticalities and the general (2+1) spatial-time dimensional CFTs for quantum topological phase transitions., Comment: 17 pages, 16 figures, including three appendices, revised version