Quantum phase transitions and string orders in the spin-1/2 Heisenberg–Ising alternating chain with Dzyaloshinskii–Moriya interaction

Guang-Hua Liu1, Wen-Long You2, Wei Li3,4 and Gang Su

Hua Liu1, Wen-Long You2, Wei Li3,4 and Gang Su5

1 Department of Physics, Tianjin Polytechnic University, Tianjin 300387, People’s Republic of China

2 College of Physics, Optoelectronics and Energy, Soochow University, Suzhou, Jiangsu 215006, People’s Republic of China

3 Physics Department, Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universitat, 80333 Munich, Germany ¨

4 Department of Physics, Beihang University, Beijing 100191, People’s Republic of China

5 Theoretical Condensed Matter Physics and Computational Materials Physics Laboratory, College of Physical Sciences, University of Chinese Academy of Sciences, PO Box 4588, Beijing 100049, People’s Republic of China

Quantum phase transitions (QPTs) and the ground-state phase diagram of the spin-1/2 Heisenberg–Ising alternating chain (HIAC) with uniform Dzyaloshinskii–Moriya (DM) interaction are investigated by a matrix-product-state (MPS) method. By calculating the oddand even-string order parameters, we recognize two kinds of Haldane phases, i.e. the odd- and even-Haldane phases. Furthermore, doubly degenerate entanglement spectra on odd and even bonds are observed in odd- and even-Haldane phases, respectively. A rich phase diagram including four different phases, i.e. an antiferromagnetic (AF), AF stripe, odd- and even-Haldane phases, is obtained. These phases are found to be separated by continuous QPTs: the topological QPT between the odd- and even-Haldane phases is verified to be continuous and corresponds to conformal field theory with central charge c = 1; while the rest of the phase transitions in the phase diagram are found to be c = 1/2. We also revisit, with our MPS method, the exactly solvable case of HIAC model with DM interactions only on odd bonds and find that the even-Haldane phase disappears, but the other three phases, i.e. the AF, AF stripe and odd-Haldane phases, still remain in the phase diagram. We exhibit the evolution of the even-Haldane phase by tuning the DM interactions on the even bonds gradually.