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arXiv:2211.05618

Phase diagram of a one-dimensional Ising model with an anomalous Z_2 symmetry

Jin-Xiang Hao, Wei Li, Yang Qi

Published Nov 10, 2022• 0 citations
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Abstract

Anomalous global symmetries, which can be realized on the boundary of symmetry-protected topological phases, bring\wlx{s} new phases and phase transitions to condensed matter physics. In this work, we study a one-dimensional model with an anomalous $\mathbb Z_2$ symmetry, using the density-matrix renormalization group method. Besides a symmetry-breaking ferromagnetic phase, we find a gapless phase described by the Tomonaga-Luttinger Liquid theory with the central charge $c=1$. Our numerical finding is compatible of theoretical constraints on possible phases resulting from the symmetry anomaly.

DOI
10.48550/arxiv.2211.05618

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Anomalous global symmetries, which can be realized on the boundary of symmetry-protected topological phases, bring\wlx{s} new phases and phase transitions to condensed matter physics. In this work, we study a one-dimensional model with an anomalous Z2\mathbb Z_2Z2​ symmetry, using the density-matrix renormalization group method. Besides a symmetry-breaking ferromagnetic phase, we find a gapless phase described by the Tomonaga-Luttinger Liquid theory with the central charge c=1c=1c=1. Our numerical finding is compatible of theoretical constraints on possible phases resulting from the symmetry anomaly.
Emergent Symmetry and Phase Transitions on the Domain Wall of Z2\mathbb{Z}_{2}Z2​ Topological Orders
The one-dimensional (1D) domain wall of 2D Z2\mathbb{Z}_{2}Z2​ topological orders is studied theoretically. The Ising domain wall model is shown to have an emergent SU(2)1_{1}1​ conformal symmetry because of a hidden nonsymmorphic octahedral symmetry. While a weak magnetic field is an irrelevant perturbation to the bulk topological orders, it induces a domain wall transition from the Tomonaga-Luttinger liquid to a ferromagnetic order, which spontaneously breaks the anomalous Z2\mathbb{Z}_{2}Z2​ symmetry and the time-reversal symmetry on the domain wall. Moreover, the gapless domain wall state also realizes a 1D topological quantum critical point between a Z2T\mathbb{Z}_{2}^{T}Z2T​-symmetry-protected topological phase and a trivial phase, thus demonstrating the holographic construction of topological transitions.
We report that a spin-1/2 tetrahedral Heisenberg chain realizes a gapless symmetry-protected topological (gSPT) phase characterized by the coexistence of the Tomonaga-Luttinger-liquid criticality due to chirality degrees of freedom and the symmetry-protected edge state due to spin degrees of freedom. This gSPT phase has an interesting feature that no symmetry forbids the trivial spin gap opening but a discrete symmetry, Z3×Z2T\mathbb Z_3\times\mathbb Z_2^TZ3​×Z2T​, forbids the unique gapped ground state. In the first part of the paper, we numerically show the coexistence of a critical entanglement entropy and a nontrivially degenerate entanglement spectrum based on the density-matrix renormalization group (DMRG) method.Next, we clarify that chirality degrees of freedom form the Tomonaga-Luttinger liquid while spin degrees of freedom form the spin-1 Haldane state based on a degenerate perturbation theory. Last but not least, we discuss the Lieb-Schultz-Mattis-type ingappability in the gSPT phase, using a local Z3\mathbb{Z}_3Z3​ rotation. We can thus characterize our gSPT phase as a symmetry-protected critical phase protected by the Z3\mathbb{Z}_3Z3​ on-site symmetry, the Z2T\mathbb{Z}_2^TZ2T​ time-reversal symmetry, the lattice translation symmetry, and the U(1) spin-rotation symmetry.
We construct a one-dimensional local spin Hamiltonian with an intrinsically nonlocal, and therefore anomalous, global Z2{\mathbb{Z}}_{2}Z2​ symmetry. The model is closely related to the quantum Ising model in a transverse magnetic field, and contains a parameter that can be tuned to spontaneously break the nonlocal Z2{\mathbb{Z}}_{2}Z2​ symmetry. The Hamiltonian is constructed to capture the unconventional properties of the domain walls in the symmetry-broken phase. Using uniform matrix product states, we obtain the phase diagram that results from condensing the domain walls. We find that the complete phase diagram includes a gapless phase that is separated from the ordered ferromagnetic phase by a Berezinskii-Kosterlitz-Thouless transition, and from the ordered antiferromagnetic phase by a first-order phase transition.
We revisit the phase diagram of the dimerized XXZ spin-12\frac{1}{2}21​ chain with nearest-neighbor couplings which was studied numerically by Mondal et al. [Phys. Rev. B 106, L201106 (2022)]. The model has isotropic XYXYXY couplings which have a uniform value and ZZZZZZ couplings which have a dimerized form, with strengths Ja{J}_{a}Ja​ and Jb{J}_{b}Jb​ on alternate bonds. We find a rich phase diagram in the region of positive Ja,Jb{J}_{a},\phantom{\rule{0.28em}{0ex}}{J}_{b}Ja​,Jb​. We provide a detailed understanding of the different phases and associated quantum phase transitions using a combination of mean-field theory, low-energy effective Hamiltonians, renormalization group calculations employing the technique of bosonization, and numerical calculations using the density-matrix renormalization group (DMRG) method. The phase diagram consists of two Ising paramagnetic phases called IPM0{\mathrm{IPM}}_{0}IPM0​ and ${\mathrm{IPM}}_{\ensuremath{\pi}}$, and a phase with Ising Neel (IN) order; all these phases are gapped. The phases IPM0{\mathrm{IPM}}_{0}IPM0​ and ${\mathrm{IPM}}_{\ensuremath{\pi}}$ are separated by a gapless phase transition line $0\ensuremath{\le}{J}_{a}={J}_{b}\ensuremath{\le}1$---the well-known Tomonaga Luttinger liquid captured by a conformal field theory (CFT) with central charge c=1c=1c=1. In contrast, there are two phase transition lines separating IPM0{\mathrm{IPM}}_{0}IPM0​ or the ${\mathrm{IPM}}_{\ensuremath{\pi}}$ from IN that are described by c=12c=\frac{1}{2}c=21​ CFTs and correspond to quantum Ising transitions. In fact, the c=1c=1c=1 line bifurcates into the two c=12c=\frac{1}{2}c=21​ lines at the point Ja=Jb=1{J}_{a}={J}_{b}=1Ja​=Jb​=1; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that IPM0{\mathrm{IPM}}_{0}IPM0​ is a topologically trivial phase while ${\mathrm{IPM}}_{\ensuremath{\pi}}$ is a time-reversal symmetry-protected topological phase (SPT) with spin-12\frac{1}{2}21​ states at the two ends of an open system. A gap opens in the low-energy spectrum whenever one moves away from one of the phase transition lines; the scaling of the gap with the distance from a transition line is found analytically using the RG method. The numerical results obtained by the DMRG method are in good agreement with the analytical results. Further, we show that useful insights into the nature of the phase diagram of the above Hamiltonian, particularly the SPT as well as the phase transitions, is obtained via Kramers-Wannier duality to a deformed quantum Ashkin-Teller model. Finally, we propose experimental platforms for testing our results.
Effective field theory is a very useful technique for understanding quantum many body systems. We use this approach to study a certain class of critical quantum spin-1 chains and symmetry protected topological phases of matter in two spatial dimensions. In the first context we consider the SU(3) symmetric spin-1 chain with additional interactions. We use bosonization to demonstrate that the effective field theories describing the critical behaviour of these spin chains can be mapped to free compact boson conformal field theories (CFTs) with central charge c=2. We also describe how some predictions from field theory can be verified in numerical calculations using exact diagonalization (ED) and the density matrix renormalization group (DMRG) algorithm. In particular, the bosonization method gives a formula for the evolution of four Tomonaga-Luttinger liquid (TLL) parameters as a function of the lattice parameters. Using the analytic formulae for the scaling dimensions in terms of the TLL parameters and matching of lowest scaling dimensions, we numerically calculate these field-theoretic parameters and confirm that their evolution agrees with the prediction using bosonization. We also tackle aspects of the challenging problem of understanding interacting topological phases. In this context, we aim to understand the effects if interactions in certain classes of symmetry protected topological (SPT) phases of matter. We consider non-chiral SPT phases in two spatial dimensions protected by a discrete symmetry such as Z_K or Z_K x Z_K symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons K-matrix theories and fermionic phases relevant to topological superconductors as examples, we demonstrate explicitly that when modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge. We also briefly discuss preliminary results of a numerical approach for simulating the 3 dimensional Landau Level problem. This includes a brief description of a potential application of variational Monte Carlo to spin-orbit coupled ab initio systems.
The Hubbard and spinless fermion chains are paradigms of strongly correlated systems, very well understood using the Bethe ansatz, density matrix renormalization group (DMRG), and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher-dimensional cases. Recently, an interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve four sites. We present a thorough analysis of the phase diagram of this model in one dimension using field-theory/RG and DMRG methods. It includes a gapped supersymmetric region and a gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first-order transition, three critical points occur: tricritical Ising, Lifshitz, and a generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1, and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions.