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arXiv:2010.12179 · Physical review. B./Physical review. B

Continuous phase transition between Néel and valence bond solid phases in a J−Q -like spin ladder system

Takuhiro Ogino, Ryui Kaneko, Satoshi Morita, Shunsuke Furukawa, Naoki Kawashima

Published Feb 11, 2021• 18 citations
arXivHTMLPDF

Abstract

We investigate a quantum phase transition between a N\'eel phase and a valence bond solid (VBS) phase, each of which breaks a different ${\mathbb{Z}}_{2}$ symmetry, in a spin-$1/2$ two-leg XXZ ladder with a four-spin interaction. The model can be viewed as a one-dimensional variant of the celebrated $J\text{\ensuremath{-}}Q$ model on a square lattice. By means of variational uniform matrix product state calculations and an effective field theory, we determine the phase diagram of the model and present evidence that the N\'eel-VBS transition is continuous and belongs to the Gaussian universality class with the central charge $c=1$. In particular, the critical exponents $\ensuremath{\beta},\ensuremath{\eta},$ and, $\ensuremath{\nu}$ are found to satisfy the constraints expected for a Gaussian transition within numerical accuracy. These exponents do not detectably change along the phase boundary while they are in general allowed to do so for the Gaussian class.

DOI
10.1103/physrevb.103.085117

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We investigate a quantum phase transition between a N'eel phase and a valence bond solid (VBS) phase, each of which breaks a different Z2{\mathbb{Z}}_{2}Z2​ symmetry, in a spin-1/21/21/2 two-leg XXZ ladder with a four-spin interaction. The model can be viewed as a one-dimensional variant of the celebrated J\ensuremath-QJ\text{\ensuremath{-}}QJ\ensuremath-Q model on a square lattice. By means of variational uniform matrix product state calculations and an effective field theory, we determine the phase diagram of the model and present evidence that the N'eel-VBS transition is continuous and belongs to the Gaussian universality class with the central charge c=1c=1c=1. In particular, the critical exponents \ensuremathβ,\ensuremathη,\ensuremath{\beta},\ensuremath{\eta},\ensuremathβ,\ensuremathη, and, \ensuremathν\ensuremath{\nu}\ensuremathν are found to satisfy the constraints expected for a Gaussian transition within numerical accuracy. These exponents do not detectably change along the phase boundary while they are in general allowed to do so for the Gaussian class.
We discuss zero‐temperature phase transititions from an antiferromagnetic ground state into a valence‐bond‐solid (VBS) in two different square‐lattice quantum spin models. In one limit these models reduce to the standard S = 1/2 XY and Heisenberg models, respectively, which have long‐range antiferromagnetic order at T = 0. Introducing particular types of four‐spin interactions, amenable to quantum Monte Carlo (QMC) simulation without negative‐sign problems, drives phase transitions into VBS states with broken Z4 symmetry. It has recently been argued that continuous quantum phase transitions of this nature represent a new class of deconfined quantum‐critical points [Senthil et al., Science 303, 1490 (2004)], which are not captured within the standard Ginzburg‐Landau framework of phase transitions. Here we discuss a QMC study aimed at accurately characterizing the transition in the XY model. We find some evidence that the transition is actually weakly first‐order, but other scenarios, such as a continuous transition violating hyperscaling, also cannot be completely ruled out. For the Heisenberg model with four‐spin interactions we have confirmed that a VBS state exists. Preliminary finite‐size scaling results show consistency with a continuous transition into the Néel state, with dynamic exponent z = 1. This model is thus a candidate for deconfined quantum criticality.
We perform a numerical study of a spin-1/2 model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry in one dimension which demonstrates an interesting similarity to the physics of two-dimensional deconfined quantum critical points (DQCP). Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working directly in the thermodynamic limit using uniform matrix product states, we find evidence for a direct continuous phase transition that lies outside of the Landau-Ginzburg-Wilson paradigm. In our model, the continuous transition is found everywhere on the phase boundary. We find that the magnetic and VBS correlations show very close power-law exponents, which is expected from the self-duality of the parton description of this DQCP. Critical exponents vary continuously along the phase boundary in a manner consistent with the predictions of the field theory for this transition. We also find a regime where the phase boundary splits, as suggested by the theory, introducing an intermediate phase of coexisting ferromagnetic and VBS order parameters. Interestingly, we discover a transition involving this coexistence phase which is similar to the DQCP, being also disallowed by the Landau-Ginzburg-Wilson symmetry-breaking theory.
Recent numerical studies of the J1{J}_{1}J1​-J2{J}_{2}J2​ model on a square lattice suggest a possible continuous phase transition between the N\'eel state and a gapped spin-liquid state with Z2{Z}_{2}Z2​ topological order. We show that such a phase transition can be realized through two steps: First bring the N\'eel state to the U(1) deconfined quantum critical point, which has been studied in the context of N\'eel--valence bond solid (VBS) state phase transition. Then condense the spinon pair--skyrmion/antiskyrmion bound state, which carries both gauge charge and flux of the U(1) gauge field emerging at the deconfined quantum critical point. We also propose a Schwinger boson projective wave function to realize such a Z2{Z}_{2}Z2​ spin liquid state and find that it has a relatively low variational energy ($\ensuremath{-}0.4893{J}_{1}$/site) for the J1{J}_{1}J1​-J2{J}_{2}J2​ model at J2=0.5J1{J}_{2}=0.5{J}_{1}J2​=0.5J1​. The spin liquid state we obtain breaks the fourfold rotational symmetry of the square lattice and therefore is a nematic spin liquid state. This direct continuous phase transition from the N\'eel state to a spin liquid state may be realized in the J1{J}_{1}J1​-J2{J}_{2}J2​ model, or the anisotropic J1x{J}_{1x}J1x​-J1y{J}_{1y}J1y​-J2{J}_{2}J2​ model.
We perform an in-depth investigation of the phase diagram of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice. We take advantage of density matrix renormalization group and fully augmented matrix product states methods and reach unprecedented accuracy with large bond dimensions. We utilize excited-level crossing analysis to pinpoint the phase transition points. It was believed before that there exists a narrow spin liquid phase sandwiched by the N\'eel antiferromagnetic (AFM) and valence bond solid (VBS) phases. Through careful finite-size scaling of the level crossing points, we find a direct phase transition between the N\'eel AFM and VBS phases at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3), suggesting the absence of an intermediate spin liquid phase. We also provide accurate results for ground-state energies for a variety of sizes, from which we find that the transition between the N\'eel AFM and VBS phases is continuous. These results indicate the existence of a deconfined quantum critical point at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3) in the model. From the crossing of the first derivative of the energies with J2{J}_{2}J2​ for different sizes, we also determine the precise location of the first-order phase transition between the VBS and stripe AFM phases at J2/J1=0.610(5){J}_{2}/{J}_{1}=0.610(5)J2​/J1​=0.610(5).
We study the S=1/2S=1/2S=1/2 Heisenberg (JJJ) model on the two-dimensional (2D) square lattice in the presence of additional higher-order spin interactions (QQQ) which lead to a valence-bond-solid (VBS) ground state. Using quantum Monte Carlo simulations, we analyze the thermal VBS transition. We find continuously varying exponents, with the correlation-length exponent $\ensuremath{\nu}$ close to the Ising value for large Q/JQ/JQ/J and diverging when Q/JQ/JQ/J approaches the quantum-critical point (the critical temperature ${T}_{c}\ensuremath{\rightarrow}0$). We identify the transition with a class of conformal field theories with charge c=1c=1c=1 and critical exponents varying between those of the 2D Ising model and the Kosterlitz-Thouless (KT) fixed point. We find explicit evidence for KT physics by studying the emergence of U(1)U(1)U(1) symmetry of the order parameter at T=TcT={T}_{c}T=Tc​ when ${T}_{c}\ensuremath{\rightarrow}0$.
We study a spin-1/2 SU(2) model on the honeycomb lattice with nearest-neighbor antiferromagnetic exchange J that favors Néel order and competing six-spin interactions Q that favor a valence-bond-solid (VBS) state in which the bond energies order at the "columnar" wave vector K=(2π/3,-2π/3). We present quantum Monte Carlo evidence for a direct continuous quantum phase transition between Néel and VBS states, with exponents and logarithmic violations of scaling consistent with those at analogous deconfined critical points on the square lattice. Although this strongly suggests a description in terms of deconfined criticality, the measured threefold anisotropy of the phase of the VBS order parameter shows unusual near-marginal behavior at the critical point.