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arXiv:2605.03700

Numerical evidence of a critical point in the (2+1)D SO(5) nonlinear sigma model with Wess-Zumino-Witten term

Yuan Da Liao, Bin-Bin Chen, Junchen Rong, Fakher F. Assaad, Lukas Janssen, Zi Yang Meng

Published May 5, 2026• 0 citations
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Abstract

We develop an optimized continuous-field quantum Monte Carlo (QMC) algorithm to investigate the projected SO(5) nonlinear sigma model with a Wess-Zumino-Witten term, which describes half-filled Dirac fermions in 2+1 space-time dimensions akin to graphene and Yukawa coupled to a quintuplet of compatible mass terms. Our algorithm reduces the computational complexity to $O(βN_{\mathbf{q}} N_ϕ^2)$, yielding a speedup of a factor of $N_ϕ$ (the number of magnetic fluxes, i.e., system size) relative to prior works [1-4]. This advance enables us to simulate system sizes up to $N_ϕ=140$ on the torus and $N_ϕ=59$ on the sphere, far exceeding the maximum sizes previously accessed, and to map out the universal phase diagram of the model on both geometries. Most notably, we identify and characterize a critical point that separates an SO(5)-broken ordered phase at small coupling from an SO(5)-symmetric disordered phase at large coupling. The critical point becomes multicritical upon the inclusion of terms that break the SO(5) symmetry down to $\mathrm{U}(1) \times \mathrm{SU}(2)$, relevant for the deconfined phase transition between Néel antiferromagnetic and valence-bond-solid orders in quantum magnets. Our finding of a multicritical point in the phase diagram of the SO(5) nonlinear sigma model with Wess-Zumino-Witten term resolves the long-standing open question of its global structure, and our QMC algorithm opens a new avenue for systematic studies of projected Hamiltonians, ranging from correlated flat bands to fractional quantum (anomalous) Hall systems.

DOI
10.48550/arxiv.2605.03700

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We develop an optimized continuous-field quantum Monte Carlo (QMC) algorithm to investigate the projected SO(5) nonlinear sigma model with a Wess-Zumino-Witten term, which describes half-filled Dirac fermions in 2+1 space-time dimensions akin to graphene and Yukawa coupled to a quintuplet of compatible mass terms. Our algorithm reduces the computational complexity to O(βNqNϕ2)O(βN_{\mathbf{q}} N_ϕ^2)O(βNq​Nϕ2​), yielding a speedup of a factor of NϕN_ϕNϕ​ (the number of magnetic fluxes, i.e., system size) relative to prior works [1-4]. This advance enables us to simulate system sizes up to Nϕ=140N_ϕ=140Nϕ​=140 on the torus and Nϕ=59N_ϕ=59Nϕ​=59 on the sphere, far exceeding the maximum sizes previously accessed, and to map out the universal phase diagram of the model on both geometries. Most notably, we identify and characterize a critical point that separates an SO(5)-broken ordered phase at small coupling from an SO(5)-symmetric disordered phase at large coupling. The critical point becomes multicritical upon the inclusion of terms that break the SO(5) symmetry down to U(1)×SU(2)\mathrm{U}(1) \times \mathrm{SU}(2)U(1)×SU(2), relevant for the deconfined phase transition between Néel antiferromagnetic and valence-bond-solid orders in quantum magnets. Our finding of a multicritical point in the phase diagram of the SO(5) nonlinear sigma model with Wess-Zumino-Witten term resolves the long-standing open question of its global structure, and our QMC algorithm opens a new avenue for systematic studies of projected Hamiltonians, ranging from correlated flat bands to fractional quantum (anomalous) Hall systems.
We use the half-filled zeroth Landau level in graphene as a regularization scheme to study the physics of the SO(5) nonlinear sigma model subject to a Wess-Zumino-Witten topological term in 2+1 dimensions. As shown by Ippoliti et al. [Phys. Rev. B 98, 235108 (2019)PRBMDO2469-995010.1103/PhysRevB.98.235108], this approach allows for negative sign free auxiliary field quantum Monte Carlo simulations. The model has a single free parameter U_{0} that monitors the stiffness. Within the parameter range accessible to negative sign free simulations, we observe an ordered phase in the large U_{0} or stiff limit. Remarkably, upon reducing U_{0} the magnetization drops substantially, and the correlation length exceeds our biggest system sizes, accommodating 100 flux quanta. The implications of our results for deconfined quantum phase transitions between valence bond solids and antiferromagnets are discussed.
Novel critical phenomena beyond the Landau-Ginzburg-Wilson paradigm have been long sought after. Among many candidate scenarios, the deconfined quantum critical point (DQCP) constitutes the most fascinating one, and its lattice model realization has been debated over the past two decades. Here we apply the spherical Landau level regularization upon the exact (2+1)D SO(5) nonlinear sigma model with a topological term to study the potential DQCP therein. We perform a density matrix renormalization group (DMRG) simulation with SU(2)_{spin}×U(1)_{charge}×U(1)_{angular-momentum} symmetries explicitly implemented. Using crossing point analysis for the critical properties of the DMRG data, accompanied by quantum Monte Carlo simulations, we accurately obtain the comprehensive phase diagram of the model and find various novel quantum phases, including Néel, ferromagnet (FM), valence bond solid (VBS), valley polarized (VP) states and a gapless quantum disordered phase occupying an extended area of the phase diagram. The VBS-disorder and Néel-disorder transitions are continuous with non-Wilson-Fisher exponents. Our results show the VBS and Néel states are separated by either a weakly first-order transition or the disordered region with a multicritical point in between, thus opening up more interesting questions on the two-decade long debate on the nature of the DQCP.
Critical phenomena beyond the Landau-Ginzburg-Wilson paradigm have been long sought after. Among many candidate scenarios, the deconfined quantum critical point (DQCP) constitutes the most fascinating one, and its lattice model realization has been debated over the past two decades. Following pioneering works with the fuzzy sphere methods [Zhu et al., Phys. Rev. X 13, 021009 (2023); Hu et al., Phys. Rev. Lett. 131, 031601 (2023); Han et al., arXiv:2312.04047; Zhou et al., Phys. Rev. X 14, 021044 (2024)], we apply the spherical Landau level regularization to study the effective (2+1)D(2+1)\mathrm{D}(2+1)D SO(5) nonlinear sigma model with a topological term and the potential DQCP therein. Utilizing the state-of-the-art density matrix renormalization group method with explicit ${\text{SU(2)}}_{\text{spin}}\ifmmode\times\else\texttimes\fi{}{\text{U(1)}}_{\text{charge}}\ifmmode\times\else\texttimes\fi{}{\text{U(1)}}_{\text{angular-momentum}}$ symmetry as well as exact diagonalization simulations, we provide a comprehensive phase diagram for the model with a SO(5) continuous transition line---extension of the previous identified SO(5) multicritical point [Chen et al., Phys. Rev. Lett. 132, 246503 (2024)]---while tuning interaction length. The state-operator correspondence with the conformal tower structure is used to identify the emergent conformal symmetry with the best scaling dimension of relevant primary fields, and they match well with the critical exponents obtained from the crossing point analysis of the correlation ratio. Our results thus further support the rich structure of the phase diagram of the SO(5) model.
Novel critical phenomena beyond the Landau-Ginzburg-Wilson paradigm have been long sought after. Among many candidate scenarios, the deconfined quantum critical point (DQCP) constitutes the most fascinating one, and its lattice model realization has been debated over the past two decades. Following the pioneering works with the fuzzy sphere methods [W. Zhu, et al, Phys. Rev. X 13, 021009], we apply the spherical Landau level regularization to study the effective (2+1)D SO(5) non-linear sigma model with a topological term and the potential DQCP therein. Utilizing the state-of-the-art density matrix renormalization group method with explicit SU(2)spin×U(1)charge×U(1)angular-momentum\text{SU(2)}_\text{spin}\times\text{U(1)}_\text{charge}\times\text{U(1)}_\text{angular-momentum}SU(2)spin​×U(1)charge​×U(1)angular-momentum​ symmetry as well as exact diagonalization simulations, we provide a comprehensive phase diagram for the model with a SO(5) continuous transition line -- extension of the previous identified SO(5) multicritical point [arXiv:2307.05307] -- while tuning interaction length. The state-operator correspondence with the conformal tower structure is used to identify the emergent conformal symmetry with the best scaling dimension of relevant primary fields and they match well with the critical exponents obtained from the crossing point analysis of the correlation ratio. Our results thus further support the rich structure of the phase diagram of the SO(5) model.
It has been proposed that the deconfined criticality in (2+1)d(2+1)d(2+1)d---the quantum phase transition between a N\'eel antiferromagnet and a valence-bond solid (VBS)---may actually be pseudocritical, in the sense that it is a weakly first-order transition with a generically long correlation length. The underlying field theory of the transition would be a slightly complex (nonunitary) fixed point as a result of fixed points annihilation. This proposal was motivated by existing numerical results from large scale Monte Carlo simulations as well as a conformal bootstrap. However, an actual theory of such a complex fixed point, incorporating key features of the transition such as the emergent SO(5) symmetry, is so far absent. Here we propose a Wess-Zumino-Witten (WZW) nonlinear sigma model with level k=1k=1k=1, defined in $2+\ensuremath{\epsilon}$ dimensions, with target space ${S}^{3+\ensuremath{\epsilon}}$ and global symmetry $\text{SO}(4+\ensuremath{\epsilon})$. This gives a formal interpolation between the deconfined criticality at d=3d=3d=3 and the SU(2)1\text{SU}{(2)}_{1}SU(2)1​ WZW theory at d=2d=2d=2 describing the spin-1/21/21/2 Heisenberg chain. The theory can be formally controlled, at least to leading order, in terms of the inverse of the WZW level 1/k1/k1/k. We show that at leading order there is a fixed point annihilation at ${d}^{*}\ensuremath{\approx}2.77$, with complex fixed points above this dimension including the physical d=3d=3d=3 case. The pseudocritical properties such as correlation length, scaling dimensions, and the drifts of scaling dimensions as the system size increases, calculated crudely to leading order, are qualitatively consistent with existing numerics.
We resolve the nature of the quantum phase transition between antiferromagnetic and valence-bond solid ground states in square-lattice spin-1/2 magnets - the putative deconfined quantum-critical point. Using quantum Monte Carlo simulations of J-Q models, in which Heisenberg exchange J competes with multi-spin interactions Qn formed by n singlet projectors, we demonstrate first-order discontinuities increasing with n. For n=2 and n=3, the first-order signatures are very weak and we extract universal scaling dimensions of a proximate multicritical point. Their values coincide closely with known bounds for an SO(5) symmetric conformal field theory (CFT) containing a relevant SO(5) singlet operator that destabilizes the generic deconfined critical point. We characterize the emergent SO(5) symmetry by its scaling dimensions of the leading irrelevant perturbations. Using the J-Q2-Q6 model, we tune the transition from weak to stronger first-order and find scaling of the coexisting order parameters with a large exponent beta ~ 0.85, in agreement with the relevant scaling dimensions of the multicritical point. The exponent that had previously been identified with the correlation length does not have a corresponding CFT level, and we explain it as pseudo-critical length scale of bubbles of one phase inside the other that are compressed in size by an SO(5)-violating irrelevant repulsive interaction between the two order parameters. The bubble scale reflects a hierarchy of excitation gaps generated by the SO(5) perturbation in the neighborhood of the first-order transition, akin to a Higgs mechanism. We argue that the multicritical point is the tip of a gapless spin liquid phase in frustrated Heisenberg models into which the J-Q models can be deformed, but the spin liquid, and perhaps the multicritical point itself, exist outside the regime amenable to sign-free simulations.