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arXiv:1912.13468 · Physical review. B./Physical review. B

Note on Wess-Zumino-Witten models and quasiuniversality in 2+1 dimensions

Adam Nahum

Published Nov 30, 2020• 80 citations
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Abstract

We suggest the possibility that the two-dimensional $\mathrm{SU}{(2)}_{k}$ Wess-Zumino-Witten (WZW) theory, which has global $\mathrm{SO}(4)$ symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with $\mathrm{SO}(5)$ symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality ${d}_{c}>2$. We suggest that ${d}_{c}<3$ for all $k$, and we compute ${d}_{c}$ in the limit of large $k$. The flows support the conjecture that the deconfined phase transition in $\mathrm{SU}(2)$ magnets is a ``pseudocritical'' point with approximate $\mathrm{SO}(5)$, controlled by a fixed point slightly outside the physical parameter space.

DOI
10.1103/physrevb.102.201116

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We suggest the possibility that the two-dimensional SU(2)k\mathrm{SU}{(2)}_{k}SU(2)k​ Wess-Zumino-Witten (WZW) theory, which has global SO(4)\mathrm{SO}(4)SO(4) symmetry, can be continued to 2+\ensuremathϵ2+\ensuremath{\epsilon}2+\ensuremathϵ dimensions by enlarging the symmetry to SO(4+\ensuremathϵ)\mathrm{SO}(4+\ensuremath{\epsilon})SO(4+\ensuremathϵ). This is motivated by the three-dimensional sigma model with SO(5)\mathrm{SO}(5)SO(5) symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ may be fixed by assuming analyticity in \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality dc>2{d}_{c}>2dc​>2. We suggest that dc<3{d}_{c}<3dc​<3 for all kkk, and we compute dc{d}_{c}dc​ in the limit of large kkk. The flows support the conjecture that the deconfined phase transition in SU(2)\mathrm{SU}(2)SU(2) magnets is a ``pseudocritical'' point with approximate SO(5)\mathrm{SO}(5)SO(5), controlled by a fixed point slightly outside the physical parameter space.
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.
I study the renormalization of D-dimensional level-k Wess-Zumino-Witten theory with Stiefel-manifold target space StN,N−D−1≅SO(N)/SO(D+1)\mathrm{St}_{N,N-D-1} \cong \mathrm{SO}(N)/\mathrm{SO}(D+1)StN,N−D−1​≅SO(N)/SO(D+1), with a particular focus on D=3D = 3D=3. I investigate in particular whether such a theory admits IR-stable fixed points of the renormalization group flow. Such fixed points have been suggested to describe conformal phases of matter that do not have a known dual (super-)renormalizable Lagrangian for N≥7N \geq 7N≥7 in D=3D = 3D=3. They are hence of interest both from the point of view of quantum phases of matter as well as pure field theory. The DDD-dimensional expressions enable the computation, by analytic computation, of beta functions in D=2+εD = 2 + εD=2+ε, at least to first non-trivial order. In D=2D = 2D=2, a stable fixed point is found, serving a generalization of the famed SU(2)k\mathrm{SU}(2)_kSU(2)k​ Wess-Zumino-Witten conformal field theory; it annihilates in D=2+εD = 2 + εD=2+ε with an unstable fixed point which splits off from the Gaussian one for ε>0ε> 0ε>0. Although the story is thus qualitatively similar to that of SO(5) deconfined (pseudo-)criticality, for N⩾6N \geqslant 6N⩾6, the annihilation appears to occur only for ε>1ε> 1ε>1, suggesting the existence of a stable phase in D=3D = 3D=3. Comparisons of the scaling dimension of the lowest singlet operator are made with known results for N=6N = 6N=6, which is dual to QED3_33​ with Nf=4N_\mathrm{f} = 4Nf​=4 fermion flavors. The predictions for the N=7N = 7N=7 Stiefel liquid represent to my knowledge the first computation of this kind for a Wess-Zumino-Witten theory without a known gauge theory dual.
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.
In this paper, we propose using the nonlinear sigma model (NLSM) with the Wess-Zumino-Witten (WZW) term as a general description of deconfined quantum critical points that separate two spontaneously symmetry-breaking (SSB) phases in arbitrary dimensions. In particular, we discuss the suitable choice of the target space of the NLSM, which is in general the homogeneous space G/K, where G G is the UV symmetry and K K is generated by {\mathfrak k}={\mathfrak h}_1\cap {\mathfrak h}_2 𝔨 = 𝔥 1 ∩ 𝔥 2 , and {\mathfrak h}_i 𝔥 i is the Lie algebra of the unbroken symmetry in each SSB phase. With this specific target space, the symmetry defects in both SSB phases are on equal footing, and their intertwinement is captured by the WZW term. The DQCP transition is then tuned by proliferating the symmetry defects. By coupling the G/K G / K NLSM with the WZW term to the background gauge field, the ’t Hooft anomaly of this theory can be determined. The bulk symmetry-protected topological (SPT) phase that cancels the anomaly is described by the relative Chern-Simons term in odd spacetime dimensions or mixed \theta θ term in even dimensions. We construct and discuss a series of models with Grassmannian symmetry defects in 3+1d. We also provide the fermionic model that reproduces the G/K G / K NLSM with the WZW term.
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.