arXiv:2509.18966

Non-Lagrangian phases of matter from Wilsonian renormalization of 3D Wess-Zumino-Witten theory on Stiefel manifolds

Shouryya Ray

Published Sep 23, 2025• 0 citations

Abstract

I study the renormalization of D-dimensional level-k Wess-Zumino-Witten theory with Stiefel-manifold target space $\mathrm{St}_{N,N-D-1} \cong \mathrm{SO}(N)/\mathrm{SO}(D+1)$, with a particular focus on $D = 3$. 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 $N \geq 7$ in $D = 3$. They are hence of interest both from the point of view of quantum phases of matter as well as pure field theory. The $D$-dimensional expressions enable the computation, by analytic computation, of beta functions in $D = 2 + ε$, at least to first non-trivial order. In $D = 2$, a stable fixed point is found, serving a generalization of the famed $\mathrm{SU}(2)_k$ Wess-Zumino-Witten conformal field theory; it annihilates in $D = 2 + ε$ with an unstable fixed point which splits off from the Gaussian one for $ε> 0$. Although the story is thus qualitatively similar to that of SO(5) deconfined (pseudo-)criticality, for $N \geqslant 6$, the annihilation appears to occur only for $ε> 1$, suggesting the existence of a stable phase in $D = 3$. Comparisons of the scaling dimension of the lowest singlet operator are made with known results for $N = 6$, which is dual to QED$_3$ with $N_\mathrm{f} = 4$ fermion flavors. The predictions for the $N = 7$ Stiefel liquid represent to my knowledge the first computation of this kind for a Wess-Zumino-Witten theory without a known gauge theory dual.

DOI: 10.48550/arxiv.2509.18966

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Functional renormalization group analysis of the O ( 3 ) nonlinear sigma model and non-Abelian bosonization duality

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Non-abelian bosonization in two and three spatial dimensions and applications

Yen-Ta Huang, D. H. Lee

In this paper, we generalize Witten's non-abelian bosonization in (1+1)-D to two and three spatial dimensions. Our theory applies to fermions with relativistic dispersion. The bosonized theories are non-linear sigma models with level-1 Wess-Zumino-Witten terms. We apply the bosonization results to the SU(2) gauge theory of the π-flux phase, critical spin liquids in 1,2,3 spatial dimensions, and twisted bilayer graphene.

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arXiv:2509.18966

Non-Lagrangian phases of matter from Wilsonian renormalization of 3D Wess-Zumino-Witten theory on Stiefel manifolds

Shouryya Ray

Published Sep 23, 2025• 0 citations

arXiv HTML PDF

Abstract

I study the renormalization of D-dimensional level-k Wess-Zumino-Witten theory with Stiefel-manifold target space

\mathrm{St}_{N,N-D-1} \cong \mathrm{SO}(N)/\mathrm{SO}(D+1)

, with a particular focus on

D = 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 \geq 7

D = 3

. They are hence of interest both from the point of view of quantum phases of matter as well as pure field theory. The

D

-dimensional expressions enable the computation, by analytic computation, of beta functions in

D = 2 + ε

, at least to first non-trivial order. In

D = 2

, a stable fixed point is found, serving a generalization of the famed

\mathrm{SU}(2)_k

Wess-Zumino-Witten conformal field theory; it annihilates in

D = 2 + ε

with an unstable fixed point which splits off from the Gaussian one for

ε> 0

. Although the story is thus qualitatively similar to that of SO(5) deconfined (pseudo-)criticality, for

N \geqslant 6

, the annihilation appears to occur only for

ε> 1

, suggesting the existence of a stable phase in

D = 3

. Comparisons of the scaling dimension of the lowest singlet operator are made with known results for

N = 6

, which is dual to QED

_3

with

N_\mathrm{f} = 4

fermion flavors. The predictions for the

N = 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.

DOI: 10.48550/arxiv.2509.18966

Related papers

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

Adam Nahum

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.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.

arXiv:1912.13468Physical review. B./Physical review. BNov 30, 202080 citations

Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders

Liujun Zou, Yin-Chen He, Chong Wang

We propose a new type of quantum liquids, dubbed Stiefel liquids, based on (2 þ 1)-dimensional nonlinear sigma models on target space SOðNÞ=SOð4Þ, supplemented with Wess-Zumino-Witten terms.We argue that the Stiefel liquids form a class of critical quantum liquids with extraordinary properties, such as large emergent symmetries, a cascade structure, and nontrivial quantum anomalies.We show that the well-known deconfined quantum critical point and Uð1Þ Dirac spin liquid are unified as two special examples of Stiefel liquids, N ¼ 5 and N ¼ 6, respectively.Furthermore, we conjecture that Stiefel liquids with N > 6 are non-Lagrangian, in the sense that under renormalization group they flow to infrared (conformally invariant) fixed points that cannot be described by any renormalizable continuum Lagrangian.Such non-Lagrangian states are beyond the paradigm of parton gauge mean-field theory familiar in the study of exotic quantum liquids in condensed matter physics.The intrinsic absence of (conventional or partonlike) mean-field construction also means that, within the traditional approaches, will be difficult to decide whether a non-Lagrangian state can actually emerge from a specific UV system (such as a lattice spin system).For this purpose we hypothesize that a quantum state is emergible from a lattice system if its quantum anomalies match with the constraints from the (generalized) Lieb-Schultz-Mattis theorems.Based on this hypothesis, we find that some of the non-Lagrangian Stiefel liquids can indeed be realized in frustrated quantum spin systems, for example, on triangular or kagome lattice, through the intertwinement between noncoplanar magnetic orders and valence-bond-solid orders.We propose a new type of quantum liquids, dubbed Stiefel liquids, based on (2 þ 1)-dimensional nonlinear sigma models on target space SOðNÞ=SOð4Þ, supplemented with Wess-Zumino-Witten terms.We argue that the Stiefel liquids form a class of critical quantum liquids with extraordinary properties, such as large emergent symmetries, a cascade structure, and nontrivial quantum anomalies.We show that the well-known deconfined quantum critical point and Uð1Þ Dirac spin liquid are unified as two special examples of Stiefel liquids, N ¼ 5 and N ¼ 6, respectively.Furthermore, we conjecture that Stiefel liquids with N > 6 are non-Lagrangian, in the sense that under renormalization group they flow to infrared (conformally invariant) fixed points that cannot be described by any renormalizable continuum Lagrangian.Such non-Lagrangian states are beyond the paradigm of parton gauge mean-field theory familiar in the study of exotic quantum liquids in condensed matter physics.The intrinsic absence of (conventional or partonlike) mean-field construction also means that, within the traditional approaches, will be difficult to decide whether a non-Lagrangian state can actually emerge from a specific UV system (such as a lattice spin system).For this purpose we hypothesize that a quantum state is emergible from a lattice system if its quantum anomalies match with the constraints from the (generalized) Lieb-Schultz-Mattis theorems.Based on this hypothesis, we find that some of the non-Lagrangian Stiefel liquids can indeed be realized in frustrated quantum spin systems, for example, on triangular or kagome lattice, through the intertwinement between noncoplanar magnetic orders and valence-bond-solid orders.

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Theory of deconfined pseudocriticality

Ruochen Ma, Chong Wang

It has been proposed that the deconfined criticality in $(2+1)d$ ---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 $k=1$ , 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 $d=3$ and the $\text{SU}{(2)}_{1}$ WZW theory at $d=2$ describing the spin- $1/2$ Heisenberg chain. The theory can be formally controlled, at least to leading order, in terms of the inverse of the WZW level $1/k$ . 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 $d=3$ 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.It has been proposed that the deconfined criticality in $(2+1)d$---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 $k=1$, 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 $d=3$ and the $\text{SU}{(2)}_{1}$ WZW theory at $d=2$ describing the spin-$1/2$ Heisenberg chain. The theory can be formally controlled, at least to leading order, in terms of the inverse of the WZW level $1/k$. 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 $d=3$ 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.

arXiv:1912.12315Physical review. B./Physical review. BJul 21, 202092 citations

Critical behavior of the QED3 -Gross-Neveu model: Duality and deconfined criticality

Lukas Janssen, Yin-Chen He

We study the critical properties of the ${\mathrm{QED}}_{3}$ -Gross-Neveu model with $2N$ flavors of two-component Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For $N=1$ , this theory has recently been suggested to be dual to the SU(2) noncompact ${\mathrm{CP}}^{1}$ model that describes the deconfined phase transition between the N\'eel antiferromagnet and the valence bond solid on the square lattice. For $N=2$ , the theory has been proposed as an effective description of a deconfined critical point between chiral and Dirac spin liquid phases, and may potentially be realizable in spin- $1/2$ systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the ${\mathrm{QED}}_{3}$ -Gross-Neveu model for all values of $N$ . This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain ${\mathrm{QED}}_{3}$ (without scalar-field coupling) unstable at low values of $N$ . The theory exhibits an upper critical space-time dimension of four, enabling us to access the critical behavior in a controlled expansion in the small parameter $\ensuremath{\epsilon}=4\ensuremath{-}D$. We compute the scalar-field anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, the correlation-length exponent $\ensuremath{\nu}$, as well as the scaling dimension of the flavor-symmetry-breaking bilinear $\overline{\ensuremath{\psi}}{\ensuremath{\sigma}}^{z}\ensuremath{\psi}$ at the critical point, and compare our leading-order estimates with predictions of the conjectured duality.We study the critical properties of the ${\mathrm{QED}}_{3}$-Gross-Neveu model with $2N$ flavors of two-component Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For $N=1$, this theory has recently been suggested to be dual to the SU(2) noncompact ${\mathrm{CP}}^{1}$ model that describes the deconfined phase transition between the N\'eel antiferromagnet and the valence bond solid on the square lattice. For $N=2$, the theory has been proposed as an effective description of a deconfined critical point between chiral and Dirac spin liquid phases, and may potentially be realizable in spin-$1/2$ systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the ${\mathrm{QED}}_{3}$-Gross-Neveu model for all values of $N$. This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain ${\mathrm{QED}}_{3}$ (without scalar-field coupling) unstable at low values of $N$. The theory exhibits an upper critical space-time dimension of four, enabling us to access the critical behavior in a controlled expansion in the small parameter $\ensuremath{\epsilon}=4\ensuremath{-}D$. We compute the scalar-field anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, the correlation-length exponent $\ensuremath{\nu}$, as well as the scaling dimension of the flavor-symmetry-breaking bilinear $\overline{\ensuremath{\psi}}{\ensuremath{\sigma}}^{z}\ensuremath{\psi}$ at the critical point, and compare our leading-order estimates with predictions of the conjectured duality.

arXiv:1708.02256Physical review. B./Physical review. BNov 8, 201746 citations

Functional renormalization group analysis of the O ( 3 ) nonlinear sigma model and non-Abelian bosonization duality

Junichi Haruna, Keito Shimizu, Masatoshi Yamada

It is known that the U ( 2 ) Wess-Zumino-Witten model is dual to the free fermion theory in two dimensions via non-Abelian bosonization. While it is decomposed into the S U ( 2 ) Wess-Zumino-Witten model and a free compact boson, the former is believed to be equivalent to the O ( 3 ) nonlinear sigma model with the theta term at θ = π . In this work, we reexamine this duality through the lens of nonperturbative renormalization group (RG) flow. We analyze the RG flow structure of the O ( 3 ) nonlinear sigma model with the theta term in two dimensions using the functional renormalization group. Our results reveal a nontrivial fixed point with a nonzero value of the topological coupling. The scaling dimensions (critical exponents) at this fixed point suggest the realization of a duality between the O ( 3 ) nonlinear sigma model with the theta term and the free fermion theory, indicating that these models belong to the same universality class.It is known that the U ( 2 ) Wess-Zumino-Witten model is dual to the free fermion theory in two dimensions via non-Abelian bosonization. While it is decomposed into the S U ( 2 ) Wess-Zumino-Witten model and a free compact boson, the former is believed to be equivalent to the O ( 3 ) nonlinear sigma model with the theta term at θ = π . In this work, we reexamine this duality through the lens of nonperturbative renormalization group (RG) flow. We analyze the RG flow structure of the O ( 3 ) nonlinear sigma model with the theta term in two dimensions using the functional renormalization group. Our results reveal a nontrivial fixed point with a nonzero value of the topological coupling. The scaling dimensions (critical exponents) at this fixed point suggest the realization of a duality between the O ( 3 ) nonlinear sigma model with the theta term and the free fermion theory, indicating that these models belong to the same universality class.

arXiv:2405.14627Physical review. D/Physical review. D.Feb 18, 20251 citation

Non-abelian bosonization in two and three spatial dimensions and applications

Yen-Ta Huang, D. H. Lee

arXiv:2104.07015Nuclear Physics BSep 27, 20219 citations

I study the renormalization of D-dimensional level-k Wess-Zumino-Witten theory with Stiefel-manifold target space

\mathrm{St}_{N,N-D-1} \cong \mathrm{SO}(N)/\mathrm{SO}(D+1)

, with a particular focus on

D = 3

N \geq 7

D = 3

. They are hence of interest both from the point of view of quantum phases of matter as well as pure field theory. The

D

-dimensional expressions enable the computation, by analytic computation, of beta functions in

D = 2 + ε

, at least to first non-trivial order. In

D = 2

, a stable fixed point is found, serving a generalization of the famed

\mathrm{SU}(2)_k

Wess-Zumino-Witten conformal field theory; it annihilates in

D = 2 + ε

with an unstable fixed point which splits off from the Gaussian one for

ε> 0

. Although the story is thus qualitatively similar to that of SO(5) deconfined (pseudo-)criticality, for

N \geqslant 6

, the annihilation appears to occur only for

ε> 1

, suggesting the existence of a stable phase in

D = 3

. Comparisons of the scaling dimension of the lowest singlet operator are made with known results for

N = 6

, which is dual to QED

_3

with

N_\mathrm{f} = 4

fermion flavors. The predictions for the

N = 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 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.

It has been proposed that the deconfined criticality in

(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=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=3

and the

\text{SU}{(2)}_{1}

WZW theory at

d=2

describing the spin-

1/2

Heisenberg chain. The theory can be formally controlled, at least to leading order, in terms of the inverse of the WZW level

1/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=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 study the critical properties of the

{\mathrm{QED}}_{3}

-Gross-Neveu model with

2N

flavors of two-component Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For

N=1

, this theory has recently been suggested to be dual to the SU(2) noncompact

{\mathrm{CP}}^{1}

model that describes the deconfined phase transition between the N\'eel antiferromagnet and the valence bond solid on the square lattice. For

N=2

, the theory has been proposed as an effective description of a deconfined critical point between chiral and Dirac spin liquid phases, and may potentially be realizable in spin-

1/2

systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the

{\mathrm{QED}}_{3}

-Gross-Neveu model for all values of

N

. This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain

{\mathrm{QED}}_{3}

(without scalar-field coupling) unstable at low values of

N

. The theory exhibits an upper critical space-time dimension of four, enabling us to access the critical behavior in a controlled expansion in the small parameter $\ensuremath{\epsilon}=4\ensuremath{-}D$. We compute the scalar-field anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, the correlation-length exponent $\ensuremath{\nu}$, as well as the scaling dimension of the flavor-symmetry-breaking bilinear $\overline{\ensuremath{\psi}}{\ensuremath{\sigma}}^{z}\ensuremath{\psi}$ at the critical point, and compare our leading-order estimates with predictions of the conjectured duality.