arXiv:2505.21611 · Progress of Theoretical and Experimental Physics

Disturbing News About the d = 2 + ϵ Expansion

Fabiana De Cesare, Slava Rychkov

Published Aug 4, 2025• 5 citations

Abstract

Abstract The $O(N)$ nonlinear sigma model (NLSM) in $d=2+\epsilon$ has long been conjectured to describe the same conformal field theory (CFT) as the Wilson–Fisher (WF) $O(N)$ fixed point obtained from the $\lambda (\phi ^2)^2$ model in $d=4-\epsilon$. In this work, we put this conjecture into question, building on the recent observation (Jones RA. Explorations in two dimensional strongly correlated quantum matter: from exactly solvable models to conformal bootstrap. PhD thesis, MIT, 2024) that the NLSM CFT possesses a protected operator with dimension $N-1$, which is instead absent in the WF $O(N)$ CFT. We investigate the possibility of lifting this operator via multiplet recombination—the only known mechanism that could resolve this mismatch while preserving a connection between the two theories. We also explore an alternative scenario in which the NLSM $O(N)$ fixed point in $d=2+\epsilon$ is not continuously connected to the WF $O(N)$ CFT, and instead corresponds to a different universality class. For $N=3$, this could be related to the hedgehog-suppressed critical point, which describes the Néel–valence band solid phase transition in three dimensions.

DOI: 10.1093/ptep/ptaf103

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arXiv:2505.21611 · Progress of Theoretical and Experimental Physics

Disturbing News About the d = 2 + ϵ Expansion

Fabiana De Cesare, Slava Rychkov

Published Aug 4, 2025• 5 citations

arXiv HTML PDF

Abstract

Abstract The

O(N)

nonlinear sigma model (NLSM) in

d=2+\epsilon

has long been conjectured to describe the same conformal field theory (CFT) as the Wilson–Fisher (WF)

O(N)

fixed point obtained from the

\lambda (\phi ^2)^2

model in

d=4-\epsilon

. In this work, we put this conjecture into question, building on the recent observation (Jones RA. Explorations in two dimensional strongly correlated quantum matter: from exactly solvable models to conformal bootstrap. PhD thesis, MIT, 2024) that the NLSM CFT possesses a protected operator with dimension

N-1

, which is instead absent in the WF

O(N)

CFT. We investigate the possibility of lifting this operator via multiplet recombination—the only known mechanism that could resolve this mismatch while preserving a connection between the two theories. We also explore an alternative scenario in which the NLSM

O(N)

fixed point in

d=2+\epsilon

is not continuously connected to the WF

O(N)

CFT, and instead corresponds to a different universality class. For

N=3

, this could be related to the hedgehog-suppressed critical point, which describes the Néel–valence band solid phase transition in three dimensions.

DOI: 10.1093/ptep/ptaf103

Related papers

Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chainsAsymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chains

Christopher Yang, Thomas Scaffidi

The two-dimensional $O(N)$ nonlinear sigma model (NLSM) is asymptotically free for $N>2$ : it exhibits neither a nontrivial fixed point nor spontaneous symmetry-breaking. Here we show that a nontrivial fixed point generically does exist in the complex coupling plane and is described by a complex conformal field theory (CCFT). This CCFT fixed point is generic in the sense that it has a single relevant singlet operator, and is thus expected to arise in any non-Hermitian model with $O(N)$ symmetry upon tuning a single complex parameter. We confirm this prediction numerically by locating the CCFT at $N = 3$ in two non-Hermitian spin-1 antiferromagnetic Heisenberg chains, and in a non-Hermitian spin- $1/2$ ladder, finding good agreement between the complex central charge and scaling dimensions and those obtained by analytic continuation of real fixed points from $N\leq 2$ . We further construct a realistic Lindbladian for a spin-1 chain whose no-click dynamics are governed by the non-Hermitian Hamiltonian realizing the CCFT. Since the CCFT vacuum is the eigenstate with the smallest decay rate, the system naturally relaxes under dissipative dynamics toward a CFT state, thus providing a route to preparing long-range entangled states through engineered dissipation.The two-dimensional $O(N)$ nonlinear sigma model (NLSM) is asymptotically free for $N>2$: it exhibits neither a nontrivial fixed point nor spontaneous symmetry-breaking. Here we show that a nontrivial fixed point generically does exist in the complex coupling plane and is described by a complex conformal field theory (CCFT). This CCFT fixed point is generic in the sense that it has a single relevant singlet operator, and is thus expected to arise in any non-Hermitian model with $O(N)$ symmetry upon tuning a single complex parameter. We confirm this prediction numerically by locating the CCFT at $N = 3$ in two non-Hermitian spin-1 antiferromagnetic Heisenberg chains, and in a non-Hermitian spin-$1/2$ ladder, finding good agreement between the complex central charge and scaling dimensions and those obtained by analytic continuation of real fixed points from $N\leq 2$. We further construct a realistic Lindbladian for a spin-1 chain whose no-click dynamics are governed by the non-Hermitian Hamiltonian realizing the CCFT. Since the CCFT vacuum is the eigenstate with the smallest decay rate, the system naturally relaxes under dissipative dynamics toward a CFT state, thus providing a route to preparing long-range entangled states through engineered dissipation.

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Motivated by the prospect of quantum simulation of quantum field theories, we formulate the $O(N)$ nonlinear sigma model as a ``qubit'' model with an ( $N+1$ )-dimensional local Hilbert space at each lattice site. Using an efficient worm algorithm in the worldline formulation, we demonstrate that the model has a second-order critical point in ( $2+1$ ) dimensions, where the continuum physics of the nontrivial $O(N)$ Wilson-Fisher fixed point is reproduced. We compute the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$ for the $O(N)$ qubit models up to $N=8$ , and find excellent agreement with known results in literature from various analytic and numerical techniques for the $O(N)$ Wilson-Fisher universality class. Our models are suited for studying $O(N)$ nonlinear sigma models on quantum computers up to $N=8$ in $d=2$ , 3 spatial dimensions.Motivated by the prospect of quantum simulation of quantum field theories, we formulate the $O(N)$ nonlinear sigma model as a ``qubit'' model with an ($N+1$)-dimensional local Hilbert space at each lattice site. Using an efficient worm algorithm in the worldline formulation, we demonstrate that the model has a second-order critical point in ($2+1$) dimensions, where the continuum physics of the nontrivial $O(N)$ Wilson-Fisher fixed point is reproduced. We compute the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$ for the $O(N)$ qubit models up to $N=8$, and find excellent agreement with known results in literature from various analytic and numerical techniques for the $O(N)$ Wilson-Fisher universality class. Our models are suited for studying $O(N)$ nonlinear sigma models on quantum computers up to $N=8$ in $d=2$, 3 spatial dimensions.

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Theory of deconfined pseudocriticality

Ruochen Ma, Chong Wang

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Qubit $O(N)$ nonlinear sigma modelsQubit $O(N)$ nonlinear sigma models

Hersh Singh

We reformulate the traditional lattice $O(N)$ nonlinear sigma models as qubit models with a $N+1$ dimensional Hilbert space at each lattice site. Using an efficient worm algorithm in the worldline formulation, we demonstrate that the model has a quantum critical point in $(2+1)$ dimensions. We compute the critical exponents $\nu$ and $\eta$ up to $N=8$ . By comparing these exponents with results from large- $N$ and $\epsilon$ -expansions, we demonstrate that these quantum critical points lie in the usual $O(N)$ Wilson-Fisher universality class. Our models are suited for studying $O(N)$ Wilson-Fisher universality class on quantum computers up to $N=8$ .We reformulate the traditional lattice $O(N)$ nonlinear sigma models as qubit models with a $N+1$ dimensional Hilbert space at each lattice site. Using an efficient worm algorithm in the worldline formulation, we demonstrate that the model has a quantum critical point in $(2+1)$ dimensions. We compute the critical exponents $\nu$ and $\eta$ up to $N=8$. By comparing these exponents with results from large-$N$ and $\epsilon$-expansions, we demonstrate that these quantum critical points lie in the usual $O(N)$ Wilson-Fisher universality class. Our models are suited for studying $O(N)$ Wilson-Fisher universality class on quantum computers up to $N=8$.

Nov 27, 20195 citations

Qubit regularization of the O(3) sigma model

Hersh Singh, Shailesh Chandrasekharan A

We construct a qubit regularization of the $O(3)$ nonlinear sigma model in two and three spatial dimensions using a quantum Hamiltonian with two qubits per lattice site. Using a worldline formulation and worm algorithms, we show that in two spatial dimensions our model has a quantum critical point where the well-known scale-invariant physics of the three-dimensional Wilson-Fisher fixed point is reproduced. In three spatial dimensions, we recover mean-field critical exponents at a similar quantum critical point. These results show that our qubit Hamiltonian is in the same universality class as the traditional classical lattice model close to the critical points. Simple modifications to our model also allow us to study the physics of traditional lattice models with $O(2)$ and ${Z}_{2}$ symmetries close to the corresponding critical points.We construct a qubit regularization of the $O(3)$ nonlinear sigma model in two and three spatial dimensions using a quantum Hamiltonian with two qubits per lattice site. Using a worldline formulation and worm algorithms, we show that in two spatial dimensions our model has a quantum critical point where the well-known scale-invariant physics of the three-dimensional Wilson-Fisher fixed point is reproduced. In three spatial dimensions, we recover mean-field critical exponents at a similar quantum critical point. These results show that our qubit Hamiltonian is in the same universality class as the traditional classical lattice model close to the critical points. Simple modifications to our model also allow us to study the physics of traditional lattice models with $O(2)$ and ${Z}_{2}$ symmetries close to the corresponding critical points.

arXiv:1905.13204Physical review. D/Physical review. D.Sep 17, 201954 citations

Conformal operator content of the Wilson-Fisher transition on fuzzy sphere bilayers

Chao Han, Liangdong Hu, Wei Kang Zhu

The Wilson-Fisher criticality provides a paradigm for a large class of phase transitions in nature (e.g., helium, ferromagnets). In three dimensions, Wilson-Fisher critical points are not exactly solvable due to the strongly correlated feature, so one has to resort to nonperturbative tools such as numerical simulations. Here, we design a microscopic model of Heisenberg magnet bilayer and study the underlying Wilson-Fisher $\mathrm{O}(3)$ transition through the lens of fuzzy sphere regularization. We uncover a wealth of crucial information which directly reveals the emergent conformal symmetry regarding this fixed point. Specifically, we accurately calculate and analyze the energy spectra at the transition, and explicitly identify the existence of a conserved Noether current, a stress tensor, and relevant primary fields. Most importantly, the primaries and their descendants form a fingerprint conformal tower structure, pointing to an almost perfect state-operator correspondence. Furthermore, by examining the leading rank-4 symmetric tensor operator, we demonstrate the cubic perturbation is relevant, implying the critical $\mathrm{O}(3)$ model is unstable to cubic anisotropy, in agreement with the renormalization group and bootstrap calculations. The successful dissection of conformal content of the Wilson-Fisher universality class extends the horizon of the fuzzy sphere method and paves the way for exploring higher-dimensional conformal field theories.The Wilson-Fisher criticality provides a paradigm for a large class of phase transitions in nature (e.g., helium, ferromagnets). In three dimensions, Wilson-Fisher critical points are not exactly solvable due to the strongly correlated feature, so one has to resort to nonperturbative tools such as numerical simulations. Here, we design a microscopic model of Heisenberg magnet bilayer and study the underlying Wilson-Fisher $\mathrm{O}(3)$ transition through the lens of fuzzy sphere regularization. We uncover a wealth of crucial information which directly reveals the emergent conformal symmetry regarding this fixed point. Specifically, we accurately calculate and analyze the energy spectra at the transition, and explicitly identify the existence of a conserved Noether current, a stress tensor, and relevant primary fields. Most importantly, the primaries and their descendants form a fingerprint conformal tower structure, pointing to an almost perfect state-operator correspondence. Furthermore, by examining the leading rank-4 symmetric tensor operator, we demonstrate the cubic perturbation is relevant, implying the critical $\mathrm{O}(3)$ model is unstable to cubic anisotropy, in agreement with the renormalization group and bootstrap calculations. The successful dissection of conformal content of the Wilson-Fisher universality class extends the horizon of the fuzzy sphere method and paves the way for exploring higher-dimensional conformal field theories.

arXiv:2312.04047Physical review. B./Physical review. BSep 9, 202425 citations

Abstract The

O(N)

nonlinear sigma model (NLSM) in

d=2+\epsilon

has long been conjectured to describe the same conformal field theory (CFT) as the Wilson–Fisher (WF)

O(N)

fixed point obtained from the

\lambda (\phi ^2)^2

model in

d=4-\epsilon

N-1

, which is instead absent in the WF

O(N)

O(N)

fixed point in

d=2+\epsilon

is not continuously connected to the WF

O(N)

CFT, and instead corresponds to a different universality class. For

N=3

, this could be related to the hedgehog-suppressed critical point, which describes the Néel–valence band solid phase transition in three dimensions.

Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional

O(N>2)

nonlinear sigma model and its realization in Heisenberg spin chains

The two-dimensional

O(N)

nonlinear sigma model (NLSM) is asymptotically free for

N>2

: it exhibits neither a nontrivial fixed point nor spontaneous symmetry-breaking. Here we show that a nontrivial fixed point generically does exist in the complex coupling plane and is described by a complex conformal field theory (CCFT). This CCFT fixed point is generic in the sense that it has a single relevant singlet operator, and is thus expected to arise in any non-Hermitian model with

O(N)

symmetry upon tuning a single complex parameter. We confirm this prediction numerically by locating the CCFT at

N = 3

in two non-Hermitian spin-1 antiferromagnetic Heisenberg chains, and in a non-Hermitian spin-

1/2

ladder, finding good agreement between the complex central charge and scaling dimensions and those obtained by analytic continuation of real fixed points from

N\leq 2

. We further construct a realistic Lindbladian for a spin-1 chain whose no-click dynamics are governed by the non-Hermitian Hamiltonian realizing the CCFT. Since the CCFT vacuum is the eigenstate with the smallest decay rate, the system naturally relaxes under dissipative dynamics toward a CFT state, thus providing a route to preparing long-range entangled states through engineered dissipation.

Motivated by the prospect of quantum simulation of quantum field theories, we formulate the

O(N)

nonlinear sigma model as a ``qubit'' model with an (

N+1

)-dimensional local Hilbert space at each lattice site. Using an efficient worm algorithm in the worldline formulation, we demonstrate that the model has a second-order critical point in (

2+1

) dimensions, where the continuum physics of the nontrivial

O(N)

Wilson-Fisher fixed point is reproduced. We compute the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$ for the

O(N)

qubit models up to

N=8

, and find excellent agreement with known results in literature from various analytic and numerical techniques for the

O(N)

Wilson-Fisher universality class. Our models are suited for studying

O(N)

nonlinear sigma models on quantum computers up to

N=8

d=2

, 3 spatial dimensions.

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.

Qubit

O(N)

nonlinear sigma models

We reformulate the traditional lattice

O(N)

nonlinear sigma models as qubit models with a

N+1

dimensional Hilbert space at each lattice site. Using an efficient worm algorithm in the worldline formulation, we demonstrate that the model has a quantum critical point in

(2+1)

dimensions. We compute the critical exponents

\nu

and

\eta

up to

N=8

. By comparing these exponents with results from large-

N

and

\epsilon

-expansions, we demonstrate that these quantum critical points lie in the usual

O(N)

Wilson-Fisher universality class. Our models are suited for studying

O(N)

Wilson-Fisher universality class on quantum computers up to

N=8

We construct a qubit regularization of the

O(3)

nonlinear sigma model in two and three spatial dimensions using a quantum Hamiltonian with two qubits per lattice site. Using a worldline formulation and worm algorithms, we show that in two spatial dimensions our model has a quantum critical point where the well-known scale-invariant physics of the three-dimensional Wilson-Fisher fixed point is reproduced. In three spatial dimensions, we recover mean-field critical exponents at a similar quantum critical point. These results show that our qubit Hamiltonian is in the same universality class as the traditional classical lattice model close to the critical points. Simple modifications to our model also allow us to study the physics of traditional lattice models with

O(2)

and

{Z}_{2}

symmetries close to the corresponding critical points.

\mathrm{O}(3)

transition through the lens of fuzzy sphere regularization. We uncover a wealth of crucial information which directly reveals the emergent conformal symmetry regarding this fixed point. Specifically, we accurately calculate and analyze the energy spectra at the transition, and explicitly identify the existence of a conserved Noether current, a stress tensor, and relevant primary fields. Most importantly, the primaries and their descendants form a fingerprint conformal tower structure, pointing to an almost perfect state-operator correspondence. Furthermore, by examining the leading rank-4 symmetric tensor operator, we demonstrate the cubic perturbation is relevant, implying the critical

\mathrm{O}(3)

model is unstable to cubic anisotropy, in agreement with the renormalization group and bootstrap calculations. The successful dissection of conformal content of the Wilson-Fisher universality class extends the horizon of the fuzzy sphere method and paves the way for exploring higher-dimensional conformal field theories.