Qubit $O(N)$ nonlinear sigma models

Hersh Singh

Published Nov 27, 2019• 5 citations

Abstract

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 $N=8$.

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

Hersh Singh

Published Nov 27, 2019• 5 citations

OpenAlex

Abstract

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

Related papers

Qubit regularized O(N) nonlinear sigma models

Hersh Singh

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.

arXiv:1911.12353Physical review. D/Physical review. D.Jun 17, 202222 citations

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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.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.

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Criticality on Rényi Defects at (2+1) $D$ O(3) Quantum Critical PointsCriticality on Rényi Defects at (2+1)$D$ O(3) Quantum Critical Points

Yanzhang Zhu, Zhe Wang, Meng Cheng +1 more

At a quantum critical point, the universal scaling behavior of Rényi entanglement entropy is controlled by the universality class of the codimension-two Rényi (or conical) defects in the infrared theory. In this work we perform a systematic study of critical correlations along Rényi defect lines in (2+1) $D$ quantum spin models realizing quantum phase transitions described by the O(3) Wilson-Fisher universality class, using large-scale quantum Monte Carlo simulations. We present numerical evidence that, for a fixed Rényi index $n$ , there exist multiple Rényi defect universality classes, with distinct critical exponents for the O(3) order parameter on the defect. These universality classes are realized by choosing microscopically different entanglement cuts in lattice models, which we classify as ordinary, special and extraordinary according to their relation to surface criticality. For the extraordinary entanglement cut, we further find evidence for a phase transition on the defect as a function of the Rényi index. Our results highlight the key role of defect universality classes in determining the universal scaling of Rényi entropy, and provide a framework for understanding the previously observed dependence of Rényi entropy scaling on microscopic lattice details.At a quantum critical point, the universal scaling behavior of Rényi entanglement entropy is controlled by the universality class of the codimension-two Rényi (or conical) defects in the infrared theory. In this work we perform a systematic study of critical correlations along Rényi defect lines in (2+1)$D$ quantum spin models realizing quantum phase transitions described by the O(3) Wilson-Fisher universality class, using large-scale quantum Monte Carlo simulations. We present numerical evidence that, for a fixed Rényi index $n$, there exist multiple Rényi defect universality classes, with distinct critical exponents for the O(3) order parameter on the defect. These universality classes are realized by choosing microscopically different entanglement cuts in lattice models, which we classify as ordinary, special and extraordinary according to their relation to surface criticality. For the extraordinary entanglement cut, we further find evidence for a phase transition on the defect as a function of the Rényi index. Our results highlight the key role of defect universality classes in determining the universal scaling of Rényi entropy, and provide a framework for understanding the previously observed dependence of Rényi entropy scaling on microscopic lattice details.

cond-mat.str-elCross-listedarXiv:2605.00104Physical Review LettersMay 4, 20260 citations

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

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.

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.

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.

Criticality on Rényi Defects at (2+1)

D

O(3) Quantum Critical Points

D

quantum spin models realizing quantum phase transitions described by the O(3) Wilson-Fisher universality class, using large-scale quantum Monte Carlo simulations. We present numerical evidence that, for a fixed Rényi index

n

, there exist multiple Rényi defect universality classes, with distinct critical exponents for the O(3) order parameter on the defect. These universality classes are realized by choosing microscopically different entanglement cuts in lattice models, which we classify as ordinary, special and extraordinary according to their relation to surface criticality. For the extraordinary entanglement cut, we further find evidence for a phase transition on the defect as a function of the Rényi index. Our results highlight the key role of defect universality classes in determining the universal scaling of Rényi entropy, and provide a framework for understanding the previously observed dependence of Rényi entropy scaling on microscopic lattice details.