Evidence for spontaneous breaking of a continuous symmetry at a non-conformal quantum critical point in one dimension

arXiv:2511.09097 · cond-mat.stat-mech, cond-mat.quant-gas, cond-mat.str-el, math-ph, math.MP · Submitted 2025-11-12 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Evidence for spontaneous breaking of a continuous symmetry at a non-conformal quantum critical point in one dimension".

Mira: The gist The numerical evidence for spontaneous breaking of a continuous U(1) symmetry in a nearest-neighbor interacting spin-1 chain at a quantum critical point separating two XY quasi-long-range ordered phases…

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, we're looking at this paper today, "Evidence for spontaneous breaking of a continuous symmetry at a non-conformal quantum critical point in one dimension." Basically, these authors are showing numerical evidence that you can get the spontaneous breaking of a continuous U(one) symmetry right at the critical point where two different XY quasi-long-range ordered phases meet <ref:2511.09097#pg1,the spontaneous breaking of a continuous U(1) symmetry>.

Mira: That's what it claims, Kai, and it's pretty striking because they say this continuous symmetry breaking emerges exactly at the critical point of that discrete order parameter, which suggests a mechanism we haven't seen before <ref:2511.09097#pg1>.

Lev: From an engineering standpoint, it’s interesting that they are focusing on the spin-one chain with this specific Hamiltonian H = X L i=one −JSi · Si+one + g1 (S z i) two + g2 S z i S z i+twelve where J equals one and g two is one <ref:2511.09097#pg4>.

Kai: Right, and what's cool is that the model itself has these global symmetries; specifically an O(two) symmetry generated by rotations around the z axis and spin flips along that same z axis <ref:2511.09097#pg4>.

Mira: And they point out that while the Heisenberg term keeps the O(two) symmetry, the single-ion and double-ion anisotropy terms only change things by a sign belonging to a Z two subgroup in O(two) = U(one) sub Z2 <ref:2511.09097#pg4>.

Lev: So they're focusing on that transition from an SSB phase to an unbroken state of the Z two subgroup, which is where the continuous symmetry breaking happens, right?

Kai: Exactly, and at this critical point, they find true long-range XY order along with a finite perpendicular magnetization and a zero-frequency Bragg peak in the transverse dynamical structure factor <ref:2511.09097#pg1>.

Mira: And they also see these sharp gapless collective excitations which are a key feature of criticality, right?

Lev: So if you look at how they measure this, they use infinite density matrix renormalization group or iDMRG for the ground state and time-evolving block decimation or TEBD to evolve the state in time <ref:2511.09097#pg4>.

Kai: They use these methods to compute static observables like the z magnetization density m z = one/L P i ⟨S z i ⟩ and they find that for parameter values g one less than approximately negative zero point eight two, they get an mz not equal to zero, which is consistent with a Z two SSB phase <ref:2511.09097#pg5>.

Mira: And at the critical point of this transition, that magnetization stays approximately constant and becomes a local maximum as you move g one less than g c <ref:2511.09097#pg5>.

Paper summary: Lev: They also check the equal-time connected longitudinal correlator defined as ⟨S z i S z zero⟩ c = ⟨S z i S z zero⟩ − ⟨S z i⟩ ⟨S z zero⟩ and they find that at the critical point, that value agrees within their error bars with what they extracted from the static correlators <ref:2511.09097#pg5>.

Kai: Then they move into the dynamical stuff to get the full set of exponents by analyzing two dynamical structure factors Szz(k, ω) and S⊥(k, ω) <ref:2511.09097#pg5>.

Mira: They observe sharp modes in both the spectrum of Szz and S⊥ at the critical point g one equals g c in Figure two for a system of L equals four hundred sites with χ max equal to one hundred fifty and twelve hundred time steps each with delta t equal to zero point one <ref:2511.09097#pg5>.

Lev: They find the relation omega proportional to k z, and the dynamical exponent is z equals one point five zero plus or minus zero point zero four <ref:2511.09097#pg5>.

Kai: And they also check that z plus eta is consistent with independent dynamical estimates they discussed later <ref:2511.09097#pg5>.

Mira: So, what we have here is a numerical determination of the critical exponents, obtaining an anomalous dimension eta equal to one point zero four plus or minus zero point zero three <ref:2511.09097#pg5>.

Lev: The implication for real hardware is that if you try to run this on actual superconducting qubits, you'd be looking at a system with these scaling properties, and the overhead for error correction would land somewhere related to these exponents <ref:2511.09097#pg4>.

Kai: They also look at the continuum limit, mapping the partition function to a theory with the Euclidean Lagrangian L1 = i ψ∂τ θ + J two (∂x ψ) two + (∂x θ) two + g1 ψ two + g2 ψ four <ref:2511.09097#pg6>.

Mira: They perform a perturbative renormalization and expansion about the upper critical dimension dc equals two <ref:2511.09097#pg7>.

Lev: The flow equations they get here differ from those of the Ising model, and the resulting critical exponents at that interacting fixed point also deviate from the Ising universality class at second order in epsilon <ref:2511.09097#pg7>.

Kai: So, what this means for us is that we're seeing something different from what we expect based on simpler models like the Ising model <ref:2511.09097#pg7>.

Mira: It points toward an interacting fixed point with critical behavior that doesn't fit into the existing Ising universality class, even at second order in epsilon <ref:2511.09097#pg7>.

Lev: The paper also discusses the non-interacting continuum limit, which they find is equivalent to a recently introduced transverse quantum fluid displaying their off-diagonal long-ranged order in one dimension <ref:2511.09097#pg2>.

Paper summary: Kai: So this suggests that even in this quantum many-body system at equilibrium, there's this off-diagonal long-range order present in the continuum limit of the non interacting theory, right?

Mira: Exactly, and complementing these numerical studies, they derive the renormalization group flow equations of the continuum theory to second order in the epsilon expansion <ref:2511.09097#pg9>.

Lev: And when you look at those RG flow equations, you identify an interacting fixed point whose critical behavior differs from the Ising universality class already at two-loop order <ref:2511.09097#pg8>.

Kai: So, what are the actual numbers we're looking at here? They obtain critical exponents eta equals one point zero four plus or minus zero point zero three, z equals one point five zero plus or minus zero point zero four, and beta equals zero point three one plus or minus zero point zero one <ref:2511.09097#pg9>.

Mira: And what's really interesting is that the dynamical exponent is remarkably close to the one-dimensional Kardar--Parisi--Zhang exponent z equals three over two <ref:2511.09097#pg9>.

Lev: That value of z being close to three over two, which is a value you see in non-equilibrium systems, is the first time it's been found in an equilibrium quantum system <ref:2511.09097#pg9>.

Kai: So it’s a significant finding because it connects this quantum critical point to something from non-equilibrium physics that we usually think of as a different regime <ref:two thousand five hundred eleven point zero nine zero nine seven#pg6.

Mira: It really suggests that the system isn't behaving like the simple Ising model near this transition, but rather something more complex in terms of its scaling behavior <ref:2511.09097#pg8>.

Lev: The limitation they mention is that their perturbative exponents remain far from the numerical values, which means we can't fully trust those theoretical predictions yet <ref:two thousand five hundred eleven point zero nine zero nine seven#pg6.

Kai: So to wrap up this paper, "Evidence for spontaneous breaking of a continuous symmetry at a non-conformal quantum critical point in one dimension," it shows that the transition is continuous and characterized by the Ising order parameter m z <ref:2511.09097#pg9>.

Mira: At criticality, they find robust numerical evidence for long-range transverse order, including a finite perpendicular magnetization m⊥ and nondecaying XY correlations in the thermodynamic limit <ref:2511.09097#pg9>.

Lev: They also found a zero-frequency Bragg peak in the transverse dynamical structure factor <ref:2511.09097#pg1>.

Kai: From the combined static and dynamical scaling analyses, they get eta equals one point zero four plus or minus zero point zero three, z equals one point five zero plus or minus zero point zero four, and beta equals zero point three one plus or minus zero point zero one <ref:2511.09097#pg9>.

Mira: And they find the value of the dynamical exponent is remarkably close to the one-dimensional KPZ value z equals three over two <ref:2511.09097#pg9>.

Lev: That's a big number because it's the first time z equals one point five zero plus or minus zero point zero four was found in a quantum system at equilibrium <ref:2511.09097#pg3>.

Conclusion: Kai: So, this paper, "Evidence for spontaneous breaking of a continuous symmetry at a non-conformal quantum critical point in one dimension," it's showing how you can get that continuous U(one) symmetry breaking right at a critical point where two different XY phases meet.

Mira: That means the continuous symmetry isn't just something that happens far away; it pops out exactly when the discrete order parameter is at its critical value, which is a pretty novel way to look at things.

Lev: From what I see on the hardware side, this kind of physics means we’re dealing with systems that have these very delicate scaling laws, and you can't just use standard Ising model predictions for error correction overhead.

Kai: Right, and they found some specific numbers, like a dynamical exponent z of one point five zero plus or minus zero point zero four.

Mira: And what I find interesting is that this value for z is remarkably close to the one-dimensional Kardar--Parisi--Zhang exponent, which usually shows up in non-equilibrium systems.

Lev: So what it changes for me on the hardware side is that if we try to implement error correction based on these scaling laws, we need to account for this specific z value, not just a generic constant.

Kai: They also showed that even in the continuum limit of this theory, there’s off-diagonal long-ranged order in one dimension.

Mira: It suggests that even when you take the system to its simplest theoretical form, it retains this kind of structure which is quite complex.

Lev: That means we need to be careful when mapping these things onto realistic lattice models because the underlying continuum behavior isn't as simple as just a free field theory.

Kai: So basically, this work gives us strong numerical evidence that continuous symmetry breaking can happen at quantum critical points in one dimension, and it hints at scaling behaviors we haven't seen in equilibrium systems before.

R. Flores-Calderon, M. Zundel

Technical University of Munich School of Natural Sciences · Munich Center for Quantum Science and Technology · Universite Grenoble Alpes CNRS Institute for Theoretical Physics, Universitat Innsbruck Institute for Theoretical Physics

cond-mat.stat-mech, cond-mat.quant-gas, cond-mat.str-el, math-ph, math.MP

Submitted: 2025-11-12

Updated: 2026-09-02

Comments: 21 pages, 9 figures

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 86/100

The gist: The gist The numerical evidence for spontaneous breaking of a continuous U(1) symmetry in a nearest-neighbor interacting spin-1 chain at a quantum critical point separating two XY quasi-long-range

Key concepts

Continuous U(1) Symmetry Breaking: This refers to a situation where a system with continuous rotational symmetry (like an XY model) spontaneously develops a non-zero expectation value for its order parameter, even though the underlying Hamiltonian doesn't explicitly break that symmetry. The paper finds this happens at the quantum critical point separating two phases.
Quantum Critical Point (QCP): A QCP is a special point in a quantum system where a phase transition occurs not by changing temperature or pressure, but by tuning a non-thermal parameter like magnetic field or interaction strength. At this point, the system exhibits universal, scale-invariant behavior that can be described by critical exponents.
Dynamical Exponent ($z$): The dynamical exponent describes how the characteristic energy (frequency $\omega$) of excitations scales with their momentum ($k$) near a critical point ($\omega \propto k^z$). The paper finds $z=1.50 ext{ to } 1.54$, which is remarkably close to the value found in the 1D Kardar-Parisi-Zhang (KPZ) universality class, linking this quantum system to a known classical growth model.
Transverse Order: This refers to long-range correlations in the direction perpendicular to the primary order parameter (the XY plane). The numerical evidence shows that at criticality, these transverse correlations develop true long-range order, accompanied by a finite perpendicular magnetization ($m_{\perp} \neq 0$), which is a key feature of the continuous symmetry breaking.

Terminology

Fixed Point — 2× in this paper · explained in 2 episodes
Phase Transition — 1× in this paper · explained in 2 episodes
Symmetry Breaking — 1× in this paper · explained in 4 episodes
Universality Class — 1× in this paper · explained in 2 episodes

Summary

The gist The numerical evidence for spontaneous breaking of a continuous U(1) symmetry in a nearest-neighbor interacting spin-1 chain at a quantum critical point separating two XY quasi-long-range ordered phases distinguished by a spontaneously broken Z2 symmetry suggests a novel mechanism beyond currently established scenarios.

Model and Symmetry

The study investigates the spin-1 Hamiltonian H = X L i=1 − JSi · Si+1 + g1 (S z i)2 + g2 S z i S z i+1 2, where J = 1 and g2 = 1 <ref:2511.09097#pg3> The first term is a ferromagnetic Heisenberg exchange interaction; the second and last terms represent easy-plane single-ion and double-ion anisotropy, respectively <ref:2511.09097#pg4> This model has several global symmetries. First, there is an O(2) symmetry which is generated by rotations around the z axis and z axis spin flips <ref:2511.09097#pg4> The remaining terms only have a zcomponent of the spin, thus are unchanged under the U(1) normal subgroup and can at most change by a sign belonging to the Z2 subgroup in O(2) = U(1) ⋊ Z2 <ref:2511.09097#pg4> It is the transition from a spontaneous symmetry-broken (SSB) phase to an unbroken state of the Z2 subgroup that we will focus on in the following <ref:2511.09097#pg4>.

Static Observables and Critical Exponents

The researchers use infinite density matrix renormalization group (iDMRG) algorithm to calculate the ground state of Eq. (1) and time-evolving block decimation (TEBD) algorithm to evolve the state in time <ref:2511.09097#pg4>. They compute the z-magnetization density mz = 1/L P i ⟨S z i ⟩ and find that for parameter values g1 less than approximately −0.82 we find mz ≠ 0 consistent with a Z2 SSB phase <ref:2511.09097#pg5>. At the critical point of a) the magnetization stays approximately constant. It becomes a local maximum as a function of g1 <ref:2511.09097#pg5>. The behavior of mz for g1 < gc is fitted to the critical exponent β by m2 z ∝ g1 − gc 2β <ref:2511.09097#pg5>. As an additional consistency check, they analyze the equal-time connected longitudinal correlator defined as ⟨S z i S z 0⟩ c = ⟨S z i S z 0⟩ − ⟨S z i⟩ ⟨S z 0⟩ and find that at the critical point, they find <ref:2511.09097#pg5>. The resulting value agrees within the errorbars with that extracted from the static correlators <ref:2511.09097#pg5>.

Dynamical Structure Factor and Critical Exponents

To obtain the complete set of critical exponents, they analyze two dynamical structure factors Szz(k, ω) and S⊥(k, ω) <ref:2511.09097#pg5>. At the critical point g1 = gc in Fig. 2 for a system of L = 400 sites, χmax = 150 with 1200 time-steps each with ∆t = 0.1 they observe sharp modes in the spectrum of both Szz and S⊥ <ref:2511.09097#pg5>. They find the relation ω ∝ k z with z = 1.50 ± 0.04 <ref:2511.09097#pg5>. The value of z + η is consistent with independent dynamical estimates discussed below <ref:2511.09097#pg5>.

Field-Theoretic Prediction and Renormalization Group Flow

In the continuum limit, they map the partition function to a theory with the Euclidean Lagrangian L1 = iψ∂τ θ + J 2 (∂xψ) 2 + (∂xθ) 2 + g1 ψ 2 + g2 ψ 4 <ref:2511.09097#pg6>. They perform a perturbative renormalization and simultaneous expansion about the upper critical dimension dc = 2 <ref:2511.09097#pg7>. The resulting flow equations differ from those of the Ising model, and the resulting critical exponents at the interacting fixed point also deviate from the Ising universality class at second order in ε <ref:2511.09097#pg7>.

Discussion and Outlook

The numerical results show that the transition described by Eq. (1) is continuous and characterized by the Ising order parameter mz <ref:2511.09097#pg9>. At criticality, they find robust numerical evidence for long-range transverse order, including a finite perpendicular magnetization m⊥, nondecaying XY correlations in the thermodynamic limit, and a zero-frequency Bragg peak in the transverse dynamical structure factor <ref:2511.09097#pg9>. From the combined static and dynamical scaling analyses, they obtain the critical exponents η = 1.04 ± 0.03, z = 1.50 ± 0.04, and β = 0.31 ± 0.01 <ref:2511.09097#pg9>. The value of the dynamical exponent is remarkably close to the one-dimensional KPZ value z = 3/2 <ref:2511.09097#pg9>. This is the first time z ≃ 3/2 was found in a quantum system at equilibrium <ref:2511.09097#pg9>.

Appendix A and B

At the Gaussian level, the continuum theory is described by the conjugate fields θ and ψ with Euclidean action S0 = Z dτ dx h iψ ∂τ θ + ρ 2 (∂xθ) 2 + κ 2 (∂xψ) 2 i <ref:2511.09097#pg2>. Integrating out the conjugate field ψ gives the effective quadratic action Seff[θ] = 1/2 Z ω,k ρk2 + ω squared κk2 θ(ω, k) 2 <ref:2511.09097#pg2>. Consequently, the equal-time phase fluctuation is infrared finite <ref:2511.09097#pg2>. This should be contrasted with an ordinary Luttinger liquid, where a local density term corresponding to a finite compressibility gives θ(k) 2 τ=0 ∼ 1/k, leading to logarithmically divergent phase fluctuations and therefore only algebraic quasi-long-range order <ref:2511.09097#pg2>.

Appendix D

To study the U(1) symmetry properties of the ground state at the critical point they use the squared of the transverse magnetization m2⊥ = 1/L X i ⟨S x i ⟩2 + ⟨S y i⟩2 <ref:2511.09097#pg9>. A non-zero value of this order parameter obtained with the iDMRG algorithm indicates spontaneous symmetry breaking m2⊥ ≠ 0 ⇒ U(1) SSB <ref:2511.09097#pg9>.

Appendix E and F

The numerical time evolution described in Sec. B allows them to obtain both the transverse response S⊥(k, ω) and the longitudinal response Szz(k, ω) <ref:2511.09097#pg9>. Panel b fits the zero-frequency momentum dependence of the longitudinal response to the scaling form Szz(k, ω = 0) ∝ k − (2−η), giving η = 1.01 ± 0.05 <ref:2511.09097#pg9>. The results are consistent with the one obtained from the static correlation function which gave ∆ψ together with the fitted value of z which is shown in panel c <ref:2511.09097#pg9>.

Appendix G and H

The final expression for the Lagrangian with S = 1 = a, dropping linear terms in the fields and everything is to order O(a2) L = iψ∂τ θ + J 2 h 1 1 − ψ2 (∂xψ)2 + 1 − ψ2 (∂xθ)2 i + g1 ψ 2 + g2 ψ 4 <ref:2511.09097#pg9>. The bare massless Euclidean action (in imaginary time τ) is given by S0[θ, ψ] = Z d x dτ h iψ ∂τ θ+ K 2 (∇θ) 2+ 1/2 (∇ψ) 2+ g 4! ψ 4 i <ref:2511.09097#pg9>. The effective action, therefore, reads SR[θR, ψR] = Z d x dτ h i ψR ∂τ θR + K 2 (∇θR) 2 + 1/2 Z (∇ψR) 2 + g 4! µ ε Zg g 4 R ψ 4 R i <ref:2511.09097#pg9>. The critical exponents are given by η = 4/243 ε squared, hence z = 2 − 2/243 ε squared <ref:2511.09097#pg9>.

Appendix K and L

The dimension of the renormalized field is given by [ψR]µ = ∆ψ = 1/2 (D−z + η) with D = d+z <ref:2511.09097#pg9>. The anomalous exponent is defined as the scaling of Z at the fixed point η ≡ η(˜g∗):= µ∂µ lnZ <ref:2511.09097#pg9>. To second order, we get β˜(˜g) = −εg˜ + 3/2 g˜ squared + γ1 g̃ cubed + O(g̃ 4) with γ1:= −25/27 − 3 ln 2 + 3/2 ln 3 <ref:2511.09097#pg9>. The positive, non-zero fixed point solution is given to second order in epsilon by g˜∗ = 2/3 ε - 8/27 γ1ε squared + O(ε 3) <ref:2511.09097#pg9>.

Improvements for AI systems

Confidence in Unconventional Criticality Determination: The system can now reliably distinguish between standard Ising universality class behavior and unconventional critical points by analyzing the resulting flow equations differ from those of the Ising model (Page 8), leading to a finite anomalous dimension η and a dynamical critical exponent z that differs from its bare value, in contrast to the one-loop result of Ref. [21] (Page 8).
Long-Range Order Characterization: The AI can perform rigorous verification of true long-range order by checking for a finite perpendicular magnetization m⊥ at the critical point, which is explicitly stated as a signature that implies U(1) SSB (Page 4).
KPZ Universality Prediction: The system can now predict and confirm the dynamical critical exponent using numerical evidence, finding that the value of z coincides with the one-dimensional Kardar–Parisi–Zhang (KPZ) exponent despite the system being an equilibrium quantum many-body system (Page 1).
RG Flow Analysis: The AI can derive and analyze the renormalization-group flow equations of the continuum theory to second order in the ε expansion, allowing for a quantitative comparison between perturbative predictions and numerical results (Page 1).
Continuum Limit Identification: The system can determine that the non-interacting continuum limit is equivalent to the recently introduced transverse quantum fluid, displaying their off-diagonal long-ranged order in one dimension (Page 1), enabling the use of TQF physics for understanding critical behavior.
Dynamical Exponent Extraction: The AI can extract robust critical exponents from complementary static and dynamical observables, specifically obtaining a dynamical exponent z = 1.50 ± 0.04 and an anomalous dimension η = 1.04 ± 0.03 (Page 1).
Field-Theoretic Mapping: The system can construct the effective Euclidean Lagrangian from the spin Hamiltonian, showing that the number of running couplings corresponds exactly to the textbook [43] massless ϕ4-theory (Page 13), allowing for systematic perturbation theory in critical phenomena.
Finite Entanglement Scaling: The AI can analyze finite-entanglement scaling using tensor network methods, which shows that the entropy follows a logarithmic dependence on bond dimension, consistent with a c = 1 Gaussian-boson description of the neighboring XY quasi-long-range-ordered phases (Page 11).
Nonconformal Behavior Detection: The AI can identify deviations from standard critical behavior by observing that the resulting critical exponents at the interacting fixed point also deviate from the Ising universality class at second order in ε, indicating an unconventional critical point outside the standard conformal paradigm (Page 8).

Abstract

In this work, we present numerical evidence for the spontaneous breaking of a continuous U(1) symmetry in a nearest-neighbor interacting spin-1 chain at a quantum critical point separating two XY quasi-long-range ordered phases distinguished by a spontaneously broken Z 2 symmetry. Remarkably, the continuous symmetry breaking emerges precisely at the critical point of the discrete order parameter, suggesting a novel mechanism beyond currently established scenarios. At criticality, the XY correlations develop true long-range order, accompanied by a finite perpendicular magnetization, a zero-frequency Bragg peak in the transverse dynamical structure factor, and sharp gapless collective excitations. From complementary static and dynamical observables, we quantitatively determine the critical exponents, obtaining a dynamical exponent z=1.50 plus or minus0.04 and an anomalous dimension η=1.04 plus or minus0.03. Remarkably, the value of z coincides with the one-dimensional Kardar--Parisi--Zhang (KPZ) exponent despite the system being an equilibrium quantum many-body system. We further show that the non-interacting continuum limit is equivalent to the recently introduced transverse quantum fluid, displaying their off-diagonal long-ranged order in one dimension. Complementing the numerical study, we derive the renormalization-group flow equations of the continuum theory to second order in the epsilon expansion. We identify an interacting fixed point whose critical behavior differs from the Ising universality class already at two-loop order, although the perturbative exponents remain far from the numerical values.