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

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.