Interface phases and dynamics in two-dimensional quantum magnets: A "holographic" approach from universality to quantum simulation

Abhishodh Prakash; Alessio Lerose; Jaydev Singh Rao; Siddharth A. Parameswaran

REVIEW 2 major objections 4 minor 108 references

Interface phases and dynamics in two-dimensional quantum magnets: A "holographic" approach from universality to quantum simulation

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 2D magnetic interfaces can emulate the full 1D quantum phase diagram

desk verdict A genuinely useful holographic dictionary for 2D interfaces with one real gap: the dynamics claims lack a full 2D benchmark. read the letter →

arxiv 2608.12312 v1 pith:A5P26J5N submitted 2026-08-12 cond-mat.stat-mech cond-mat.quant-gascond-mat.str-elquant-ph

Abhishodh Prakash , Jaydev Singh Rao , Siddharth A. Parameswaran , Alessio Lerose This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.quant-gascond-mat.str-elquant-ph

keywords interfacephasesholographicmappingexactbosonizationSchrieffer-Wolfftransformationcurvature-drivendynamicsfullcountingstatisticsquantumsimulation2Dmagnets

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that an interface (domain wall) between ordered regions in a 2D quantum magnet maps exactly onto a 1D chain of fermions or spins: minimal-length domain walls become Fock states, local interface slope becomes spin or fermion occupation, and interface height becomes the 1D charge counting field. From this holographic dictionary it derives effective 1D Hamiltonians for a menu of bulk perturbations, showing their ground states reproduce essentially the entire known phase diagram of U(1)-symmetric 1D quantum systems, with stiff and rough interfaces mapping to gapped and gapless phases. Curvature-driven non-equilibrium interface dynamics maps to 1D transport, yielding distinct universal scaling classes: diffusive, super-diffusive, and ballistic. Because the height field is a local geometric coordinate in 2D but non-local in 1D, local projective measurements in 2D directly reveal the 1D charge full counting statistics, demonstrated numerically for a neutral-atom array. If correct, this turns programmable 2D quantum simulators into tunable simulators of 1D quantum matter with unique measurement capabilities.

What carries the argument

The load-bearing object is the holographic identification of a 2D interface with a 1D system: each minimal-length domain-wall path is a directed path whose slopes become the $\tau^z$ spins (or fermion occupancies) of a fixed-magnetization chain, and the height $\varphi_j$ is the accumulated magnetization, i.e., the counting field. The transformation that carries the argument is the Schrieffer-Wolff block diagonalization, $H_{SW}=U_{SW}^\dagger(H_{2D}^{(0)}+V_{2D})U_{SW}$, truncated at finite order and projected onto the minimal-length domain-wall subspace to give $H_{1D}$; the same rotation dresses the wave function with dilute bulk and interface defects. The paper proves that the symmetries protecting the phases, namely U(1), spin reflection, lattice parity, and one- and two-site translations, are inherited exactly from the parent 2D lattice at every perturbative order, so the phase structure persists at finite couplings, while the accidental SU(2) symmetry at $J'=\pm g/2$ is broken at second order.

What would settle it

Prepare a 2D Ising-type quantum magnet with transverse field and diagonal Ising coupling tuned to the predicted stiff side of the XXZ regime, then measure the interface height variance $\langle \varphi_{L/2}^2\rangle$ as a function of system length; if the variance grows with $L$ instead of saturating to a finite value, the predicted stiff phase and its phase boundary are wrong. Likewise, at the tuned isotropic point, measuring a curvature relaxation exponent clearly outside the predicted super-diffusive window near $z=3/2$ would falsify the dynamical scaling claim.

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Extended reading notes

Core claim

The paper's central claim is that a 2D quantum Ising-type magnet with pinned boundary conditions admits an exact lattice-level bosonization of its interface sector: minimal-length domain-wall configurations are in one-to-one correspondence with the Fock space of N fermions or a fixed-magnetization spin chain, with height $\varphi_j = 2\sum_{l\le j} n_l - j$ (equivalently $\sum_{l\le j} \tau^z_l$). A Schrieffer-Wolff expansion projects the dressed 2D dynamics onto a U(1)-symmetric 1D Hamiltonian $H_{1D}=P_{1D}H_{SW}P_{1D}$. From this, a dictionary (Table I) maps bulk perturbations to 1D interactions: transverse field to XX hopping, diagonal Ising couplings to ZZ coupling, checkerboard modulations to dimerization and staggered chemical potential, and longer-range couplings to Hubbard and ladder models. Bosonization and DMRG show the resulting interface ground states realize the full catalogue of 1D phases: gapless rough phases, ferromagnetic and antiferromagnetic stiff phases, trivial and symmetry-protected topological stiff phases, a valence-bond-solid disordered-flat phase, and symmetry-enriched critical phases, all distinguishable by the statistics of 2D snapshots. For dynamics, curvature $\partial_x^2\varphi$ maps to density gradients, giving spatiotemporal scaling $m(x,t)\sim F(x/t^{1/z})$ and hence interface curvature scaling $\sim t^{-1/z}F'$, with $z=2$ diffusive, $z=3/2$ super-diffusive at the isotropic SU(2)-symmetric point, and $z=1$ ballistic in the gapless regime. Finally, because the height field is non-compact and local in 2D, the 1D charge full counting statistics is directly measurable via local projective measurements; a numerical simulation of a $65\times 12$ neutral-atom Rydberg array reconstructs the full counting statistics of an $L=64$ XXZ chain with about one percent contrast after post-selecting minimal-length domain walls.

Load-bearing premise

The load-bearing premise is that a finite-order Schrieffer-Wolff effective Hamiltonian, truncated and projected onto the minimal-length domain-wall subspace, faithfully describes the 2D ground states and long-time interface dynamics at the couplings of interest; the paper deliberately sets convergence questions aside and relies on prethermalization only for quasi-conservation of domain-wall length.

Editorial extensions

If this is right

Bulk parameter variations in a 2D quantum magnet can realize the known phase diagram of U(1)-symmetric 1D systems, with each phase appearing as a distinct pattern of height fluctuations in 2D snapshots: stiff versus rough, pinned versus fluctuating.
Curvature-driven interface dynamics follows 1D transport universality: diffusive (z=2), super-diffusive (z=3/2) at the isotropic point, and ballistic (z=1) in the gapless regime, all accessible in Rydberg-atom arrays with tunable anisotropy.
Interfaces tilted at rational slopes can be stiffened by commensurate periodic modulations of bulk couplings, following $k_F=\pi\ell/R$, while incommensurate modulations can localize and pin interfaces at arbitrary orientations.
Local projective measurements of the 2D spin configuration directly yield the charge full counting statistics and symmetry-resolved properties of the encoded 1D chain, with errors localized rather than extensive.
All predictions apply equally to confining strings in dual lattice gauge theories, so the phase catalogue and dynamical scaling laws translate directly to string phases and string dynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension is to use the same encoding to measure full counting statistics at finite temperature or after quenches, since the height field remains a local 2D observable even when the 1D state is not a ground state.
If the dictionary is as complete as claimed, essentially any U(1)-symmetric 1D Hamiltonian reachable by these bulk perturbations could be simulated in 2D, including non-stoquastic cases with no classical statistical interpretation; this suggests searching for additional bulk terms that generate longer-range or multi-species 1D models.
The protocol implies that 2D simulators may avoid the extensive noise accumulation that makes full counting statistics hard in conventional 1D simulators, so comparing full counting statistics measured on the interface versus directly on a 1D chain would be a sharp test of the claimed advantage.
The mapping of curvature-driven relaxation to spin transport suggests that interface experiments could serve as a cleaner geometric probe of the debated super-diffusive regime, since the 2D signal is measured locally without requiring site-resolved spin readout of a 1D chain.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague, read this. The core mapping is real and the paper is better than the abstract suggests. The identification between a minimal-length domain wall on a tilted square lattice and a fixed-magnetization 1D spin chain is elementary, but the authors turn it into a systematic dictionary: bulk perturbations to 1D Hamiltonians, an all-order symmetry-inheritance proof, and a parameter-free DMRG comparison of full-counting-statistics distributions in Fig. 7. That last check is the most convincing part; no fitted parameters, and the agreement is around one percent. The neutral-atom proposal is also concrete, with actual parameter maps and a post-selection rate that stays roughly constant when the Rabi frequency is scaled as L^-1/2.

The soft spots are real but localized. The authors explicitly set aside Schrieffer-Wolff convergence in Sec. IIB footnote 1. For static phases, the all-order symmetry proof carries much of the weight, and that is a genuine result. For dynamics, it does not. The curvature-driven z = 1, 3/2, 2 predictions come from the effective 1D Hamiltonian and are checked only with 1D TEBD. The stress-test note is right: the dressed initial state deviates from the ideal 1D state at order (lambda/J)^2, and prethermalization conserves domain-wall number without guaranteeing that the projected dynamics inside the subspace equals the truncated 1D evolution. There is no full 2D time-evolution benchmark, unlike the static sector. That makes A2 a well-motivated prediction rather than a demonstrated result. The full-counting-statistics protocol also relies on post-selection whose cost grows exponentially with system size at fixed perturbation strength; the authors acknowledge this and partly mitigate it with a locality argument, so I would call that a cost to state clearly rather than a fatal flaw.

Minor point: calling the lattice mapping an "exact bosonization" is a bit strong. It is an exact lattice duality that connects to standard field-theoretic bosonization; the labeling slightly oversells the field-theoretic content. The citation pattern is appropriate: Kogut et al. and Balducci et al. are credited, and the genuinely new pieces are the systematic dictionary, the all-order symmetry proof, and the measurement protocol.

Who gets value: people working on quantum interfaces, confining strings, Rydberg simulators, and 1D transport. It deserves a serious referee, and I would send it out. My own verdict would be revise-and-resubmit: the static sector is strong, and the dynamics section needs either a 2D tensor-network time-evolution benchmark at moderate J/g or an explicit statement that A2 is a conjecture with an error estimate attached.

Referee Report

2 major / 4 minor

Summary. This paper develops a "holographic" mapping between interfaces in two-dimensional quantum Ising-type magnets and one-dimensional spin/fermion chains, and uses it to predict interface phases, curvature-driven dynamics, and a new quantum-simulation protocol. Starting from the classically degenerate manifold of minimal-length domain walls, a Schrieffer-Wolff expansion in bulk perturbations yields effective 1D Hamiltonians of XXZ, dimerized XXZ, Hubbard, and Su-Schrieffer-Heeger type. The height field is reinterpreted as an exact lattice bosonization, giving a dictionary between 2D bulk perturbations and 1D interactions. The authors classify interface phases (Luttinger liquid, ferromagnetic, Néel, trivial/SPT dimer, VBS, and symmetry-enriched critical phases) and argue that the protecting symmetries are exact to all orders. They further map curvature-driven interface relaxation onto 1D charge transport, predicting diffusive (z=2), superdiffusive (z=3/2), and ballistic (z=1) scaling, and they propose that local 2D measurements of the height field constitute direct projective measurements of 1D charge full counting statistics. The static predictions are tested parameter-free against 2D DMRG on a 65x12 Rydberg ribbon; the dynamics are tested only with 1D TEBD simulations of the effective Hamiltonian.

Significance. If the framework holds, it would make programmable 2D quantum simulators a flexible platform for exploring a large family of U(1)-symmetric 1D quantum systems, with non-local 1D observables accessible through local 2D measurements. The strongest positive evidence is the genuinely parameter-free static benchmark in Fig. 7: the empirical 2D snapshot statistics and the 1D full counting statistics agree to about 1% with no fitted parameters. The exact-symmetry argument in Sec. IIIF is also a substantive result: it rules out a large class of perturbative corrections to the emergent 1D symmetries and protects many phase labels at finite coupling. The paper is ambitious, clearly written, and connects to ongoing neutral-atom experiments, including a concrete interpretation of the coarsening dynamics observed by Manovitz et al. As discussed in the major comments, the dynamical predictions currently lack a full 2D benchmark, and the finite-coupling phase catalogue rests on an explicitly uncontrolled truncation; these are the main obstacles between the present manuscript and a fully convincing central claim.

major comments (2)

[Sec. IV, Eq. (40), Fig. 5(c)] The central dynamical claim A2—universal curvature-driven exponents z=1, 3/2, or 2—is validated only by 1D TEBD simulations of the truncated effective Hamiltonian H1D, not by any full 2D time evolution. The prethermalization argument in Sec. IV controls leakage out of the minimal-length domain-wall subspace via the quasi-conserved operator Ñ_dw, but it does not control the difference between the post-selected, Schrieffer-Wolff-dressed state and the ideal 1D-evolved state. As the paper itself states in footnote 10, it merely expects "no qualitative differences" for small |g/J|. Because the exponents are sharp quantitative predictions and the experimental protocol starts from a computational-basis product state whose dressed and post-selected projection differs from the ideal 1D initial state at order (λ/J)^2, the manuscript should provide a full 2D time-evolution benchmark (for example, DMRG/TEBD on a ribbon at the parameters of Fig. 5) or an analytic bound showing that the projected dynamics is close to H1D on the scaling timescale. Without such a check, A2 remains a prediction of the effective 1D theory rather than an established property of the 2D magnet.
[Sec. IIB, footnote 1, and Sec. IIIF] The finite-coupling phase catalogue (claim A1) is not established beyond leading order. The manuscript explicitly sets aside convergence in footnote 1 ("we will not be concerned with convergence issues"), and the all-orders argument in Sec. IIIF proves only that the symmetries in Table II are preserved by every order of the Schrieffer-Wolff expansion. Symmetry preservation does not by itself prove that the phases survive: for the gapless Luttinger-liquid phase, whose sufficient symmetry set is just U(1) according to Table III, higher-order terms O(λ^n/J^{n-1}) with range n (described in Sec. IIC) could in principle shift the Luttinger parameter K or generate a relevant Umklapp-type perturbation that opens a gap. The same caveat applies to the locations of the phase boundaries in Fig. 2(a) and Fig. 4(a). The paper should either compute or bound the leading higher-order corrections for each representative perturbation, or state explicitly that the quantitative phase diagram is a leading-order prediction whose all-orders robustness has been established only for the symmetry labels, not for the phases themselves.

minor comments (4)

[Abstract and Sec. I, answer A1] The phrase "essentially including the entire known phase diagram of U(1)-symmetric 1D quantum systems" overstates the demonstrated scope; the paper derives a dictionary for a specific but finite set of 1D Hamiltonians (XXZ, dimerized XXZ, Hubbard, SSH, ladder-type). Suggest rephrasing to "a large class of" or listing the covered families.
[Sec. VI, Outlook] The outlook mentions "many-body localized behaviors" as a holographic manifestation, but no MBL regime is identified or derived in Sec. IV; either add a brief argument or citation, or soften the statement to avoid an unsupported claim.
[Fig. 3, panel (a)] The asymptotics written in the main text, ⟨φ²_{L/2}⟩ = (2/π²)[log(L/2) + log 2 + γ_E + 1 + o(1)], differs from the formula shown in the figure, which reads (2/π²)[log L + γ_E + 1]; please make the displayed formula and the curve consistent.
[Sec. II.C, paragraph after Eq. (6)] The derivation of next-nearest-neighbor hopping H1D = -g' Σ (c†_j c_{j+2} + h.c.) after choosing g'_x = -g'_y is easy to misread because the Jordan-Wigner sign reversal is stated only in words; a short sign convention or a one-line derivation would improve clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the 2D-to-1D dictionary is derived by explicit Schrieffer-Wolff perturbation theory and independently benchmarked; self-citations are not load-bearing.

full rationale

The paper's central reduction—2D interface configurations to 1D Fock space—is constructed in Sec. II from elementary directed-path enumeration, not imported from the target results. The effective 1D Hamiltonians in Table I are obtained by explicit Schrieffer-Wolff block-diagonalization (Eq. 3 and surrounding text), rather than defined as the 1D models whose phases are later 'predicted.' The phase analysis then uses standard bosonization and Bethe-ansatz results (Eqs. 18-21) that are external to this paper; the 2D-to-1D comparison in Fig. 7 is explicitly parameter-free ('with no free parameters to fit') and provides independent DMRG support for the static sector. The dynamics claim A2 follows an exact height/curvature identity (Eq. 37: κ = ∂_x m/(1+m^2)^{3/2}) and imports the 1D transport exponents z=2,3/2,1 from prior literature; Fig. 5 simulates the derived effective Hamiltonian (Eq. 40) without fitting. The acknowledged approximations—footnote 1 setting aside Schrieffer-Wolff convergence, and footnote 10 stating 'we expect no qualitative differences' for dressed initial states—are validity/robustness caveats, not circular steps. Several building blocks come from the authors' earlier works (Refs. [49,50,64]), but these are published results with independent numerical support and the present derivations rederive or use them as ingredients; no load-bearing argument reduces to an unverified self-citation. Thus the appropriate finding is no significant circularity, with a minor allowance for self-citation content.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted: the effective 1D couplings, including g, J', dimerizations, and anisotropy, are computed from microscopic 2D parameters via Schrieffer-Wolff perturbation theory and are explicitly reported in Appendices A and B. The 'holographic' mapping is an exact identity between the interface height and the accumulated 1D charge, not a new physical entity. No new particles, forces, or dimensions are introduced.

assumptions (5)

domain assumption Schrieffer-Wolff perturbation series truncated at finite order accurately describes interface ground states and dynamics for small but finite lambda/J.
Stated in Sec. IIB and footnote 1: the authors explicitly decline to address convergence and treat truncations as accurate descriptions.
domain assumption The bulk gap of the 2D Ising magnet, 8J, is large compared with all perturbations, so bulk domains remain ordered and only interface fluctuations are dynamically active.
Sec. IIB: lambda << 8J; this is the regime in which the 'holographic' projection is valid.
standard math Standard field-theoretical bosonization results for 1D spin chains, including Luttinger parameter formulas and the OYA gapping criterion, apply to the effective 1D Hamiltonians.
Used throughout Sec. III for phase classification, e.g., Eqs. (18)-(22) and the OYA condition in Sec. IIIG.
domain assumption The neutral-atom Rydberg Hamiltonian can be truncated to nearest and next-nearest neighbor van der Waals interactions of strength V and V/8.
Sec. VA; a standard approximation due to the rapid decay of van der Waals tails, used to derive Eq. (42).
standard math If a 2D symmetry preserves the minimal-length domain-wall subspace, it is preserved at every order of the Schrieffer-Wolff transformation.
Proved by induction in Sec. IIIF; used to argue phase robustness to finite couplings.

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Cite this review

Pith. "Pith review of Interface phases and dynamics in two-dimensional quantum magnets: A "holographic" approach from universality to quantum simulation." pith.science (2026). https://pith.science/paper/A5P26J5N

@misc{pith2026260812312,
  author       = {Pith},
  title        = {Pith review of: Interface phases and dynamics in two-dimensional quantum magnets: A "holographic" approach from universality to quantum simulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A5P26J5N}},
  note         = {Machine review of arXiv:2608.12312}
}

read the original abstract

We introduce a framework to classify quantum phases, phase transitions, and non-equilibrium dynamics of interfaces separating ordered bulk domains in 2D quantum magnets - equivalently, confining strings in dual lattice gauge theories - based on effective 1D Hamiltonians governing geometric fluctuations. Building on a "holographic" approach from [Phys. Rev. Lett. 129, 120601 (2022)], here reinterpreted as an exact bosonization, we uncover a rich quantum phase structure, with a variety of stiff and rough interface phases described by gapped and gapless 1D ground states, respectively, all distinguishable through the statistics of 2D wave-function snapshots. Our framework allows us to predict distinct spatiotemporal scaling laws for non-equilibrium curvature-driven interface dynamics across parameter space, which can be readily probed in existing experiments. We finally show that our approach enables the unprecedented experimental opportunity of directly measuring charge full counting statistics and symmetry-resolved properties of an encoded 1D system, as we explicitly demonstrate by numerically simulating a neutral-atom array experiment.

Figures

Figures reproduced from arXiv: 2608.12312 by the authors.

**Figure 1.** “Holographic” description of interfaces in 2D quantum magnets. For illustration, we consider an Ising square lattice tilted by 45◦ . Spins on the vertical sides are frozen as shown, pinning the endpoints of a domain wall. Panel (a): Typical computational-basis snapshot of the ground-state wave function of a quantum Isingtype ferromagnet with the considered boundary conditions, with white/gray plaquettes depicting u… view at source ↗

**Figure 3.** Roughness scaling law (21). Panels (a,b): Mid-point variance of the interface height ⟨φ 2 L/2 ⟩ vs interface length, numerically computed from the effective 1D XXZ spin chain ground state, for (a) the free-fermion point J ′ = 0 and (b) general values of the effective anisotropy ∆ = 2J ′ /g. Panel (c): illustration of the scaling law in Eq. (21). For the free-fermion point, we overlay the numerically exact result of … view at source ↗

**Figure 4.** Interface quantum phase structure (II). Panel (a): Schematic phase diagram of the effective 1D Hamiltonian (25) for g ′ /g = α ∼ 2.5. Panels (b,c): Interface fluctuation statistics computed from the 1D ground-state wave function in the rough trival (XY2) (b) and non-trivial (XY∗ 2) (c) symmetry-enriched critical phases. XY∗ 2 has degenerate ground states with distinct edge modes that are shown. Here L = 128 and the … view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: Curvature-driven quantum dynamics. Panel (a): Illustration of curvature-driven evolution starting from a corner-like configuration at t = 0. Panel (b): Evolution of the interface height variation ∆φ(t) = φL/2(t) − φL/2(0) from initial slopes tan Θ = ±m = ±0.01, compute…

**Figure 6.** Figure 6: Neutral-atom interface engineering. Panel (a): Checkerboard (black-white) and anticheckerboard (gray-white) ordered regions, composed of interleaved ground (empty dots) and Rydberg (full dots) atoms, separated by a low-energy interface. Such interface configurations …

**Figure 7.** Figure 7: “Holographic” reconstruction of full counting statistics in an engineered neutral-atom array. Panel (a): Comparison between numerically computed charge full counting statistics in the ground state of a L = 64 XXZ spin chain with ∆ = 0.4 (left panel), and the correspond…

**Figure 8.** Figure 8: Iso-dimerization curves εXY (δ0, ζ±) = εXY (δ0, ζ±) for various fixed values of the detuning δ0 in the allowed range (top panel) and the values of the common dimerization parameter ε on these curves (bottom panel). The dashed diagonal line highlights the trivial iso-di…

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Reviewed August 16, 2026 · model on record in the stance chip above.

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Kinematically, each interface configura- tion is a path in two dimensions and is therefore spe- cifiedbyitstransversecoordinate, theinteger-valued heightφj

From the compact boson to the physical height There are two complementary descriptions of the interface. Kinematically, each interface configura- tion is a path in two dimensions and is therefore spe- cifiedbyitstransversecoordinate, theinteger-valued heightφj. Dynamically, theelementaryslopesofthis path are spin-1/2variables, φj−φ 0 = jX l=1 τz l, φ j−φj...

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disordered flat

In this regime the Luttinger liquid of the XXZ chain is unstable to this perturbation, and the RG flow leads to a gapped symmetry preserving phase. The phase diagram of Eq. (11) is shown in Fig. 2(a). Depend- ing on the sign ofε, the gapped phase is either trivial (ε <0) or a symmetry protected topological (SPT) phase (ε>0). For|ε|= 1, these ground states...

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holographic

(c) symmetry-enriched critical phases. XY∗ 2 has degenerate ground states with distinct edge modes that are shown. HereL= 128and the plots are made by overlapping 200 wave-function snapshots. tion of this phase in the interface variables is shown in Fig. 4(b). The underlying ferromagnetic cor- relations which occurs in the spin chain through alignment of ...

2020

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We relax the integer-valued height to a real variable φj∈Rand introduce its conjugate momentumΠ j

Softening the height constraint A complementary way to obtain the continuum dynamics is to work directly with the height field. We relax the integer-valued height to a real variable φj∈Rand introduce its conjugate momentumΠ j. The Gaussian height Hamiltonian is Hheight = X j " 2vK π Π2 j + πv 8K (φj+1−φj)2 # +..., (A7) where[φj,Πl] =iδ jl. Thefirsttermist...

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The slopes of this interface are the spins of a zero- magnetization spin-1/2chain, so the allowed locking operators inherit the Lieb–Schultz–Mattis (LSM) constraint

The built-in LSM constraint and ground-state counting The periodic locking term has a more specific physical meaning here than in a generic soft-height or Restricted-Solid-On-Solid (RSOS) model. The slopes of this interface are the spins of a zero- magnetization spin-1/2chain, so the allowed locking operators inherit the Lieb–Schultz–Mattis (LSM) constrai...

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