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Gauge-Symmetry Violation Quantum Phase Transition in Lattice Gauge Theories

Maarten Van Damme, Jad C. Halimeh, Philipp Hauke

arXiv:2010.07338v1cond-mat.quant-gascond-mat.stat-mechcond-mat.str-elhep-latquant-ph

TL;DR

Approximate gauge symmetry in quantum simulators raises whether the resulting systems still reproduce gauge-theory physics. The paper combines thermodynamic-limit iMPS simulations with analytic gauge-protection and Schrieffer–Wolff constructions, finding an exponentially accurate renormalized symmetry and retention of long-distance behavior until a sharp quantum phase transition.

Abstract

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Gauge symmetry plays a key role in our description of subatomic matter. The vanishing photon mass, the long-ranged Coulomb law, and asymptotic freedom are all due to gauge invariance. Recent years have seen tantalizing progress in the microscopic reconstruction of gauge theories in engineered quantum simulators. Yet, many of these are plagued by a fundamental question: When gauge symmetry is only approximate in the quantum device, do we actually quantum-simulate a gauge theory? Here, we answer this question in the affirmative for a paradigm gauge theory akin to quantum electrodynamics. Analytically, we derive a renormalized gauge symmetry that is at least exponentially accurate. Further, numerically computing the phase diagram in the thermodynamic limit, we find that the long-distance behavior of the gauge theory is only compromised upon reaching a sharp quantum phase transition. This behavior is enabled by an energy penalty term, which lends a mass to the Higgs boson to which the coherent gauge breaking couples. Our results not only lend validity to ongoing gauge-theory quantum simulations, they also probe the fundamental question of how gauge symmetry could emerge in nature.

iMPS method

The study uses matrix product states to approximate ground states directly in the thermodynamic limit, with fidelity susceptibility locating quantum phase transitions.

iMPS method

  • Matrix product states enable efficient treatment of one-dimensional local Hamiltonians directly in the thermodynamic limit.Ground states are obtained variationally using VUMPS and Riemannian-manifold gradient descent.Source: iMPS method · p. 5
  • Fidelity susceptibility divergences identify the quantum phase transition along cuts through parameter space.The susceptibility is evaluated for both λ and µ; the derivative direction must not be parallel to the transition line.Source: iMPS method · p. 5, Figure 4 · p. 5

Renormalized gauge symmetry of physical subspace

The analysis combines gauge-protection results with a Schrieffer–Wolff construction to obtain a renormalized symmetry for the low-error phase.

Renormalized gauge symmetry of physical subspace

  • The renormalized symmetry analysis starts from an effective Hamiltonian that almost commutes with the gauge-protection term.It combines a gauge-protection theorem with a perturbative Schrieffer–Wolff transformation.Source: Renormalized gauge symmetry of… · pp. 5–6
  • Fidelity susceptibility in χµ detects most transition lines, while a transition nearly parallel to µ appears through a nonanalytic derivative instead.The complementary χλ susceptibility shows the strong divergence for that cut.Source: Figure 5 · p. 6
  • The microscopic Hamiltonian assumes a dominant protection scale V, with H0 gauge invariant and λH1 coupling different gauge sectors.The physical subspace is defined by Gj |ψ⟩phys = 0 and coincides with the zero-eigenvalue subspace of HG.Source: Renormalized gauge symmetry of… · p. 6

Gauge protection theorem

The gauge-protection theorem constructs a unitarily renormalized symmetry whose violation is exponentially small in the protection strength, although rigorous bounds may be quantitatively loose.

Gauge protection theorem

  • The theorem is formulated using local operator algebras, decompositions into finite connected supports, and weighted norms controlled by a decay rate κ.These definitions provide the locality framework for the protection bound.Source: Gauge protection theorem · p. 6
  • The theorem guarantees a renormalized physical sector containing the ground state while the Schrieffer–Wolff transformation converges.Residual terms commuting with or coupling different HG eigenspaces are separated by the unitary transformation.Source: Gauge protection theorem · p. 6
  • A unitary transformation produces a renormalized theory whose gauge-breaking contribution is exponentially small in V/V0.The theorem applies to a protection Hamiltonian with integer spectrum under stated locality and energy-scale assumptions.Source: Gauge protection theorem · p. 6
  • The transformed generator G′ approximately commutes with H with an exponentially small error independent of system size.This yields a renormalized gauge symmetry and physical subspace in the thermodynamic limit.Source: Gauge protection theorem · p. 6
  • The rigorous bounds are often not tight and may involve very large constants, despite establishing exponential accuracy in the thermodynamic limit.Numerics show that moderate protection strength can suffice and that exponentially small symmetry breaking does not immediately force a gauge-violating phase.Source: Gauge protection theorem · pp. 6–7

Schrieffer–Wolff transformation

A Schrieffer–Wolff transformation constructs a perturbatively renormalized gauge generator and physical subspace when gauge protection dominates gauge breaking. The resulting symmetry approximately commutes with the Hamiltonian and is strongly suppressed in the ground state.

Schrieffer–Wolff transformation

  • The transformation expands in powers of λ/V to cancel processes leaving the physical subspace order by order.Its generator is anti-Hermitian, S = Σ_k S_k, with S_k = O((λ/V)^k).Source: Schrieffer–Wolff transformation · p. 7
  • The effective Hamiltonian approximately commutes with the gauge generator within the physical subspace after truncating the transformation.The construction uses diagonal and off-diagonal perturbations relative to the protected subspace.Source: Schrieffer–Wolff transformation · p. 7
  • The renormalized generator defines a dressed physical subspace containing the ground state while the perturbative transformation converges.This gives ⟨ψ0|˜HG|ψ0⟩ = 0 in gauge-symmetry-retaining phases.Source: Schrieffer–Wolff transformation · p. 7
  • The dressed gauge violation is much smaller than the bare violation and is a local few-body observable accessible in experiments.The ground-state expectation of ˜HG is evaluated to the displayed perturbative order.Source: Schrieffer–Wolff transformation · p. 7
  • The renormalized generator yields dressed Gauss-law generators and can target other gauge sectors by shifting the protection operator.Within the dressed physical subspace, ˜HG is equivalent to renormalized local Gauss-law constraints.Source: Schrieffer–Wolff transformation · p. 7

Gutzwiller-type mean-field treatment

A Gutzwiller decomposition separates matter and gauge fields to provide qualitative insight into the phase transitions at large gauge-breaking strength. The resulting sectors map to coupled Ising and XY spin models with distinct transition lines and a possible Peierls instability.

Gauge fields: Ising model in transverse and longitudinal field

  • The gauge-field sector maps to an antiferromagnetic Ising model in transverse and longitudinal fields.In the µ → ∞ limit, the longitudinal field vanishes and the transition occurs at λ = V/2.Source: Gutzwiller-type mean-field treatment · p. 8
  • For finite µ, the gauge-field transition is expected at λ ≤ V/2 because a sufficiently strong longitudinal field suppresses symmetry breaking.This produces the nearly horizontal transition line in the phase diagram.Source: Gutzwiller-type mean-field treatment · p. 8
  • The gauge-field mean-field approximation cannot readily explain the transition at small µ because its longitudinal field becomes large.The transition’s origin is instead clarified through the matter fields.Source: Gutzwiller-type mean-field treatment · p. 8

Matter fields: XY spin chain in transverse field

  • The matter-field sector maps, after a staggered rotation, to an XY spin model in a staggered transverse field.In the λ → ∞ limit, the mean-field transition is expected and observed at µ = −V.Source: Gutzwiller-type mean-field treatment · p. 9

Gutzwiller-type mean-field treatment

  • The coupled mean-field transitions form two lines connecting the ideal Coleman transition, the gauge-field point, and the matter-field point.The transition lines become strongly correlated and inflected rather than crossing in the intermediate region.Source: Gutzwiller-type mean-field treatment · p. 9

Matter fields: XY spin chain in transverse field

  • A nonzero staggered gauge-field magnetization can dimerize the effective matter coupling, producing Peierls-type physics enabled by loss of the bare gauge symmetry.This behavior appears in cuts at fixed µ = 1.2J and V = 2J.Source: Gutzwiller-type mean-field treatment · p. 9

Coupled self consistent mean field results

Self-consistent mean-field iterations determine transition lines across the phase diagram and reproduce the qualitative structure seen in the main results. At fixed µ = 1.2J and V = 2J, local observables become nonanalytic near the quantum phase transition.

Coupled self consistent mean field results

  • Self-consistent coupled mean-field equations yield a transition line smoothly connected to µ = ∞, λ = V/2.The procedure alternates between gauge- and matter-field ground states until convergence.Source: Coupled self consistent mean field… · pp. 9–10
  • The mean-field transition driven by matter fields connects smoothly to the λ = ∞, µ = −V point.The critical condition is obtained by solving the matter-sector equations together with δz = 0.Source: Coupled self consistent mean field… · p. 9
  • Figure 6 tracks staggered and homogeneous magnetizations along λ at fixed µ = 1.2J and V = 2J.The observables use two-site translational invariance and distinguish staggered from homogeneous components.Source: Figure 6 · p. 9, Coupled self consistent mean field… · p. 9
  • At λ/J ≈ 0.9, the Coleman order parameter drops through a quantum phase transition while all displayed observables show nonanalyticities.The staggered magnetization Mz remains nonzero over a broad parameter window before the transition.Source: Coupled self consistent mean field… · p. 9
  • At large µ, the mean-field gauge sector is an antiferromagnetic Ising model whose numerically determined critical point is reported in the analysis.The supplied passage introduces the critical point but does not include its numerical value.Source: Coupled self consistent mean field… · p. 9
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