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Gauss Law, Minimal Coupling and Fermionic PEPS for Lattice Gauge Theories

Patrick Emonts, Erez Zohar

arXiv:1807.01294v4quant-phcond-mat.str-elhep-lathep-th

TL;DR

These notes address how the Gauss law’s different forms for gauge fields and matter constrain unitary gauging and matter elimination in Hamiltonian lattice gauge theories. They review these approaches alongside fermionic PEPS, concluding that general local unitary gauging fails, while separately gauging decoupled pieces can work and solving for matter is algebraically simple.

  • Problem

    In general, solving the Gauss law for the gauge field does not uniquely determine its configuration from matter, limiting whether minimal coupling can be implemented by a local unitary transformation.

    Source: §2.3 · p. 8

  • Method

    The lecture compares the Gauss law as a differential equation for gauge fields and an explicit algebraic equation for matter, connecting both perspectives to unitary gauging, fermion elimination, and PEPS.

    Source: §1 · pp. 2–3, §4.1 · p. 22, §4.3 · p. 26

  • Results

    General local-unitary gauging does not work, but it can be done for separate decoupled local parts that are uniquely gaugeable; solving the law for matter gives an explicit, unique charge from electric-field divergence.

    Source: §2.3 · p. 8, §4.1 · p. 22

  • Takeaways & Limitations

    The matter-side algebraic form supports eliminating fermionic matter in eligible gauge groups, with an auxiliary Z2 gauge field available for other cases described in the notes.

    Source: §4.1 · p. 22, §4.3 · p. 26

    Local unitary gauging is limited to Hamiltonians or states built from separate decoupled parts that can each be gauged uniquely; the described PEPS method truncates physical electric fields to 0, ±1 unless its bond dimension is increased.

    Source: §2.3 · p. 8, §3.4 · p. 18

Abstract

from arXiv · show

In these lecture notes, we review some recent works on Hamiltonian lattice gauge theories, that involve, in particular, tensor network methods. The results reviewed here are tailored together in a slightly different way from the one used in the contexts where they were first introduced, by looking at the Gauss law from two different points of view: for the gauge field it is a differential equation, while from the matter point of view, on the other hand, it is a simple, explicit algebraic equation. We will review and discuss what these two points of view allow and do not allow us to do, in terms of unitarily gauging a pure-matter theory and eliminating the matter from a gauge theory, and relate that to the construction of PEPS (Projected Entangled Pair States) for lattice gauge theories.

1 Introduction

The introduction motivates Hamiltonian lattice gauge theory and tensor-network methods as approaches to non-perturbative questions that Euclidean Monte Carlo generally cannot address in real time or in the presence of the fermionic sign problem. It frames the Gauss law as a constraint linking matter and gauge fields, and previews the lecture’s two perspectives on solving it and constructing gauge-invariant states.

1 Introduction

  • Euclidean Monte Carlo does not generally address real-time evolution and encounters the fermionic sign problem in scenarios relevant to exotic QCD phases.The notes present lattice gauge theories as a successful approach to non-perturbative physics, including calculations of the hadronic spectrum.Source: §1 · p. 2
  • The lecture reviews quantum simulation and tensor-network methods in the Hamiltonian formulation to address lattice gauge-theory problems.It focuses on enforcing gauge invariance in quantum states through the Gauss law and tensor-network construction.Source: §1 · p. 2
  • The Gauss law relates gauge and matter fields through local constraints, with matter acting as the source of electric fields.Its form can be an eigenvalue equation after quantization or a difference equation on a lattice.Source: §1 · p. 2
  • As a static constraint commuting with the Hamiltonian, the Gauss law can be solved and substituted into other equations to reduce degrees of freedom.It expresses constants of motion rather than an equation of motion.Source: §1 · p. 2
  • The lecture contrasts solving the Gauss law for gauge fields with solving it for matter, then connects these perspectives to unitary gauging, matter elimination, and PEPS.It previews Trotter-Suzuki decomposition and PEPS construction, as well as replacing fermionic matter by spins in some cases.Source: §1 · pp. 2–3

2 Gauging Hamiltonians and Quantum States

This section examines how minimal coupling and the Gauss law shape the possibility of gauging locally and unitarily, including how Trotterized dynamics can gauge local contributions before they are combined. It also covers exact gauge invariance in Trotterized gauge dynamics and the construction of locally invariant PEPS with dynamical gauge fields by gauging local ingredients before contraction.

2.1 Minimal Coupling and its Compact Lattice Formulation

  • Minimal coupling replaces the matter derivative with a covariant derivative and introduces a gauge connection; on the lattice, gauge fields are added to the links to couple matter hopping to them.The initial matter kinetic term is not invariant under local transformations, and the gauge field can later acquire dynamics through additional Hamiltonian or Lagrangian terms.Source: §2.1 · pp. 3–4

2.2 Gauge Symmetry and the Physical Hilbert Space

  • The physical Hilbert space is an embedded, gauge-invariant sector of the gauge-field and matter product space, and its states are eigenstates of every Gauss-law operator with static-charge eigenvalues.The static charges label superselection sectors that are not mixed by the dynamics.Source: §2.2 · p. 5

2.3 Can Minimal Coupling be Unitary and Local?

  • A single link admits unitary gauging, and a one-dimensional system admits unitary gauging only nonlocally; in more dimensions, unitary gauging is not possible in general.In one dimension, the inverse transformation can eliminate gauge-field degrees of freedom, but introduces nonlocal interactions from the electric Hamiltonian.Source: §2.3 · pp. 7–8
  • A Gauss-law solution for the electric field is generally nonunique, so gauging is generally not unitary; local unitary gauging works when separate, decoupled blocks can each be gauged uniquely before being connected.The Gauss law relates matter to the electric field through one differential or lattice difference equation for a vector field, allowing multiple gauge configurations for fixed matter charges.Source: §2.3 · p. 8

2.4 The Case of Hamiltonians: Trotterized Gauge Invariant Dynamics

  • Trotter-Suzuki steps gauge separate link contributions using simple one-dimensional Gauss laws, which become symmetries of the complete time evolution when the steps are sequenced.Finite-step evolution is approximate, while the decomposition is exact for infinitesimally short steps.Source: §2.4 · pp. 8–9
  • Trotterized evolution can be approximate because intersecting-link Hamiltonian terms do not commute, while every step remains exactly gauge invariant; this was used for digital simulations.Source: §2.4 · p. 9

2.5 The Case of Quantum States: Gauging PEPS

  • Gauged PEPS are built by gauging local product-state ingredients before contracting virtual bonds into an interacting, locally gauge-invariant state.The construction introduces dynamical gauge fields and removes the virtual degrees of freedom through projection onto maximally entangled pairs.Source: §2.5 · pp. 9, 11
  • Spatial symmetries of the original globally invariant PEPS can survive gauging with modified gauge-field transformation laws, while its global symmetry becomes local.Source: §2.5 · p. 11

2.5.1 Example

  • In the simple U(1) example, Kronecker-delta tensor elements enforce Gauss law, and finite virtual-field cutoff j also truncates physical fields to |E| ≤ j.Source: §2.5.1 · p. 13

3 Construction of Gauge Invariant Fermionic PEPS (fPEPS)

The construction starts from globally invariant Gaussian fermionic PEPS and then gauges them. The notes focus on U(1) in 2+1 dimensions, chosen to capture qualitative features without complications from more general groups or higher dimensions.

3 Construction of Gauge Invariant Fermionic PEPS (fPEPS)

  • Gaussian fPEPS are the starting states to be gauged, and are ground states of quadratic, free, non-interacting Hamiltonians.They are described by second moments, with other correlation functions obtainable through Wick's theorem.Source: §3 · p. 13
  • The derivation focuses on U(1) in 2+1 dimensions, while SU(2) and general-gauge-group extensions are cited elsewhere.The stated motivation is that this setting captures qualitative features without the mathematical complications of more complicated groups or higher dimensions.Source: §3 · p. 13
  • The construction proceeds by first building globally invariant fPEPS and then gauging them.Source: §3 · p. 13

3.1 The Physical Setting

A particle-hole transformation on the odd sublattice changes how fermion presence represents antiparticles and permits a more translationally invariant state construction. The section also imposes global U(1) invariance and notes that translation and lattice-rotation symmetries may be required.

3.1 The Physical Setting

  • A particle-hole transformation on the odd sublattice makes fermion presence represent an antiparticle, enabling a more translationally invariant state construction.Previously, absence of a fermion on an odd vertex represented an antiparticle.Source: §3.1 · p. 13
  • Under the particle-hole transformation, the Hamiltonian Hf takes superconducting form up to a constant.Source: §3.1 · p. 14
  • The target state is required to be invariant under global U(1) transformations generated by the total fermionic number.Source: §3.1 · p. 14
  • Translation and lattice-rotation invariance are additional symmetries that may also be demanded.Source: §3.1 · p. 14

3.2 The Local Ingredients

The local construction uses operators involving physical and virtual fermions, rather than a tensor product of fermionic states. Even local parity makes their product well-defined, while virtual modes implement the local Gauss-law constraint.

3.2 The Local Ingredients

  • Each vertex uses an operator A(x) involving physical and virtual degrees of freedom because fermionic Fock space lacks a well-defined tensor-product factorization.Source: §3.2 · p. 14
  • Requiring each A(x) to have even fermionic parity makes operators on distinct vertices commute and avoids specifying a lattice ordering.The virtual degrees of freedom must also be fermionic; otherwise, with one physical mode per vertex, even local parity would prevent physical fermion excitations.Source: §3.2 · p. 14
  • The example introduces eight virtual fermionic modes at each vertex and defines a virtual Gauss-law operator from their electric fields and the physical fermion number.The stated operator is G0 = E0(r) + E0(u) − E0(l) + E0(d) − ψ†ψ.Source: §3.2 · p. 14
  • A Gaussian state is built from creation operators, and the virtual Gauss-law symmetry condition restricts its allowed form.The passages describe sorting modes according to the sign of their phase under the transformation and state that the symmetry condition holds if and only if the constructed state has the specified form.Source: §3.2 · pp. 14–15
  • The local coupling matrix Tij has a first row coupling virtual and physical modes and a square virtual block τ that must be antisymmetric for translation by one staggered-fermion site.A minimal choice is Tij(x) = Tij; the one-site translation acts as charge conjugation on the virtual modes.Source: §3.2 · p. 15
  • The resulting construction satisfies the PEPS condition for global symmetry.Source: §3.2 · p. 15

3.3 Contracting the Globally Invariant fPEPS

The globally invariant fPEPS is contracted using Gaussian projectors on horizontal and vertical links. The staggering is accounted for so the resulting state is globally invariant.

3.3 Contracting the Globally Invariant fPEPS

  • Gaussian projectors contract the virtual modes on horizontal and vertical links, with Ωℓ projecting the link-edge modes to vacuum.Source: §3.3 · p. 15
  • The horizontal-link conditions are recognized as PEPS symmetry projectors, modified from previous forms because of staggering.Source: §3.3 · pp. 15–16
  • The contraction yields an fPEPS, and the staggering is already incorporated so the state is invariant under the global transformation.Source: §3.3 · p. 16

3.4 Gauging the PEPS

The gauging construction adds link gauge-field spaces and adapts the gauged PEPS to fermions. It yields a gauge-invariant state, with the virtual fermion construction restricting the physical electric-field values.

3.4 Gauging the PEPS

  • The gauging transformation entangles a link’s initial virtual fermion with its physical gauge field.The local tensor is formed by applying the link gauging transformations to the tensor and a gauge-field vacuum.Source: §3.4 · p. 16
  • The constructed state |ψ⟩ is gauge invariant under transformations at arbitrary vertices.The symmetry check distinguishes even and odd vertices.Source: §3.4 · pp. 17–18
  • The physical electric field is restricted to 0, ±1 because the virtual electric fields are differences of two fermion-number operators.Adding virtual fermions on each leg can increase the truncation and the number of PEPS parameters.Source: §3.4 · p. 18

3.5 Other Symmetries

The gauged state inherits symmetry conditions from its fermionic construction. Staggered translation acts as charge conjugation on the gauge field, while translation by two sites is an invariance.

3.5 Other Symmetries

  • For staggered fermions, translation by one site combines with gauge-field sign inversion, giving charge conjugation; translation by two sites is a simple invariance.The one-site translation does not represent simple translational invariance because the staggered unit cell contains multiple vertices.Source: §3.5 · p. 19
  • A π/2 rotation of the fermions requires a sign on the physical fermion and exchanges virtual modes while preserving positive and negative modes.The virtual-mode transformations use phases and rotation matrices with one nonzero element per row and column.Source: §3.5 · p. 19
  • In the local case, setting t = 0 gives pure gauge states.The passage contrasts this with the global case, where t = 0 is meaningless.Source: §3.5 · p. 19

3.6 Generalizations

The gauging prescription extends to other spatial dimensions, geometries, and gauge groups. Larger bond dimensions can expand the virtual and physical electric-field truncations, subject to symmetry constraints.

3.6 Generalizations

  • In higher spatial dimensions, the construction adds similar virtual legs and gauges only outgoing directions; analogous procedures apply to other geometries.Source: §3.6 · p. 19
  • The prescription also applies to other gauge groups, though non-Abelian groups require more complex symmetry conditions involving left and right transformations.The electric field on a non-Abelian link has left and right fields, whose difference reflects that the gauge field carries charge.Source: §3.6 · p. 19
  • Adding virtual fermions enlarges the bond dimension and parameter space, but symmetry requires adding complete irreducible multiplets or their multiple copies.This also increases the virtual-space truncation and the physical electric-field truncation on links.Source: §3.6 · p. 20

3.7 Analytical (Toy Model) Results

Transfer-matrix spectra and virtual symmetries provide ways to analyze phases of PEPS states. The reviewed studies report confining and non-confining gapped phases, and virtual-symmetry fixed lines that agree with transfer-matrix results.

3.7 Analytical (Toy Model) Results

  • On a cylinder, a gapped PEPS transfer-matrix spectrum corresponds to exponentially decaying correlation functions.The transfer matrix propagates information between layers along the cylinder’s open direction.Source: §3.7 · p. 20
  • Transfer-matrix eigenvalues across PEPS parameter choices can map gapped regions and suspected phase boundaries, where the spectrum becomes gapless.The resulting state phase diagram concerns parent Hamiltonians for which the PEPS are ground states.Source: §3.7 · p. 20
  • U(1) PEPS studies find confining and non-confining gapped phases for static charges, while SU(2) studies find a gapped deconfining phase and a gapless phase with perimeter-law decay.The SU(2) gapless phase is described as a hint of a possible Higgs phase.Source: §3.7 · p. 20
  • In SU(2), virtual particle-hole symmetry connects parameter choices for physically identical states, and its fixed lines agree with transfer-matrix phase-boundary results.Source: §3.7 · p. 20

3.8 Contracting Gauged Gaussian Fermionic PEPS: Combining Monte-Carlo with Tensor Network Methods

Gauged Gaussian fermionic PEPS provide a route to higher-dimensional tensor-network calculations that avoid the sign problem. Their observables reduce to Monte Carlo integrals over gauge-field configurations, with fermionic calculations handled efficiently by Gaussian methods.

3.8 Contracting Gauged Gaussian Fermionic PEPS: Combining Monte-Carlo with Tensor Network Methods

  • Gauged Gaussian fermionic PEPS support dimension-independent observable calculations without the sign problem, addressing two limitations of higher-dimensional tensor networks and Monte Carlo methods.Calculations use Hamiltonian Hilbert-space expectation values rather than a Wick-rotated path integral.Source: §3.8 · p. 20
  • The fermionic states used in the construction are Gaussian states, for which physical-operator expectation values have a simple analytical treatment.The described state is coupled to an external static U(1) field configuration.Source: §3.8 · p. 21
  • The state norm becomes an integral over positive squares of Gaussian-state norms, defining a probability density over gauge-field configurations.The associated partition function is Z.Source: §3.8 · p. 21
  • Wilson-loop expectation values are computed by integrating over gauge configurations, with link orientation determining whether to use U or U †.The resulting integral can be evaluated using Monte Carlo.Source: §3.8 · p. 21
  • Mesonic-string expectation values reduce to Gaussian fermion calculations for an open path, followed by Monte Carlo integration over gauge configurations.The fermionic part involves a quadratic operator and can be calculated using covariance-matrix elements.Source: §3.8 · pp. 21–22
  • For these states, gauge-field configurations serve as integration variables, while Gaussian fermion methods make the configuration-dependent calculations efficient.The resulting integrals can be evaluated with Monte Carlo and can support higher-bond-dimension variational guesses for lattice gauge theory ground states.Source: §3.8 · p. 22

4 The Opposite Process: Eliminating the Fermions

The Gauss law is difficult to solve for gauge fields but can explicitly determine matter charge, enabling conditional matter decoupling. For fermions, auxiliary modes and parity-preserving transformations replace fermions with hard-core bosons, with further matter elimination depending on the gauge group.

4.1 Solving Gauss Law for the Matter Field

  • The Gauss law does not uniquely determine gauge fields in dimensions above one, but explicitly fixes matter charge from electric-field divergence.In one dimension, the gauge-field solution is unique but nonlocal.Source: §4.1 · p. 22
  • When each local charge eigenstate uniquely identifies a matter configuration, a controlled unitary can decouple matter while preserving physical information and removing Gauss-law constraints.The operation is controlled by the electric fields on links around each vertex.Source: §4.1 · p. 22

4.2 The Unitary Gauge of Higgs Theories

  • In the Higgs unitary gauge, a vertex-dependent transformation controlled by surrounding electric fields drives matter charge to zero, leaving a trivial matter state and gauge-field state with the same physical information but no symmetry.For Z2 gauge theory with spin-half gauge and matter fields, the transformation is a CNOT.Source: §4.2 · p. 23

4.3 Eliminating Fermionic Matter

  • Direct local transformations that rotate links by fermionic phases violate local fermionic parity superselection; their globally parity-preserving product requires an ordering that yields nonlocal results.The construction therefore introduces auxiliary Majorana modes at link ends, absent from the original Hamiltonian, whose initial state can be fixed.Source: §4.3 · p. 25
  • Even-parity local transformations using auxiliary link modes preserve local fermionic superselection and map fermionic matter to hard-core bosons, which can be represented by spins.The ξ_i phases maintain the required statistics and commutation relations but break lattice rotations.Source: §4.3 · pp. 25–26
  • The fermion-to-boson transformation applies when the gauge group has a Z2 normal subgroup; otherwise, an auxiliary Z2 gauge field can be added without adding its dynamics.For multiple spinor components per site, representing hard-core bosons by spins requires local Jordan-Wigner transformations.Source: §4.3 · p. 26
  • With a U(1) normal subgroup, matter can also be completely eliminated, while the inverse unitary procedure couples matter to gauge-like fields without gauge symmetry.For bounded matter, this adds local projectors and constraints; the notes contrast this with unitarily gauging pure matter, which cannot be done in the general setting.Source: §4.3 · p. 26

A Tensor Network Notation

The notes introduce tensor-network notation by expressing a spin state with an MPS ansatz, then generalizing the construction to PEPS for higher-dimensional systems. Physical indices label state coefficients, while virtual indices enable tensor contractions.

A Tensor Network Notation

  • The MPS ansatz rewrites state coefficients as products of matrices, whose tensors form the construction’s elementary building blocks.The coefficients initially depend on the configuration of all spins.Source: A Tensor Network Notation · p. 27
  • Each tensor carries physical indices corresponding to state coefficients and virtual indices added to define contractions.Connected virtual legs indicate that their indices are summed over.Source: A Tensor Network Notation · p. 27
  • Adding virtual indices extends the MPS description to higher-dimensional systems using PEPS.Figure 4 depicts virtual connections in the x- and y-directions for a three-dimensional system.Source: A Tensor Network Notation · p. 27, Figure 4 · p. 28
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