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Eliminating fermionic matter fields in lattice gauge theories
Erez Zohar, J. Ignacio Cirac
TL;DR
Lattice gauge theories are difficult to simulate when fermionic matter must be represented by nonlocal mappings or incompatible hardware. The paper introduces a local unitary transformation using auxiliary fermions and gauge fields, yielding hard-core bosons with the same spectrum and gauge symmetry; the construction applies directly to U(N) and SU(2N), while SU(2N+1) requires an extra Z2 gauge field.
Problem
Fermionic degrees of freedom create nonlocality in standard lattice-gauge-theory computations and impose overhead or hardware restrictions in quantum simulation.
Source: §I · p. 1Method
A local unitary transformation combines the original fermions with auxiliary fermions and gauge fields, replacing matter operators by hard-core bosonic ones.
Source: §I · p. 1Results
The transformed theory is local, gauge invariant with the same group, and spectrally identical, applying directly to U(N) and SU(2N); SU(2N+1) requires an extra Z2 gauge field.
Source: §I · p. 1, D. Transformed Hamiltonian · p. 6, A. Introducing Auxiliary Z2 Gauge Fields · p. 8Takeaways & Limitations
The construction provides a local bosonic formulation of dynamical-fermion lattice gauge theories for quantum-simulation and computational settings.
Source: §V · p. 9
Abstract
from arXiv · showhide
We devise a unitary transformation that replaces the fermionic degrees of freedom of lattice gauge theories by (hard-core) bosonic ones. The resulting theory is local and gauge invariant, with the same symmetry group. The method works in any spatial dimensions and can be directly applied, among others, to the gauge groups $G=U(N)$ and $SU(2N)$, where $N\in\mathbb{N}$. For $SU(2N+1)$ one can also carry out the transformation after introducing an extra idle $\mathbb{Z}_2$ gauge field, so that the resulting symmetry group trivially contains $\mathbb{Z}_2$ as a normal subgroup. Those results have implications in the field of quantum simulations of high-energy physics models.
I. INTRODUCTION
The paper motivates replacing fermions in lattice gauge theories because standard computational mappings introduce nonlocality or hardware overhead. It formulates the 2+1-dimensional U(N) and SU(N) Hamiltonians on a square lattice as the setting for this transformation.
I. INTRODUCTION
- Fermionic fields are essential in lattice gauge theories but complicate computation because path-integral elimination is temporally nonlocal and Jordan-Wigner mappings are spatially nonlocal beyond 1+1 dimensions.Quantum and analog simulators also face qubit overhead or hardware restrictions when representing fermionic degrees of freedom.Source: §I · p. 1
- Auxiliary fermions can cancel Jordan-Wigner strings while local constraints preserve the Hamiltonian spectrum, motivating a local fermion-to-boson construction for gauge theories.The paper adopts a different unitary-transformation approach using gauge fields explicitly.Source: §I · p. 1
II. LATTICE GAUGE THEORY MODELS
- The framework concentrates on 2+1-dimensional lattice gauge theories with gauge groups U(N) or SU(N), using N fermionic modes per vertex in the fundamental representation.The gauge fields are represented by link operators L, R, and U, while matter hopping is separated into horizontal and vertical interactions.Source: §II · p. 2, A. Hamiltonians · pp. 2–3
A. Hamiltonians
- The models considered use fermionic modes on lattice vertices and gauge fields on links, with the physical Hilbert space selected by the Gauss law.The Hamiltonian includes matter, electric, magnetic, and gauge-field–matter interactions on a square lattice.Source: A. Hamiltonians · pp. 2–3, Figure 1 · p. 2
- The Hamiltonian’s gauge-field sector comprises electric and magnetic terms, while matter couples to gauge links through horizontal and vertical hopping interactions.The magnetic term is associated with plaquettes, and the interaction conventions are illustrated in Fig. 1.Source: A. Hamiltonians · pp. 2–3, Figure 1 · p. 2
B. Gauge invariance
Gauge invariance is imposed through local transformations and Gauss-law constraints defining the physical space. The U(1) and SU(2) examples identify the link operator structure needed for the later fermion-elimination method.
B. Gauge invariance
- The Hamiltonians are invariant under local unitary transformations, and states satisfying the corresponding Gauss law span the physical Hilbert space.The local generators act on matter and the surrounding links at each vertex.Source: B. Gauge invariance · p. 3
C. Examples: U(1) and SU(2) lattice gauge theories
- For U(1), the link electric fields satisfy L=R=E, the local generator is the electric-field divergence, and one fermionic mode occupies each lattice site.The link operator U obeys [E,U]=U.Source: C. Examples: U(1) and SU(2) lattice… · p. 3
- For SU(2), three generators act on two fermionic modes per site, with 2Tα equal to the Pauli matrices in the fundamental representation.The non-Abelian structure is encoded by the Pauli-matrix generators and their commutation relations.Source: C. Examples: U(1) and SU(2) lattice… · p. 3
- An operator P satisfying the required parity relations can be constructed for SU(2) using a half-integer representation, providing the basis for the transformation.The construction extends to U(N) and SU(2N) in the fundamental representation.Source: C. Examples: U(1) and SU(2) lattice… · p. 3
III. ELIMINATION OF THE FERMIONIC DEGREES OF FREEDOM: U(N) AND SU(2N)
The method introduces auxiliary fermions and a local unitary transformation that converts matter operators into hard-core bosonic operators while preserving locality and spectral equivalence. Gauge-field-dependent signs retain the required fermionic information in the transformed Hamiltonian.
A. Method
- Type I and type II auxiliary fermions are introduced locally; type II modes remain in a factorized vacuum sector after transformation, leaving only type I modes in the effective Hamiltonian.The original model is extended by one-dimensional auxiliary vacuum spaces before applying the unitary.Source: A. Method · pp. 3–4
- The transformation replaces each fermionic operator with ηx,m=cxψx,m, whose operators commute across sites and therefore represent hard-core bosonic modes.The local Hilbert space is generated by the η† operators acting on the vacuum.Source: A. Method · p. 4
- The transformed link operators acquire local gauge-field-dependent sign factors, which preserve the fermionic signs but break 90-degree rotational invariance of the Hamiltonian.These modifications affect horizontal and vertical hopping as well as the magnetic term.Source: A. Method · p. 4
- The transformed Hamiltonian has the same spectrum as the original, while gauge-invariant observables can be evaluated through locally transformed operators.This avoids reconstructing the original wavefunction with nontrivial type II phases.Source: A. Method · p. 4
B. Auxiliary fermions and the transformation
- The construction assumes a link operator E satisfying the required parity relations, which holds for U(N) and SU(2N) in the considered representations.The unitary is assembled from local fermionic operators and E operators on links adjacent to each vertex.Source: B. Auxiliary fermions and the… · pp. 4–5
- The local unitary is well-defined despite noncommuting vertex factors because the complete vertex operators commute across vertices, although the chosen ordering of local factors affects the transformed Hamiltonian.The auxiliary Majorana modes are assigned to links around each vertex and combine into link fermionic modes.Source: B. Auxiliary fermions and the… · p. 5
C. The Transformation
The transformation uses link-dependent phase factors and auxiliary fermions to preserve local commutation relations while converting the relevant modes into bosonic ones.
C. The Transformation
- The unitary transformation distinguishes horizontal and vertical links and incorporates phase factors to preserve neighboring transformed operators’ commutation relations.The factors are assigned according to the link orientation and its two connected vertices.Source: C. The Transformation · p. 5
- Type-II fermions enter only through commuting number-like operators, so they remain in the vacuum and can be projected out of the transformed Hamiltonian.After projection, the corresponding factors can be replaced by 1.Source: C. The Transformation · p. 5
- The construction defines auxiliary operators at each lattice site and distinguishes their same-site relations as part of the unitary mapping.Source: C. The Transformation · p. 5
- For G = U(1), the auxiliary operators can be represented with Pauli operators, while higher-rank cases use N spins and a local Jordan-Wigner transformation.Source: C. The Transformation · p. 5
D. Transformed Hamiltonian
After projection onto the type-II vacuum, the transformed Hamiltonian remains decomposable into matter, electric, magnetic, and interaction parts, while gauge invariance and the original symmetry group are preserved.
D. Transformed Hamiltonian
- The transformed Hamiltonian decomposes into matter, electric, magnetic, and interaction contributions, with the electric and magnetic gauge-field parts treated separately.Source: D. Transformed Hamiltonian · pp. 5–6
- The interaction term is organized as contributions from horizontal and vertical links, whose transformations depend on the link geometry.Source: D. Transformed Hamiltonian · p. 6
- The transformed Hamiltonian remains gauge invariant with the same group G as the original theory, and the Gauss law is expressed using transformed variables.Relevant observables include electric fields, Wilson loops, and mesonic operators in the transformed picture.Source: D. Transformed Hamiltonian · pp. 5–6
- Transformed Wilson loops and mesonic operators provide the corresponding physical observables, including operators defined over links joining two points.Source: D. Transformed Hamiltonian · p. 6
E. Discussion: U, Statistics and the Unitary Gauge
The unitary transformation can be interpreted as a fermionic analogue of unitary gauge fixing, with auxiliary fermions resolving local parity issues. Its applicability depends on gauge-group structure, while an auxiliary gauge field extends the construction to otherwise excluded groups.
E. Discussion: U, Statistics and the Unitary Gauge
- The transformation parallels unitary gauge fixing: a fermionic Z2 phase operator c_x replaces the bosonic U(1) phase e^−iθ_x.The analogy identifies π(c_x−1)/2 as the Z2 counterpart of the U(1) phase θ_x.Source: E. Discussion: U, Statistics and the… · p. 6
- The naive local transformation can be parity-odd, so transformations at different vertices generally fail to commute and require an ordering choice.The parity depends on the electric-field divergence E_h+E_v−E_h′−E_v′.Source: E. Discussion: U, Statistics and the… · p. 6
- Type II auxiliary fermions solve this problem by completing the local transformation into an even operator that preserves parity.They balance fermionic parity, have no physical role, and can eventually be factored out.Source: E. Discussion: U, Statistics and the… · p. 6
- The transformed state retains a local Z2 symmetry linking type I auxiliary-fermion parity to the parity of physical excitations.The construction requires a gauge group containing Z2 as a normal subgroup and a representation including −1.Source: E. Discussion: U, Statistics and the… · pp. 6–7
IV. ELIMINATION OF THE FERMIONIC DEGREES OF FREEDOM: OTHER CASES
- For groups without the required normal Z2 subgroup, an auxiliary gauge field supplies the needed operators, extra gauge invariance, and Gauss laws before applying the elimination method.This extension makes the new Hamiltonian unitarily equivalent to the original one and covers cases such as SU(2N+1).Source: §IV · p. 7
- Compared with earlier auxiliary-field constructions, this approach uses vertex-resident type I fermions and avoids link-based fusion constraints, while adding an extra Gauss law for equivalence.The auxiliary fermions emerge from a unitary transformation involving local gauge fields.Source: §IV · p. 7
A. Introducing Auxiliary Z2 Gauge Fields
An auxiliary Z2 gauge field extends the theory so fermion elimination can be applied while preserving equivalence on the physical space, including a sector restriction for periodic boundaries.
A. Introducing Auxiliary Z2 Gauge Fields
- The construction adds a qubit-valued auxiliary gauge field on every link and imposes three Gauss-law constraints, producing an extended gauge structure.The auxiliary field uses Pauli operators X and Z; the resulting group is G × Z2 × Z2, with the latter factor on the dual lattice.Source: A. Introducing Auxiliary Z2 Gauge Fields · pp. 7–8
B. H and H′ are unitarily equivalent on the physical space
- Allowed auxiliary-qubit configurations contain an even number of |1⟩ states per plaquette, forming closed-loop patterns generated by vertex operators.These loop states form an orthonormal basis, and the auxiliary space has dimension 2^Nv.Source: B. H and H′ are unitarily equivalent… · p. 8
- The extended Hamiltonian is unitarily equivalent to the original Hamiltonian on their corresponding physical spaces.The equivalence also preserves expectation values of Z2 gauge-invariant observables when operators are transformed appropriately.Source: B. H and H′ are unitarily equivalent… · pp. 8–9
- For periodic boundaries, additional noncontractible loop operators create four topological sectors, but equivalence holds only in the nh = nv = 0 sector.The extra operators wrap around the torus and are needed to generate the full auxiliary loop space.Source: B. H and H′ are unitarily equivalent… · p. 9
V. CONCLUSION
The paper maps fermionic matter in lattice gauge theories to hard-core bosonic degrees of freedom through a local Z2-based construction. It presents the method as applicable to broad classes of lattice gauge theories and dimensions.
V. CONCLUSION
- The method maps fermions to hard-core bosons locally in lattice gauge theories with gauge groups U(N) or SU(N), using a local Z2 symmetry.The transformed theory retains the local character of the construction.Source: §V · p. 9
- The construction applies to dynamical-fermion lattice gauge theories and pure fermionic Hamiltonians with a well-defined minimal-coupling procedure.Although presented for SU(N) and U(N) theories in 2 + 1 dimensions, it can be generalized to other dimensions and gauge groups.Source: §V · p. 9
Appendix A: Construction of P Operators
The appendix identifies when an operator P can implement the required sign reversal of gauge-link operators. Its existence depends on a normal Z2 subgroup represented by −1 in the chosen representation.
Appendix A: Construction of P Operators
- The required relation PUP = −U requires a group element represented by −1 in the working irreducible representation.This element must square to the identity and form a normal Z2 subgroup with the identity.Source: Appendix A · p. 9
- For U(N), the needed operator exists in any representation, while for SU(N) in the fundamental representation it exists only when N is even.For SU(2N), the relevant element is represented by −1 in the fundamental representation.Source: Appendix A · p. 9
- For U(N), P can be constructed from the abelian U(1) electric field, whereas for SU(2N) it can be built from a suitable left or right generator.The construction uses P = exp(iπE) for the U(1) component and an appropriate generator for SU(2N).Source: Appendix A · p. 9
- The required relation is representation-dependent; for SU(2), it applies only to even-dimensional, half-integer-spin representations.Thus the group condition alone does not guarantee that the construction works in every representation.Source: Appendix A · p. 9
- For finite Z2N models, the construction gives P = X^(N/2), with P = X when N = 2.This operator produces the phase −1 on the link operator through the ZN relations.Source: Appendix A · p. 10
Appendix B: Properties of U
Appendix B explains how the local unitary transformation is assembled from auxiliary Majorana modes and link electric fields. Sign factors ensure that transformed link operators retain the required commutation relations.
1. The Transformation’s Building Blocks
- The local building blocks are V_i = C_i^E_i, where C_i is formed from a type II auxiliary Majorana mode and E_i is a link electric-field operator.The spectrum of C_i is ±1, so (1 − C_i)/2 has eigenvalues 0 and 1.Source: §1 · p. 10
- Only local transformations at the endpoints of a link affect its U operator, and the ordering of vertex factors determines horizontal and vertical sign factors.The endpoint transformations commute with one another, but the constituent V_i operators at a vertex need not commute.Source: §1 · p. 10
- The ξr factors preserve commutation between transformed operators on intersecting links by converting induced anticommutation back into commutation.Without these factors, the added fermionic operators would make transformed U operators on intersecting links anticommute.Source: §1 · p. 10
2. Transformation of the Auxiliary Fermions
The auxiliary-fermion transformation establishes invariance relations for type II modes and incorporates a constraint tied to gauge invariance. The magnetic Hamiltonian is then transformed by collecting auxiliary and ξ operators into the transformed plaquette expression.
2. Transformation of the Auxiliary Fermions
- Type II auxiliary modes are transformed and combined on links to obtain gauge-invariant relations for horizontal and vertical link modes.The construction uses relations for paired modes such as α, γ on horizontal links and analogous modes on vertical links.Source: §2 · p. 10
- Gauge invariance, including its Z2 normal subgroup, imposes a constraint that reduces the enlarged Hilbert space while retaining type I auxiliary modes.The text states that a later local unitary transformation can remove this constraint together with the type I auxiliary degrees of freedom.Source: §2 · p. 10
3. Transformation of the Magnetic Hamiltonian
- The magnetic Hamiltonian is transformed plaquette by plaquette, with auxiliary c modes contributing trivially and the remaining ξ operators collected into the transformed Hamiltonian ˜HB (27).The plaquette links are labeled a, b, c, d, and the type I auxiliary fermions are associated with vertex intersections.Source: §3 · pp. 10–11
4. Representing the Hard-Core Bosons by Spins
The transformed fermions become hard-core bosonic operators, which can be represented locally by spin operators through a vertex-wise Jordan-Wigner transformation.
4. Representing the Hard-Core Bosons by Spins
- The unitary transformation replaces the fermions with N hard-core bosonic operators ηx,m per vertex.These operators are introduced as the hard-core bosonic representation before conversion to spins.Source: §4 · p. 11
- A local Jordan-Wigner transformation numerates the fermionic modes at each vertex and replaces them with spins without non-local strings for local even-parity operators.The construction assigns a logical order to the modes associated with each vertex.Source: §4 · p. 11