Source-linked AI summary
Transfer matrix for Kogut-Susskind fermions in the spin basis
Sergio Caracciolo, Fabrizio Palumbo
TL;DR
In gauge fields, the Kogut-Susskind spin and flavour formulations are not equivalent, complicating transfer-matrix construction in the spin basis. The paper constructs the spin-basis transfer matrix through a unitary transformation from the flavour basis, using an explicit block Fock space. The resulting transfer matrix is positive definite and translates by two lattice spacings.
Problem
The spin and flavour Kogut-Susskind formulations are equivalent without gauge fields but differ with gauge-field coupling, affecting transfer-matrix construction.
Source: §1 · p. 2Method
The paper constructs the spin-basis transfer matrix by a unitary transformation from the flavour basis, with the Fock space built on blocks.
Source: §5 · p. 14Results
The spin-basis transfer matrix is positive definite and performs time translations by two lattice spacings.
Source: §4.1 · p. 12Takeaways & Limitations
For nilpotency-expansion simulations, the spin-basis expression uses N′ in place of N, with gauge fields defined on blocks.
Source: §5 · p. 14The new transfer matrix cannot be related simply to the earlier construction in [11], which keeps gauge variables on original-lattice links.
Source: §5 · p. 14
Abstract
from arXiv · showhide
In the absence of interaction it is well known that the Kogut-Susskind regularizations of fermions in the spin and flavor basis are equivalent to each other. In this paper we clarify the difference between the two formulations in the presence of interaction with gauge fields. We then derive an explicit expression of the transfer matrix in the spin basis by a unitary transformation on that one in the flavor basis which is known. The essential key ingredient is the explicit construction of the fermion Fock space for variables which live on blocks. Therefore the transfer matrix generates time translations of two lattice units.
1 Introduction
Kogut-Susskind fermions treat doublers as physical fields, but the spin and flavour formulations cease to be equivalent in gauge fields, affecting transfer-matrix construction. The paper addresses the spin-basis construction using a transformation from the flavour basis, with translations by one block as the target.
1 Introduction
- Kogut-Susskind formulations interpret doublers as physical fields with additional quantum numbers, assigned differently in the spin and flavour bases.The spin basis associates spin and flavour with hypercube corners, while the flavour basis associates spin and taste with blocks.Source: §1 · p. 2
- With gauge fields, the spin and flavour formulations are not equivalent, and their difference affects construction of the corresponding transfer matrices.Without gauge-field coupling, a linear transformation relates the fermion fields in the two formulations.Source: §1 · p. 2
- Spin-basis transfer-matrix construction is complicated by the lack of a positive-definite operator for one-lattice-spacing translations and the missing explicit block Fock-space construction.For a Fock space on one time slice, a square root of the two-spacing matrix translates by one spacing; with a block Fock space, one-spacing translations are undefined.Source: §1 · p. 2
- The paper derives the spin-basis transfer matrix by transforming the known flavour-basis expression, targeting translations by one block.The motivation includes using the transfer matrix in nilpotency-expansion simulations, for which prior results were expressed only in the flavour basis despite faster spin-basis simulations.Source: §1 · pp. 2–3
2 Kogut-Susskind fermions
The formulations use block variables alongside lattice-site fields, with the flavour basis providing a known transfer matrix and the spin basis presenting a construction difficulty. The spin and flavour fields are related classically, enabling a transformation between their actions and transfer matrices.
2 Kogut-Susskind fermions
- Block fields q_y carry Dirac-spinor and taste indices, while lattice-site fields ψ_x and block coordinates distinguish the two descriptions.The lattice extents are twice the block extents, with η specifying positions within each block.Source: §2 · p. 3
2.1 The flavour basis
- In the flavour basis, particle and antiparticle projections propagate forward and backward in time, and their creation and annihilation operators define the Fock-space description.The known flavour-basis transfer matrix is expressed using N_t, which depends on block-time gauge links, and includes the chemical potential.Source: §2.1 · p. 5
2.2 The spin basis
- The spin-basis Lagrangian lacks a direct identification of forward and backward movers, complicating an operator transfer-matrix construction.The classical relation between spin- and flavour-basis fields is used to derive the spin-basis action and transfer matrix from the flavour basis.Source: §2.2 · pp. 6–7
3 Transformation of the Lagrangian
The spin- and flavor-basis Lagrangians are generally inequivalent with gauge fields, although they coincide without gauge interaction. In the transformed spin-basis action, gauge fields couple only sites in different blocks, subject to a common gauge transformation for fields within each block.
3 Transformation of the Lagrangian
- The spin-basis Lagrangian is obtained using the basis transformations and tracing over spinor and flavour indices.The derivation uses relations for the mass and kinetic terms.Source: §3 · pp. 7–8
- In the transformed kinetic term, the gauge field couples only sites belonging to different blocks.The link-field definition specifies when the gauge field is present.Source: §3 · pp. 8–9
- Fields within each block are constrained to transform in the same way under gauge transformations.The proposed generalization is only a pure-gauge transformation because curvature vanishes on plaquettes contained within one block.Source: §3 · p. 9
- With a generic gauge-field configuration, the spin- and flavor-basis Lagrangians are not equivalent.Without gauge interaction, the transformed spin-basis Lagrangian coincides with the original spin-basis one.Source: §3 · pp. 7, 9
- With chemical potential included, it can be attached to all links in the transformed Lagrangian at half its flavor-basis value.The construction described is for vanishing chemical potential; its inclusion is stated to be straightforward.Source: §3 · p. 9
4 Transformation of transfer matrix and coherent states
The paper transforms the known flavor-basis transfer matrix and coherent states into the spin basis. The resulting transfer matrix is positive definite and advances time by two lattice spacings; the transformed coherent states also recover the spin-basis Lagrangian.
4 Transformation of transfer matrix and coherent states
- The spin-basis temporal index η0 corresponds to the flavor-basis ± projection, but spin-basis fields cannot be identified as forward or backward movers.As time advances by one original lattice unit, η0 changes, turning a particle into the hole of an antiparticle.Source: §4 · p. 11
4.1 Transfer matrix
- In the transformed transfer-matrix operator, gauge-variable terms connect sites in different blocks, while the other terms involve sites within the same blocks.The link-field definition controls the presence of the gauge field.Source: §4.1 · p. 12
- The spin-basis transfer matrix is positive definite and performs time translations by two lattice spacings.Its construction transforms the flavor-basis creation-annihilation operators into the spin basis.Source: §4 · p. 10, §4.1 · p. 12
4.2 Coherent states
- Transforming the coherent states provides a cross-check: evaluating the partition function recovers the spin-basis Lagrangian.The associated Grassmann variables are assigned to even or odd times according to their definitions.Source: §4.2 · p. 13
5 Conclusion
The paper derives the spin-basis transfer matrix through a unitary transformation of the flavor-basis matrix, with the Fock space built on blocks. The construction supports nilpotency-expansion simulations while translating by two lattice spacings.
5 Conclusion
- Spin-basis simulations are faster than flavor-basis simulations, and a positive-definite spin-basis transfer matrix is of interest because it is related to unitarity.The transfer matrix is needed for simulations using the nilpotency expansion.Source: §5 · p. 14
- An alternative spin-basis formulation had a fermion determinant that was not positive definite even at zero chemical potential, making it less suitable for numerical simulations.Source: §5 · p. 14
- The earlier positive-definite spin-basis matrix T 2 also translates by two lattice spacings, but its block-based Fock space was not explicitly constructed.Source: §5 · p. 14
- A unitary transformation yields the spin-basis transfer matrix using an explicitly constructed block Fock space, with N replaced by N′ and gauge fields defined on blocks.For nilpotency-expansion simulations, the paper says these are the required changes to the equations.Source: §5 · p. 14
- The result cannot be simply related to the earlier transfer matrix [11], which leaves gauge variables on original links and includes creation and annihilation operators as powers.Source: §5 · p. 14