  Spring-antispring systems have been investigated as possible low-frequency
seismic isolation in high-precision optical experiments. These systems provide
the possibility to tune the fundamental resonance frequency to, in principle,
arbitrarily low values, and at the same time maintain a compact design of the
isolation system. It was argued though that thermal noise in spring-antispring
systems would not be as small as one may naively expect from lowering the
fundamental resonance frequency. In this paper, we present a detailed
calculation of the suspension thermal noise for a specific spring-antispring
system, namely the Roberts linkage. We find a concise expression of the
suspension thermal noise spectrum, which assumes a form very similar to the
well-known expression for a simple pendulum. It is found that while the Roberts
linkage can provide strong seismic isolation due to a very low fundamental
resonance frequency, its thermal noise is rather determined by the dimension of
the system. We argue that this is true for all horizontal mechanical isolation
systems with spring-antispring dynamics. This imposes strict requirements on
mechanical spring-antispring systems for the seismic isolation in potential
future low-frequency gravitational-wave detectors as we discuss for the four
main concepts: atom-interferometric, superconducting, torsion-bars, and
conventional laser interferometer.
