Abstract
A central tenant in the classification of phases is that boundary conditions cannot affect the bulk properties of a system. In this work, we show striking, yet puzzling, evidence of a clear violation of this assumption. We use the prototypical example of an XYZ chain with no external field in a ring geometry with an odd number of sites and both ferromagnetic and antiferromagnetic interactions. In such a setting, even at finite sizes, we are able to calculate directly the spontaneous magnetizations that are traditionally used as order parameters to characterize the system’s phases. When ferromagnetic interactions dominate, we recover magnetizations that in the thermodynamic limit lose any knowledge about the boundary conditions and are in complete agreement with standard expectations. On the contrary, when the system is governed by antiferromagnetic interactions, the magnetizations decay algebraically to zero with the system size and are not staggered, despite the antiferromagnetic coupling. We term this behavior ferromagnetic mesoscopic magnetization. Hence, in the antiferromagnetic regime, our results show an unexpected dependence of a local, one-spin expectation values on the boundary conditions, which is in contrast with predictions from the general theory.
Highlights
Landau theory is one of the most impactful constructions of the last century
The new century has taught us that this classification is not complete, because certain phases of quantum matter are characterized by non–local order, Landau theory remains a cornerstone to interpret phases, directly borrowed from classical statistical mechanics
For the XY chain, we can express the one-point function as the determinant of a Toeplitz matrix and evaluate it analytically, while for the interacting cases we can numerically diagonalize the model and calculate the expectation values. We benchmarked these procedures on a ferromagnetic phase with FBC to show that they reproduce the expected results eq (6), while in an AFM phase the magnetizations, while finite in a finite chain, decay toward zero algebraically in the thermodynamic limit
Summary
Landau theory is one of the most impactful constructions of the last century. It allows distinguishing between different phases through different local order parameters, quantities which are finite or vanish depending on the phase of a system [1,2,3,4]. After a Jordan-Wigner Transformation [44, 51], the RHS of eq (7) can be written again as the determinant of a Toeplitz matrix, whose asymptotic behavior can be studied analytically, to what has been done in [50] This novel “trick” can be understood as originating from the fact that, at zero external fields, the chain eq (1) has particle/hole duality and that, on a chain with an odd number of sites, this symmetry relates states with different parities. The result of such analysis reproduces eq (6), proving the consistency of the two methods of evaluation for the order parameters. We remark that FBC seems to somewhat spoil the cluster decomposition, since the non–staggered mesoscopic magnetization we find is not compatible with (8), both of them vanish in the thermodynamic limit
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