We consider the finite-temperature phase diagram of the S = 1/2 frustrated Heisenberg bilayer. Although this two-dimensional system may show magnetic order only at zero temperature, we demonstrate the presence of a line of finite-temperature critical points related to the line of firstorder transitions between the dimer-singlet and -triplet regimes. We show by high-precision quantum Monte Carlo simulations, which are sign-free in the fully frustrated limit, that this critical point is in the Ising universality class. At zero temperature, the continuous transition between the ordered bilayer and the dimer-singlet state terminates on the first-order line, giving a quantum critical end point, and we use tensor-network calculations to follow the first-order discontinuities in its vicinity. possible at T = 0, on top of which thermal fluctuations may cause qualitatively different physics to set in. The model we investigate is represented schematically in the insets of Fig. 1. In addition to the interaction, J ⊥ , defining the dimer unit and the intralayer interaction, J , defining the two planes of the system, we include a symmetrical, diagonal, and frustrating interlayer coupling, J × . Only antiferromagnetic couplings are considered. The Hamiltonian for any quantum spin S iswhere i is the dimer bond index, j denotes the nearestneighbor dimers in the bilayer, m = 1 and 2 denote the two layers, and m is the layer opposite to m.Our initial focus is the fully frustrated bilayer (FFB), J × = J . In this situation, Eq. (1) can be reexpressed as
We investigate the minus-sign problem that afflicts quantum Monte Carlo (QMC) simulations of frustrated quantum spin systems, focusing on spin S = 1/2, two spatial dimensions, and the extended Shastry-Sutherland model. We show that formulating the Hamiltonian in the diagonal dimer basis leads to a sign problem that becomes negligible at low temperatures for small and intermediate values of the ratio of the inter-and intra-dimer couplings. This is a consequence of the fact that the product state of dimer singlets is the exact ground state both of the extended Shastry-Sutherland model and of a corresponding "sign-problem-free" model, obtained by changing the signs of all positive off-diagonal matrix elements in the dimer basis. By exploiting this insight, we map the sign problem throughout the extended parameter space from the Shastry-Sutherland to the fully frustrated bilayer model and compare it with the phase diagram computed by tensor-network methods. We use QMC to compute with high accuracy the temperature dependence of the magnetic specific heat and susceptibility of the Shastry-Sutherland model for large systems up to a coupling ratio of 0.526(1) and down to zero temperature. For larger coupling ratios, our QMC results assist us in benchmarking the evolution of the thermodynamic properties by systematic comparison with exact diagonalization calculations and interpolated high-temperature series expansions. arXiv:1808.02043v2 [cond-mat.str-el] FIG. 1. Schematic representations of the extended Shastry-Sutherland model [Eq. (2)] in (a) single-plane and (b) bilayer format.
Neuronal networks are interesting physical systems in various respects: they operate outside thermodynamic equilibrium [1], a consequence of directed synaptic connections that prohibit detailed balance [2]; they show relaxational dynamics and hence do not conserve but rather constantly dissipate energy; and they show collective behavior that self-organizes as a result of exposure to structured, correlated inputs and the interaction among their constituents. But their analysis is complicated by three fundamental properties: Neuronal activity is stochastic, the input-output transfer function of single neurons is non-linear, and networks show massive recurrence [3] that gives rise to strong interaction effects. They hence bear similarity with systems that are investigated in the field of (quantum) many particle systems. Here, as well, (quantum) fluctuations need to be taken into account and the challenge is to understand collective phenomena that arise from the non-linear interaction of their constituents. Not surprisingly, similar methods can in principle be used to study these two a priori distinct system classes [4][5][6][7][8].But so far, the techniques employed within theoretical neuroscience just begin to harvest this potential. Here we will take essential steps towards this goal. Concretely, we adapt methods from statistical field theory and functional renormalization group techniques to the study of neuronal dynamics.A central motivation for this work is a coherent presentation of the technical machinery, which is well-developed in other fields of physics [9], to study the statistics and in particular phase transitions in stochastic neuronal systems and to provide a bridge between the stochastic dynamics and effective descriptions with reduced complexity.The large number of synaptic inputs to each neuron in a network allows the application of mean-field theory [10-12] to explain many dynamical phenomena, among them first order phase transitions. The transition from a quiescent to a highly active state in a bistable neuronal network is a prime example of a first order phase transition in neuronal networks [12]. The activation of attractors embedded into the connectivity of a Hopfield network is a second [13]. Combined with linear response theory, network fluctuations can quantitatively be described in binary [14][15][16] and in spiking networks [17][18][19][20][21][22]. Also transitions into oscillatory states by Andronov-Hopf bifurcations are in the realm of linear response theory around a mean-field solution [23][24][25][26].Second order phase transitions in neuronal networks are more challenging because the behavior of the system is dominated by fluctuations on all length scales, so that mean-field theory and its systematic correction by loopwise expansion break down [4]. But understanding these transitions is highly interesting from a neuroscientific point of view, because networks then show large susceptibility to signals. Moreover, signatures of critical states are found ubiquitously in experiments: Par...
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