Spherical model and quantum phase transitions

This thesis is divided into two parts. In Part I, we give an explicit construction for a class of lattices with effectively non-integral dimensionality. A reasonable definition of dimensionality applicable to lattice systems is proposed. The construction is illustrated by several examples. We calculate the effective dimensionality of some of these lattices. The attainable values of the dimensionality d, using our construction, are densely distributed in the interval 1 The variation of critical exponents with dimensionality is studied for a variety of Hamiltonians. It is shown that the critical exponents for the spherical model, for all d, agree with the values derived in literature using formal arguments only. We also study the critical behavior of the classical p-vector Heisenberg model and the Fortuin-Kasteleyn cluster model for lattices with d<2. It is shown that no phase transition occurs at nonzero temperatures. The renormalization procedure is used to determine the exact values of the connectivity constants and the critical exponents α, γ, v for the self-avoiding walk problem on some multiply connected lattices with d<2. It is shown by explicit construction that the critical exponents are not functions of dimensionality alone, but depend on detailed connectivity properties of the lattice. In Part II, we investigate a model of the melting transition in solids. Melting is treated as a layer phenomenon, the onset of melting being characterized by the ability of layers to slip past each other. We study the variation of the root-mean-square deviation of atoms in one layer as the temperature is increased. The adjacent layers are assumed held fixed and provide an external periodic potential. The coupling between atoms within the layer is assumed to be simple harmonic. The model is thus equivalent to a lattice version of the Sine-Gordon field theory in two dimensions. Using an exact equivalence, the partition function for this problem is shown to be related to the grand partition function of a two-species classical lattice Coulomb gas. We use the renormalization procedure to determine the critical behavior of the lattice Coulomb gas problem. Translating the results back to the original problem, it is shown that there exists a phase transition in the model at a finite temperature Tc. Below Tc, the root mean square deviation of atoms in the layer is finite, and varies as (Tc-T)-1/4 near the phase transition. Above Tc, the root mean square deviation is infinite. The specific heat shows an essential singularity at the phase transition, varying as exp(-|Tc -T|-1/2) near Tc.

The spherical model can be regarded either as an exactly-soluble approximation to the Ising model or as a model of isotropically interacting spins in which the spin dimensionality v approaches infinity. In this note we cal- culate 80 terms in the high-temperature expansions for the spherical model and compare with predictions which would have been obtained if only - 10 terms were known (as in the case for finite v). We find that the critical point exponents deduced from short serles are extremely doubtful when d > 3, where d 1s the lattlce dimens~onality. In particular, we note that it is quite possible that the mean field exponents are valid for all v if d > 3. There is neither general proof of the existence of the critical point exponents nor clear evidences of a set of physical parameters on which these exponents are mainly dependent. The facts are that the charac- teristics of phase transitions predominantly depend on the dimensionality d of the lattices and that so far the scaling law hypothesis (l) 121 is the only unifying theory of the critical point exponents for various physical systems. Hence, it is desirable to investigate these facts for mathematically tractable model systems. The two-dimensional Ising model and the three- dimensional spherical model (3) are the only two soluble models of magnetic systems which exibit phase transition. The spherical model is a modification of the Ising model but at high temperatures and at high- density limit the properties of these two are very similar (4) (5). Recently it has been conjectured the breakdown of one of the predictions of the scaling law hypothesis for -the three-dimensional Ising model (6) (7). Therefore we are led to examine the similar predictions for the spherical model and because of the mentioned similarity of the two models we will consi- der the critical exponents above the transition point T, of the spherical model with nearest-neihbor inter- actions for d 2 3 (for d > of the scaling hypothesis was established on the basis of high- temperature expansions we have firstly performed this type of calculations. For the Ising model Fisher and Gaunt (8) have calculated 11 coefficients in the corres- ponding expansions, but for the spherical model we have been able to calculate 80 coefficients. The results for the zero-field susceptibility are displayed in figure 1. The similarity in the asymptotic behavior of the two

The characterization of the quantum phase transition in a lowdimensional system has attracted a considerable amount of attention in quantum manybody systems. As one of the fundamental models in quantum magnetism, spin-1 models have richer phase diagrams and show more complex physical phenomena. In the spin-1 antiferromagnetic XXZ model, the Haldane phase and the Nel phase are the gapped topologic phases which cannot be characterized by the local order parameters. To characterize the nature in such phases, one has to calculate the non-local long range order parameters. Normally, the non-local order parameter in the topological phase is obtained from the extrapolation of finite-sized system in numerical study. However, it is difficult to extract the critical exponents with such an extrapolated non-local order parameter due to the numerical accuracy. In a recently developed tensor network representation, i.e., the infinite matrix product state (iMPS) algorithm, it was shown that the non-local order can be directly calculated from a very large lattice distance in an infinite-sized system rather than an extrapolated order parameter in a finite-sized system. Therefore, it is worthwhile using this convenient technique to study the non-local orders in the topological phases and characterize the quantum criticalities in the topological quantum phase transitions. In this paper, by utilizing the infinite matrix product state algorithm based on the tensor network representation and infinite time evolving block decimation method, the quantum entanglement, fidelity, and critical exponents of the topological phase transition are investigated in the one-dimensional infinite spin-1 bond-alternating XXZ Heisenberg model. It is found that there is always a local dimerization order existing in the whole parameter range when the bond-alternative strength parameter changes from 0 to 1. Also, due to the effect of the bond-alternating, there appears a quantum phase transition from the long-rang ordering topological Nel phase to the local ordering dimerization phase. The von Neumann entropy, fidelity per lattice site, and order parameters all give the same phase transition point at c = 0.638. To identify the type of quantum phase transition, the central charge c 0.5 is extracted from the ground state von Neumann entropy and the finite correlation length, which indicates that the phase transition belongs to the two-dimensional Ising universality class. Furthermore, it is found that the Nel order and the susceptibility of Nel order have power-law relations to |-c|. From the numerical fitting of the Nel order and its susceptibility, we obtain the characteristic critical exponents ' = 0.236 and ' = 0.838. It indicates that such critical exponents from our method characterize the nature of the quantum phase transition. Our critical exponents from the iMPS method can provide guidance for studying the properties of the phase transition in quantum spin systems.

By using the infinite time evolving block decimation in the presentation of infinite matrix product states, we study an extended quantum compass model (EQCM). This model does not only include extremely rich phase diagrams due to competitions of orbital degrees of freedom and anisotropic couplings between pseudospin-1/2 operators but also have the capacity to describe property of protected qubits for quantum computation which leads to lots of attentions paid to the phase boundaries of the EQCM. However, few attentions are paid to long-range topological string correlation order parameters of the EQCM. To study order parameters, one should understand spontaneous symmetry breaking which relates to Landau quantum phase transitions theory. Once spontaneous symmetry breaking happens, there should exist some local order which can be described by a local order parameter. This order parameter can be used to distinguish the phase from others. For continuous quantum phase transitions, in the critical regime, critical exponents can be extracted. Unfortunately, the long-range topological string correlation orders are beyond Landau quantum phase transitions theory, one can not directly use two paradigms of Landau-Ginzburg-Wilson. Usually, one can define a local order parameter by local magnetization. Naturally, one can also refer to this way to define the long-range topological string correlation order parameters by long-range topological string correlations on the following conditions, i.e. the quantum system undergoes a hidden spontaneous symmetry breaking; the long-range topological string correlation order parameter can be used to distinguish the phase from others; for continuous quantum phase transitions, the long-range topological string correlation order parameter satisfies scaling law when control parameter getting close to critical point. Based on above idea, in order to characterize the topological ordered phases and quantum phase transitions in the EQCM, even/odd long-range topological string correlations are introduced based on even/odd bonds. Hereafter, fidelity per lattice site, even/odd long-range topological string correlations, the saturation behavior of odd long-range topological string correlations and order parameters are calculated. The long-range topological string correlations show three distinguished behaviors which include decaying to zero, monotonic saturation and oscillatory saturation. By the above characterizations, oscillatory/monotonic odd long-range topological string correlation order parameter is derived. Then ground-state phase diagram of order parameters is computed which includes oscillatory/monotonic odd long-range topological string correlation order phase and antiferromagnetic phase. In the critical regime, critical exponent β=1/8 extracted from monotonic odd long-range topological string correlation order parameter and local magnetization shows the phase transition belongs to Ising universality. In addition, the phase transition points, the order of the phase transitions of fidelity show consistent with the results of order parameters.

The study of low-dimensional, strongly-interacting quantum systems has been of interest in the past few decades due to new and exotic physics that show great potential in the advancement of fundamental physics as well as technological applications. In order to investigate such systems, a vast number of techniques have been developed throughout the years, all of which have their respective strengths and weaknesses. In the world of one-dimensional (1D) quantum systems, the density-matrix renormalization group (DMRG) algorithm and matrix product states (MPS) have proven to be powerful and reliable tools that provide invaluable information while providing a wide range of data that support both theoretical and experimental studies. Coupled with the development and refinement of tensor networks - an ansatz for quantum many-body wavefunctions, in particular MPS, the DMRG algorithm has seen a spike of improvements that has pushed the boundaries of its application into more sophisticated and complex systems.This thesis broadly covers the study of several 1D many-body quantum systems represented by infinite MPS (iMPS) - a class of ansatz that represent translationally-invariant, 1D many-body quantum states. More specifically, it is divided into two main parts.The first part investigates the 1D topological Kondo insulator (TKI). This is an effective model designed to investigate how strong interactions between conduction electrons and localized moments result in a symmetry-protected topological (SPT) phase, specifically the Haldane phase. By studying the effect of the electron repulsion as well as the competition between the local and non-local couplings between the electrons and the local moments, it is found that a topological phase transition from the SPT phase to a topologically trivial phase occurs with the assistance of electron interaction. The properties of both topologically trivial and nontrivial groundstates were revealed via means of quantum information theoretic quantities such as the entanglement entropy, entanglement spectrum and non-local order parameters. These quantities unveil the nature, structure and the effect that the different interactions have on the groundstate properties. Besides the phases, the properties of the critical point was studied and classified. By obtaining the critical exponents and central charge, the universality class of this transition was found to be that of 1D free fermions/bosons. Furthermore, the presence of electron interactions was found not to alter this universality class.The second part of this thesis is a project that stems from the necessity to overcome the inherent limitation of an iMPS in representing a system at a cusp of a quantum phase transition (QPT), specifically, in determining the critical point and exponents. In the thermodynamic limit, the laws of statistical mechanics dictates that the system’s correlation length diverges at the critical point. However, when representing a critical state with an iMPS, the finite bond dimension implies that the correlation length of the iMPS remains finite, albeit large. This presents a fundamental challenge in extracting critical data since the critical exponents are defined for systems in the thermodynamic limit. To overcome this shortcoming, finite-entanglement scaling (FES) - the analog of finite-size scaling, has previously been used to locate the critical point and extract the critical exponents for systems represented by finite-bond dimensional iMPS. While FES serves its purpose sufficiently well, the computational cost is high for just about any moderately complex, real-world system. To overcome such difficulties, the second part of this thesis combines new ideas of using higher-order cumulants of the order parameter with previously-known iMPS techniques to determine the critical point and extract the exponents, thus relating statistical mechanics of QPTs and tensor networks of critical states. Some of the main methods used and developed here are FES, the application of the Binder cumulant in iMPS and the new cumulant exponent relation. Putting these tools together provides a scheme that can be used to efficiently locate the critical point and extract the critical exponents at a lower computational cost than previously-known methods. This is shown through comparisons of several exemplary models such as the 1D and quasi-1D transverse field Ising model, the 1D TKI, the S = 1 Heisenberg chain with single-ion anisotropy and the 1D Bose-Hubbard model.

Epsilon expansion is an important part of the renormalization group approach; it allows substantiating the scaling paradigm and is used in the modern theory of critical phenomena. It is shown that at a critical temperature equal to zero, the expression for the critical heat capacity exponent already in the first order in epsilon d − = 4 – (d – is the dimension of space) changes its form. This, however, does not indicate a violation of the scaling approach, although the classical Essam-Fisher equation also changes, as does the Rushbrook inequality. The generalized Essam-Fisher interpolation equality and the generalized Rushbrook inequality are given, which are valid for any value of the critical temperature, in particular, for the critical temperature equal to zero. The generalized relations are consistent with the renormalization-group approach, - expansion and scaling paradigm. The fluctuations are considered to be classical (thermodynamic), the conditions are determined when this takes place.

The spherical version of Dyson's hierarchical model is analyzed. A particular case which is designed to simulate the long-range Ising problem is dealt with in detail. A phase transition is found with critical temperature $$\beta _c = \tfrac{1}{2}(2^\alpha - 2)(4 - 2^\alpha )^{ - 1} $$ wherenth neighbor spins interact with a strength ofn−α. Critical exponents are calculated for this particular case and are found to be identical with the critical exponents of the long-range spherical Ising model.

In our everyday lives, we experience three spatial dimensions and a fourth dimension of time. Neverthe-less, several intricate problems of modern physics may be mastered with the introduction of additional dimensions, including the hierarchy problem and the unification of the fundamental forces. Furthermore, dualities between certain strongly coupled quantum field theories and particular gravitational theories in higher dimensions were conjectured based on string theory, which generically comes with the premise of extra dimensions. Specifically, the AdS/CFT correspondence or rather gauge/gravity duality motivated the study of a wide variety of higher dimensional gravitational theories with additional matter. The first part of this thesis covers the numerical construction of localized black holes in five, six and ten dimensional Kaluza-Klein theories. We focus on static, asymptotically flat vacuum solutions of Einsteins field equations with one periodic compact dimension. Our study concentrates on the critical regime, where the poles of the localized black holes are about to merge. We utilize a well adapted multi-domain pseudo-spectral scheme for obtaining high accuracy results and investigate the phase diagram of the localized solutions far beyond previous results. A spiral phase space structure is found for the five and six dimensional setups which matches to the results that were recently obtained for non-uniform black strings. On the contrary, the phase space structure of the ten dimensional configuration exhibits no spiraling behavior of the thermodynamical quantities. These critical exponents were extracted from the numerical data of the aforementioned configurations and show an excellent agreement with the theoretical predictions. In the second part of this thesis, the AdS/CFT correspondence is employed for studying strongly coupled Weyl semimetals. More concretely, we numerically investigate the effects of inhomogenities within a holographic Weyl semimetal by using a pseudo-spectral scheme, including interfaces of Weyl semimetals and the impact of time independent disorder. When studying interfaces between differentWeyl semimetal phases, we observe the appearance of an electric current, that is restricted to the interface in the presence of an electric chemical potential. The related integrated current is universal in the sense that it only depends on the topology of the phases. These results may shed some light on anomalous transport for inhomogeneous magnetic fields. As another point, we study the effects of time independent one-dimensional disorder on the holographic Weyl semimetal quantum phase transition (QPT), with a particular focus on the quantum critical region. We observe the smearing of the sharp QPT linked to the appearance of rare regions where the order parameter is locally non-zero. We discuss the role of the disorder correlation and we compare our results to expectations from condensed matter theory at weak coupling. We also analyze the interplay of finite temperature and disorder.

Anatoli Polkovnikov Department of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215, U.S.A.Vladimir Gritsev Physics Department, University of Fribourg, Chemin du Musee 3, 1700 Fribourg, SwitzerlandContinuous quantum phase transitions (QPTs) have been a subject of intense theoretical research in recent decades (see, e.g., Refs. [1-3] for overview). Unlike usual phase transitions driven by temperature, QPTs are driven entirely by quantum fluctuations. They are believed to occur in many situations as described later in this book. Quite recently a second order QPT was observed in a cold atom system of interacting bosons in an optical lattice. There a system of interacting bosons was driven in real time from the superfluid to the insulating phase [4], confirming an earlier theoretical prediction [5]. Up till now, Ref. [4] has provided probably the cleanest experimental confirmation of a QPT. The unifying property of all continuous (second order) phase transitions is the emergent universality and scale invariance of the long-distance low energy properties of the system near the quantum critical point (QCP) [1]. This universality implies that low-energy properties of the system can be described by very few parameters, like the correlation length or the gap, which typically have power-law scaling with the tuning parameter characterized by critical exponents. These exponents are not sensitive to the microscopic details of the Hamiltonian describing the system, but rather depend only on the universality class to which a given phase transition belongs [1].

A new energy-enstrophy model for the equilibrium statistical mechanics of barotropic flow on a sphere is introduced and solved exactly for phase transitions to quadrupolar vortices when the kinetic energy level is high. Unlike the Kraichnan theory, which is a Gaussian model, we substitute a microcanonical enstrophy constraint for the usual canonical one, a step which is based on sound physical principles. This yields a spherical model with zero total circulation, a microcanonical enstrophy constraint and a canonical constraint on energy, with angular momentum fixed to zero. A closed-form solution of this spherical model, obtained by the Kac – Berlin method of steepest descent, provides critical temperatures and amplitudes of the symmetry-breaking quadrupolar vortices. This model and its results differ from previous solvable models for related phenomena in the sense that they are not based on a mean-field assumption.

Study of the critical behavior of a three-dimensional Heisenberg XXZ model via entanglement

Quantum phases (QPs) and quantum phase transitions (QPTs) are very important parts of the strongly correlated quantum many-body systems in condensed matter. To study the QPs and QPTs, the systems should include rich quantum phase diagram. In this sense, the corresponding quantum spin models should have strong quantum fluctuation, strong geometric frustration, complicated spin-spin exchange or orbital degrees of freedom, which induces a variety of spontaneous symmetry breaking (SSB) or hidden spontaneous symmetry breaking. The QPs induced by the SSB can be characterized by local order parameters, a concept that originates from Landau-Ginzburg-Wilson paradigm (LGW). However, there is also a novel class of topological QPs beyond LGW, which has aroused one’s great interest since the Haldane phase was found. Such QPs can be characterized only by topological long-range nonlocal string correlation order parameters instead of local order parameters. In this paper, we investigate a spin-1/2 quantum compass chain model (QCC) with orbital degrees of freedom in &lt;i&gt;x&lt;/i&gt;, &lt;i&gt;y&lt;/i&gt; and &lt;i&gt;z&lt;/i&gt; components. The prototype of QCC is the quantum compass model including novel topological QPs beyond LGW, and consequently one can also anticipate the existence of novel topological QPs in QCC. However, very little attention has been paid to the QPs and QPTs for QCC, which deserves to be further investigated. By using the infinite time evolving block decimation in the presentation of matrix product states, we study the QPs and QPTs of QCC. To characterize QPs and QPTs of QCC, the ground state energy, local order parameter, topological long-range nonlocal string correlation order parameters, critical exponent, correlation length and central charge are calculated. The results show the phase diagram of QCC including local antiferromagnetic phase, local stripe antiferromagnetic phase, oscillatory odd Haldane phase and monotonic odd Haldane phase. The QPTs from oscillatory odd Haldane phase to local stripe antiferromagnetic phase and from local antiferromagnetic phase to monotonic odd Haldane phase are continuous; on the contrary, QPTs from local stripe antiferromagnetic phase to local antiferromagnetic phase and from oscillatory odd Haldane phase to monotonic odd Haldane phase are discontinuous. The crossing point where the line of continuous QPTs meets with the line of discontinuous QPTs is the multiple critical point. The critical exponents &lt;i&gt;β&lt;/i&gt; of local antiferromagnetic order parameter, local stripe antiferromagnetic order parameter, topological long-range nonlocal oscillatory odd string correlation order parameter, and topological long-range nonlocal monotonic odd string correlation order parameter are all equal to 1/8. Moreover, &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$\beta =1/8$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and the central charges &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$c = 1/2$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M4.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M4.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; at the critical points show that the QPTs from local phases to nonlocal phases belong to the Ising-type universality class.

At a coverage of $5∕3$ monolayer (ML), Pb adsorbed on Ge(001) forms a ground phase displaying a $(\genfrac{}{}{0}{}{2\phantom{\rule{0.3em}{0ex}}1}{0\phantom{\rule{0.3em}{0ex}}6})$ symmetry. This phase undergoes two reversible phase transitions $(\genfrac{}{}{0}{}{2\phantom{\rule{0.3em}{0ex}}1}{0\phantom{\rule{0.3em}{0ex}}6})\ensuremath{\leftrightarrow}(\genfrac{}{}{0}{}{2\phantom{\rule{0.3em}{0ex}}1}{0\phantom{\rule{0.3em}{0ex}}3})\ensuremath{\leftrightarrow}(2\ifmmode\times\else\texttimes\fi{}1)$ at the critical temperatures ${T}_{{c}_{1}}\ensuremath{\sim}178\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and ${T}_{{c}_{2}}\ensuremath{\sim}375\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, respectively. We investigated the behavior of the relevant order parameters at the critical temperatures by means of He and in-plane x-ray diffraction (HAS and XRD, respectively). Both phase transitions at the critical temperature put in evidence a clear order-disorder behavior, in agreement with the universality class expected for the corresponding symmetry group transformation. The low-temperature transition yields the critical exponent of the two-dimensional (2-D) Ising universality class, whereas the three-state Potts' critical exponents are found for the high-temperature transition. By out-of-plane XRD measurements, the low-temperature phase transition is observed to be accompanied by a static surface distortion at room temperature. A complementary HAS study of the temperature evolution of the surface charge corrugation reveals that the complete $(\genfrac{}{}{0}{}{2\phantom{\rule{0.3em}{0ex}}1}{0\phantom{\rule{0.3em}{0ex}}6})\ensuremath{\leftrightarrow}(\genfrac{}{}{0}{}{2\phantom{\rule{0.3em}{0ex}}1}{0\phantom{\rule{0.3em}{0ex}}3})$ transition is of the displacive type. On the contrary, the high-temperature phase transition does not show any change of the surface corrugation up to its irreversible decomposition, thus pointing to a pure order-disorder character.

In the past decades complex networks and their behavior have attracted much attention. In the real world many of such networks can be found, for instance as social, information, technological and biological networks. An interesting property that many of them share is that they are scale free. Such networks have many nodes with a moderate amount of links, but also a significant amount of nodes with a very high number of links. The latter type of nodes are called hubs and play an important role in the behavior of the network. To model scale free networks, we use power-law random graphs. This means that their degree sequences obey a power law, i.e., the fraction of vertices that have k neighbors is proportional to k- for some &gt; 1. Not only the structure of these networks is interesting, also the behavior of processes living on these networks is a fascinating subject. Processes one can think of are opinion formation, the spread of information and the spread of viruses. It is especially interesting if these processes undergo a so-called phase transition, i.e., a minor change in the circumstances suddenly results in completely different behavior. Hubs in scale free networks again have a large influence on processes living on them. The relation between the structure of the network and processes living on the network is the main topic of this thesis. We focus on spin models, i.e., Ising and Potts models. In physics, these are traditionally used as simple models to study magnetism. When studied on a random graph, the spins can, for example, be considered as opinions. In that case the ferromagnetic or antiferromagnetic interactions can be seen as the tendency of two connected persons in a social network to agree or disagree, respectively. In this thesis we study two models: the ferromagnetic Ising model on power-law random graphs and the antiferromagnetic Potts model on the Erd¿os-Rényi random graph. For the first model we derive an explicit formula for the thermodynamic limit of the pressure, generalizing a result of Dembo and Montanari to random graphs with power-law exponent &gt; 2, for which the variance of degrees is potentially infinite. We furthermore identify the thermodynamic limit of the magnetization, internal energy and susceptibility. For this same model, we also study the phase transition. We identify the critical temperature and compute the critical exponents of the magnetization and susceptibility. These exponents are universal in the sense that they only depend on the power-law exponent and not on any other detail of the degree distribution. The proofs rely on the locally tree-like structure of the random graph. This means that the local neighborhood of a randomly chosen vertex behaves like a branching process. Correlation inequalities are used to show that it suffices to study the behavior of the Ising model on these branching processes to obtain the results for the random graph. To compute the critical temperature and critical exponents we derive upper and lower bounds on the magnetization and susceptibility. These bounds are essentially Taylor approximations, but for power-law exponents 5 a more detailed analysis is necessary. We also study the case where the power-law exponent 2 (1, 2) for which the mean degree is infinite and the graph is no longer locally tree-like. We can, however, still say something about the magnetization of this model. For the antiferromagnetic Potts model we use an interpolation scheme to show that the thermodynamic limit exists. For this model the correlation inequalities do not hold, thus making it more difficult to study. We derive an extended variational principle and use to it give upper bounds on the pressure. Furthermore, we use a constrained secondmoment method to show that the high-temperature solution is correct for high enough temperature. We also show that this solution cannot be correct for low temperatures by showing that the entropy becomes negative if it were to be correct, thus identifying a phase transition.

Quantum phase transitions occur in non-Hermitian systems. In this work we show that density functional theory, for the first time, uncovers universal critical behaviors for quantum phase transitions and quantum entanglement in non-Hermitian many-body systems. To be specific, we first prove that the non-degenerate steady state of a non-Hermitian quantum many body system is a universal function of the first derivative of the steady state energy with respect to the control parameter. This finding has far-reaching consequences for non-Hermitian systems. First, it bridges the non-analytic behavior of physical observable and no-analytic behavior of steady state energy, which explains why the quantum phase transitions in non-Hermitian systems occur for finite systems. Second, it predicts universal scaling behaviors of any physical observable at non-Hermitian phase transition point with scaling exponent being (1 − 1/p) with p being the number of coalesced states at the exceptional point. Third, it reveals that quantum entanglement in non-Hermitian phase transition point presents universal scaling behaviors with critical exponents being (1 − 1/p). These results uncover universal critical behaviors in non-Hermitian phase transitions and provide profound connections between entanglement and phase transition in non-Hermitian quantum many-body physics.