Abstract We analyze a wave function of a tensor model in the canonical formalism, when the argument of the wave function takes Lie group invariant or nearby values. Numerical computations show that there are two phases, which we call the quantum and the classical phases. In the classical phase fluctuations are suppressed, and configurations emerge which are discretizations of classical geometric spaces invariant under Lie group symmetries. This is explicitly demonstrated for emergence of Sn (n = 1, 2, 3) for SO(n + 1) symmetries by checking the topological and geometric (Laplacian) properties of the emerging configurations. The transition between the two phases has the form of splitting/merging of distributions of variables, resembling a matrix model counterpart, namely the transition between one-cut and two-cut solutions. However, this resemblance is obscured by a difference in the mechanism of distribution in our setup from that in the matrix model. We also discuss this transition as a replica symmetry breaking. We perform various preliminary studies of the properties of the phases and the transition for such values of the argument.
Matrix models have phase transitions in which distributions of variables change topologically like the Gross-Witten-Wadia transition. In a recent study, similar splitting-merging behavior of distributions of dynamical variables was observed in a tensor-vectors system by numerical simulations. In this paper, we study the system exactly in some large-
N limits, in which the distributions are discrete sets of configurations rather than continuous. We find cascades of first-order phase transitions for fixed tensors, and first- and second-order phase transitions for random tensors, being characterized by breaking patterns of replica symmetries. The system is of interest across three different subjects at least: The splitting dynamics plays essential roles in the emergence of classical spacetimes in a tensor model of quantum gravity. The splitting dynamics automatically detects the rank of a tensor in the tensor rank decomposition in data analysis. The system provides a variant of the spherical
p-spin model for spin glasses with a new nontrivial parameter. We discuss some implications of the results from these perspectives. The results are compared with some numerical simulations to check the large-
N convergence and the assumptions made in the analysis.
Advances in the study of relativistic quantum phase space have established the set of Linear Canonical Transformations (LCTs) as a candidate for the fundamental symmetry group associated with relativistic quantum physics. In this framework, for a spacetime of signature
(N+,N−), the symmetry of the relativistic quantum phase space is described by the LCT group, isomorphic to the symplectic Lie group
Sp(2N+,2N−), which preserves the canonical commutation relations (CCRs) and treats spacetime coordinates and momenta operators on an equal footing. In this work, we investigate the contraction structure of the Lie algebra associated with the LCT group for signature
(1,4), clarifying how familiar spacetime symmetry groups emerge from this more fundamental quantum phase space symmetry. Using the Inönü-Wigner group contraction formalism, we examine each limit case corresponding to the possible combinations of asymptotic values of two fundamental length scale parameters associated with the theory, namely a minimum length
ℓ and a maximum length
L, which may be identified respectively with the Planck length and the de Sitter radius. We explicitly analyze how contractions of the LCT Lie algebra lead to the physically relevant de Sitter algebra
so(1,4) and, in the flat-curvature limit, to the Poincaré algebra
iso(1,3) of four-dimensional spacetime. This provides an explicit mechanism through which relativistic spacetime symmetry can emerge from a deeper symplectic structure of quantum phase space.
The canonical tensor model, which is a tensor model in the Hamilton formalism, can be straightforwardly quantized and has an exactly solved physical state. The state is expressed by a wave function with a generalized form of the Airy function. The simplest observable on the state can be expressed by a matrix model which contains non-pairwise index contractions. This matrix model has the same form as the one that appears when the replica trick is applied to the spherical
p-spin model for spin glasses, but our case has different ranges of variables and parameters from the spin glass case. We analyze the matrix model analytically and numerically. We show some evidences for the presence of a continuous phase transition at the location required by a consistency condition of the canonical tensor model. We also show that there are dimensional transitions of configurations around the transition region. Implications of the results to the canonical tensor model are discussed.
Traditional mean-field theory is a generic variational approach for analyzing symmetry breaking phases. However, this simple approach only applies to symmetry breaking states with short-range entanglement. In this paper, we describe a generic approach for studying two-dimensional (2D) quantum phases with long-range entanglement (such as topological phases). The method is based on (a) a general class of trial wave functions known as tensor-product states and (b) a 2D real-space renormalization group algorithm for efficiently calculating expectation values for these states. We demonstrate our method by studying several simple 2D quantum spin models exhibiting both symmetry breaking phase transitions and topological phase transitions. Our approach can be viewed as a unified mean-field theory for both symmetry breaking phases and topological phases.
This paper establishes an emergent theoretical framework for Lie group topological orders with continuous symmetry, based on computable criteria. Traditionally, research on the emergence of topological order has primarily focused on discrete symmetric systems, lacking quantitative criteria for judging the stability of Lie group topological orders. By introducing group-theoretic and topological invariants such as Lie group dimension, fundamental group order, and quantum dimension, this study extends the three core criteria—structural stability, information efficiency, and dynamical accessibility—to continuous Lie group systems. This framework provides a quantifiable triangular standard for determining whether a topological order “emerges” under realistic physical conditions. Systematic numerical validation using three typical topological orders—SU(2), SU(2), and U(1)— demonstrates that under reasonable physical parameters, all criterion values stably exceed the preset threshold (>0.5) once the system size surpasses a minimal scale, confirming the effectiveness and robustness of this criteria system. This work not only provides a theoretical tool for studying topological phases protected by continuous symmetry but also points the way for their experimental realization and detection on quantum simulation platforms
The quest to reconcile quantum mechanics with gravitational theory motivates the exploration of frameworks that treat quantum uncertainty and spacetime geometry under a unified approach. A promising candidate that emerges from this pursuit is the relativistic quantum phase space (QPS) formalism, which extends classical phase space by incorporating both mean values and variance-covariance matrices of quantum states, providing a unified setting where the uncertainty principle and relativistic covariance coexist. For the signature
(1,4), we construct a scalar from the mean values and the inverse variance-covariance matrix and prove its invariance under linear canonical transformations (LCTs). Motivated by the form of the variance-covariance matrix in a particular reference frame, we identify this invariant as
Γ=L2/ℓ2 for states that saturate the uncertainty relations, where
L and
ℓ are two fundamental length scales that can be identified with the de Sitter radius and the Planck length, respectively. From this invariant, we obtain a geometric equation that unifies mean values and quantum fluctuations. In the limit
ℓ→0, the equation reduces to the de Sitter spacetime equation; in the limit
L→∞, it yields a curved momentum space reminiscent of Born reciprocity. In the Minkowski limit (both
ℓ→0 and
L→∞), the familiar relativistic relations for rest mass and proper time emerge. These limiting cases show how the Planck length and the cosmological constant can be unified within a single geometric constraint, establishing the QPS geometry as a promising framework for exploring the interplay between quantum mechanics and gravity.