Bosonic Quantum Mechanics Research Articles

Early-time and late-time quantum chaos

We show the relation between the Heisenberg averaging of regularized 2-point out-of-time ordered correlation function and the 2-point spectral form factor in bosonic quantum mechanics. The generalization to all even-point is also discussed. We also do the direct extension from the bosonic quantum mechanics to the noninteracting scalar field theory. Finally, we find that the coherent state and large-[Formula: see text] approaches are useful in the late-time study. We find that the computation of the coherent state can be simplified by the Heisenberg averaging. Therefore, this provides a simplified way to probe the late-time quantum chaos through a coherent state. The large-[Formula: see text] result is also comparable to the [Formula: see text] numerical result in the large-[Formula: see text] quantum mechanics. This can justify that large-[Formula: see text] technique in bosonic quantum mechanics can probe the late time, not the early time. Because the quantitative behavior of large-[Formula: see text] can be captured from the [Formula: see text] numerical result, the realization in experiments should be possible.

Spectral form factor as an OTOC averaged over the Heisenberg group

We prove that in bosonic quantum mechanics the two-point spectral form factor can be obtained as an average of the two-point out-of-time ordered correlation function, with the average taken over the Heisenberg group. In quantum field theory, there is an analogous result with the average taken over the tensor product of many copies of the Heisenberg group, one copy for each field mode. The resulting formula is expressed as a path integral over two fields, providing a promising approach to the computation of the spectral form factor. We develop the formula that we have obtained using a coherent state description from the JC model and also in the context of the large-N limit of CFT and Yang-Mills theory from the large-N matrix quantum mechanics.

Log-gamma directed polymer with fixed endpoints via the replica Bethe Ansatz

We study the model of a discrete directed polymer (DP) on a square lattice with homogeneous inverse gamma distribution of site random Boltzmann weights, introduced by Seppalainen (2012 Ann. Probab. 40 19–73). The integer moments of the partition sum, , are studied using a transfer matrix formulation, which appears as a generalization of the Lieb–Liniger quantum mechanics of bosons to discrete time and space. In the present case of the inverse gamma distribution the model is integrable in terms of a coordinate Bethe Ansatz, as discovered by Brunet. Using the Brunet-Bethe eigenstates we obtain an exact expression for the integer moments of for polymers of arbitrary lengths and fixed endpoint positions. Although these moments do not exist for all integer n, we are nevertheless able to construct a generating function which reproduces all existing integer moments and which takes the form of a Fredholm determinant (FD). This suggests an analytic continuation via a Mellin–Barnes transform and we thereby propose a FD ansatz representation for the probability distribution function (PDF) of Z and its Laplace transform. In the limit of a very long DP, this ansatz yields that the distribution of the free energy converges to the Gaussian unitary ensemble (GUE) Tracy-Widom distribution up to a non-trivial average and variance that we calculate. Our asymptotic predictions coincide with a result by Borodin et al (2013 Commun. Math. Phys. 324 215–32) based on a formula obtained by Corwin et al (2011 arXiv:1110.3489) using the geometric Robinson–Schensted–Knuth (gRSK) correspondence. In addition we obtain the dependence on the endpoint position and the exact elastic coefficient at a large time. We argue the equivalence between our formula and that of Borodin et al. As we will discuss, this provides a connection between quantum integrability and tropical combinatorics.

Parasupersymmetry in quantum graphs

We study hidden parasupersymmetry structures in purely bosonic quantum mechanics on compact equilateral graphs. We consider a single free spinless particle on the graphs and show that the Huang–Su parasupersymmetry algebra is hidden behind degenerate spectra.

Twist Positivity

We study a heat kernel e−βH defined by a self-adjoint Hamiltonian H acting on a Hilbert space H, and a unitary representation U(g) of a symmetry group G of H, normalized so that the ground vector of H is invariant under U(g). The triple {H, U(g), H} defines a twisted partition function Zg and a twisted Gibbs expectation 〈·〉g, Zg=TrH(U(g−1)e−βH) and 〈·〉g=TrH(U(g−1)·e−βH)/TrH(U(g−1)e−βH). We say that {H, U(g), H} is twist positive if Zg>0. We say that {H, U(g), H} has a Feynman–Kac representation with a twist U(g), if one can construct a function space and a probability measure dμg on that space yielding (in the usual sense on products of coordinates) 〈·〉g=∫·dμg. Bosonic quantum mechanics provides a class of specific examples that we discuss. We also consider a complex bosonic quantum field ϕ(x) defined on a spatial s-torus Ts and with a translation-invariant Hamiltonian. This system has an (s+1)-parameter abelian twist group Ts×R that is twist positive and that has a Feynman–Kac representation. Given τ∈Ts and θ∈R, the corresponding paths are random fields Φ(x, t) that satisfy the twist relationΦ(x, t+β)=eiΩθΦ(x−τ, t). We also utilize the twist symmetry to understand some properties of “zero-mass” limits, when the twist τ, θ lies in the complement of a set ϒsing of singular twists.

Directed polymer melts and quantum critical phenomena

The statistical mechanics of directed linelike objects, such as directed polymers in an external field, strands of dipoles in both ferro- and electrorheological fluids, and flux lines in high-Tc superconductors, bears a close resemblance to the quantum mechanics of bosons in 2+1 dimensions. We show that single-component and binary mixture critical phenomena in these systems are in the universality class of three-dimensional uniaxial dipolar ferromagnets and ferroelectrics. Our results also apply to films of two superfluid species undergoing phase separation well below their λ-points near T=0. In the case of directed polymers and electrorheological fluids we analyze the effects of free ends occurring in the sample as well as a novel directionally-dependent compressibility.

An algebraic version of the central limit theorem

A non-commutative analogue of the central limit theorem and the weak law of large numbers has been derived, the analogues of integrable functions being non-commutative polynomials. Without the assumption of positivity higher central limit theorems hold which have no analogy in the classical probabilistic case. The treatment includes this classical case and the convergence to so-called “quasi-free states” in the quantum mechanics of bosons [3, 4].