<p>This paper explores how magnetic fields affect the Casimir effect within the context of a simple quasi-1D interacting fermionic system. A novel phenomenon emerges, resulting from the interaction between external magnetic fields and boundary conditions, which alters the ground state in complex ways and leads to first-order phase transitions among various ground states, specifically inhomogeneous solutions with differing node counts. We calculate the Casimir force, which exhibits discontinuities (jumps) at the transition points between the different ground states.</p>

Japan

Antonino Flachi

Muneto Nitta

Satoshi Takada

Ryosuke Yoshii

Fermion Casimir effect and magnetic Larkin-Ovchinnikov phases

We investigate spatially inhomogeneous solutions in a top-down holographic model: the D3/D7 model which provides a holographic description of the chiral phase transition for a finite external magnetic field, chemical potential, and temperature. We numerically find a domain wall (or kink) solution in the three dimensional space, which incorporates between the chiral symmetry broken phase at the spatial infinity, under the homogeneous sources. Along with the inhomogeneity of the chiral condensate, the charge density is also spatially modulated. The modulated charge density and finite magnetic field lead to the chiral edge current close to the domain wall. We explore the dependences of those profiles on the chemical potential and temperature near the first and second order phase transition points. Our results indicate that the inhomogeneous solutions we found are in good agreement with those obtained by the Ginzburg-Landau theory in the vicinity of the transition points.

China

Shuta Ishigaki

Masataka Matsumoto

A domain wall and chiral edge currents in holographic chiral phase transitions

Holographic superconductor model represents various inhomogeneous solutions with homogeneous sources. In this paper, we study inhomogeneous structures in the presence of the homogeneous current and the chemical potential. We find single complex kink condensates, multiple complex kinks condensates, and twisted kink crystal condensates in which both the amplitude and the phase of the order parameter modulate in space. Analyzing the gauge-invariant phase difference in the single complex kink condensates, we find the nonmonotonic behavior with respect to the current. We confirm that the multiple complex kinks condensates and the twisted kink crystal condensates are well described by the analytic solutions obtained from the Gross-Neveu model or the Nambu-Jona-Lasinio model. We also analyze the thermodynamic stability of the complex kink(s) condensates by computing the free energy. Our results imply that holographic superconductor model provides the boundary physics which is effectively represented by the Ginzburg-Landau theory with higher corrections.

Twisted kink crystal in holographic superconductor

The theory of superconductivity can be divided into two groups depending on whether it has multi-kink solutions. For example, the BCS theory and the Gross-Neveu model have metastable multi-kink solutions whereas the conventional Ginzburg-Landau theory without higher-derivative interactions does not have any multi-kink solutions. In this paper, we systematically examine the solutions of the holographic superconductor model to find out which group the model falls into. We show that the holographic superconductor model has metastable multi-kink solutions. In this sense, we find that the holographic superconductor model falls into the category of the BCS theory and the Gross- Neveu model. We also find that the holographic superconductor model has kink crystalline condensates which are well-fitted by the Jacobi elliptic functions.

Shin Nakamura

Kink crystalline condensate and multi-kink solution in holographic superconductor

In this work we examine a system consisting of a confined one-dimensional arrangement of atoms that we describe by using the 2-dimensional <math><mi>C</mi><msup><mi>P</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math> model, restricted to an interval and at finite temperature. We develop a method to obtain the bulk and boundary parts of the one-loop effective action as a function of the effective mass of the fluctuations. The formalism has the advantage of allowing for a systematic analysis of a large class of boundary conditions and to model the (adiabatic) response of the ground state to changes in the boundary conditions. In the case of periodic boundary conditions, we find that inhomogeneous phases are disfavored for intervals of large size. Away from periodic boundary conditions, our numerical results show that the ground state has a generic crystal-like structure that can be modulated by variations of the boundary conditions. The results presented here could be relevant for experimental implementations of nonlinear sigma models and could be tested by lattice numerical simulations.

United States

Guglielmo Fucci

Ground state modulations in the <math><mi>C</mi><msup><mi>P</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math> model

In this work, we derive exact self-consistent solutions to the gap equations of the <math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math> model on a finite interval with Dirichlet boundary conditions in the large-N approximation. The solution reproduces the confining phase in the infinite system by taking the appropriate limit. We compute the vacuum energy and the Casimir force and observe that the sign of the force is always attractive.

Casimir force for the Model

We construct self-consistent analytic solutions in the ℂP  N −1 model in the large-N limit, in which more than one Higgs scalar component take values inside a single or multiple soliton on an infinite space or on a ring, or around boundaries of a finite interval.

Self-consistent analytic solutions in twisted ℂPN−1 model in the large-N limit

The quantum ℂP  N −1 model is in the confining (or unbroken) phase with a full mass gap in an infinite space, while it is in the Higgs (broken or deconfinement) phase accompanied with Nambu-Goldstone modes in a finite space such as a ring or finite interval smaller than a certain critical size. We find a new self-consistent exact solution describing a soliton in the Higgs phase of the ℂP  N −1 model in the large-N limit on a ring. We call it a confining soliton. We show that all eigenmodes have real and positive energy and thus it is stable.

Confining solitons in the Higgs phase of ℂP  N −1 model: self-consistent exact solutions in large-N limit

We give, for the first time, self-consistent large- N analytical solutions of inhomogeneous condensates in the quantum ℂP N  − 1 model in the large- N limit. We find a map from a set of gap equations of the ℂP N  − 1 model to those of the Gross-Neveu (GN) model (or the gap equation and the Bogoliubov-de Gennes equation), which enables us to find the self-consistent solutions. We find that the Higgs field of the ℂP N  − 1 model is given as a zero mode of solutions of the GN model, and consequently only topologically non-trivial solutions of the GN model yield nontrivial solutions of the ℂP N  − 1 model. A stable single soliton is constructed from an anti-kink of the GN model and has a broken (Higgs) phase inside its core, in which ℂP N  − 1 modes are localized, with a symmetric (confining) phase outside. We further find a stable periodic soliton lattice constructed from a real kink crystal in the GN model, while the Ablowitz-Kaup-Newell-Segur hierarchy yields multiple solitons at arbitrary separations.

Self-consistent large- N analytical solutions of inhomogeneous condensates in quantum ℂP N  − 1 model

The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time, which could be the reason of the difficulty for the numerical analyses to exhibit the exponential growth.