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Tetsuo Deguchi

Publications and source records attributed to Tetsuo Deguchi.

At least 37 records · Page 2Linked to original sources
arXiv2015

Crossover temperature of the spin-1/2 XXZ chain with an impurity

We study exactly the effect of an impurity in the interacting quantum spin chain at low temperature by solving the integrable spin-1/2 XXZ periodic chain with an impurity through the algebraic and thermal Bethe ansatz methods. In particular, we investigate how the crossover temperature for the impurity specific heat depends on the impurity parameter, i.e. the coupling of the impurity to other spins, and show that it is consistent with the analytic expression that is obtained by setting the impurity susceptibility to be proportional to the inverse of the crossover temperature. In the model, two types of crossover behavior appear: one from the high-temperature regime to the low-temperature Kondo regime and another from the N-site homogeneous chain to the (N-1)-site chain with a decoupled free impurity spin, with respect to the temperature and the impurity parameter, respectively.

cond-mat.stat-mech↗
arXiv2015

Exact quantum numbers of collapsed and non-collapsed 2-string solutions in the Heisenberg spin chain

Every solution of the Bethe-ansatz equations (BAE) is characterized by a set of quantum numbers, by which we can evaluate it numerically. However, no general rule is known how to give quantum numbers for the physical solutions of BAE. For the spin-1/2 XXX chain we rigorously derive all the quantum numbers for the complete set of the Bethe-ansatz eigenvectors in the two down-spin sector with any chain length $N$. Here we obtain them both for real and complex solutions. Consequently, we prove the completeness of the Bethe ansatz and give an exact expression for the number of real solutions which correspond to collapsed bound-state solutions (i.e., 2-string solutions) in the sector: $2[ (N-1)/2 - (N/π) \tan^{-1}(\sqrt{N-1})]$ in terms of Gauss' symbol. Moreover, we prove in the sector the scheme conjectured by Takahashi for solving BAE systematically. We also suggest that by applying the present method we can derive the quantum numbers for the spin-1/2 XXZ chain.

cond-mat.stat-mech↗
arXiv2015

Power-law relaxation behavior of an initially localized state in the spin-1/2 Heisenberg chain

We present power-law relaxation behavior of the local magnetizations in the equilibration dynamics of the spin-1/2 Heisenberg spin chain as an isolated integrable quantum system. We perform the exact time evolution of the expectation values of the local spin operators by evaluating them with the determinant formula of the form factors. We construct such an initial quantum state that has a localized profile of the local magnetizations, and perform the exact time evolution over a very long period of time. We show that the local magnetization relaxes as some power of the time variable with no definite time scale, while the fidelity relaxes very fast with its relaxation time being proportional to the inverse of the energy width, i.e. the Boltzmann time.

cond-mat.stat-mech↗
arXiv2015

Recurrence time in the quantum dynamics of the 1D Bose gas

Recurrence time is evaluated for some initial quantum states in the one-dimensional Bose gas with repulsive short-range interactions. In the relatively strong and weak coupling cases some different types of initial states show almost complete recurrence and the estimates of recurrence time are proportional to some powers of the system size at least in some range of the system size. They are much longer than in the case of free particles such as 100 times. In the free-bosonic and free-fermionic regimes we evaluate the recurrence time rigorously, which is proportional to the square of the system size. The estimate of recurrence time is given by the order of ten milliseconds in the corresponding experimental systems of cold atoms trapped in one dimension of ten micrometers in length. It is much shorter than the estimate in a generic quantum many-body system, which may be as long as the age of the universe.

cond-mat.quant-gas↗
arXiv2015

Singular eigenstates in the even(odd) length Heisenberg spin chain

We study the implications of the regularization for the singular solutions on the even(odd) length spin-1/2 XXX chains in some specific down-spin sectors. In particular, the analytic expressions of the Bethe eigenstates for three down-spin sector have been obtained along with their numerical forms in some fixed length chains. For an even-length chain if the singular solutions \{λ_α\} are invariant under the sign changes of their rapidities {λ_α\}=\{-λ_α\} , then the Bethe ansatz equations are reduced to a system of (M-2)/2 ((M-3)/2) equations in an even (odd) down-spin sector. For an odd N length chain in the three down-spin sector, it has been analytically shown that there exist singular solutions in any finite length of the spin chain of the form N= 3\left(2k+1\right) with k=1, 2, 3, \cdots. It is also shown that there exist no singular solutions in the four down-spin sector for some odd-length spin-1/2 XXX chains.

hep-th↗
arXiv2015

Non Self-conjugate Strings, Singular Strings and Rigged Configurations in the Heisenberg Model

We observe a different type of complex solutions in the isotropic spin-1/2 Heisenberg chain starting from N=12, where the central rapidity of some of the odd-length strings becomes complex making not all the strings self-conjugate individually. We show that there are at most (N-2)/2 singular solutions for M=4, M=5 down-spins and at most (N^2-6N+8)/8 singular solutions for M=6, M=7 down-spins in an even-length chain with N \geq 2M. Correspondence of the non self-conjugate string solutions and the singular string solutions to the rigged configurations has also been shown.

hep-th↗
arXiv2015

Heisenberg model and Rigged Configurations

We show a correspondence of all the solutions of the spin-1/2 isotropic Heisenberg model for N=12 to the rigged configurations based on the comparison of the set of Takahashi quantum numbers in lexicographical order with the set of riggings of the rigged configurations in co-lexicographical order.

hep-th↗
arXiv2014

Finite-temperature behavior of an impurity in the spin-1/2 XXZ chain

We study the zero- and the finite-temperature behavior of the integrable spin-1/2 XXZ periodic chain with an impurity by the algebraic and thermal Bethe ansatz methods. We evaluate the impurity local magnetization at zero temperature analytically and derive the impurity susceptibility exactly from it. In the graphs of the impurity specific heat versus temperature, we show how the impurity spin becomes more liberated from the bulk many-body effect as the exchange coupling between the impurity spin and other spins decreases, and also that in low temperature it couples strongly to them such as the Kondo effect. Thus, we observe not only the crossover behavior from the high- to the low-temperature regime but also another one from the $N$-site chain to the $(N-1)$-site chain with a free impurity spin. We also show that the estimate of the Wilson ratio at a given low temperature is independent of the impurity parameter if its absolute value is small enough with respect to the temperature, and the universality class is described by the XXZ anisotropy in terms of the dressed charge.

cond-mat.stat-mech↗
arXiv2014

Exact quantum dynamics of yrast states in the finite 1D Bose gas

We demonstrate that the quantum dynamics of yrast states in the one-dimensional (1D) Bose gas gives an illustrative example to equilibration of an isolated quantum many-body system. We first formulate the energy spectrum of yrast states in terms of the dressed energy by applying the method of finite-size corrections. We then review the exact time evolution of quantum states constructed from yrast states shown by the Bethe ansatz. In time evolution the density profile of an initially localized quantum state constructed from yrast states collapses into a flat profile in the case of a large particle number such as N=1000, while recurrence of the localized state occurs in the case of a small particle number such as N=20. We suggest that the dynamical relaxation behavior for the large N case is consistent with the viewpoint of typicality for generic quantum states: the expectation values of local operators valuated in most of quantum states are very close to those of the micro-canonical ensemble.

cond-mat.quant-gas↗
arXiv2013

Exponents of interchain correlation for self-avoiding walks and knotted self-avoiding polygons

We show numerically that critical exponents for two-point interchain correlation of an infinite chain characterize those of finite chains in Self-Avoiding Walk (SAW) and Self-Avoiding Polygon (SAP) under a topological constraint. We evaluate short-distance exponents $θ(i,j)$ through the probability distribution functions of the distance between the $i$th and $j$th vertices of $N$-step SAW (or SAP with a knot) for all pairs ($1 \le i, j \le N$). We construct the contour plot of $θ(i,j)$, and express it as a function of $i$ and $j$. We suggest that it has quite a simple structure. Here exponents $θ(i,j)$ generalize des Cloizeaux's three critical exponents for short-distance interchain correlation of SAW, and we show the crossover among them. We also evaluate the diffusion coefficient of knotted SAP for a few knot types, which can be calculated with the probability distribution functions of the distance between two nodes.

cond-mat.stat-mech↗
arXiv2013

Finite-size scaling behavior of Bose-Einstein condensation in the 1D Bose gas

Through exact numerical solutions we show Bose-Einstein condensation (BEC) for the one-dimensional (1D) bosons with repulsive short-range interactions at zero temperature by taking a particular large size limit. Following the Penrose-Onsager criterion of BEC, we define condensate fraction by the fraction of the largest eigenvalue of the one-particle reduced density matrix. % We show the finite-size scaling behavior such that condensate fraction is given by a scaling function of one-variable: interaction parameter multiplied by a power of particle number. Condensate fraction is nonzero and constant for any large value of particle number or system size, if the interaction parameter is proportional to the negative power of particle number. %Here the interaction parameter is defined by the coupling constant of the delta-function potentials devided by the density. %With the scaling behavior we derive various themodynamic limits where condensate fraction is constant for any large system size; for instance, it is the case even in the system of a finite particle number.

cond-mat.quant-gas↗
arXiv2012

Probability Theory of Random Polygons from the Quaternionic Viewpoint

We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Knutson and Hausmann using the Hopf map on quaternions, from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons which comes from a real Stiefel manifold. The edgelengths of polygons sampled according to our measures obey beta distributions. This makes our polygon measures different from those usually studied, which have Gaussian or fixed edgelengths. One advantage of our measures is that we can explicitly compute expectations and moments for chordlengths and radii of gyration. Another is that direct sampling according to our measures is fast (linear in the number of edges) and easy to code. Some of our methods will be of independent interest in studying other probability measures on polygon spaces. We define an edge set ensemble (ESE) to be the set of polygons created by rearranging a given set of n edges. A key theorem gives a formula for the average over an ESE of the squared lengths of chords skipping k vertices in terms of k, n, and the edgelengths of the ensemble. This allows one to easily compute expected values of squared chordlengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges.

math.DG↗
arXiv2012

Quantum dark solitons in the 1D Bose gas and the superfluid velocity

We give explicit connections of quantum one-hole excited states to classical solitons for the one-dimensional Bose gas with repulsive short-range interactions. We call the quantum states connected to classical solitons the quantum soliton states. We show that the matrix element of the canonical field operator between quantum soliton states with $N-1$ and $N$ particles is given by a dark soliton of the Gross-Pitaevskii equation in the weak coupling case. We suggest that the matrix element corresponds to the order parameter of BEC in the quantum soliton state. The result should be useful in the study of many-body effects in Bose-Einstein condensation and superfluids. For instance, we derive the superfluid velocity for a quantum soliton state.

cond-mat.quant-gas↗
arXiv2012

Exact relaxation dynamics of a localized many-body state in the 1D bose gas

Through an exact method we numerically solve the time evolution of the density profile for an initially localized state in the one-dimensional bosons with repulsive short-range interactions. We show that a localized state with a density notch is constructed by superposing one-hole excitations. The initial density profile overlaps the plot of the squared amplitude of a dark soliton in the weak coupling regime. We observe the localized state collapsing into a flat profile in equilibrium for a large number of particles such as N=1000. The relaxation time increases as the coupling constant decreases, which suggests the existence of off-diagonal long-range order. We show a recurrence phenomenon for a small number of particles such as N=20.

cond-mat.quant-gas↗
arXiv2011

Reduction formula of form factors for the integrable spin-s XXZ chains and application to the correlation functions

For the integrable spin-s XXZ chain we express explicitly any given spin-$s$ form factor in terms of a sum over the scalar products of the spin-1/2 operators. Here they are given by the operator-valued matrix elements of the monodromy matrix of the spin-1/2 XXZ spin chain. In the paper we call an arbitrary matrix element of a local operator between two Bethe eigenstates a form factor of the operator. We derive all important formulas of the fusion method in detail. We thus revise the derivation of the higher-spin XXZ form factors given in a previous paper. The revised method has several interesting applications in mathematical physics. For instance, we express the spin-$s$ XXZ correlation function of an arbitrary entry at zero temperature in terms of a sum of multiple integrals.

cond-mat.stat-mech↗
arXiv2011

Quantum Group U_q(sl(2)) Symmetry and Explicit Evaluation of the One-Point Functions of the Integrable Spin-1 XXZ Chain

We show some symmetry relations among the correlation functions of the integrable higher-spin XXX and XXZ spin chains, where we explicitly evaluate the multiple integrals representing the one-point functions in the spin-1 case. We review the multiple-integral representations of correlation functions for the integrable higher-spin XXZ chains derived in a region of the massless regime including the anti-ferromagnetic point. Here we make use of the gauge transformations between the symmetric and asymmetric R-matrices, which correspond to the principal and homogeneous gradings, respectively, and we send the inhomogeneous parameters to the set of complete 2s-strings. We also give a numerical support for the analytical expression of the one-point functions in the spin-1 case.

cond-mat.stat-mech↗
arXiv2010

Exact Yrast Spectra of Cold Atoms on a Ring

We propose a methodology to construct excited states with a fixed angular momentum, namely, "yrast excited states" of finite-size one-dimensional bosonic systems with periodic boundary conditions. The excitation energies such as the first yrast excited energy are calculated through the system-size asymptotic expansion and expressed analytically by dressed energy. Interestingly, they are grouped into sets of almost degenerate energy levels. The low-lying excitation spectrum near the yrast state is consistent with the $ U(1)$ conformal field theories if the total angular momentum is given by an integral multiple of particle number; i.e., if the system is supercurrent.

cond-mat.quant-gas↗