Kosterlitz-Thouless Research Articles

Kosterlitz-Thouless phase and Zd topological quantum phase

It has been known that encoding Boltzmann weights of a classical spin model in amplitudes of a many-body wave function can provide quantum models whose phase structure is characterized by using classical phase transitions. In particular, such correspondence can lead to find new quantum phases corresponding to well-known classical phases. Here, we investigate this problem for Kosterlitz-Thouless (KT) phase in the d-state clock model where we find a corresponding quantum model constructed by applying a local invertible transformation on a d-level version of Kitaev's Toric code. In particular, we show the ground state fidelity in such quantum model is mapped to the heat capacity of the clock model. Accordingly, we identify an extended topological phase transition in our model in a sense that, for $d \geq 5$, a KT-like quantum phase emerges between a $Z_d$ topological phase and a trivial phase. Then, using a mapping to the correlation function in the clock model, we introduce a non-local (string) observable for the quantum model which exponentially decays in terms of distance between two endpoints of the corresponding string in the $Z_d$ topological phase while it shows a power law behavior in the KT-like phase. Finally, using well-known transition temperatures for d-state clock model we give evidences to show that while stability of both $Z_d$ topological phase and the KT-like phase increases by increasing d, the KT-like phase is even more stable than the $Z_d$ topological phase for large d.

In praise of theorists who build bridges

For Physics Today’s August 2006 issue I wrote a news story about the observation of the Berezinskii-Kosterlitz-Thouless topological phase transition in flattened clouds of ultracold rubidium atoms. As conceived in the early 1970s by Vadim Berezinskii, J. Michael Kosterlitz, and David Thouless, the BKT transition occurs in a two-dimensional lattice model known as XY. When I wrote the story, the transition had already been observed in superfluid films of helium-4 and superconducting films of mercury–xenon alloy.The 2006 experiment was conducted at the École Normale Supérieure (ENS) in Paris by Zoran Hadzibabic, Peter Krüger, Marc Cheneau, Baptiste Battelier, and Jean Dalibard. In summarizing its significance, I wrote, “Their experiment not only confirms BKT theory in a new system, but also reveals for the first time the transition’s microscopic instigators: local topological defects or vortices.”Then as now, the editors who write for Physics Today’s Search and Discovery department choose papers to cover based primarily on the advice of experts. We don’t learn a paper’s backstory until we interview its authors. When I phoned Hadzibabic to ask about his experiment, I found out something unexpected and interesting. To connect the predictions of BKT to the measurements made in the lab, he and his collaborators relied on a method devised by an independent trio of theorists, Anatoli Polkovnikov, Ehud Altman, and Eugene Demler.11. A. Polkovnikov, E. Altman, E. Demler, Proc. Natl. Acad. Sci. USA 103, 6125 (2006). https://doi.org/10.1073/pnas.0510276103 As in the plot of a B movie, a preprint from the three theorists arrived at ENS just as the experimenters were wondering how to interpret their data. “Within a minute, we wrote back to say we have the same equation on the board!” Hadzibabic told me.Polkovnikov, Altman, and Demler took up the challenge after reading a previous paper by the ENS group. The correlations that embody the BKT transition are manifest when two pancakes of ultracold atoms are released from their traps and allowed to interfere with each other. Those correlations, Polkovnikov, Altman, and Demler realized, depend on system size in an experimentally accessible way that can be tied directly to BKT physics.One of my favorite examples of theorists building bridges between experiment and theory comes from solar physics. Our star’s photosphere is so dense that a photon emitted at the core takes 170 000 years of repeated scattering off electrons and ions before it escapes. Some of the sunlight that fell on your face today was born when Neanderthals hunted and gathered in Europe! If astronomers relied only on solar photons, they’d have little to test their theories of the Sun’s composition and structure. Fortunately, prompt, direct information from the Sun’s interior reaches us in two other forms: acoustic oscillations, which I discuss here, and solar neutrinos.The Sun is a self-gravitating, differentially rotating ball of plasma that quivers in myriad acoustic modes, some of which entrain matter deep in the solar interior. Those helioseismic signals are manifest as Doppler shifts of certain spectral lines at localized patches in the photosphere. Remarkably, those seemingly limited data are sufficient to constrain models of the Sun—provided someone goes to the trouble of collating, among other things, all the nuclear reactions of all the chemical elements at all levels of the Sun and then creating models that predict what can be detected.The effort of helioseismologists is prodigious. To give one example, in a 1991 paper, Jørgen Christensen-Dalsgaard, Douglas Gough, and Michael Thompson derived the depth of the Sun’s convection zone.22. J. Christensen-Dalsgaard, D. O. Gough, M. J. Thompson, Astrophys. J. 378, 413 (1991). https://doi.org/10.1086/170441 To get to their answer of 0.287 ± 0.003 solar radii, they started with the Schwarzschild criterion, which specifies when convention ensues. Over the course of 25 pages, the three theorists described how they built two models of the Sun and then computed the sound speed, which can be inferred from helioseismic observations.I mention the work of those bridge-building theorists not just to praise them and their kind. When I look back at my own physics education, I don’t recall being taught what might be called applied theory. Polkovnikov has recognized the same gap. When I sought his comments about this editorial, he told me that he, Marcos Rigol, and Pieter Claeys are writing a new quantum mechanics textbook that will include realistic, nonideal examples. “When the formalism is too abstract and is related to very particular experiments,” he wrote, “it creates a gap in intuition and a gap in connecting theory and experiment.”ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. A. Polkovnikov, E. Altman, E. Demler, Proc. Natl. Acad. Sci. USA 103, 6125 (2006). https://doi.org/10.1073/pnas.0510276103, Google ScholarCrossref2. J. Christensen-Dalsgaard, D. O. Gough, M. J. Thompson, Astrophys. J. 378, 413 (1991). https://doi.org/10.1086/170441, Google ScholarCrossref© 2020 American Institute of Physics.

Nonlinear current–voltage characteristics in electron doped YBCO superconductor below the critical temperature

Below the critical temperature, several Ce substituted YBa2Cu3O7−δ superconductors exhibit nonlinear current–voltage (IV) characteristics. We have extracted an exponent following the idea of the Kosterlitz–Thouless transition using IV over a wide range of temperatures around the critical temperature. The superfluid phase stiffness has been extracted within the framework of the Ambegaokar–Halperin–Nelson–Siggia theory. In a combined plot of the superfluid density and resistivity as a function of temperature, we observe that the superfluid density varies widely corresponding to the zero resistive states. The shift of the temperature corresponding to the crossing point with the electron doping level has been found to be very strong.

Strong pairing in two dimensions: pseudogaps, domes, and other implications

This paper addresses the transition from the normal to the superfluid state in strongly correlated two dimensional fermionic superconductors and Fermi gases. We arrive at the Berezinskii–Kosterlitz–Thouless (BKT) temperature TBKT as a function of attractive pairing strength by associating it with the onset of ‘quasi-condensation’ in the normal phase. Our approach builds on a criterion for determining the BKT transition temperature for atomic gases which is based on a well established quantum Monte Carlo analysis of the phase space density. This latter quantity, when derived from BCS–BEC crossover theory for fermions, leads to non-monotonic behavior for TBKT as a function of the attractive interaction or inverse scattering length. In Fermi gases, this implies a robust superconducting dome followed by a long tail from the flat BEC asymptote, rather similar to what is observed experimentally. For lattice systems we find that TBKT has an absolute maximum of the order of 0.1EF. We discuss how our results compare with those derived from the Nelson–Kosterlitz criterion based on the mean field superfluid density and the approach to the transition from below. While there is agreement in the strict mean-field BCS regime at weak coupling, we find that at moderate pairing strength bosonic excitations cause a substantial increase in TBKT followed by an often dramatic decrease before the system enters the BEC regime.

The free energy of the two-dimensional dilute Bose gas. II. Upper bound

We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for a2ρ ≪ 1 and βρ ≳ 1, the free energy per unit volume differs from the one of the non-interacting system by at most 4πρ2|lna2ρ|−1(2−[1−βc/β]+2) to leading order, where a is the scattering length of the two-body interaction potential, ρ is the density, β is the inverse temperature, and βc is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. In combination with the corresponding matching lower bound proved by Deuchert et al. [Forum Math. Sigma 8, e20 (2020)], this shows equality in the asymptotic expansion.

Thermal Kosterlitz–Thouless transitions in the 1/r2 long-range ferromagnetic quantum Ising chain revisited

For the inverse square long-range ferromagnetic Ising chain in a transverse field, the thermal phase boundary of the floating Kosterlitz–Thouless phase is obtained for several values of the transverse field down to the quantum critical point. The sharp domain walls in the classical model are increasingly smeared out by the transverse field, which is evidenced by a pronounced broadening of the non-universal bump in the specific heat. The discernability of KT critical scaling in finite-size simulations is discussed.

Dynamical scaling analysis of symmetry breaking for the antiferromagnetic triangular Heisenberg model in a uniform magnetic field

Phase transitions for the two-dimensional antiferromagnetic triangular Heisenberg model in a uniform field are primarily analyzed using dynamical scaling analysis. The system has ${Z}_{3}$ and $O(2)$ symmetries. The transition temperatures and types of transition are investigated using the relaxation of order parameters and the asymptotic behaviors of relaxation time around transition temperatures. In the low-field range, the order parameters for the ${Z}_{3}$ and $O(2)$ symmetries exhibit the behavior typical of a second-order transition and Kosterlitz-Thouless (KT) transition, respectively, with distinct transition temperatures. In the high-field range, an order parameter for the ${Z}_{3}$ symmetry exhibits the behavior of a second-order transition, while another order parameter related to the components perpendicular to the field demonstrates that the relaxation time diverges with an asymptotic form, indicating a second-order transition rather than a KT transition. These transition temperatures are estimated to be close. Critical exponents for these two order parameters are estimated separately by calculating the relaxation of fluctuations in the high-field range. It was confirmed that the critical exponents for both of these order parameters change continuously as a function of field, and they are distinct from each other.

Quantum Berezinskii–Kosterltz–Thouless transition for topological insulator

ABSTRACTWe consider the interacting helical liquid system at the one-dimensional edge of a two-dimensional topological insulator, coupled to an external magnetic field and s-wave superconductor and map it to an XYZ spin chain system. This model undergoes quantum Berezinskii–Kosterlitz–Thouless (BKT) transition with two limiting conditions. We derive the renormalization group (RG) equations explicitly and also present the flow lines behavior. We also present the behavior of RG flow lines based on the exact solution. We observe that the physics of Majorana fermion zero modes and the gaped Ising-ferromagnetic phase, which appears in a different context. We observe that the evidence of gapless helical Luttinger liquid phase as a common non-topological quantum phase for both quantum BKT transitions. We explain analytically and physically that there is no Majorana-Ising transition. In the presence of chemical potential, the system shows the commensurate to incommensurate transition.

Enhancement of Coherence Length and Nonlinear Current-Voltage (IV) Characteristics of Ag-Doped GBCO

The resistivity as a function of temperature of GdBa2Cu3-xAgxO7-δ superconductors reveals that the critical temperature, Tc, increases with the concentration of Ag (x). An equation has been suggested to understand the behaviour of Tc (x). The coherence length has been extracted on the basis of the three dimensional Aslamazov-Larkin (3DAL) equation to understand the enhancement in the critical temperature. We have studied current-voltage (IV) curves below Tc to understand the nonlinearity observed in doped samples. An exponent has been extracted to explore the possibility of Kosterlitz-Thouless (KT) transition and the role of the coherence length in the nonlinear behaviour.

Entanglement-Entropy Study of Phase Transitions in Six-State Clock Model

The Berezinskii-Kosterlitz-Thouless (BKT) transitions of the six-state clock model on the square lattice are investigated by means of the corner-transfer matrix renormalization group method. A classical analog of the entanglement entropy $S( L, T )$ is calculated for $L \times L$ square system up to $L = 129$, as a function of temperature $T$. The entropy exhibits a peak at $T = T^*_{~}( L )$, where the temperature depends on both $L$ and the boundary conditions. Applying the finite-size scaling to $T^*_{~}( L )$ and assuming presence of the BKT transitions, the two distinct phase-transition temperatures are estimated to be $T_1^{~} = 0.70$ and $T_2^{~} = 0.88$. The results are in agreement with earlier studies. It should be noted that no thermodynamic functions have been used in this study.

The Berezinskii–Kosterlitz–Thouless topological phase transition in the Josephson medium of granular high-temperature superconductors under the effect of magnetic and electric fields

This study establishes the identity of the topological phases that emerge in the Josephson medium of granular high-temperature superconductors YBa2Cu3O7–δ in the process of the Berezinskii–Kosterlitz–Thouless (BKT) phase transitions, under the influence of an external magnetic field and transport current. The nature of the topological phases arising as a result of the BKT phase transition was found to be independent of the type of external effect.

Dynamical scaling analysis on critical universality class for fully frustrated XY models in two dimensions

Critical properties of the fully frustrated XY models on square and triangular lattices are investigated by means of the nonequilibrium relaxation (NER) method. We examine the validity of the conclusion on the universality class of the chiral transition previously obtained by the NER method, in which that belongs to a non-Ising-type one. To clarify it, we analyze the NER of fluctuations for longer Monte Carlo steps on lager lattices than those performed previously. The calculation is made at the chiral transition temperatures estimated carefully by the use of recently improved dynamical scaling analysis. The result indicates that the asymptotic behavior of the time-dependent exponents, which should converge to the critical exponents, show the same tendency as those obtained previously in shorter times. We also apply the improved dynamical scaling analysis to the estimation of Kosterlitz-Thouless (KT) transitions and confirm the existence of the double transitions for the chiral and KT phases with more reliable estimations.

Strange metal state near quantum superconductor-metal transition in thin films

We develop a theory of quantum T=0 phase transition (q–SMT) between metal and superconducting ground states in a two-dimensional metal with frozen-in spatial fluctuations δλ(r) of the Cooper attraction constant. When strength of fluctuations δλ(r) exceeds some critical magnitude, usual mean-field-like scenario of the q–SMT breaks down due to spontaneous formation of local droplets of superconducting phase. The density of these droplets grows exponentially with the increase of average attraction constant λ. Interaction between the droplet’s order parameters is due to proximity effect via normal metal and scales with distance ∝1∕rβ , with 2<β≤3. We account for this interaction by means of a real-space strong-disorder renormalization group (RG). Near the q–SMT the RG flow is, formally, a dual equivalent of the Kosterlitz–Thouless RG. The corresponding line of fixed points describes a Griffiths phase of a metal with large fractal clusters of superconducting islands. Typical number of islands in a cluster grows as Nδ∼1∕δ, where 0<δ≪1 is the distance to the critical point. Superconducting side is described by a runaway of RG trajectories into the strong-coupling region. Close to the transition point on the SC side, 0<−δ≪1, RG trajectories possess an extremum as function of the RG parameter |δ|1∕2ln(1∕Tτ). It results in a wide temperature range where physical properties are nearly T-independent. This observation may be relevant to the understanding of a strange metal state frequently observed near q–SMT.

Superfluids as higher-form anomalies

We recast superfluid hydrodynamics as the hydrodynamic theory of a system with an emergent anomalous higher-form symmetry. The higher-form charge counts the winding planes of the superfluid - its constitutive relation replaces the Josephson relation of conventional superfluid hydrodynamics. This formulation puts all hydrodynamic equations on equal footing. The anomalous Ward identity can be used as an alternative starting point to prove the existence of a Goldstone boson, without reference to spontaneous symmetry breaking. This provides an alternative characterization of Landau phase transitions in terms of higher-form symmetries and their anomalies instead of how the symmetries are realized. This treatment is more general and, in particular, includes the case of BKT transitions. As an application of this formalism we construct the hydrodynamic theories of conventional (0-form) and 1-form superfluids.

Berezinskii–Kosterlitz–Thouless transition in an Al superconducting nanofilm grown on GaAs by molecular beam epitaxy

We have performed extensive transport experiments on a 4 nm thick aluminum (Al) superconducting film grown on a GaAs substrate by molecular beam epitaxy (MBE). Nonlinear current–voltage (I–V) measurements on such a MBE-grown superconducting nanofilm show that V ∼ I3, which is evidence for the Berezinskii–Kosterlitz–Thouless (BKT) transition, both in the low-voltage (TBKT ≈ 1.97 K) and high-voltage regions (TBKT ≈ 2.17 K). In order to further study the two regions where the I–V curves are BKT-like, our experimental data are fitted to the temperature-induced vortices/antivortices unbinding model as well as the dynamical scaling theory. It is found that the transition temperature obtained in the high-voltage region is the correct TBKT as confirmed by fitting the data to the aforementioned models. Our experimental results unequivocally show that I–V measurements alone may not allow one to determine TBKT for superconducting transition. Therefore, one should try to fit one’s results to the temperature-induced vortices/antivortices unbinding model and the dynamical scaling theory to accurately determine TBKT in a two-dimensional superconductor.

Isotropic Lifshitz scaling in four dimensions

The presence of isotropic Lifshitz points for a [Formula: see text]-symmetric scalar theory is investigated with the help of the Functional Renormalization Group. In particular, at the supposed lower critical dimension [Formula: see text], evidence for a continuous line of fixed points is found for the [Formula: see text] theory, and the observed structure presents clear similarities with the properties observed in the two-dimensional Berezinskii–Kosterlitz–Thouless phase.

Nonlinear Current–Voltage Characteristics in Composite Systems Consisting of YBCO Superconductor and Y2CoMnO6

The resistive properties of several (1) pure YBCO superconductors and (2) composite systems consisting of YBCO and the low-density multiferroic component of Y2CoMnO6 (YCMO) have been studied. The current–voltage (IV) characteristics of underdoped YBCO superconductors have been studied by adding low-density multiferroic YCMO as an intergranular network. The nonlinear nature of IV below the critical temperature has been used to extract an exponent related to the superfluid stiffness. Variations of the exponent with temperature have been studied within the framework of Kosterlitz–Thouless (KT) transition. Within the framework of the Ambegaokar–Halperin–Nelson–Siggia (AHNS) theory, we have discussed controlling the superfluid phase stiffness as a function of temperature in different composite systems. The magnetic nature of the intergranular networks in YBCO strongly influences the superfluid phase stiffness.

The Tsallis entropy and the BKT-like phase transition in the impact parameter space for pp and collisions

In this paper, one uses the Tsallis entropy (TE) in the impact parameter space to study pp and inelastic overlap function and the energy density filling up mechanism responsible by the so-called black disk limit as the energy increases. The TE is non-additive and non-extensive and these features are of fundamental importance since the internal constituents of pp and are strongly correlated and also the existence of the multifractal character of the total cross-section. The entropy approach presented here takes into account a phase transition occurring inside the hadrons as the energy increases. This phase transition in the impact parameter space is quite similar to the Berezinskii–Kosterlitz–Thouless phase transition, possessing also a topological feature due to the multifractal dimension of the total cross-sections in pp and scattering.

THE FREE ENERGY OF THE TWO-DIMENSIONAL DILUTE BOSE GAS. I. LOWER BOUND

We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density$\unicode[STIX]{x1D70C}$and inverse temperature$\unicode[STIX]{x1D6FD}$differs from the one of the noninteracting system by the correction term$4\unicode[STIX]{x1D70B}\unicode[STIX]{x1D70C}^{2}|\ln \,a^{2}\unicode[STIX]{x1D70C}|^{-1}(2-[1-\unicode[STIX]{x1D6FD}_{\text{c}}/\unicode[STIX]{x1D6FD}]_{+}^{2})$. Here,$a$is the scattering length of the interaction potential,$[\cdot ]_{+}=\max \{0,\cdot \}$and$\unicode[STIX]{x1D6FD}_{\text{c}}$is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. The result is valid in the dilute limit$a^{2}\unicode[STIX]{x1D70C}\ll 1$and if$\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70C}\gtrsim 1$.

Topological Transition in a 3 nm Thick Al Film Grown by Molecular Beam Epitaxy

We have performed detailed transport measurements on a 3 nm thick (as-grown) Al film on GaAs prepared by molecular beam epitaxy (MBE). Such an epitaxial film grown on a GaAs substrate shows the Berezinskii-Kosterlitz-Thouless (BKT) transition, a topological transition in two dimensions. Our experimental data shows that the MBE-grown Al nanofilm is an ideal system for probing interesting physical phenomena such as the BKT transition and superconductivity. The increased superconductor transition temperature (~2.4 K) compared to that of bulk Al (1.2 K), together with the ultrathin film quality, may be advantageous for future superconductor-based quantum devices and quantum information technology.