In classical finite-range spin systems, especially those with disorder such as spin glasses, a low-temperature Gibbs state may be a mixture of a number of pure or ordered states; the complexity of the Gibbs state has been defined in the past roughly
as the logarithm of this number, assuming the question is meaningful in a finite system. As non-trivial pure-state structure is lost in finite size, in a recent paper [Phys. Rev. E 101, 042114 (2020)] Holler and the author introduced a definition of the complexity of an infinite-size Gibbs state as the mutual information between the pure state and the spin configuration in a finite region, and applied this also within a metastate construction. (A metastate is a probability distribution on Gibbs states.) They found an upper bound on the complexity for models of Ising spins in which each spin interacts with only a finite number of others, in terms of the surface area of the region, for all $Tgeq 0$. In the present paper, the complexity of a metastate is defined likewise in terms of the mutual information between the Gibbs state and the spin configuration. Upper bounds are found for each of these complexities for general finite-range (i.e. short- or long-range, in a sense we define) mixed $p$-spin interactions of discrete or continuous spins (such as $m$-vector models), but only for $T>0$. For short-range models, the bound reduces to the surface area. For long-range interactions, the definition of a Gibbs state has to be modified, and for these models we also prove that the states obtained within the metastate constructions are Gibbs states under the modified definition. All results are valid for a large class of disorder distributions.
Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations follo
wing a quench from infinite temperature to a temperature well below the spin-glass transition temperature $T_c$ for a one-dimensional Ising spin glass model with diluted long-range interactions. In this model, the probability $P_{ij}$ that an edge ${i,j}$ has nonvanishing interaction falls as a power-law with chord distance, $P_{ij}propto1/R_{ij}^{2sigma}$, and we study a range of values of $sigma$ with $1/2<sigma<1$. We consider a correlation function $C_{4}(r,t)$. A dynamic correlation length that shows power-law growth with time $xi(t)propto t^{1/z}$ can be identified in the data and, for large time $t$, $C_{4}(r,t)$ decays as a power law $r^{-alpha_d}$ with distance $r$ when $rll xi(t)$. The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent $alpha_d$ differentiates between a trivial MMAS ($alpha_d=0$), as expected in the droplet picture of spin glasses, and a nontrivial MMAS ($alpha_d e 0$), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero $alpha_d$ even in the regime $sigma >2/3$ which corresponds to short-range systems below six dimensions. For $sigma < 2/3$, the decay exponent $alpha_d$ follows the RSB prediction for the decay exponent $alpha_s = 3 - 4 sigma$ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches $alpha_d=1-sigma$ in the regime $2/3<sigma<1$; however, it deviates from both lines in the vicinity of $sigma=2/3$.
The de Almeida-Thouless (AT) line in Ising spin glasses is the phase boundary in the temperature $T$ and magnetic field $h$ plane below which replica symmetry is broken. Using perturbative renormalization group (RG) methods, we show that when the dim
ension $d$ of space is just above $6$ there is a multicritical point (MCP) on the AT line, which separates a low-field regime, in which the critical exponents have mean-field values, from a high-field regime where the RG flows run away to infinite coupling strength; as $d$ approaches $6$ from above, the location of the MCP approaches the zero-field critical point exponentially in $1/(d-6)$. Thus on the AT line perturbation theory for the critical properties breaks down at sufficiently large magnetic field even above $6$ dimensions, as well as for all non-zero fields when $dleq 6$ as was known previously. We calculate the exponents at the MCP to first order in $varepsilon=d-6>0$. The fate of the MCP as $d$ increases from just above 6 to infinity is not known.
In a tight-binding lattice model with $n$ orbitals (single-particle states) per site, Wannier functions are $n$-component vector functions of position that fall off rapidly away from some location, and such that a set of them in some sense span all s
tates in a given energy band or set of bands; compactly-supported Wannier functions are such functions that vanish outside a bounded region. They arise not only in band theory, but also in connection with tensor-network states for non-interacting fermion systems, and for flat-band Hamiltonians with strictly short-range hopping matrix elements. In earlier work, it was proved that for general complex band structures (vector bundles) or general complex Hamiltonians---that is, class A in the ten-fold classification of Hamiltonians and band structures---a set of compactly-supported Wannier functions can span the vector bundle only if the bundle is topologically trivial, in any dimension $d$ of space, even when use of an overcomplete set of such functions is permitted. This implied that, for a free-fermion tensor network state with a non-trivial bundle in class A, any strictly short-range parent Hamiltonian must be gapless. Here, this result is extended to all ten symmetry classes of band structures without additional crystallographic symmetries, with the result that in general the non-trivial bundles that can arise from compactly-supported Wannier-type functions are those that may possess, in each of $d$ directions, the non-trivial winding that can occur in the same symmetry class in one dimension, but nothing else. The results are obtained from a very natural usage of algebraic $K$-theory, based on a ring of polynomials in $e^{pm ik_x}$, $e^{pm ik_y}$, . . . , which occur as entries in the Fourier-transformed Wannier functions.
Even if a noninteracting system has zero Berry curvature everywhere in the Brillouin zone, it is possible to introduce interactions that stabilise a fractional Chern insulator. These interactions necessarily break time-reversal symmetry (either spont
aneously or explicitly) and have the effect of altering the underlying band structure. We outline a number of ways in which this may be achieved, and show how similar interactions may also be used to create a (time-reversal symmetric) fractional topological insulator. While our approach is rigorous in the limit of long range interactions, we show numerically that even for short range interactions a fractional Chern insulator can be stabilised in a band with zero Berry curvature.
We show that the topological central charge of a topological phase can be directly accessed from the ground-state wavefunctions for a system on a surface as a Berry curvature produced by adiabatic variation of the metric on the surface, at least up t
o addition of another topological invariant that arises in some cases. For trial wavefunctions that are given by conformal blocks (chiral correlation functions) in a conformal field theory (CFT), we carry out this calculation analytically, using the hypothesis of generalized screening. The topological central charge is found to be that of the underlying CFT used in the construction, as expected. The calculation makes use of the gravitational anomaly in the chiral CFT. It is also shown that the Hall conductivity can be obtained in an analogous way from the U($1$) gauge anomaly.
The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the continuum limit by a logarithmic conformal field theory (LCFT) at central charge c=0. This theory corresponds to th
e strong coupling regime of a sigma model on the complex projective superspace $mathbb{CP}^{1|1} = mathrm{U}(2|1) / (mathrm{U}(1) times mathrm{U}(1|1))$, and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this continuum limit by extending to the periodic case the approach of [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the continuum limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.
Parisis formal replica-symmetry--breaking (RSB) scheme for mean-field spin glasses has long been interpreted in terms of many pure states organized ultrametrically. However, the early version of this interpretation, as applied to the short-range Edwa
rds-Anderson model, runs into problems because as shown by Newman and Stein (NS) it does not allow for chaotic size dependence, and predicts non-self-averaging that cannot occur. NS proposed the concept of the metastate (a probability distribution over infinite-size Gibbs states in a given sample that captures the effects of chaotic size dependence) and a non-standard interpretation of the RSB results in which the metastate is non-trivial and is responsible for what was called non-self-averaging. Here we use the effective field theory of RSB, in conjunction with the rigorous definitions of pure states and the metastate in infinite-size systems, to show that the non-standard picture follows directly from the RSB mean-field theory. In addition, the metastate-averaged state possesses power-law correlations throughout the low temperature phase; the corresponding exponent $zeta$ takes the value $4$ according to the field theory in high dimensions $d$, and describes the effective fractal dimension of clusters of spins. Further, the logarithm of the number of pure states in the decomposition of the metastate-averaged state that can be distinguished if only correlations in a window of size $W$ can be observed is of order $W^{d-zeta}$. These results extend the non-standard picture quantitatively; we show that arguments against this scenario are inconclusive.
We construct a low-energy effective action for a two-dimensional non-relativistic topological (i.e. gapped) phase of matter in a continuum, which completely describes all of its bulk electrical, thermal, and stress-related properties in the limit of
low frequencies, long distances, and zero temperature, without assuming either Lorentz or Galilean invariance. This is done by generalizing Luttingers approach to thermoelectric phenomena, via the introduction of a background vielbein (i.e. gravitational) field and spin connection a la Cartan, in addition to the electromagnetic vector potential, in the action for the microscopic degrees of freedom (the matter fields). Crucially, the geometry of spacetime is allowed to have timelike and spacelike torsion. These background fields make all natural invariances--- under U(1) gauge transformations, translations in both space and time, and spatial rotations---appear locally, and corresponding conserved currents and the stress tensor can be obtained, which obey natural continuity equations. On integrating out the matter fields, we derive the most general form of a local bulk induced action to first order in derivatives of the background fields, from which thermodynamic and transport properties can be obtained. We show that the gapped bulk cannot contribute to low-temperature thermoelectric transport other than the ordinary Hall conductivity; the other thermoelectric effects (if they occur) are thus purely edge effects. The coupling to reduced spacelike torsion is found to be absent in minimally-coupled models, and using a generalized Belinfante stress tensor, the stress response to time-dependent vielbeins (i.e. strains) is the Hall viscosity, which is robust against perturbations and related to the spin current as in earlier work.
In a recent paper by Neupert, Santos, Chamon, and Mudry [Phys. Rev. B 86, 165133 (2012)] it is claimed that there is an elementary formula for the Hall conductivity of fractional Chern insulators. We show that the proposed formula cannot generally be
correct, and we suggest one possible source of the error. Our reasoning can be generalized to show no quantity (such as Hall conductivity) expected to be constant throughout an entire phase of matter can possibly be given as the expectation of any time independent short ranged operator.
