Teaching Renormalization, Scaling, and Universality with an Example from Quantum Mechanics

Teaching renormalization, scaling, and universality with an example from quantum mechanics

Steve T. Paik∗ Physical Science Department, Santa Monica College, Santa Monica, CA 90405 (Dated: February 2, 2018) We discuss the quantum mechanics of a particle restricted to the half-line x > 0 with potential energy V = α/x2 for −1/4 < α < 0. It is known that two scale-invariant theories may be defined. By regularizing the near-origin behavior of the potential by a finite square well with variable width b and depth g, it is shown how these two scale-invariant theories occupy fixed points in the resulting (b, g)-space of Hamiltonians. A renormalization group flow exists in this space and scaling variables are shown to exist in a neighborhood of the fixed points. Consequently, the propagator of the regulated theory enjoys homogeneous scaling laws close to the fixed points. Using renormalization group arguments it is possible to discern the functional form of the propagator for long distances and long imaginary times, thus demonstrating the extent to which fixed points control the behavior of the cut-off theory. By keeping the width fixed and varying only the well depth, we show how the mean position of a bound state diverges as g approaches a critical value. It is proven that the exponent characterizing the divergence is universal in the sense that its value is independent of the choice of regulator. Two classical interpretations of the results are discussed: standard Brownian motion on the real line, and the free energy of a certain one-dimensional chain of particles with prescribed boundary conditions. In the former example, V appears as part of an expectation value in the Feynman–Kac formula. In the latter example, V appears as the background potential for the chain, and the loss of extensivity is dictated by a universal power law.

I. INTRODUCTION should mention that other pedagogical presentations of the renormalization of the inverse-square potential exist. Readers may wish to consult Refs. 4 and 5 for additional The inverse-square potential in quantum mechanics background. However, in these works they study the case has a rich history that continues to be updated in light α < −1/4. of new connections to diverse physical phenomena in- The other purpose of the article is to continue ex- cluding electron capture by neutral polar molecules,1 ploring the implications of the renormalization group re- the Efimov effect in a system of three identical bosons,2 sults uncovered in the work of Kaplan, Lee, Son, and the transition between asymptotically free and confor- Stephanov in Ref. 3. Although their work is certainly not mal phases in QCD-like theories as a function of the the first instance in which the inverse-square potential is ratio of the number of quark flavors to colors, and the discussed, it is notable for, among other things, discus- AdS/CFT correspondence.3 A common thread running sions of the beta function, operator anomalous dimen- through these applications, and the reason for our inter- sions at the fixed points, and the general phenomenon est, is that the attractive 1/x2 potential is a fascinating of conformal to non-conformal phase transitions. In par- case study that naturally calls upon the framework of ticular, we follow up their understanding of the fixed- the renormalization group (RG). We will see that the point structure with a natural extension to the quantum renormalization group approach mirrors, in many ways, mechanical propagator, and we use well-known quantum- the modern treatment of quantum effective field theories classical equivalences to extract statistical lessons for spe- whereby one demands that long-distance observables re- cific one-dimensional classical systems. main insensitive to the adjustment of fine details at short Consider a particle in one spatial dimension subject to arXiv:1707.04388v3 [quant-ph] 1 Feb 2018 distances. the potential The purpose of this article is twofold. It is primarily a pedagogical treatment of renormalization in the context n ∞ x ≤ 0 V (x) = (1) of single-particle quantum mechanics and it is intended α/x2 x > 0 for teachers of quantum field theory. We feel that the example presented herein can provide an instructive in- for −1/4 < α < 0. We work in units where ~ = 1 so troduction to basic RG ideas and terminology. Because that energy is the reciprocal of time, and ~2/2m = 1 our renormalization group analysis involves a quantum so that energy is also the reciprocal of length-squared. theory where one maintains full nonperturbative control, This means that time and length-squared have equivalent teachers may find that it serves as a useful aid for the be- dimensions. In these units, α is a dimensionless number ginning graduate student who is learning field theoretic whose value we do not imagine changing in any of our renormalization, but having difficulty separating the core analyses. principles from the technology required to do perturba- The paper is organized as follows. In Sec. II we ex- tive renormalization with many degrees of freedom. We plain why the choice −1/4 < α < 0, although giving 2

∗ ∞ perfectly consistent dynamics, is still peculiar. We then adjoint if and only if [f g]0 = 0 for any f, g ∈ H. explain that two distinct Hilbert spaces may be defined. The energy operator H is self-adjoint if and only if ∗ 0 ∗0 ∞ In Sec. III the Hamiltonian is modified at short-distances [f g − f g]0 = 0. Both of these conditions arise as at the expense of a dimensionful length scale and a cou- boundary terms resulting from an integration by parts in pling constant so that these two Hilbert spaces meld into trying to establish the equality of (f, Og) and (Of, g). one. In Sec. IV we discuss how the process of renor- The general solution to the eigenvalue equation malization allows one to continuously vary the short- distance modification without affecting a long-distance ψ00 = (α/x2 − E)ψ (2) observable. This naturally leads to the construction of the propagator in Sec. V and analysis of its properties in may be constructed as a linear combination of two Frobe- −1/2 Sec. VI when there is a continuous spectrum. In Sec. VII nius series. For E 6= 0 and x |E| , the solution is ν+ ν− we analyze the low-energy discrete spectrum when it ex- ψ ∼ C+x + C−x , where ists. Classical applications of quantum mechanics are r given in Sec. VIII. Finding these applications were, in 1 1 ν± = ± ω, ω = + α, fact, the original source of motivation for this work. In 2 4 Sec. IX remarks that generalize the inverse-square potential to three dimensions and α < −1/4 are given. are the roots of the characteristic equation ν(ν − 1) = α. We note that the range −1/4 < α < 0 implies 0 < Since this paper is part pedagogical guide, part re- ω < 1/2, so both solutions satisfy ψ(0) = 0. We will search, a remark about which portions are novel is in see that the regularity of both solutions is what makes order. Secs. II, III, and IV follow the path set forth in the quantum mechanics of the inverse-square potential Ref. 3, although we provide a different interpretation of so interesting. the fixed points than in that work. To the best of our It is straightforward to construct eigenfunctions of H knowledge, Secs. V, VI, VII, and VIII are new. Lastly, using series, and it turns out that they define ordinary some of the remarks made in Sec. IX briefly summarize or modified Bessel functions. The simplest way to see what is already well-established in the literature regard- this is to define a dimensionless variable x = |E|−1/2ξ ing the renormalization of the α < −1/4 case. and let ψ = ξ1/2ϕ(ξ). The resulting ode for ϕ is a vari- ant of Bessel’s equation. For E < 0 there is a unique linear combination of the independent solutions (with II. PURE INVERSE-SQUARE POTENTIAL 0 < |C+/C−| < ∞) that exhibits asymptotic exponential decay and is therefore square-integrable, namely 2 1/2 1/2 Does the Hamiltonian given by H = P + V (X) de- ψE = x Kω(|E| x). Here K is a modified Bessel fine a sensible quantum theory? To answer this ques- function of the second kind. Such an eigenfunction is tion one may proceed to construct the physical Hilbert inadmissible as an element of H for several related rea- space of states H over which the operators X, P , and sons. The scale invariance of the eigenvalue equation H are self-adjoint. It is important to remember that implies that if ψE is a normalized state, then so is 1/2 the self-adjointness property is not inherent to a differen- ψλ2E = λ ψE(λx) for λ > 0. But that would indicate a tial expression for an operator—one must also consider continuum of square-integrable states, a situation which the vector space of functions on which it acts and the directly contradicts the Feynman–Hellmann theorem as 7 boundary conditions satisfied by those functions. And so λ is not an explicit parameter of H. Furthermore, ψE it should be stressed that the physical Hilbert space is and ψλ2E are not orthogonal when λ 6= 1. Most damning not necessarily the space of square-integrable complex- of all, by scaling we can make E arbitrarily negative so valued functions L2(0, ∞) equipped with the standard there is no ground state! inner product (·, ·). While this is a bonafide Hilbert The root of these problems lie in the fact that ψE fails space in the functional analysis sense and is consistent to satisfy the required boundary conditions. Generally, a with the degrees of freedom available to a single spin- solution to Eq. (2) that is also square-integrable vanishes less particle moving along a line, it is not necessarily at spatial infinity and has vanishing first derivative. In the physical Hilbert space because certain functions ex- addition, ψE also vanishes at the origin. Therefore, P is ist whose behavior at x = 0 or x = ∞ ruin the hermitic- self-adjoint with regard to such eigenfunctions. However, ity of observables. Nevertheless, it is certainly true that self-adjointness of H requires H ⊂ L2(0, ∞). We shall construct H by forming linear combinations of the eigenfunctions of H. It is crucial f ∗(0)g0(0) − f ∗0(0)g(0) = 0. (3) that H be complete with respect to the distance function inherited from the inner product so that any Cauchy se- But 2 quence built from elements of H converge in L -norm h d d i to a limit also in H. This is guaranteed, according to lim ψE(x) ψλ2E(x) − ψλ2E(x) ψE(x) x↓0 dx dx Sturm–Liouville theory, for a self-adjoint Hamiltonian.6 C+ The position operator X is already self-adjoint in the ∝ ω(λν+ − λν− ) 6= 0. space L2(0, ∞). The momentum operator P is self- C− 3

Essentially, hermiticity fails because both near-origin so- Before moving on it is worth mentioning what happens lutions xν± are acceptable. One can force the self- if V is modified at long distances instead of short dis- adjointness of H by artificially introducing another tances. For example, one analytically attractive method boundary condition to select one solution or the other, or is to add a harmonic trap, Ω2x2. Here Ω−1/2 serves as by modifying the near-origin behavior of the potential. an explicit length scale. Miraculously, there exist raising For E > 0 a continuum of eigenfunctions exist, and lowering operators that create two entirely independent ladders of states with equally spaced rungs.7 Or, 1/2 1/2 ek(x) = (kx) J±ω(kx), k = E . imagine a hard wall at some position x = L. This im- poses a quantization of energies related to the zeros of Both signs satisfy the important closure relation the Bessel J function. The zeros may be those of J or R ∞ e (x)e (y)dk = δ(x − y) since the order of the Bessel +ω 0 k k J , which are, in general, distinct. function is greater than −1/2.8 Since both positive and −ω negative orders give independent representations of the identity operator, we have the freedom to use either sign in forming physical states. That is, consider the subspace A. Regularizing with a square well of L2(0, ∞) whose elements may be expressed as Z ∞ Modify the potential in Eq. (1) so that it reads f(x) = fe(k)ek(x)dk. 0 2 −g/(bx0) 0 < x < bx0 V (x) = 2 , (4) Here we assume that kfk = kfek = 1 so that the integral α/x x > bx0 exists by Plancherel’s theorem. This describes an inverse Hankel transform. A function constructed by this kind where g > 0 and b > 0 are dimensionless parameters, and of superposition inherits the same near-origin behavior we regard x0 as a fixed length scale. As usual, hermiticity as that of the Bessel J function. To wit, near the origin, of the momentum and kinetic energy operators −id/dx 1/2 ν± ek(x) ∼ (E x) . Notice that the dependence on E and −d2/dx2 constrain a wavefunction and its derivative and x is separable. Therefore, in an integral of the form R ∞ to exist and be everywhere continuous, even at the jump 0 g(E)ek(x)dE, where the x-dependence is paramet- discontinuity in V .10 ric, the result will be proportional to xν± . This ensures Our expressions are naturally stated in terms of a di- that H is self-adjoint and that one obtains two possible mensionless wavenumber, Hilbert spaces H± depending on the sign chosen in the order of the Bessel function. A final remark: E = 0 is the 1/2 infimum of the eigenvalues, but strictly speaking there is ξ = bx0|E| . no zero-energy eigenstate. There is a simple physical argument that explains why the Hilbert space is not unique.9 Imagine scattering a de Broglie wave of fixed energy E > 0 coming from x = ∞. B. Discrete eigenfunctions This is somewhat artificial since we should really speak in terms of a normalizable wave packet, but such extra rigor Let E < 0. The exact bound state eigenfunction is does not change the essential conclusion. For x k−1 −ikx ikx the wavefunction takes the form e + re . It should A sin(xpE + g/b2x2) 0 ≤ x < bx ψ = 0 0 . be possible to express the reflection amplitude r in terms E<0 p 1/2 1/2 C |E| xKω(|E| x) x > bx0 of C+/C−, however there is no condition that determines the value of C+/C− itself! Ordinarily, a boundary condition like ψ(0) = 0 suffices to fix the ratio of coeffi- The allowed energies follow from cients of linearly independent solutions. In this situation, 0 ν+ ν− p p 1 K (ξ) ψ(0) = 0 rules out neither x nor x . g − ξ2 cot g − ξ2 = + ξ ω . (5) 2 Kω(ξ)

III. BREAKING SCALE INVARIANCE Constants A and C are ﬁxed by continuity and normalization. Imagine graphing each side of Eq. (5) with re- Our goal is to connect the two scale-invariant theories spect to ξ: the left side is monotone increasing but the H± in the Wilsonian sense by linking them in a continu- right side is monotone decreasing. Thus, a root occurs ous space of Hamiltonians. However, the kind of explicit only if the left side starts somewhere below, or at, the symmetry breaking needed in V must take place at x = 0 starting point of the right side. By making use of the and must be suﬃcient to resolve the singularity. When identity lim ξK0 (ξ)/K (ξ) = −ω, the root ξ = 0 ob- ξ√↓0 ω√ ω framed as a two-body problem in three dimensions, it is tains when g cot g = ν−. Let us denote the solution obvious that “near-origin” is synonymous with “short- to this equation by g−. Therefore, a valid bound state distance” (i.e., nearly coincident particles). exists when g > g−. 4

C. Continuum eigenfunctions preserving C+/C− is, as will be discussed later, its direct relation to the scattering phase shift and indirect relation Let E > 0. The exact eigenfunction is to the binding energy. These are low-energy observables that may be measured at spatial infinity. p 2 2 ( A sin(x E + g/b x0) 0 ≤ x < bx0 Take Eq. (7) for fixed E and small nominal b. As b de- √ −2ω 1/2 1/2 creases further, the factor ξ increases, and so C /C ψE>0 = C+ √E xJω(E x) . √ √ √ +√ − 1/2 1/2 remains constant only if ( g cot g − ν−)/( g cot g − +C− E xJ−ω(E x) x > bx0 ν+) approaches zero from below. The implicit energy equation is Shrinking b ought to be understood as a flow to the infrared. Why? The position of a particle at x can be 0 0 p p 1 C+Jω(ξ) + C−J−ω(ξ) said to be neither close to the origin nor far unless a g + ξ2 cot g + ξ2 = +ξ , (6) 2 C+Jω(ξ) + C−J−ω(ξ) comparison is made to some length scale. V as given by Eq. (1) lacks such a scale, but introducing the cutoff bx0 where primes denote derivatives with respect to the whole in Eq. (4) makes it possible to judge whether a physical argument. It is important to note that although we have distance is small or large. For instance, for a nominal chosen to write these expressions using the notation of value of b = 1, x/x0 1 indicates that the particle is Bessel functions, the analyses in this article rely on little “close” to the origin, while x/x0 1 indicates that it more than the first couple terms in their power series for is “far.” In three dimensions, for a fixed spatial separa- small argument. tion x between two particles, rescaling the cutoff from x0 Suppose we wish either C+ or C− to vanish. Using the to bx0 makes the distance between the particles larger identity lim ξJ 0 (ξ)/J (ξ) = ω, Eq. (6) becomes, as in units of the cutoff. That is, x/(bx ) increases as b de- √ ξ√↓0 ω ω 0 b ↓ 0, g cot g = ν±. Denote the roots of this equation creases. This is precisely the regime one must examine to as g±. That is, understand the long-distance behavior of the interaction. √ √ The infrared flow described by b ↓ 0 takes a coupling g C− = 0 : g+ cot g+ = ν+ within the interval (g , g ) and makes it tend toward the √ √ √ √ + − C+ = 0 : g− cot g− = ν−. root of g cot g = ν−. In other words, g ↑ g−. The pre- √ √ cise manner by which g needs adjustment is called renor- Since g cot g is monotonically decreasing for 0 < g < malization. This indicates that (b, g) = (0, g−) is the 2 π it follows that g+ < g−. For instance, if α = −3/16, infrared-attractive fixed point (IRFP) of the flow. This is then g+ ≈ 0.7136 and g− ≈ 1.9411. depicted in Fig. 1. The explicit formula for the ratio C+/C− is p p C −ξJ 0 (ξ) + J (ξ) g + ξ2 cot g + ξ2 − 1 + (g, ξ) = −ω −ω 2 . 0 p 2 p 2 1 = 0 C− ξJ (ξ) − Jω(ξ) g + ξ cot g + ξ − /C − ω 2 C + increasing C Assume that E is fixed. For any desired positive value IRFP g of C+/C− there is a corresponding value of g ∈ (g+, g−). − We will primarily be interested in the analytical form of the ratio for small b, or equivalently, ξ 1. Then

2ω √ √ C −2 Γ(1 + ω) g cot g − ν− + = ξ−2ω √ √ [1+o(b)]. (7) + C− Γ(1 − ω) g cot g − ν+ /C

− We have established that for g+ < g < g− every eigenfunction ψE>0 will involve a well-defined linear combina- g tion of the fundamental solutions characterized by a fi- + UVFP nite and nonzero value for C+/C−. In particular, C+/C− −ω scales as E for fixed b and g. 1/2 bx0E

IV. RENORMALIZING THE COUPLING 1/2 FIG. 1. Contours of equal C+/C− in the (bx0E , g)-plane. The infrared ﬂow (b ↓ 0) is indicated on the contours by ar- A. C+/C− rows. The values of g± shown are speciﬁc to the example α = −3/16, and the values of C+/C− increase toward the Consider a point in the space of Hamiltonians bottom by integer powers of 2. parametrized by the (b, g)-plane. We would like to preserve the quantity C+/C− as b is taken to zero. In order We have learned that the IRFP corresponds to the 11 to do this, g must vary too. The physical motivation for eigenfunction when C+ = 0. Why does this make sense? 5

Recall that both of the fundamental solutions to the fixed point. Specifically, β0(γ) = 1 − 2γ so y = ∓2ω for Schrödingerequation with V given by Eq. (1) satisfy the γ = ν±. boundary condition ψ(0) = 0. Thus, any linear combina- A nice way to summarize what we have found is to tion of the fundamental set is acceptable near the origin. define a reduced coupling in the vicinity of each fixed Once a regulator is introduced (take b = 1, say) and a point. Note that g+ < g < g− maps to ν+ > γ > ν−. unique linear combination is chosen (up to scalar mul- Near the IRFP define u = γ − ν−. Then tiplication), we may ask, in generic terms: How should 0 2ω the coefficients of these solutions be adjusted so that the u = u(1 − ) . (9) appropriate C1-matching can be done at the cutoff scale Since u0 < u, we learn that u is an irrelevant variable x ? Assuming that x E−1/2, the solution xν− domi- 0 0 that tends to shrink as one enlarges the system. Near nates over xν+ near the cutoff because ν < ν . In other − + the UVFP define u = ν − γ. Then words, xν− changes more rapidly than xν+ does for small + x. As such, if one desires to have a scattering phase shift u0 = u(1 − )−2ω. (10) dictated by the xν+ solution, then one must finely tune 0 the coefficient C− to be zero. Therefore, from the point Since u > u, we learn that u is a relevant variable that of view of x x0, the phase shift is generically dictated tends to grow. by the xν− solution, whereas the xν+ solution is rather The assignment of the descriptions “irrelevant” and special. “relevant” to Eqs. (9) and (10) might sound backwards Let us be precise about how g must change as b does. to the reader familiar with real-space renormalization, We work in the limit of small b. A certain change of but are, in fact, consistent. For instance, the real-space variables makes the analysis elegant:3 approach assigns irrelevant scaling variables negative RG √ √ eigenvalues, not positive ones like ours. This is because γ(g) = g cot g. the sense by which one progresses to long distances in, say, a discrete lattice model, is a bit different. On the Then for a given E and b 1, lattice there is a spacing a which cannot be adjusted. In- C γ − ν stead, one coarse-grains over successively larger chunks + ∼ b−2ω − C γ − ν of the (infinite) lattice, each step producing an interme- − + diate lattice with larger effective size a0 = (1 + )a for up to a multiplicative factor independent of b. From > 0. Such procedure reduces the measure of the di- the invariance condition d(C+/C−)/db = 0 we obtain mensionless correlation length and is a way of probing a renormalization group equation: successively longer physical distances. A coupling obey- 0 yLAT ing the relation u = u(1 + ) for yLAT < 0 is then dγ said to be irrelevant because it shrinks. This is equiva- b = −(γ − ν−)(γ − ν+), (8) db lent to the reduction b0 = (1 − )b used in our analysis of which is exact in the b = 0 limit. Consider db < 0. the inverse-square potential since any physical distance x grows bigger in units of the cutoff scale bx0. For ν− < γ < ν+, the right-hand side of Eq. (8), de- noted β(γ) and called the beta function, is positive and Lastly, it is worth reframing the scaling in terms of the original coupling g. By the same logic as above it should so dγ < 0. Thus, γ = ν− is the IRFP, while γ = ν+ 0 y˜ be that, near a fixed point, g − g∗ = (g − g∗)(1 − ) is an infrared-repulsive fixed point, or, equivalently, an 0 ultraviolet-attractive fixed point (UVFP). This is consis- wherey ˜ = β (g∗). One should not confuse β(g) with dg dg dγ 0 tent with what we discovered in terms of the parameter β(γ(g)). Rather, β(g) = b db = b dγ db = β(γ)/γ (g) g. There are two fixed points.12 by the inverse function theorem as long as γ0(g) 6= 0. It is both interesting and particularly simple to study Differentiating with respect to g yields β0(g) = β0(γ) − 00 0 2 the behavior of γ close to a zero of the beta function. β(γ)γ (g)/[γ (g)] . However, at a fixed point β(γ∗) = 0 0 0 Here a scaling behavior emerges. Let γ = γ(b) indicate so the extra term vanishes and we have β (g∗) = β (γ∗), 0 the coupling associated to some choice of b. Let b = ory ˜ = y. Hence, the reduced coupling g− − g is irrele- 0 b(1−) for some infinitesimal > 0. It follows that b < b. vant, and g − g+ is relevant with exactly the same RG 0 0 0 dγ eigenvalues as in Eqs. (9) and (10). Let γ = γ(b ). So γ ≈ γ(b) − b db b = γ − β(γ). Differ- entiate with respect to γ to obtain dγ0/dγ ≈ 1 − β0(γ). At a zero γ∗ of the beta function, the coupling does not change. One hypothesizes that the reduced coupling B. Scattering phase shift γ − γ∗ obeys a simple scaling in the vicinity of the zero: 0 y γ − γ∗ = (γ − γ∗)(1 − ) . In the language of the renor- One may also frame the renormalization condition as malization group, the difference γ − γ∗ is called a scaling the requirement that the relative phase between incoming variable and the exponent y is called an RG eigenvalue.13 and outgoing plane waves remain invariant as one varies But this implies dγ0/dγ ≈ 1 − y. Equating both expres- the wavelength. Let µ = kbx0. For small µ we find that 0 0 dγ sions for dγ /dγ implies that y = β (γ∗)—the RG eigen- µ dµ = β(γ) with exactly the same beta function as in value is equal to the slope of the beta function at the Eq. (8). The salient details are presented in an appendix. 6

There are two ways to interpret the findings: (i) One V. PROPAGATOR could regard the cutoff b as held fixed at some nominal value and imagine varying k. Observe the phase shift A choice of (b, g) with b > 0 and g+ < g < g− selects experimentally for some k. The value so obtained lo- a particular Hamiltonian Hb,g. From this we will now cates a unique point in the (µ, g)-plane. Taking the long- construct the Green’s function—the position-space real- wavelength limit k ↓ 0 (and hence µ ↓ 0) requires ad- ization of unitary time evolution. However, we shall work justing g so that one remains on a certain integral curve with pure negative imaginary times, Gb,g(x, −it; y) = in this plane. (ii) Another approach is to regard the hx|e−tH |yi, t ≥ 0. This imaginary-time propagator is wavenumber k of the incoming plane wave as held fixed, ∂ ∂2 the solution to ∂t G = ∂x2 − V (x) G with boundary but allow the freedom to adjust b. A choice of (b, g) condition G(x, −it ↓ 0; y) = δ(x − y). The completeness uniquely specifies a Hamiltonian. For this Hamiltonian, property of eigenfunctions of H means that there will be a definite phase shift. Now as we take b ↓ 0, b,g it is possible to adjust g so that the phase shift does not Z ∞ −tE change if, once again, we follow an integral curve in the Gb,g(x, −it; y) = dE e ψE(x)ψE(y). (µ, g)-plane. In this interpretation there is a flow between 0 Hamiltonians that preserves a long-distance observable. We are motivated to demonstrate the following: two Thus, taking b to zero is an equivalent way of reaching a Hamiltonians, one at scale b and the other at scale b0, long-wavelength approximation. but both with couplings close to g±, will give equivalent long-distance behavior (as measured by G at fixed x, y > bx0 and fixed t) provided that the coupling g is C. Binding energy renormalized from b to b0 according to the scaling laws found in Eqs. (9) and (10). Although our discussion of renormalization has been At this point we remember that C+/C− is not the only limited to g ∈ (g+, g−), we may also consider g > g−. ratio needed in order to fully specify eigenfunctions. We At least one bound state will be present with ground also need state energy E. One might suspect that dE/db = 0 p 2 C− sin g + ξ leads to the same renormalization group equation for (g, ξ) = √ √ C+ A (g, ξ) ξJω(ξ) + ξJ−ω(ξ) g, namely Eq. (8), as the previously studied conditions C− d(C+/C−)/db = 0 and dr/db = 0 in Secs. IV A, IV B, 2 and the appendix. Our expectation is that this should and the normalization factor A . This can be fixed by be true at least in the limit b ↓ 0. remembering that the set of eigenfunctions {ψE>0} must On general grounds, the phase shift is essentially the satisfy orthonormality and closure relations. Let us focus phase angle of the reflection amplitude r, whereas the on the closure property which must hold for any choice of ground state energy E is the pole of r (more generally, the x, y and g. Therefore,√ √ take x, y < bx0 and g = 0 so that R ∞ 2 S-matrix) in the complex k-plane. Since r has modulus 0 A sin(x E) sin(y E)dE = δ(x − y). We recognize one, it is possible to express it as r = (s − ik)/(s + ik), here an identity√ of Fourier sine transforms so it is clear 2 where s is some real constant and k the real wavenumber. that A = 1/π E. This is consistent with dimensional 1/2 However, there is a simple pole at k = is, and so E = analysis since ψE should have dimensions of length . k2 = −s2. Requiring that r remain constant as b changes For g > 0 we may write implies that the value of k/s remains constant as b is B(g, ξ) adjusted. Thus, the renormalization group equations are A2 = √ identical. π E > The bound state energy for g ∼ g− is given by solving 0 for some dimensionless function B which is strictly posi- Eq. (5). For 0 < ω < 1/2 and ξ 1, ξKω(ξ)/Kω(ξ) = tive. The propagator may be written as Γ(1−ω) 2ω 2 −ω −2 Γ(ω) (ξ/2) +O(ξ ). Expanding to lowest order G (x, −it; y) in both ξ and g − g−, b,g Z ∞ h BdE −tE C− C+ p 1/2 1/2 1/ω = √ e E xJω(E x) (g − g−) 0 π E A C− E ∼ −C 2 , (11) (bx0) i C− p 1/2 1/2 + E xJ−ω(E x) × [x ↔ y], x, y > bx0. 2ω−2 1/ω A where C = [2 (1+α/g−)Γ(ω)/Γ(1−ω)] , a positive (12) constant that depends only on the value of α. Eq. (11) shows once more that E remains invariant as long as the Without explicitly evaluating the integral, we are in- 2ω reduced coupling g − g− scales as b , the same scaling terested in proving that Gb,g is simply related to Gb0,g0 0 found when g was just below g−. Note that as b ↓ 0, as for b < b. The simple relation we seek is an equivalence long as g − g− follows this scaling rule, a single bound of the two propagators up to an overall scale factor with state (of arbitrary energy) remains! This is an example physical lengths and time held fixed. This is a homoge- of dimensional transmutation. neous transformation with respect to the parameters b 7 and g. The existence of such a relation would place a alence of g0 and g00 can be satisfied. The crux of our severe constraint on the form of the ratio C+/C−. The argument is this: it is only close to a fixed point that f reason is that the contours in Fig. 1 show exactly how takes the asymptotic form to preserve the propagator as b ↓ 0. However, individual 1/2 β 1/2 γ points in Fig. 1 are not representative of a choice of (b, g). f(g, bE ) ∼ α(g − g∗) (bE ) Rather, a given (b, g) corresponds to a horizontal line in the plot—all of the uncountably many eigenfunctions of for some real numbers α, β, and γ which may be ex- Hb,g correspond to points on this line. Under a shrinking tracted by studying Eq. (7). Then Eqs. (13) and (14) 0 0 γ/β 00 of b, one can preserve the propagator by following the become g − g∗ = (b/b ) (g − g∗) and g − g∗ = 0 γ/β contour passing through each point to its left. We man- (b/b ) (g − g∗), respectively. Together they imply age, on an individual basis, to keep all eigenfunctions un- g0 = g00 as desired. changed if we can adjust g so that each eigenfunction’s C+/C− ratio remains the same. However, there is no guarantee that the new value of the coupling for one en- A. Homogeneous transformation laws ergy will coincide with the new coupling required for a different energy! More precisely, recall that C+/C− is a The scaling relation we seek for the propagator may function of two variables g and ξ. Write ξ = bE1/2 (sup- 1/2 be obtained from the following two transformation laws. pressing x0) so that C+/C− = f(g, bE ). Consider any 0 two distinct energies E1 and E2. By changing b to b it is possible to find some g0 such that 1. An exact one 1/2 0 0 1/2 f(g, bE1 ) = f(g , b E1 ). (13) And it must be possible to find some g00 such that The following equation is exact. Suppose x, y > bx0 and b0 = b/λ for some λ > 1. Then 1/2 00 0 1/2 f(g, bE2 ) = f(g , b E2 ). (14) 2 −1 G (λx, −iλ t; λy) = λ G 0 (x, −it; y). (15) These statements are illustrated schematically in Fig. 2. b,g b ,g 0 00 However, G 0 0 ∝ G requires g = g , a highly non- b ,g b,g Proof: In Eq. (12) make a change of variables E˜ = trivial condition! There is no obvious reason why, start- λ2E. Within B, C /C , and C /A this can be perfectly ing from generic g, the renormalized couplings g0 and g00 + − − compensated by a redefinition of the scale factor b since ought to be the same. Therefore, this appears to be an ξ = bx E1/2 = (b/λ)x E˜1/2 = b0x E˜1/2. Since b0 < b, obstruction to finding a simple scaling law for the prop- 0 0 0 the appropriate eigenfunctions are still Bessel J. agator.

2. One valid only close to a ﬁxed point g′ The following equation is correct only asymptotically close to a ﬁxed point. Suppose x, y > bx0, b 1, and g′′ λ > 1. Then

2 2ν±−1 Gb,u(λx, −iλ t; λy) ∼ λ Gb,u0 (x, −it; y), (16)

0 −2ω for (upper sign) u = g − g+ 1 and u = λ u, or 0 2ω (lower sign) u = g− − g 1 and u = λ u. g Proof of upper sign case: Once again, in Eq. (12), make a change of variables to E˜ = λ2E. Near a fixed point 1/2 1/2 b′E 1/2 bE 1/2 it is possible to absorb factors of λ into a redefinition b′E1 bE1 2 2 of the reduced coupling rather than a redefinition of b. Before doing this, replace C± by their asymptotic forms FIG. 2. A Hamiltonian Hb,g has a spectrum corresponding 1/2 for ξ 1. (This is justified by taking the upper limit of to a horizontal line in the (bx0E , g)-plane. Changing to 2 some b0 < b does not necessarily guarantee that all values integration to be some large, but finite, value M/x0 for of C /C remain the same for all energies if we adjust g to M 1. We shall take M√→ ∞ later. This means√ that + − 1/2 a new value. As we illustrate here, different values of the E is bounded above√ by M/x0, and hence, ξ < b M. 0 00 coupling, g 6= g , may be needed to keep C+/C− unchanged Let us choose b 1/ M so that ξ is small over the entire for different energies E1 and E2. integration region.) So in the small-b regime, √ √ √ However, it is easy to see that in the vicinity of a fixed C Γ(±ω) sin g( g cot g − ν ) ± (g, ξ) ∼ ∓ . (17) point (0, g∗) the nontrivial condition requiring the equiv- A 21∓ω ξν± 8

Also, we replace B(g, ξ) by its limiting value B(g+, 0) A. At the fixed points assuming it exists. Thus, We wish to discuss the propagator at the fixed points. G (λx, −iλ2t; λy) b,g We have seen that g = g± corresponds to C∓ = 0 only in 1 Z ∞ B(g , 0)dE˜ h cst.λν+ p the limit that b equals 0. Taking this limit squeezes the + −tE˜ ˜1/2 ˜1/2 = p e E xJω(E x) sinusoidal part of the eigenfunction into an infinitesimally λ ˜ (bx E˜1/2)ν+ 0 π E 0 narrow region around the origin. Adapting an integral ν− p i cst.uλ 1/2 1/2 from Ref. 14, we obtain + E˜ xJ−ω(E˜ x) × [x ↔ y]. ˜1/2 ν− (bx0E ) √ xy 2 2 xy G (x, −it; y) = e−(x +y )/4tI , The constants “cst.” appearing above result from ex- 0,g± 2t ±ω 2t panding Expr. (17) around g = g . They are nonzero. + where I is a modified Bessel function of the first kind. ν+−ν− 0 √ By writing u = λ u , it is seen that two factors of z Using the well-known properties Iν (z) ≈ e / 2πz for λν+ may be pulled out of the integral. Although this |z| 1, and I (z) ≈ (z/2)ν /Γ(ν + 1) for |z| 1, the concludes the proof, it is worth explaining it heuristi- ν asymptotic behavior is given by cally. For g close to g+, we can set C− = 0 so that the 2 eigenfunctions are, for small x, power laws of the form √1 e−(x−y) /4t t ↓ 0, ν+ 4πt x . Therefore, in the propagator we have two wave- G0,g (x, −it; y) ∼ . ± 1 √1 ( xy )ν± t → ∞ functions of the form (λx)ν+ and (λy)ν+ . This is how a 21±2ω Γ(1±ω) t t factor of λ2ν+ is obtained. A proof for the lower sign case The fixed-point theories both reproduce free-particle be- is similar. havior at short times, but have differing power-laws at long times.

VI. SCALING B. Close to the fixed points By combining Eq. (15) with (16) we are able to de- rive the scaling relation for the propagators evaluated at For (b, g) in the vicinity of (0, g±), but not strictly the same values of x, y, t, but corresponding to different at those points, the functional form of the propaga- Hamiltonians Hb,g and Hb0,g0 . We shall suppress depen- tor may be uncovered by renormalization group scaling dence on the variables x, y, t which are imagined to be analysis.13 We illustrate these standard techniques near held fixed, and instead highlight the dependence of the the UVFP. propagator on the parameters b and g. Then

G(b, u) ∼ λ−2ν± G(b/λ, uλ±2ω). (18) 1. RG for u > 0

> The upper sign is for g ∼ g+ and the lower sign for For u = g − g 1 but strictly greater than zero, < + g ∼ g−. Eq. (18) is a central result of this article. One G(b, u) ∼ λ−2ν+ G(b/λ, λ2ωu). Iterating n times gives can recognize the RG eigenvalues of u found earlier in G(b, u) ∼ λ−2ν+nG(b/λn, λ2ωnu). Since u is relevant, Eqs. (9) and (10). the effective reduced coupling grows under iteration so n An equivalent way to state Eq. (18) is that the propa- cannot be taken arbitrarily large or else the asymptotic gator satisfies a certain first-order pde which is obtained approximation breaks down. So we stop the iteration 2ωn by imagining an infinitesimal change in parameters. Let at the point where λ u = u0, where u0 is an arbi- λ = 1 + δλ for infinitesimal positive δλ. Expanding to trary but fixed constant that is sufficiently small. Solv- n 1/2ω first order in δλ, we obtain ing yields λ = (u0/u) . Plugging this back in gives ν+/ω 1/2ω G(b, u) ∼ (u/u0) G(b(u/u0) , u0). Observe that h ∂ ∂ i the parametric dependence of G on b and u has simpli- b ∓ 2ωu + 2ν± G = 0. (19) ∂b ∂u fied to the point where we may now express it as

1+1/2ω 1/2ω The physical interpretation of Eq. (19) is that a change of G(b, u) ∼ (u/u0) Φ(b(u/u0) ), (20) b in G can be absorbed through a compensating change in u. In other words, scale dependency may be exchanged for some function Φ. At first sight it might ap- for coupling dependency, a fact already clear from our pear that the functional form of Φ depends on the derivation of Eq. (16). Notice that the coefficients ∓2ω specific value of u0. Indeed, we could have imag- ∂ 0 of the operator u ∂u are precisely the slope of the beta ined the iteration stopping at some u0 6= u0. Along function at the fixed points γ = ν±. Eq. (19) is reminis- the lines of Eq. (20) we could then write G(b, u) ∼ 0 1+1/2ω 0 1/2ω cent of the Callan–Symanzik equation in quantum field (u/u0) Ψ(b(u/u0) ) for some apparently unre- theory which describes how correlation functions change lated function Ψ. But the important fact is that the with the renormalization scale. left-hand side of Eq. (20) is insensitive to whether the 9

0 scaling variable halts at u0 or u0. This implies an equal- alent to Eq. (20) when u > 0. Consider 0 1+1/2ω 0 1/2ω ity: Ψ(z) = (u0/u0) Φ((u0/u0) z), and means 0 b−ν+ + cub−ν− that the ratio u0/u0, and any powers thereof, may be absorbed into a redeﬁnition of the scaling variable u. Thus, = (bu1/2ωu−1/2ω)−ν+ + cu(bu1/2ωu−1/2ω)−ν− the form of Φ must be independent of u . In other words, 0 1/2ω −ν 1/4ω+1/2 1/2ω −ν 1+1/4ω−1/2 Φ is truly a function with a single argument. In such in- = (bu ) + u + c(bu ) − u stance, Φ is referred to as a scaling function. = u1/2+1/4ωΦ(bu1/2ω),

where Φ(z) = z−ν+ + cz−ν− . Of course, there is also a similar factor involving y. Thus, the factor u1/2+1/4ω 2. RG for u = 0 gets squared and becomes u1+1/2ω as desired.

For u = 0 our n-times-iterated homogeneous transformation law for the propagator may be written VII. DISAPPEARING BOUND STATE −n G √ b , 0 ∼ (λ−n)2ν+ G √bλ , 0. We have rein- xy/x0 xy/x0 troduced the spatial variables x and y using dimensional In this section we set b = 1 in the regulator Eq. (4) analysis. This may be thoroughly justified by redefin- and do not allow the well width to vary. For clarity we ing the integration variable in Eq. (12) from E to E/t. also take x0 = 1 (it can always be restored by looking at > Without explicitly evaluating the integral it is easy to see dimensions). For g ∼ g− the ground state energy scales −1/2 2 2 1/ω that G = t f(x/bx0, x /t, y/bx0, y /t) for some func- as E ∝ (g − g−) . As g ↓ g−, this binding energy van- tion f. Our recursion relation is not sensitive to terms ishes at a rate described by the exponent 1/ω. It must like x2/t or y2/t; it must be correct only for t x2, y2. also be the case that the probability to find the particle Furthermore, the propagator must be symmetric in x within any finite interval tends to zero—the mean po- and y. Since λ > 1 one may imagine n being so large sition of the particle (and all higher moments) should √ that bλ−n/( xy/x ) = is an exceedingly small fixed run off to infinity in a continuous fashion. The rate at 0 √ number. It could be so small that xy/x0 1 (recall which this occurs is given by computing the divergent R 2 that x and y are fixed). This requires that we choose part of x|ψ| dx. In fact, we need only focus on the n log b/ log λ, which is always possible for a given b exponentially-falling tail of the wavefunction. We find R ∞ 2 R ∞ 2 1 −1/2 and λ. So that 1 x|ψ| dx 1 |ψ| dx ∼ 2 |E| . Up to a constant factor, √ b xy 2ν+ 2ν+ −1/2ω G √ , 0 ∼ G(, 0). hxi ∼ (g − g−) . xy/x0 bx0 The rate of vanishing of E, or the degree of divergence Note that 2ν+ G(, 0) is a constant. Including an overall of hxi, is controlled by the long-distance part of V . The t−1/2 factor from dimensional analysis, we obtain specific scheme chosen for the short-distance part (e.g., square well) has no effect other than to modify the lo- b 1 xy ν+ cation of the critical value of g. This independence of √ G √ , 0 ∼ 2 2 . (21) xy/x0 t b x0 regulator scheme is an example of universality: physical systems with very different ultraviolet behavior (in this case, Hamiltonians with very different excited spectra) may have similar low-energy behavior. 3. Verification A nice exercise is to try a regulator like the linear well, V = −gx for 0 < x < 1. Like the square well, it has Fortunately, analytical evaluation of G is rather sim- a single dimensionless parameter g > 0 responsible for ple for asymptotically large t. This allows us to check controlling the well slope. The same scaling exponent Eqs. (20) and (21). For large t the integral in Eq. (12) is 1/ω emerges for the ground state energy as in Eq. (11), but the coefficient C and the location of the fixed point dominated√ by values of E near zero which allows us to re- 1/2 1/2 1/2 ν± g− are not the same as in the square well case. One says place E xJ±ω(E x) by (E x) . After rescaling the dummy integration variable to extract t, we obtain, that 1/ω is universal, but C and g− are not. up to an overall constant factor, It is readily proven that the same exponent results for a generic regulator. Consider a scheme given by

1 h x ν+ x ν− i G ∼ √ + cu × [x ↔ y], (22) −gf(x) 0 < x < 1 b,g∼g+ V (x) = , t bx0 bx0 α/x2 x > 1 where u = g − g+ 1 and c is some constant. where f is a function defined on [0, 1] with additional Setting u = 0 in Eq. (22) gives precisely Eq. (21). It properties to be imposed below. First we prove that for takes a little more effort to show that Eq. (22) is equiv- sufficiently large g the potential binds. It is sufficient to 10 show that there exists some ψ ∈ H with (ψ, Hψ) < 0 process, a real random variable x(t) varies with a time since this implies that at least one of the discrete eigen- parameter t. The map x(t) is continuous, satisfies an ini- values of H is negative. Note that ψ need not be a state of tial condition and may satisfy a final condition as well. definite energy. A convenient choice is ψ = x exp(−x/2). Crucially, the process is such that increments in x are So independent and identically distributed (i.e., the probability distribution for x(t2)−x(t1) is independent of x(t1) Z ∞ 1 Z 1 α ψ(−ψ00 + V ψ)dx = − g f(x)x2e−xdx + . for t2 > t1 > 0, but they are identical in the sense that 0 2 0 e P(x(t2)−x(t1) < a) = P(x(t1) < a)). These distributions are normal with mean zero and variance equal to twice Assume f is integrable. A negative expectation value is the time interval. It is well known how to rigorously as- 1 α R 1 2 −x guaranteed for g > ( 2 + e )/ 0 f(x)x e dx. sign a probabilistic measure to such a set of real-valued Next we prove that for sufficiently small g there is no continuous functions. bound state. Recall the fact that the square well regula- We are interested in the following conditional expecta- 15 tor, Eq. (4), does not produce binding if g < g−, where tion value, g− is entirely determined by α. So take g < g−/f(x) for all x < 1. We assume f is bounded. This means that our 0 h − R t V (x(τ))dτ i W (x, t; y) = Ex,t;y,0 e 0 potential V is everywhere bounded below by the square Z ∞ well. −N/2 = lim dx1 ··· dxN−1 (4π) The solution to the eigenvalue equation is N→∞ −∞ N−1 2 Aϕ(1)(x) + Bϕ(2)(x) 0 ≤ x < 1 X (xj+1 − xj) ψ = √g, g, , × exp − + V (xj) , E<0 1/2 1/2 4 C xKω( x) x > 1 j=0

(i) 00 where ϕg, are the linearly independent solutions to ϕ + with = t/N, x0 = y, and xN = x. Since V is infinite for gf(x)ϕ − ϕ = 0, with = −E. Suppose that g is large all x ≤ 0, any path that “dips” into x ≤ 0 at any time −∞ enough to admit at least one bound state so that is a will be weighted by e = 0. This effectively eliminates real positive number. Call the critical value of g needed from consideration all paths that travel left of the origin. Thus, we are considering Brownian motion on the half- for this to be true some g∗. By continuity, line x > 0. See Fig. 3. 0 0 ϕ(1) (1)ϕ(2)(0) − ϕ(1)(0)ϕ(2) (1) 1 K0 (1/2) g, g, g, g, = + 1/2 ω . (1) (2) (1) (2) 2 K (1/2) ϕg,(1)ϕg,(0) − ϕg,(0)ϕg,(1) ω x (23) Call the left side of Eq. (23), L(g, ). Since x = 0, 1 are (i) ordinary points of the ode, ϕg, and their derivatives are y finite when evaluated at x = 0, 1. Now let us consider how L depends on the parameters. Expanding in energy 1, we claim that L(g, ) equals L(g, 0) plus a term V = /x2 whose order is no larger than 1/2. (This is because non- bx 0 2 analyticity in the -dependence of the solutions to the V = −g/(bx0) ode develops when f vanishes. For example, if f = 0 √ √ 0 near x = 0, then ϕ ∼ e± x ≈ 1 ± x.) Call the right t side of Eq. (23), R(). Expanding in 1, we obtain disallowed region R(0) + O(ω). Since ω < 1/2, this O(ω) term dominates 1/2 over the O( ) term obtained by expanding the left side. FIG. 3. The quantity W is given by evaluating the functional − R t V (x(τ))dτ Expanding L once more, but this time in g around g∗, e 0 over all paths x(τ) that start at y at time 0 we get L(g∗, 0) + O(g − g∗). Finally, if g∗ is such that and end at x by time t. Only those that do not “dip” into ω L(g∗, 0) = R(0), then we have O(g − g∗) = O( ). It negative positions are counted. For instance, the dashed path 1/ω follows that ∼ (g − g∗) . above does not contribute to the expectation.

2 Kac proved that W satisfies the pde ∂ W = ∂ −V W VIII. SOME CLASSICAL APPLICATIONS ∂t ∂x2 with boundary condition limt↓0 W (x, t; y) = δ(x − y). This is mathematically identical to the problem of finding A. Brownian motion the quantum mechanical Green’s function in imaginary time.16 Brownian motion is a stochastic process which may be Consider V given by Eq. (4) and let g = g±. According regarded as the symmetric random walk in the limit of in- to the quantum result Eq. (21), for x, y > bx0, b 1, and −1/2 2 2 ν± finitesimally small time increments. In a one-dimensional asymptotically large t, W (x, t; y) ∼ t (xy/b x0) . 11

Consider two Brownian expectations: one for all contin- where the proportionality constant is independent of g. uous paths from y to x, and another from λ0y to λx, The limit we have taken is a continuum limit. where all positions are strictly greater than bx0. Then in Now we wish to take a thermodynamic limit by mak- the large-t limit, ing the “volume” t arbitrarily large. In this limit we are interested in extracting the free energy density defined by W (λx, t; λ0y) ∼ (λλ0)ν± W (x, t; y). −1 fxy = limt→∞ t log Zxy(t). According to the discussion in Sec. VII, for g > g∗ a bound state exists so the domi- In particular, observe that λ0 = 1/λ leads to asymptotic nant behavior of W (x, t; y) is exponential in t for large t. −E0t equivalence. This is rather suprising from the stochastic That is, W (x, t; y) ∼ e ψ0(x)ψ0(y) plus exponentially point of view because the interval where the potential suppressed corrections. Here E0 is the ground state en- depends on g can be made arbitrarily narrow, neverth- ergy. Therefore, fxy ∼ E0. Terms like (log ψ0(x))/t van- less it exerts an outsized influence on the sum over paths! ish in the large-t limit so the specific choice of x and y do More precisely, consider the set of continuous functions not affect the free energy density in the thermodynamic that begin at (y, 0) and end at (x, t) while always satis- limit. Thus, fying x(τ) > 0 for all τ ∈ (0, t). Sampling only from this 1/ω > space, what is the conditional probability that a particle fxy ∝ (g − g∗) , g ∼ g∗. passes through the portal 0 < x < bx0 at some interme- 17 diate time t1? There is a finite answer to this question This is finite and so an extensive phase exists if g > g∗. and it can be made arbitrarily small by taking b → 0. A However, if g < g∗, then W (x, t; y) is asymptotically a naive conclusion might be then that the value of g has power law in t so the system is no longer extensive. In little effect on W . However, the analysis shows the op- other words, posite: W exhibits very different scaling at two special values of g. fxy ∼ 0, g < g∗.

We see that the loss of extensivity as the parameter g ↓ g∗ B. Statistical mechanics of a chain is characterized by a universal critical exponent 1/ω. The phase transition described by the crossing of g Yet another way to interpret the imaginary-time across g∗ is different from the kind of finite-temperature Green’s function is as a classical partition function. Con- phase transition familiar from lattice models. One promi- sider a discrete and finite one-dimensional system with nent difference is that in short-range lattice spin systems, N real degrees of freedom x . Impose fixed boundary the free energy is extensive on both sides of the critical j point. Nevertheless, similar extensive-to-nonextensive conditions x0 = y and xN+1 = x. On the set of {xj} define a probability measure by the Boltzmann factor, phase transitions are described in Ref. 19 for the case µ = exp(−S)/Z, where the energy of a configuration is α < −1/4. These authors treat α as a variable ther- mal parameter. In one experimentally realizable sys- N tem it is explained how the approach α ↑ −1/4 re- Xh 1 i S = (x − x )2 + V (x ) + V (x ) . produces Berezinskii–Kosterlitz–Thouless scaling related 4 j+1 j 2 j 2 j+1 j=0 to a topological transition between winding states of a floppy polymer circling a defect. is a nearest-neighbor coupling and V is an external potential. We take > 0 so that the site-to-site coupling is attractive—it is energetically favorable for xj to be IX. OTHER ASPECTS OF THE similar in value to xj±1. Lastly, Z is a constant chosen INVERSE-SQUARE POTENTIAL to satisfy the normalization condition. Explicitly, Z While there are other features and applications of the −S Zxy(N, ) = dx1 ··· dxN e . inverse-square potential, here we make two brief remarks that extend our analysis and bridge our work with other pedagogical discussions of the inverse-square potential.4,5 A priori there is no relation between N and . One is typi- cally interested in evaluating expectations in the measure µ in the infinite volume limit. A. Three dimensions It is well known that, in the double scaling limit N → ∞, → 0 such that N = t fixed, there is a correspondence between the classical statistical mechanics Our results apply to two particles interacting in the 2 5 problem we have just defined and quantum mechanics of s-wave via a central potential that is α/r at large r. a particle propagating in negative imaginary time.18 In Since the potential is time-independent the nontrivial fact, part of the two-particle wavefunction is obtained by solv- 2 ~ 2 ing − 2m ∇rψ + V ψ = Eψ, where m is the reduced mass. W (x, t; y) ∝ Zxy(t), In a zero-orbital-angular-momentum state, ψ ∝ U(r)/r. 12

It is well known that the radial function U obeys an equa- This describes a renormalization group limit cycle—the tion that looks just like the Schrödingerequation on the coupling g cycles through an interval of values. 2 ~ 2 2 half-line, − 2m d U/dr + V (r)U = EU, r ≥ 0. It turns out that the appropriate boundary condition Note added: After publication we became aware of is that U(0) = 0, precisely analogous to that of our one- a previous RG study in which all possible universality dimensional example. In Ref. 20 a proof is given by de- classes are exhibited for the line-depinning phase tran- 2 2 manding that −d /dr be hermitian with respect to the sition in arbitrary dimensions for an attractive pinning space obtained from all square-integrable linear combi- potential with an inverse-square tail.21 In fact, Ref. 21 nations of functions U satisfying the eigenvalue equation. predates Ref. 3. The scope of Ref. 21 is far more broad This argument is repeated below. than the simple question taken up in our Secs. VII and Let U1 and U2 be two such functions. The combina- VIIIB, although the criterion used to signal a phase tion of square integrability and satisfaction of the ode transition—the disappearance of a bound state—is pre- 0 implies that Ui and Ui limit to 0 as r goes to ∞. Then cisely the same. Our conclusion in Sec. VIIIB may be 0 ∗ operator hermiticity necessitates that (U1(0)/U1(0)) = found in Sec. IVB of Ref. 21. Among the many detailed 0 U2(0)/U2(0), which is precisely Eq. (3) again. From this discussions in that work is a very natural example of we see that U1(0) and U2(0) are equal up to a multiplica- a classical statistical mechanical application of the one- tive constant for which the only self-consistent choice is particle quantum problem, namely the attraction of a 1. Thus, all Ui(0) must equal the same constant c. How- (one-dimensional) wetting liquid interface to an impene- ever, if c 6= 0, then ψ ∼ 1/r which is problematic because trable substrate. We thank Eugene Kolomeisky for bring- 2 3 −∇r(1/r) ∝ δ (r) yet there is no corresponding delta ing this to our attention. function in V . The only way to avoid this situation is to have c = 0. Appendix: Integral curves of constant phase shift B. α < −1/4 For k = E1/2 > 0, the time-independent part of the While the case α < −1/4 has been well-studied in the solution to the Schrödingerequation with V given by literature (see, for example, Refs. 4 and 9) it is worth Eq. (4) may be written highlighting its renormalization group analysis for its rπ qualitatively different long-distance behavior. ψ = e−iωπ/2−iπ/4 The pure α/x2 potential with α < −1/4 has a spec- 2 trum unbounded from below so one really does need to ( A sin(xpk2 + g/b2x2) 0 ≤ x < bx regulate the Hamiltonian with a cutoff in order to obtain √ 0 √ 0 × (2) iωπ+iπ/2 (1) . a healthy theory. Suppose this is done with the finite kxHω (kx) + re kxHω (kx) x > bx0 2 square well, Eq. (4). Let = −Ex0. Then for the shal- (i) lowest bound states satisfying 1, the implicit energy Hω are Hankel functions. The asymptotic form of this equation is solution is ψ ∼ e−ikx +reikx. Doing the matching at x = √ √ bx yields the following explicit formula for the reflection g cot g ∼ 1 − |ω| tan(|ω| log b + 1 |ω| log + φ), (24) 0 2 2 amplitude, where the phase φ is determined by the normalizable so- 1 − ic lution and is not a free parameter. This regulator breaks r = ie−iωπ , the continuous scale symmetry to a discrete subgroup 1 + ic (notice that → e−2π/|ω| leaves Eq. (24) invariant). Fix and consider the locus of points in the (log b, g)-plane where obeying Eq. (24). If (log b0, g0) is any such point, then a 0 Y (µ) − dYω(µ) continuous curve of points exist in its vicinity. By follow- c = ω , J 0 (µ) − dJ (µ) ing the curve to more negative log b, g tends toward zero. ω ω p p Eventually, there will come a time when g attains zero, 2 µ2 + g cot µ2 + g − 1 d = . but at this point we may jump to another curve and start 2µ the flow all over again with a finite value of g. This is possible because there are an infinite number of solutions It is obvious that |r| = 1 so this is pure phase. g for a given b. The periodicity of tangent means that Demand that dr/dµ = 0. This is only possible if g is the coupling g is renormalized, but never approaches a allowed to “run” with µ. Therefore, the µ dependence limit, as b decreases. Instead, each time log b is shifted by of the phase r is present both explicitly and via implicit −π/|ω|, g may be identified with its starting value. Such dependence through g. Write r = R(µ, g(µ)). Then identification is natural because it keeps g positive. The process may be replicated indefinitely; b remains strictly dr ∂R ∂R ∂g 0 = = + . positive because it is reduced by a multiplicative factor. dµ ∂µ ∂g ∂µ 13

This is a linear first-order pde of the form Rµ + case of scattering off a hard wall. When the de Broglie f(µ, g)Rg = 0, with wavelength of the incident particle is much longer than −1 the width of the square well, k bx0 (i.e., µ 1), −αζ sin2 ζ + gζ cos2 ζ − g sin ζ cos ζ p f = , ζ = µ2 + g. √ √ (µ/2)(ζ − sin ζ cos ζ) π π g cot g − ν+ δ = (1−2ω)− √ √ µ2ω+O(µ4ω). 4 ω(2ωΓ(ω))2 g cot g − ν The directional derivative of R in the direction of the vec- − tor field (1, f(µ, g)) is zero. Therefore, in the (µ, g)-plane there is a one-parameter family of solutions φ(µ, g) = c for constant c. Since R is constant along each integral curve, this means that R depends only on the value of c, ACKNOWLEDGMENTS or, equivalently, all the µ and g dependence in R appears as r = R(c) = R(φ(µ, g)). We are indebted to Laurence Yaffe for a careful reading Define the phase shift δ by r = −e2iδ. The rationale for of the manuscript and for helping to clarify the use of RG this definition is that for scattering off an attractive po- arguments that lead to scaling functions. We gratefully tential, δ will be positive and the wave function is “drawn acknowledge helpful correspondence with David Kaplan, into” the well by a distance given by δ/k relative to the Cristiano Nisoli, and Lawrence Schulman.

∗ [email protected] 12 Note that γ(g) is monotonically decreasing for 0 < g < π2. 1 H. E. Camblong, L. N. Epele, H. Fanchiotti, and However, it is quasiperiodic in the sense that the shape of C. A. Garc´ıa Canal, “Quantum Anomaly in Molecular its graph repeats for intervals (π2, (2π)2), ((2π)2, (3π)2), Physics,” Phys. Rev. Lett. 87, 220402 (2001). etc. Since 0 < ν± < 1, there are a pair of fixed points in 2 T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, each of these intervals. C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, 13 See, for example, J. Cardy, Scaling and Renormalization H.-C. Nägerl,and R. Grimm, “Evidence for Efimov quan- in Statistical Physics, (Cambridge, 1996). tum states in an ultracold gas of cesium atoms,” Nature 14 D. Peak, A. Inomata, “Summation over Feynman Histories 440, 315–318 (2006). in Polar Coordinates,” J. Math. Phys. 10, 1422 (1969). See 3 D. B. Kaplan, J.-W. Lee, D. T. Son, and M. A. Stephanov, Appendix A. “Conformality Lost,” Phys. Rev. D 80, 125005 (2009). 15 It should be noted that W is not quite a true expectation Many topics beyond the scope of our article are discussed value because it is not properly normalized. The correctly in this paper; the regularization and renormalization of the normalized expression turns out to be W (x, t; y)/ρ(x, t), 2 √ inverse-square potential occupies Sec. III. where ρ(x, t) = e−x /4t/ 4πt. See Ref. 16. 4 A. M. Essin and D. J. Griffiths, “Quantum mechanics of 16 L. S. Schulman, Techniques and Applications of Path In- 2 the 1/x potential,” Am. J. Phys. 74, 109–117 (2006). tegration, (Dover, 2005), Ch. 9. 5 17 S. A. Coon and B. R. Holstein, “Anomalies in Quantum For some 0 < t1 < t, the probability is given by 2 Mechanics: the 1/r Potential,” Am. J. Phys. 70, 513 R bx0 R ∞ ρ(x1, t1)ρ(x − x1, t − t1)dx1/ ρ(x1, t1)ρ(x − x1, t − (2002). 0 0 6 t1)dx1, where ρ is given in Ref. 15. P. M. Morse and H. Feshbach, Methods of Theoretical 18 Here is a reminder of how that correspon- Physics, Part I (McGraw-Hill, 1953), pp. 736–739. 7 dence works. Consider the operator mechanics F. Werner and Y. Castin, “The unitary gas in an isotropic given by the hermitian transfer matrix T = harmonic trap: symmetry properties and applications,” exp[−V (X)/2] exp[−P 2] exp[−V (X)/2], where [P,X] = Phys. Rev. A 74, 053604 (2006). N+1 8 −i. Then hx|T |yi equals, using the completeness prop- G. B. Arfken and H. J. Weber, Mathematical Methods for −(N+1)/2 R −S erty for P and X, (2π) dx1 ··· dxN e . But Physicists, 6th edition (Elsevier, 2005), p. 696. 2 −t(P 2+V (X)) 9 −1 S. R. Beane, P. F. Bedaque, L. Childress. A Kryjevski, lim→0 log T = P +V (X). Thus, hx|e |yi = −(N+1)/2 J. McGuire, and U. van Kolck, “Singular potentials and (2π) Zxy(t). We recognize the left-hand side as limit cycles,” Phys. Rev. A 64, 042103 (2001). G(x, −it; y). A trace over positions is not taken. 10 D. Branson, “Continuity conditions on Schrödingerwave 19 C. Nisoli and A. R. Bishop, “Attractive Inverse Square functions at discontinuities of the potential,” Am. J. Phys. Potential, U(1) Gauge, and Winding Transitions,” 47, 1000–1003 (1979). Phys. Rev. Lett. 112, 070401 (2014). 11 20 Strangely, in Ref. 3 it is claimed that the C− = 0 solution R. Shankar, Principles of Quantum Mechanics, 2nd edition should be identified with the IRFP. See p. 6 in the para- (Springer, 1994), Ch. 12.6. graph below Eq. (13) and also p. 8 in the paragraph below 21 E. B. Kolomeisky and J. P. Straley, “Universality classes Eq. (19). for line-depinning transitions,” Phys. Rev. B 46, 12664 (1992).