Evolution Operators in Conformal Field Theories and Conformal

Home , Spin model

Evolution operators in conformal ﬁeld theories and conformal mappings: the entanglement Hamiltonian, the sine-square deformation, and others

Xueda Wen,1 Shinsei Ryu,1 and Andreas W.W. Ludwig2 1Institute for Condensed Matter Theory and Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green St, Urbana IL 61801 2Department of Physics, University of California, Santa Barbara, CA 93106, USA (ΩDated: April 13, 2016) By making use of conformal mapping, we construct various time-evolution operators in (1+1) dimensional conformal field theories (CFTs), which take the form R dx f(x)H(x), where H(x) is the Hamiltonian density of the CFT, and f(x) is an envelope function. Examples of such deformed evolution operators include the entanglement Hamiltonian, and the so-called sine-square deformation of the CFT. Within our construction, the spectrum and the (finite-size) scaling of the level spacing of the deformed evolution operator are known exactly. Based on our construction, we also propose a regularized version of the sine-square deformation, which, in contrast to the original sine-square deformation, has the spectrum of the CFT defined on a spatial circle of finite circumference L, and for which the level spacing scales as 1/L2, once the circumference of the circle and the regularization parameter are suitably adjusted.

I. INTRODUCTION be periodic, infinite, or even open, but, we are interested in the Hamiltonian density.) We ”deform” this lattice Many classical statistical mechanical systems and Hamiltonian by introducing an ”envelope” function f(x) quantum many-body systems at criticality enjoy confor- as mal invariance – invariance under scale as well as special X xi + xi+1 conformal transformations. Combined with translations H[f] = f h . (4) 2 i,i+1 and spatial rotations (or spacetime Lorentz boosts), they i are invariant under the conformal group. That critical systems are conformally invariant can be exploited to put There are various problems that fit into the above class some constraints on the operator content of the critical of deformations. For example, let us consider the ground theory. Such constraints are most restrictive and power- state |Ψi of a CFT defined on infinite one-dimensional ful in 2 or (1+1) dimensions, and in some cases can fully space, and then define the reduced density matrix ρA as- specify1 the critical theory.2 sociated with a region x ∈ (−R,R) by ρA = TrB |ΨihΨ|, In this work, we consider various kinds of ”defor- where the partial trace TrB is taken over the all degrees mations” of (1+1) dimensional conformal field theories of freedom associated with the region outside of the inter- (CFTs). By ”deformation” we mean the following. Let val (−R,R). Then, the entanglement Hamiltonian HE, H(x) be the Hamiltonian density of a CFT where x is the defined by ρA = exp(−HE), is of the form (2) where the spatial coordinate. Then, the ordinary time-evolution is envelope function is generated by R2 − x2 f(x) = , x ∈ (−R,R), (5) Z 2R H = dx H(x). (1) and f(x) = 0 otherwise,3–7 i.e., We ”deform” this Hamiltonian by introducing an ”enve- R lope function” f(x) as Z R2 − x2 H = dx H(x). (6) E 2R Z −R H[f] = dx f(x) H(x). (2) Another example is the so-called sine-square deformation (SSD) of quantum many-body Hamiltonians in arXiv:1604.01085v2 [cond-mat.str-el] 12 Apr 2016 Similarly, suppose we have a lattice model, which is crit- (1+1) dimensions.8–21 In the SSD, one chooses the enve- ical and described by a CFT. Schematically, its Hamilto- lope function as nian is given by πx X f(x) = sin2 , x ∈ (0,L), (7) H = hi,i+1 (3) L i and f(x) = 0 otherwise. It was discovered that, for where hi,i+1 is the lattice analogue of the Hamiltonian CFTs, the ground state of the SSD Hamiltonian is identi- density. (We have restricted ourselves to the case of cal to the ground state of the CFT defined on an infinite nearest-neighbor interactions, and neglected for simplic- one-dimensional space. This has practical implications ity further neighbor interactions. The lattice here can as the SSD Hamiltonian allows us to study the CFT in 2 the thermodynamic limit by studying a finite system of (in the continuum limit and at criticality). For this rea- length L (in numerical simulations, say). son, it is rather subtle to discuss the scaling of the finite There are various other examples. For example, yet size spectrum of the SSD Hamiltonian on a lattice. Nev- another context where such a deformation has been dis- ertheless, it has been shown numerically that the spec- cussed is the quantum energy inequalities.22 trum of the SSD Hamiltonian on a finite lattice shows Obviously, there are infinitely many ways to deform (1/length)2 scaling. (Once again, this should be con- CFTs in this way. As an attempt to find a systematic trasted with the ordinary (1/length) scaling of ordinary and controlled construction of such deformations, we will CFT put on a finite cylinder. ) The regularized SSD does make use of conformal mapping. Our construction can not have such subtle issues. The (1/length)2 scaling of be described as follows: (i) We start from a reference the regularized SSD may shed some light on the scaling (1+1)-dimensional spacetime, parameterized by a com- of the original SSD on a lattice, by taking the limit where plex coordinate which is denoted in the following by w, the regularization parameter goes to zero. and an evolution operator H˜ . (ii) We next pick a suit- By using the same idea, we also generated other defor- able conformal map which maps the reference space-time mations of CFTs. For example, we obtain the ”square (coordinate w) to the “target” spacetime, parameterized root” deformation of CFTs, defined by the envelope func- by a complex coordinate which we denote in the follow- tion ing by z. The conformal map maps the set of trajecto- f(x) = p(R − x)(R + x) (8) ries generated by H˜ (determined by the Killing vectors) in the reference spacetime to some (potentially compli- for x ∈ (−R,R) and f(x) = 0 otherwise (see Eq. (57)). cated) trajectories in the complex z-plane. (iii) Finally, We will show that the level spacing of the deformed evo- we transform H˜ and express it in terms of the energy- lution operator with the envelope function (8) does not momentum tensor on the complex z-plane. If we choose depend on R. This deformation was previously discussed the reference evolution H˜ to be something simple, by in the context of quantum information transport in quan- construction, the spectrum of the deformed Hamiltonian tum spin chains. In particular, so-called ‘perfect state is known exactly, and so is its level spacing as a function transfer’ can be achieved in the XX model with inhomo- of the parameters on which the conformal map depends geneous nearest neighbor couplings which are modulated 24–2728 (e.g., the system size). Put differently, in our construc- according to the envelope function (8). tion we deal with the set of envelope functions, which we can “undo” by choosing a suitable conformal map. II. SINGLE VORTEX The construction described above has been used, for example, to obtain the entanglement Hamiltonian in a number of cases.6,7,23 In this paper, by making use of We will consider conformal maps from the Euclidean conformal mapping, we describe various deformations of spacetime to another (“target”) spacetime. The “target” the CFT with various envelope functions, and also dis- spacetime is parameterized by the complex coordinate cuss the finite size scaling of their spectra. As a particular (z, z¯), and we write the real and imaginary parts of z as example, we obtain a “regularized version” of the SSD. z = x + iy. (9) The regularized SSD is closed related to the entanglement Hamiltonian (defined for a finite interval), in that the en- The coordinate of the “reference” spacetime is denoted tanglement Hamiltonian and the regularized SSD can be by (w, w¯), and we write the real and imaginary parts of obtained from the same conformal mapping. However, w as the direction of the evolutions generated by them are or- w = u + iv. (10) thogonal to each other. (In the fluid dynamics language, the flows generated by these two evolution operators cor- As a warm up, we start by illustrating our strategy by respond to the equipotential lines and the streamlines, taking the well-known example of the radial and angular respectively.) quantization of CFTs in the complex plane. Consider the As compared to the original SSD, the regularized SSD conformal mapping has the following properties: The spectrum of the regu- w(z) = log z (11) larized SSD Hamiltonian matches the spectrum of a CFT with periodic boundary conditions (PBC). However, the which maps the entire complex z-plane (“target” space- level spacing of the regularized SSD Hamiltonian shows time) into an infinitely long cylinder (w-coordinates; “ref- (1/length)2 scaling, as opposed to the familiar (1/length) erence spacetime”). This conformal map also transforms scaling of a CFT with PBC. (To be more precise, the an annulus in the z-plane into a finite cylinder in the length here means the length in the complex plane – in w-plane. In the following, we will consider a ‘flow’, or the actual Hamiltonian, one needs to scale simultane- ‘time-evolution’ along the u or the v direction in the “ref- ously both, the size of the system and the regularization erence” spacetime. We then consider the corresponding parameter.) evolution in the “target” spacetime by ‘mapping back’ On the contrary, the spectrum of the original SSD the evolution operator from the “reference” space time Hamiltonian is known to possess a continuous spectrum into the z-plane. 3

For later use, let us consider polar coordinates (r, θ) in 3 the z-plane deﬁned by

2 z = reiθ. (15)

1 From the tensorial transformation law of the energy- momentum tensor, -3 -2 -1 1 2 3 ∂uµ ∂uν -1 T = T˜ , (16) ij µν ∂xi ∂xj -2 T˜uu can be expressed as -3 ∂u ∂u 1 T (r, θ) = T˜ = T˜ (u, v) (17) rr uu ∂r ∂r r2 uu (up to the Schwartzian derivative term). Hence, H˜ can also be expressed as Z 2π Z 2π ˜ 2 2 H = r0 dv Trr(r0, θ) = r0 dθ Trr(r0, θ), (18) 0 0

where r0 is deﬁned by u0 = log r0. Since the circumference of the circle in the z-plane is L := 2πr0, it is natural to introduce -3 -2 -1 0 1 2 3 2π θ = s, s ∈ [0,L],L = 2πr0. (19) FIG. 1. Conformal map w = log z. L Then, H˜ can be written as

A. Radial flow (”radial quantization”) L Z L L 2πs H˜ = ds Trr , . (20) 2π 0 2π L The radial flow in the z-plane is mapped onto a ”straight” flow in the w-plane (flow along the u- Comparing with Eq. (13), the spectrum of the operator direction). This simple evolution is generated by the Z L L 2πs evolution operator in the u-direction, ds Trr , (21) 0 2π L Z 2π H˜ = dv T˜ (u , v), (12) scales as 1/L, and hence this operator can be uu 0 29 0 considered as the Hamiltonian of a CFT defined on a circle of circumference L.30 where T˜uu is the uu component of the stress (energy- momentum) tensor, and we choose a particular “time” u = u0 to define the evolution operator. This evolution B. Angular flow (”Rindler Hamiltonian”) operator is the Hamiltonian of a CFT on a circle of circumference 2π. Since it is defined on a finite space, it The conformal transformation in Eq. (11) also maps has a discrete spectrum. the angular flow in the z-plane into the ”straight” flow ˜ Mapping back to the z-plane, H can be expressed in along the v-direction in the w-plane. This simple time- terms of the dilatation operator (a Virasoro generator) evolution is generated by the evolution operator in v in the plane as direction31 Z +∞ H˜ c ˜ ˜ = L0 + L¯0 − (13) H = du Tvv(u, v). (22) 2π 24 −∞ We next transform this evolution operator back into the where c is the central charge and L0 and L¯0 are given by z-plane. Using the tensorial transformation law (16), 1 I −1 I the energy-momentum tensor on the cylinder and in the L = dz z T (z), L¯ = dz¯ z¯ T¯(¯z). (14) 0 2πi 0 2πi plane are related by 1 h i The dilatation operator generates translations in the ra- 2 ˜ 2 ˜ ˜ 2 ˜ Tyy = 2 2 2 y Tuu + (xy) (Tuv + Tvu) + x Tvv . dial direction in the complex plane. This is the well- (x + y ) known radial quantization. (23) 4

log(z + R), and a sink with the same strength located at z = +R. 2 Taking the limit R/z → 0, the vortex-anti-vortex pair reduces to a dipole, 1 z + R 2R log ∼ . (26) -2 -1 1 2 z − R z It is also convenient to consider a pair of a vortex of -1 strength k and an anti-vortex of strength −k. Then, by letting R → 0 and k → ∞ such that 2kR is ﬁnite, -2 z + R 2Rk w(z) = k log ∼ . (27) z − R z This is the complex potential due to a dipole, i.e., the combination of a source and sink of equal strengths sep- arated by a very small distance. The quantity 2kR is the dipole moment. - Note that the inverse of the left equa- tion in Eq. (27) reads z(w) = R coth(w/2k), which shows that the period of v in the “reference” spacetime tends -2 -1 0 1 2 to inﬁnity in the dipolar limit, k → ∞. This is consistent TABLE I. Conformal map w = log(z + 1)/(z − 1). with the fact that ‘dipolar map’ w = 1/z maps the entire complex plane into itself. (See also SectionIV.)

In particular, when y = 0 we have the relationship Tyy = −2 A. v-evolution: Entanglement Hamiltonian x T˜vv, and hence when expressed in the z-plane, H˜ is given by As in the simple exercise we did in Sec.II, we now Z Z ∞ consider two kinds of time-evolutions associated to the ˜ ˜ H = duTvv = dx xTyy. (24) conformal map (25). 0 Let us first take v as time, and consider the evolution 33 This is the Hamiltonian in Rindler spacetime. The operator at v = v0 = π, Rindler Hamiltonian Eq. (24) generates the angular flow Z + log[(2R−)/] (the Lorentz-boost in Minkowski signature), and corre- H˜ = du T˜vv(u, v0 = π). (28) sponds to “angular quantization” in the z-plane. It is − log[(2R−)/) also nothing but the entanglement Hamiltonian of the Here, we have cut off the integral over u by introducing34 reduced density matrix associated to the semi-infinite in- an UV cutoff > 0 in position space x (so that −R + < terval x ∈ (0, ∞) (where we consider the ground state |Ψi x < +R − ). of a CFT defined on infinite one-dimensional space, and The (time-)evolution operator in (28) generates the then take the partial trace over all degrees of freedom evolution along the constant u-trajectories. In the fluid for x ∈ (−∞, 0).) - Since the entanglement or Rindler dynamic terminology, these are equipotential lines and Hamiltonian (or Lorentz-boost) in Eq. (24) is equal to are given in the z-plane by the Hamiltonian Eq. (22) of the CFT defined on an infinite space, its spectrum is continuous.32 R2 [x ± R coth(u)]2 + y2 = . (29) sinh2(u) III. VORTEX-ANTI-VORTEX PAIR Thus, the constant u trajectories are, for different values of u, circles having centers at (x, y) = (±R coth(u), 0) In this section, we consider the conformal map and radii equal to R/| sinh(u)|. (Compare TABLEI.) The evolution operator (28) can be now mapped into w(z) = log(z + R) − log(z − R), (25) the z-plane. Focusing on y = 0, T˜vv is transformed as 2 2 with inverse z(w) = R coth(w/2). This maps the entire ∂v 2R T = T˜ = T˜ , (30) complex z-plane into an infinitely long cylinder. The yy vv ∂y vv (x − R)(x + R) coordinate v in the “reference” spacetime (coordinates w) is periodic with period = 2π. This conformal map and hence H˜ , when mapped into the z-plane, reads can be thought of as describing the complex potential Z +R− (x − R)(x + R) of a flow which consists of a source with unit strength H˜ = dx Tyy. (31) located at z = −R, represented by the complex potential −R+ 2R 5

This is the entanglement Hamiltonian obtained from a From the discussion in the paragraph containing Eq. (21), CFT defined on an infinite line, after tracing out de- R L the operator 0 ds Trr (L/2π, 2πs/L) is the Hamiltonian grees of freedom living outside of the finite interval of a CFT defined on a circle of circumference L. Thus, 3–7 x ∈ [−R,R]. the part of H˜ which we call HrSSD, defined by Z L 2πs L 2πs B. u-evolution: “Regularized” SSD HrSSD = ds cos + cosh u0 Trr , , 0 L 2π L (40) Let us now move on to consider the evolution operator along the u-direction in w-space which is given by is the ”deformed Hamiltonian” with the envelope function (cos(2πs/L) + cosh u ). Because of the presence of Z π 0 the term cosh u0, this deformation is different from the H˜ = dv T˜uu(u0, v) (32) 0 ordinary SSD, and can be regarded as a ”regularized” version of the SSD. The limit u → 0 corresponds to the where we fix u = u . The constant-v trajectories un- 0 0 ordinary SSD, which for fixed L, or equivalently for fixed der this evolution, which are ‘streamlines’ in the fluid r , corresponds to R → 0. (See Eq. (35).) As discussed dynamics language, are given in the z-plane by 0 in the paragraph containing Eq. (27) this is the dipolar R2 limit. x2 + [y + R cot(v)]2 = . (33) sin2(v) By construction, when u0 6=0, the spectrum of the u- evolution operator is the spectrum of a CFT defined on For different values of v, these are circles with centers at a finite circle (i.e., with PBC), which is discrete. This (0, −R cot(v)), and radii R/| sin(v)|. These circles pass should be contrasted with the ordinary SSD, for which through (±R, 0). - See TABLEI. the spectrum of the evolution operator is a continuum. Turning now to the constant-u “time”-slices (the a. Finite Size Scaling We now turn to the finite-size Cauchy surfaces for the current choice of “time”- scaling of the spectrum of the regularized SSD evolution evolution), we see from Eq. (29) that their (x, y)- operator, Eq. (40). (Once again, in the SSD limit u0 → coordinates in the z-plane satisfy, for a fixed u = u0, 0, the spectrum is continuous, and hence there is no finite size scaling to discuss.) We can in principle discuss the 2 2 cosh u0 2 R following two kinds of finite-size scaling behaviors. x + R + y = 2 . (34) sinh u0 (sinh u0) First, we fix u0 and change the distance R between the two monopoles which controls the (spatial) size of the These are circles of radius r and circumference L, where 0 system. Since H˜ has a level spacing of order one, recalling R (35), the level spacing of HrSSD = (2π sinh u0/L)H˜ scales r0 := , and L = 2πr0. (35) sinh u0 as 2 The evolution operator (32), defined at u = u , acts on sinh u0 1 (sinh u0) 1 0 ∼ = ∼ . (41) quantum states defined on the circle. L 2π R R Making use of the transformation law of the energy- Since R is proportional to L when u is fixed, this means momentum tensor, 0 1/L scaling. 2 2 On the other hand, we can fix R and change u which, ˜ ∂u ˜ 1 sinh u0 0 Trr = Tuu = Tuu , (36) due to (35), also controls the size L of the system. In ∂r r2 cos θ + cosh u 0 0 this case, the level spacing scales as H˜ can be mapped into the z-plane and can be written as sinh u 1 (sinh u )2 1 ∼ 0 = 0 ∼ . (42) Z 2π L 2π R L2 ˜ 2 cos θ + cosh u0 H = r0 dθ Trr(r, θ). (37) 0 sinh u0 This should be contrasted with the regular 1/L scaling of By further introducing CFTs put on a finite spatial circle of circumference L. It should also be noted that for the original SSD, previous 2π 2 9,17 2 θ = s, s ∈ [0,L], (38) numerical studies reported 1/L scaling. The 1/L L scaling of the regularized SSD may be related to this ob- servation. Finally, as we will discuss momentarily, for H˜ is written as the original SSD the spectrum consists of a continuum, Z L when we consider the continuum field theory (CFT) for- ˜ L 1 H = ds mulation of the system, and hence there is no finite size 2π sinh u0 0 scaling to discuss. 2πs L 2πs × cos + cosh u0 Trr r = , θ = . b. The Dipolar Limit Let us now consider the dipo- L 2π L lar limit R → 0. In the dipolar limit, the spacetime cylin- (39) der in w-space shrinks as it is bounded in the u-direction 6 by the cut off ± log[(2R − )/]. - See Eq. (28). (Note that in the section containing Eq. (28), we discussed the 2 entanglement Hamiltonian where u represented the “spatial” coordinate, and v the (imaginary time) “temporal” 1 coordinate. In contrast, in the present section, we have chosen the u-direction as our (imaginary time) “temporal”, and the v-direction as our ”spatial” coordinate.) -2 -1 1 2 Thus, in the limit R → , the (“temporal”) u-direction shrinks to zero. Hence, the ”modular parameter” of the -1 CFT, which depends on the aspect ratio of space and

(imaginary) time directions, is given by -2

(total space length)/(total time length) → ∞. (43) 2 This can be interpreted as achieving the infinite size limit, 20,21 as noted by Ishibashi and Tada. 1 Before closing this section, let us discuss the special case u = 0. This means that we consider the evolution in the z-plane right on the imaginary axis, x = 0, rather -2 -1 1 2 than on a finite circle that we considered when u 6= 0. -1 Hence, we consider the evolution (flow) that brings the infinite line to a point (eventually, at asymptotically long -2 times). - See TABLEI. The evolution operator H˜ = R dvT˜uu(u = 0, v) is mapped into FIG. 2. Conformal map w = 1/z. Z +∞ y2 + R2 H˜ = dy Txx. (44) −∞ 2R We will focus on the evolution in u-direction. Fur- thermore, we use the parametrization u = 1/(2r0). For ˜ By construction, this Hamiltonian H has a spectrum u = 1/(2r0), Eq.(46) implies which is described by a CFT with PBC, although the 1 x system is defined on infinite one-dimensional space. In 2 2 2 = 2 2 ⇒ (x − r0) + y = r0. (47) the dipolar limit, R y, this yields 2r0 x + y Thus, the constant u trajectories are circles of radius r Z 2 Z 0 1 1 + (R/y) 1 2 centered at (r , 0). H˜ = dy Txx ∼ dy y Txx. (45) 0 2R y−2 2R We consider the evolution operator in w-space given by This evolution operator can be considered as derived Z +∞ from the decompactification limit of the SSD Hamilto- ˜ ˜ R L 2 πx R 2 H = dv Tuu(u0, v), (48) nian: 0 dx sin L H ∼ dx x H. Observe that while −∞ before taking the dipolar limit, the system is defined on ˜ the whole imaginary axis; the limit R → 0 ”cuts” the and then map H to the z-plane. In the z-plane, we work imaginary axis into two halves, y > 0 and y < 0. with the polar coordinate (r, θ) defined by

x − r0 =: r0 cos θ, y =: r0 sin θ. (49) IV. DIPOLAR MAP By transforming H˜ , the evolution operator in the r direction is generated by35 The conformal map Z 2π 3 2 H˜ = 4r dθ cos (θ/2) Trr(r0, θ). (50) 1 0 w = (46) 0 z Shifting the angular variable θ ≡ φ + π for convenience, describes the dipolar flow, which maps the entire complex and introducing L = 2πr0 as well as φ ≡ 2πs/L, this z-plane into the entire complex w-plane. As described reads in the paragraph containing Eq. (27), the corresponding Z 2π ˜ 3 2 flow can be obtained from a pair of a vortex and an anti- H = 4r0 dφ sin (φ/2) Trr(r0, θ) 0 vortex, by taking the limit where their separation goes to 2 Z L zero. Here, we directly deal with the dipolar flow without L 2 πs L 2πs = 2 ds sin Trr , . (51) taking the limit. π 0 L 2π L 7

(−π/2, +π/2), x ranges over the interval (−R,R). When x is close to its upper limit, x = R − where → 0+, 2 which suggests

1 R − = R sin(π/2 − δ) = R(1 − δ2/2 + ...), (55)

p + -2 -1 1 2 and hence δ ∼ 2/R → 0 . In terms of , the evolution operator is then written as -1 √ Z +π/2− 2/R ˜ ˜ -2 H = √ du Tvv(u, v = 0). (56) −π/2+ 2/R

By mapping T˜vv(u, v) to Tyy(x, y) we obtain 2 Z +R− p 2 2 1 H˜ = dx R − x Tyy(x, y = 0). (57) −R+ We call this evolution operator ”the square root deforma- -1 tion” (SRD). This evolution operator is somewhat similar to the entanglement Hamiltonian. However, this evolu- -2 tion brings (eventually, at asymptotically long times) the interval x ∈ (−R,R) to to infinite space x ∈ (−∞, +∞), FIG. 3. Conformal map z = sin(w). unlike the entanglement Hamiltonian which takes the interval (−R,R) to its compliment on the real axis. Note ˜ that, interestingly, by construction the spectrum of the Thus, we have related H to square root deformation, Eq. (57), does not scale with Z L R. 2 πs L 2πs HSSD = ds sin Trr , . (52) 0 L 2π L By construction, we expect this SSD Hamiltonian has VI. NUMERICS a CFT spectrum on the infinite line, although HSSD is defined for a circle of circumference L. The prefactor L2 In this section, we use specific lattice models, such as 2 relating H˜ and HSSD is indicative of the 1/L scaling. the s=1/2 XX quantum spin model, to study the deformed Hamiltonians. We have also checked numerically However, since both, HSSD as well as H˜ (defined on infinite space - see Eq. (48)) have a continuum spectrum, the transverse Ising model and the XXZ model, but the there is no finite size scaling that we can discuss for the results for these models are qualitatively similar. We level spacing. therefore focus here on the XX model. The spin 1/2 XX model is defined by:

X x x y y X V. INVERSE SINE MAP H = Si Si+1 + Si Si+1 = hi,i+1, (58) i i As yet another conformal transformation, let us con- x,y,z where Si is the spin 1/2 operator defined at site i of a sider one-dimensional lattice. The finite-size spectra of the XX model, both for PBC and for open boundary conditions z = R sin w. (53) (OBC), are shown in Fig.4. The low-energy part of By this transformation, the infinite strip defined by these spectra are described by the c = 1 compactified free −π/2 < u < π/2 and −∞ < v < +∞ is mapped onto boson theory. We will use these spectra as our reference the complex z-plane. when discussing the spectra of the deformed evolution Consider the evolution operator in the v-direction. At operators. “time” v = v0 this reads Z +π/2−δ + A. SSD H˜ = du T˜vv(u, v0), (δ → 0 ). (54) −π/2+δ Let us now consider the deformation of the XX model In the following, we focus on y = 0, which translates into by the envelope function: v0 = 0. Then, x and u are related by x/R = sin u. 2πx In (54), the upper and lower limit of the integral f(x) = cos + 1 (59) should be suitably cut off. As u ranges over the interval L 8

10 20

8 15

10 4

5 2

0 0 5 10 15 20 25 30 5 10 15 20

0.05 4

0 0 5 10 15 20

FIG. 4. The finite size spectra of the XX model (18 sites) FIG. 5. The finite size spectra of the XX model with the SSD with PBC and OBC. (18 sites) and the finite size scaling analysis of its low-lying spectrum. The resulting SSD Hamiltonian is given by

L where we impose the PBC, h = h . X L,L+1 L,1 HSSD = f(xi + 1/2)hi,i+1 (60) In the continuum limit, we expect that this model ex- i=1 hibits the spectrum of a CFT defined on a spatial circle. The low-lying part of the numerical exact-diagonalization where we impose the PBC, hL,L+1 = hL,1. For previous spectrum of the model (Fig.6) compares reasonably with analytical studies of the SSD of the XX model, see Ref. the expected spectrum of the CFT with PBC in Fig.4. 18. As for the finite size scaling, we scale the system size In the continuum Hamiltonian, we expect that this as well as the second term in the enveloping function. model exhibits a continuum spectrum even when the sys- From the finite size scaling analysis within continuum tem is put on a circle of finite circumference. The numer- field theories, we expect the level spacing scales as ∼ ical, exact-diagonalization spectrum of the model shows 1/L2. The numerical analysis in Fig.4 up to L ∼ 20 sites, a spectrum which does not compare well with the CFT where we choose R = 20, is in reasonable agreement with spectrum, neither with PBC nor with OBC (Fig.5), at the expected 1/L2 scaling. On the other hand, we have least for the system sizes we studied. checked that, if we choose a smaller value of R, R ∼ 1, On the other hand, finite size scaling analysis shows 2 the low-lying spectrum does not look like the CFT with the level spacing scales as ∼ 1/L (Fig.5). This finding PBC. agrees with a previous numerical study.17

C. Square root deformation (SRD) B. Regularized SSD Finally, the square root deformation of the XX model Next, we turn to the regularized SSD. The regularized is given by the envelope function SSD deformation of the XX model is given by the envelope function f(x) = p(L/2)2 − (x − 1/2 − L/2)2 2πx p f(x) = cos + p1 − R2/L2. (61) = (x − 1/2)(L − x + 1/2). (63) L Observe that we have a ”shift” by 1/2, which ensures Here R here is a parameter which serves as a regulariza- f(L + 1/2) = 0. The resulting SRD Hamiltonian is given tion. The envelope function reduces to the SSD envelope by function by taking the limit R → 0. The resulting SSD Hamiltonian is given by L X H = f(x + 1/2)h (64) L SRD i i,i+1 X i=1 HrSSD = f(xi + 1/2)hi,i+1 (62) i=1 where f(L + 1/2)hL,L+1 = 0. 9

8 10

8 6

4 4

2 2

0 0 5 10 15 20 25 30 5 10 15 20

0.05 0 -1

FIG. 6. The finite size spectra of the XX model with the FIG. 7. The finite size spectra of the XX model with the regularized SSD (18 sites) and the finite size scaling analysis square root deformation (18 sites) and the finite size scaling of its low-lying spectrum. We chose R = 20. analysis of its low-lying spectrum.

In the continuum limit, we expect that this model ex- of the original SSD Hamiltonian. For the latter, within hibits the spectrum of a CFT with boundary. The low- the continuum field theory description, the spectrum con- lying part of the numerical exact-diagonalization spec- sists of a continuum (and hence there is no scaling to be trum of the model (Fig.7) compares very well with the discussed for the level spacing). When the SSD Hamil- tonians are studied on discrete one-dimensional lattices, spectrum of the XX model with OBC (Fig.4). As for 2 the level spacing, the finite size scaling analysis shows 1/L level spacing has been observed, which seems closely 0 related to the scaling of the regularized SSD. the level spacing scales as ∼ 1/L as expected. In fact, 36 already from the very small system size (L = 4), the Generalization to conformal maps (e.g. , z = (2/π)arctan w ) other than those we considered in this low-lying spectrum of HSRD (64) the finite size spectra of the XX model with OBC agrees surprisingly well (al- paper should be straightforward. To give a broader per- most perfectly). spective, it is worth pointing out that our construction, namely, the construction of the deformed evolution oper- In fact, this deformation is quite peculiar. When ap- ator on the complex z-plane from some reference evo- plied to a non-interacting fermion system, it completely lution operator on the w-plane (cylinder or strip), is ”straightens” out the spectrum, and hence makes the closely related to the classification scheme of conformal spectrum perfectly relativistic, as analytically shown in vacua37,38 in curved spacetime. In that classification, we Ref. 24. are interested in a curved spacetime M, which is conformally mapped into the flat spacetime M˜ . We suppose that Σ, a global Cauchy hypersurface of M, is mapped VII. CONCLUSION under the conformal transformation to a global Cauchy hypersurface Σ˜ of M˜ . Then, for a timelike conformal Summarizing, we have constructed various deformed Killing vector field in M˜ , there exists a global timelike evolution operators of type (2). In particular, we have conformal Killing vector field in M. Thus, we can clas- constructed a regularized version of the SSD. From our sify the conformal vacua defined with respect to the latter construction, it is also obvious that the regularized SSD conformal Killing vector field by reference to the vacua Hamiltonian has a very close connection with the en- defined in M˜ .37 In Ref. 38, various (1+1) dimensional tanglement Hamiltonian; the evolution generated by the spacetimes are conformally mapped into the Einstein regularized SSD and the entanglement Hamiltonian are static universe (which can be represented as a spacetime orthogonal to each other. cylinder.) We have also studied the scaling of the level spacing Taking the z-plane (the w-space) as M (M˜ ), the only of the spectra of the deformed evolution operators. The minor difference is that here we have considered Eu- regularized SSD shows 1/L2 scaling as opposed to (i) the clidean conformal field theories, while in Ref. 38 con- regular 1/L scaling of CFTs put on a finite spatial circle formal field theories in curved spacetime with Minkowski of circumference L and (ii) the scaling of the spectrum signature are studied. At any rate, the global consid- 10 erations as well as the classification scheme developed in ACKNOWLEDGMENTS Ref. 38 can all be applied to the set of deformed evolution operators discussed in this paper. We thank Tom Faulkner and Hosho Katsura for use- ful discussion. We are grateful to the KITP Program “Entanglement in Strongly-Correlated Quantum Mat- ter” (Apr 6 - Jul 2, 2015). This work is supported by the NSF under Grants No. DMR-1455296 (X.W. and Note added: Upon completion of this work, we became S.R.), No. NSF PHY11-25915, and No. DMR-1309667 aware of the recent preprint Ref. 39, in which the regu- (A.W.W.L.), as well as by the Alfred P. Sloan founda- larized SSD is also studied. tion.

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